Digital processing of shallow seismic refraction data with the refraction convolution section

$Page 1: Digital processing of shallow seismic refraction data with the refraction convolution section$
Digital Processing

Of

Shallow Seismic Refraction Data

With

The Refraction Convolution Section

by

Derecke Palmer M Sc

A Thesis Submitted in Fulfillmentof the Requirements for the Degree of

Doctor of Philosophy

School of Geology,The University of New South Wales,

Sydney, Australia.

September, 2001

2

Declaration of Originality

I hereby declare that this submission is my own work and to the best of my

knowledge it contains no materials previously published or written by another

person, nor material which to a substantial extent has been accepted for the

award of any other degree or diploma at UNSW or any other educational

institution, except where due acknowledgement is made in the thesis. Any

contribution made to the research by others, with whom I have worked at UNSW

or elsewhere, is explicitly acknowledged in the thesis.

I declare that the intellectual content of this thesis is the product of my own work,

except to the extent that assistance from others in the project’s design and

conception or in style, presentation and linguistic expression is acknowledged.

Derecke Palmer

26 September, 2001

3

Abstract

The refraction convolution section (RCS) is a new method for imaging shallow

seismic refraction data. It is a simple and efficient approach to full trace

processing which generates a time cross-section similar to the familiar reflection

cross-section. The RCS advances the interpretation of shallow seismic refraction

data through the inclusion of time structure and amplitudes within a single

presentation.

The RCS is generated by the convolution of forward and reverse shot records.

The convolution operation effectively adds the first arrival traveltimes of each pair

of forward and reverse traces and produces a measure of the depth to the

refracting interface in units of time which is equivalent to the time-depth function

of the generalized reciprocal method (GRM).

Convolution also multiplies the amplitudes of first arrival signals. To a good

approximation, this operation compensates for the large effects of geometric

spreading, with the result that the convolved amplitude is essentially proportional

to the square of the head coefficient. The signal-to-noise (S/N) ratios of the RCS

show much less variation than those on the original shot records.

The head coefficient is approximately proportional to the ratio of the specific

acoustic impedances in the upper layer and in the refractor, where there is a

reasonable contrast between the specific acoustic impedances in the layers. The

convolved amplitudes or the equivalent shot amplitude products can be useful in

resolving ambiguities in the determination of wavespeeds.

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The RCS can also include a separation between each pair of forward and

reverse traces in order to accommodate the offset distance in a manner similar to

the XY spacing of the GRM. The use of finite XY values improves the resolution

of lateral variations in both amplitudes and time-depths.

Lateral variations in the near-surface soil layers can affect amplitudes thereby

causing “amplitude statics”. Increases in the thickness of the surface soil layer

correlate with increases in refraction amplitudes. These increases are

adequately described and corrected with the transmission coefficients of the

Zoeppritz equations. The minimum amplitudes, rather than an average, should

be used where it is not possible to map the near surface layers in detail.

The use of amplitudes with 3D data effectively improves the spatial resolution of

wavespeeds by almost an order of magnitude. Amplitudes provide a measure of

refractor wavespeeds at each detector, whereas the analysis of traveltimes

provides a measure over several detectors, commonly a minimum of six. The

ratio of amplitudes obtained with different shot azimuths provides a detailed

qualitative measure of azimuthal anisotropy.

Dip filtering of the RCS removes “cross-convolution” artifacts and provides a

convenient approach to the study of later events.

The RCS facilitates the stacking of refraction data in a manner similar to the CMP

methods of reflection seismology. It can significantly improve S/N ratios.

The RCS is a simple extension of the GRM, which in turn is a generalization from

which most of the standard refraction inversion methods can be derived. The

RCS advances refraction interpretation through the inclusion of time structure

and amplitudes within a single presentation, which is similar to seismic reflection

data. Accordingly, the RCS facilitates the application of current seismic reflection

acquisition, processing and interpretation technology to refraction seismology.

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Acknowledgements

This work would not have been possible without the support and encouragement

of my supervisor Geoff Taylor, and our head of school, Colin Ward. My focus on

the thesis in the last few years has resulted in some of my academic duties

receiving less than my full attention.

Much of the work for this thesis was carried out between 4:00 am and 6:00 am in

the morning, and it resulted in a number of innocent victims. My wife Coori, and

our two sons, Evan and Heath have had to accommodate an often sleep-

deprived out-of-sorts partner or parent on more than one occasion.

The processing of this and other refraction data has been made possible by

Seismic Un*x developed by the Centre for Wave Propagation Studies at the

Colorado School of Mines. My sincere appreciation to John Stockwell and the

late Jack Cohen for its development, and to Ken Larner for introducing me to SU.

Jacques Jenny of W_Geosoft has generously provided a copy of Visual_SUNT.

Much of the data were acquired when I was an employee of the Geological

Survey of New South Wales. The data for the Mt Bulga 3D survey were acquired

with the assistance of Ross Spencer during a week in the spring of 1986 which

rapidly turned cold and damp. My memory of the survey is of two bedraggled

geophysicists who had forgotten their wet weather clothing wallowing in ankle

deep mud and becoming increasingly frustrated with a temperamental drill rig.

Ian Grierson of Encom Technologies demuxed many of the older field tapes.

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Contents

Declaration of Originality ________________________________________ 2Abstract ______________________________________________________ 3Acknowledgements_____________________________________________ 5Contents______________________________________________________ 6

Chapter 1______________________________________________________ 10Introduction __________________________________________________ 10

1.1 - Recent Innovations in Reflection Seismology ___________________ 101.2 - Recent Innovations in Shallow Refraction Seismology ____________ 111.3 - Digital Processing with the Refraction Convolution Section_________ 141.4 – Outline of Thesis _________________________________________ 191.5 - References______________________________________________ 21

Chapter 2______________________________________________________ 24Inversion of Shallow Seismic Refraction Data – A Review ____________ 24

2.1 - Summary _______________________________________________ 242.2 - Introduction _____________________________________________ 252.3 - Field Data Requirements ___________________________________ 262.4 - Undetected Layers ________________________________________ 272.5 - Incomplete Sampling of Each Layer __________________________ 272.6 - Implications for Model-Based Methods of Inversion ______________ 282.7 - Anisotropy ______________________________________________ 302.8 - The Need to Employ Realistic Models for Refraction Inversion______ 302.9 - The Large Number of Refraction Inversion Methods ______________ 312.10 - Wavefront Reconstruction Methods__________________________ 312.11 - The Intercept Time Method ________________________________ 322.12 - The Reciprocal Methods __________________________________ 332.13 - Data Processing in the Time Domain_________________________ 332.14 - Accommodation of the Offset Distance with Refraction Migration ___ 352.15 - Using Refraction Migration to Recognize Artifacts_______________ 362.16 - Non-uniqueness in Determining Refractor Wavespeeds __________ 372.17 - Fundamental Requirements for Refraction Inversion_____________ 38References __________________________________________________ 39

Chapter 3______________________________________________________ 47Imaging Refractors with the Convolution Section___________________ 47

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3.1 - Summary _______________________________________________ 473.2 - Introduction _____________________________________________ 483.3 - The Large Variations in Signal-to-Noise Ratios with Refraction Data _ 503.4 - Full Trace Processing Of Refraction Data ______________________ 553.5 - Imaging The Refractor Interface Through The Addition of Forward AndReverse Traveltimes __________________________________________ 583.6 - The Addition of Traveltimes With Convolution ___________________ 613.7 - The Effects of Geometrical Spreading on the Convolution SectionAmplitudes __________________________________________________ 653.8 - Effects Of Refractor Dip On Convolution Amplitudes______________ 693.9 - Conclusions _____________________________________________ 703.10 - References_____________________________________________ 72

Chapter 4______________________________________________________ 75Starting Models For Refraction Inversion__________________________ 75

4.1 - Summary _______________________________________________ 754.2 - Introduction _____________________________________________ 764.3 - Inversion Of A Two Layer Model With The GRM Algorithms________ 784.4 - Time Differences Between Starting Models_____________________ 834.5 - Agreement Between Starting Models And Traveltime Data_________ 864.6 - Discussion ______________________________________________ 874.7 - Conclusions _____________________________________________ 894.8 - References______________________________________________ 90

Chapter 5______________________________________________________ 93Resolving Refractor Ambiguities With Amplitudes __________________ 93

5.1 - Summary _______________________________________________ 935.2 - Introduction _____________________________________________ 945.3 - Amplitude and Wavespeed Relationships ______________________ 955.5 - Mt Bulga Case History _____________________________________ 975.5 - Conclusions ____________________________________________ 1045.6 - References_____________________________________________ 106

Chapter 6_____________________________________________________ 107Efficient Mapping Of Structure And Azimuthal Anisotropy With ThreeDimensional Shallow Seismic Refraction Methods _________________ 107

6.1 - Summary ______________________________________________ 1076.2 - Introduction ____________________________________________ 1086.3 - Data Processing With The GRM ____________________________ 1106.4 - Survey Details __________________________________________ 1116.5 - Analysis of the In-line Traveltime Data________________________ 1136.6 - Analysis of the In-line Amplitude Data ________________________ 1216.7 - Analysis of the Cross-line Traveltime Data ____________________ 1246.8 - The Cross-line Amplitude Data _____________________________ 1286.9 - Discussion and Conclusions _______________________________ 1326.10 - References____________________________________________ 134

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Chapter 7_____________________________________________________ 137Effects Of Near-Surface Lateral Variations On Refraction Amplitudes _ 137

7.1 - Summary ______________________________________________ 1377.2 - Introduction ____________________________________________ 1387.3 - Traveltime Results _______________________________________ 1397.4 - Effects of Near-surface Lateral Variations on Amplitudes _________ 1447.5 - Relationships Between Amplitudes and Refractor Wavespeeds ____ 1517.6 - Discussion and Conclusions _______________________________ 1537.7 - References_____________________________________________ 155

Chapter 8_____________________________________________________ 157Enhancement of Later Events in the RCS with Dip Filtering _________ 157

8.1 - Summary ______________________________________________ 1578.2 - Introduction ____________________________________________ 1588.3 - Generation of Useful Events and Artifacts in the RCS____________ 1598.4 - Removal of Cross-convolution Artifacts with Dip Filtering _________ 1638.5 - Times for Near-surface Events in the Uncorrected RCS __________ 1668.6 - Near-surface Wavespeeds from the Uncorrected RCS ___________ 1688.7 - Conclusions ____________________________________________ 1728.8 - References_____________________________________________ 172

Chapter 9_____________________________________________________ 173Stacking Seismic Refraction Data in the Convolution Section________ 173

9.1 - Summary ______________________________________________ 1739.2 - Introduction ____________________________________________ 1749.3 – The Cobar Stacked RCS Section ___________________________ 1769.4 - The Static Geophone Spread_______________________________ 1829.4 - Continuous Acquisition of Shallow Seismic Refraction Data _______ 1839.5 – Determination of Fold with RCS Data ________________________ 1859.6 - Discussion and Conclusions _______________________________ 1869.7 - References_____________________________________________ 188

Chapter 10____________________________________________________ 190Discussion and Conclusions ___________________________________ 190

10.1 - Shallow Refraction Seismology for the New Millenium: A PersonalPerspective_________________________________________________ 19010.2 - Conclusions ___________________________________________ 193

Appendix 1 ___________________________________________________ 198Comments on “A brief study of the generalized reciprocal method andsome of the limitations of the method” by Bengt Sjögren.___________ 198

A.1 - Introduction ____________________________________________ 198A.2 - The Use of Average Wavespeeds___________________________ 199A.3 - The Similarities Between The GRM and Sjogren’s Approach ______ 201A.4 - Recognizing And Defining Narrow Zones With Low Wavespeeds InRefractors__________________________________________________ 203

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A.5 - Use Of Alternative Presentations And Amplitudes For DeterminingWavespeeds In Refractors _____________________________________ 205A.6 - A Systematic Approach With The GRM_______________________ 211A.7 - The Need To Promote Innovation In Shallow Refraction Seismology 212A.8 - References ____________________________________________ 213

Appendix 2 ___________________________________________________ 216Model Determination For Refraction Inversion ____________________ 216

A.1 - Summary ______________________________________________ 216A.2 - Introduction ____________________________________________ 217A.3 - Model and Inversion Strategies _____________________________ 219A.4 - Single Layer Constant Wavespeed Inversion Model_____________ 226A.5 - Two Layer Constant Wavespeed Inversion Model ______________ 229A.6 - Two Layer Wavespeed Reversal Inversion Model ______________ 231A.7 - The Evjen Inversion Model ________________________________ 232A.8 - Transverse Isotropy Inversion Model_________________________ 238A.9 - Errors Related to the Optimum XY Value _____________________ 241A.10 - Discussion and Conclusions ______________________________ 244A.11 - References ___________________________________________ 247A.12 - Appendix: Definition of Variable Wavespeed Media with the GRM 250

Appendix 3 ___________________________________________________ 252Surefcon.c __________________________________________________ 252

Appendix 4 ___________________________________________________ 256The Effects of Spatial Sampling on Refraction Statics ______________ 256

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Chapter 1

Introduction

1.1 - Recent Innovations in Reflection Seismology

In the last fifty years, there have been major advances in the acquisition,

processing and interpretation of seismic reflection data. These advances have

been driven largely by the spectacular developments in the electronic and

computer industries.

The first was the common midpoint (CMP) method for acquiring data (Mayne,

1962). CMP methods improve the signal-to-noise (S/N) ratios of primary

reflections through stacking redundant data.

The second was the application of signal processing with digital computers

(Yilmaz, 1988). Digital processing achieves improvements in S/N ratios through

the attenuation of coherent and random noise and some types of multiple energy,

with CMP stacking and velocity filtering. It can also improve vertical resolution

with deconvolution, and lateral resolution with migration or imaging.

In the last twenty five years, three dimensional (3D) seismic reflection methods

have revolutionized the exploration for, and production of petroleum resources.

Where seismic data were once acquired along single profiles, they are now

obtained over densely sampled grids in the great majority of surveys (Weimer

and Davis, 1996). The improved images of the subsurface geology are a result

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of the recognition that most geological targets are in fact three dimensional, and

that it is essential to employ spatial sampling densities and processing methods

which recognize and accommodate this reality. It is now generally accepted that

in many cases, two dimensional seismic reflection methods give an incorrect

rather than an incomplete picture of the sub-surface (Nestvold, 1992).

An integral component in the interpretation of the increased volumes of data is

the use of computer-based interpretation programs. Most software includes a

range of presentation facilities to change vertical and horizontal scales, gain, and

colour palettes or traditional wiggle trace options; interpretation facilities such as

automatic horizon picking of times and amplitudes; and post-processing

capabilities such as attribute processing and phase rotation. These programs

facilitate the extraction of more and greater detail and therefore, the generation of

more complex geological models.

There have been similar advances in the airborne magnetic and radiometric

methods used in the geological mapping of fold belts (Gunn, 1997). The

advances have occurred in the improved resolution of the instrumentation, the

higher density of spatial sampling, the quality of the processing and the greater

detail of geological interpretation of the data with image processing methods.

1.2 - Recent Innovations in Shallow Refraction Seismology

By contrast, the advances in shallow seismic refraction methods have been

much more modest. There have been few developments comparable to the

common midpoint method, digital processing, or the 3D methods of reflection

seismology.

Most research has focused on the inversion of scalar first arrival times. They

include the standard approaches, such as wavefront construction methods

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(Thornburg, 1930; Rockwell, 1967; Aldridge and Oldenburg, 1992); the

conventional reciprocal method (CRM), (Hawkins, 1961), which is also known as

the ABC method in the Americas, (Nettleton, 1940; Dobrin, 1976), Hagiwara's

method in Japan, (Hagiwara and Omote, 1939), and the plus-minus method in

Europe, (Hagedoorn, 1959); Hales' method, (Hales, 1958; Sjogren, 1979;

Sjogren, 1984); and the generalized reciprocal method (GRM), (Palmer, 1980;

Palmer, 1986). In recent decades, model-based inversion or tomography (Zhang

and Toksoz, 1998; Lanz et al, 1998) has become popular.

Many of the standard methods for inverting shallow seismic refraction data share

fundamental similarities through the addition of forward and reverse traveltimes

to obtain a measure of the depth to the refracting interface in units of time, and

the differencing of the same traveltimes to obtain a measure of refractor

wavespeeds. Many methods also employ refraction migration in order to

accommodate the offset distance, which is the horizontal separation between the

point of refraction on the interface and the point of detection at the surface.

Furthermore, most of these methods can be demonstrated to be special cases of

the generalized reciprocal method (GRM), (Palmer, 1980; Palmer, 1986).

Nevertheless, there are still publications (Whiteley, 1992; Sjogren, 2000) which

seek to emphasize differences between the disparate inversion methods, rather

than to reach a consensus on the intrinsic similarities. They represent a

defensive and backward-looking culture which has done little to promote

innovation in shallow refraction seismology (see Appendix 1).

There have been few advances in the acquisition of shallow refraction data. This

can be largely attributed to the limited capabilities of most field systems, together

with the use of traditional field operations. While the channel capacity of most

reflection field crews has increased from about 96 in 1980, to in excess of 1000

in 2000, the equivalent increase for most shallow refraction crews has been from

12 to 24 channels. In addition, few if any, shallow refraction field crews in

Australia employ radio shot firing systems. Such systems have been available

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for many decades and they represent the application of relatively simple and

readily available technology for improving the efficiency of field operations.

Standard field operations are still largely based on the static geophone spread

with multiple shot points (Walker and Win, 1997). With this approach, 15 or more

collinear shots which are located both within the geophone spread and at various

offset positions on either side, are recorded with a linear pattern of 12 or 24

geophones. The geophone spread is then re-deployed beside the previous

setup, commonly with an overlap of 2 geophones. A more efficient roll along

approach, which is the norm for acquiring CMP reflection data, produces more

data from the critical near surface layers but less shot points per unit distance.

Commonly, there can be a reduction of up to 40% in the number of shot points.

Continuous single-pass roll-along acquisition methods can result in more reliable

interpretations, less environmental impact and lower unit costs (Palmer, 2000).

In the last two decades, the roles of most geophysicists in the petroleum and

mineral exploration industries have changed from having a significant data

acquisition and processing component, to being largely an interpretation role in

conjunction with other geoscientists. This has been made possible through the

extensive use of specialist seismic contractors who have maintained competitive

costs and continual advancement of their products and services. Similar

changes in emphasis from acquisition and processing towards interpretation and

the generation of more complex geological models have yet to occur with most

groups using shallow seismic refraction methods.

In many cases, the shallow seismic methods are applied to geotechnical

investigations, and as a result, they reflect an engineering culture which is

characterized by conservative approaches, risk minimization and standard

practices. It contrasts with the culture of the exploration industry which is

characterized by experimentation, risk taking and innovation.

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In summary, most shallow seismic refraction operations have not taken

advantage of advances in technology for acquisition, processing or interpretation,

they are under-capitalized and they are relatively inefficient. Where shallow

refraction technology was once perceived to be twenty years behind reflection

methods, the difference is now nearer half a century.

1.3 - Digital Processing with the Refraction Convolution Section

The point of departure for this study is that the current methods of acquiring,

processing, and interpreting seismic reflection data provide compelling models

for the advancement of shallow refraction seismology. Of these, the most critical

aspect is the development of an efficacious method for digital processing using

the complete seismic trace. Digital processing is an essential requirement for

deriving more information from existing data as well as for efficient handling of

the increased volumes of data which are typical of most 3D surveys. The

development of digital processing techniques suitable for use in routine seismic

surveys has been my objective in this thesis.

1.3.1 - The refraction convolution section

This study describes a new method of digital processing for shallow seismic

refraction data with the refraction convolution section (RCS). It seeks to

demonstrate that the RCS results in more detailed geological models of the

subsurface through the convenient use of amplitudes as well as traveltimes, and

that it provides an effective domain for the advancement of shallow refraction

seismology using the model provided by existing seismic reflection technology.

The RCS generates a time cross-section similar to the familiar reflection cross-

section through the convolution of forward and reverse traces. It is simple in

concept, and very rapid in execution. The addition of the traveltimes with

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convolution is equivalent to that achieved graphically with Hales’ and wavefront

methods and arithmetically with the GRM. Accordingly, the RCS shows the

same structure on the refracting interface in units of time as do many of the

standard methods of inversion. The convolution process also multiplies the

amplitudes and to a very good approximation, it compensates for the effects of

geometrical spreading and dipping interfaces. The RCS facilitates the

examination of important issues such as S/N ratios, the resolution of ambiguities

in refractor models, 3D refraction methods and azimuthal anisotropy, signal

processing to enhance second and later events and stacking data in a manner

similar to CMP reflection methods. I investigate all of these issues in this study.

1.3.2 - Interpretation using travel times and amplitudes

Past use of amplitudes in shallow refraction seismology has been virtually non-

existent, mainly because of the very large geometric spreading component. It

can be much larger than the theoretically derived reciprocal of the distance

squared function and it dominates any geological effects. The geometric

spreading component also results in varying S/N ratios across the refraction

spread, and therefore varying accuracies with measured traveltimes. The

compensation for the geometric effect with convolution equalizes S/N ratios, and

results in RCS amplitudes which vary as the square of the head coefficient, the

expression relating head wave amplitudes to the petrophysical parameters.

In addition to the large geometric spreading component, the use of head wave

amplitudes has been limited by the lack of a convenient quantitative relationship

with petrophysical parameters. Although the original formulations of the head

coefficient were first published more than forty years ago, they are sufficiently

unwieldy to prevent their use in most applications. Just as the normal incidence

approximations of the Zoeppritz equations are used widely in reflection

seismology, so there is a need to develop a convenient form of the head

coefficient, for use in routine shallow refraction seismology.

16

The approximation of the head coefficient presented in this study is the ratio of

the specific acoustic impedance in the upper layer to that in the refractor. This

approximation facilitates the application of head wave amplitudes to a number of

important problems.

The first is the fundamental issue of the non-uniqueness which is not adequately

addressed with most current approaches to refraction inversion. This study

demonstrates that amplitudes can be very useful in addressing many ambiguities

in determining wavespeeds in the refractor.

Secondly, amplitudes provide an efficient means of improving spatial resolution,

particularly with 3D sets of data, because they provide a measure of wavespeeds

at each point whereas the use of traveltimes generally provides a measure over

several detectors. The improved resolution is comparable with that achieved

with tomographic inversion, but without the need to acquire almost an order of

magnitude of additional data.

The third application of amplitudes is in the qualitative measurement of azimuthal

anisotropy using 3D acquisition methods. Azimuthal anisotropy, which can be

caused by foliation, fracture porosity, etc. is a measure of rock fabric which can

be of considerable importance in environmental, groundwater and geotechnical

investigations. Although there has been a small number of studies of azimuthal

anisotropy, mainly with series of 2D profiles of varying azimuth over relatively

uniform refractors, none has sought to resolve refractors exhibiting both complex

3D structure and anisotropy. This study demonstrates that significant variations

in depths, wavespeeds and azimuthal anisotropy can occur in the refractor in the

cross-line as well as the in-line directions, and that each be resolved with the

application of simple processing methods to relatively small volumes of data,

using standard methods such as the GRM.

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The use of amplitudes has also been limited by the ubiquitous concerns about

the effects of coupling of the geophone with the ground on the observed

amplitudes. This study demonstrates that the major cause of “amplitude statics”

is variations in the petrophysical properties, usually the wavespeed, of the near

surface layers, and that there are relatively simple methods for recognizing and

accommodating these effects.

1.3.3 - Processing of the full waveform

Another long-standing limitation of traditional shallow seismic refraction

processing methods, which I address in this study, has been the almost complete

reliance on the first arrival signal. Although the potential value of later events to

assist in the resolution of undetected layers or in the use shear wave studies has

often been noted, nevertheless there are no widely accepted approaches to

efficacious use of the complete seismic refraction trace. This study

demonstrates that the convolution operation also generates a relative time-depth

profile for any later events and that it can be highlighted with simple processing

methods such as dip filtering in the f-k domain.

An important advantage of convolution is the preservation of the phase

relationships. The most common energy sources for shallow seismic refraction

surveys are impulsive sources such as explosives or dropping weights, which

generate minimum phase wavelets. When two such minimum phase wavelets

are convolved with one another, as is the case with the generation of the RCS,

then the resultant is also minimum phase. Accordingly, the time structure

determined in the RCS correlates with that computed with the traveltimes

measured on the shot records. It also is facilitates further processing in order to

improve vertical resolution using, for example, deconvolution.

Perhaps one of the most important implications of the compensation for the large

geometric effect and the equalization of S/N ratios with the RCS is that it

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facilitates stacking in a manner similar to the CMP methods of reflection

seismology. Stacking may eventually achieve improvements in S/N ratios

sufficient to reduce the relatively large source energy requirements of acquisition,

which traditionally have limited the application of refraction methods because of

cost and environmental impact. Furthermore, it is possible that stacking in the

RCS may promote fundamental changes in data acquisition which are necessary

to achieve much needed efficiencies in field operations, as well as to generate

data with suitable fold or redundancy for stacking.

1.3.4 - Thesis aims

In summary, my major aims in this study are to demonstrate that the use of head

wave amplitudes results in more detailed geological models of the subsurface,

and that the RCS provides an effective and convenient domain for processing

and interpreting shallow seismic refraction data in order to obtain the amplitude

information. Furthermore, this study also demonstrates that many of the benefits

of the RCS can be maximized with acquisition programs which resemble those

used in current seismic reflection surveys. Accordingly, the RCS provides a

suitable domain for the continued advancement of shallow refraction seismology

using the model provided by current seismic reflection technology.

This study focuses on the near surface region for geotechnical, groundwater and

environmental applications. However, there should be few problems in applying

many of the results of this study to the deeper targets of petroleum exploration,

where current reflection methods are not efficacious, and even to regional

geological studies of the Earth’s crust.

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1.4 – Outline of Thesis

Each chapter is presented in the format of a manuscript for publication. Chapter

2 is part of an invited manuscript which has been accepted for publication in a

special edition of Geophysical Prospecting to commemorate the late J G

Hagedoorn. Chapters 3 and 5 have undergone extensive review and have been

accepted for publication in Geophysics, while chapters 4, 6, 7 ,8, 9 and Appendix

2 have been submitted for review. This format results in concise chapters,

although there can be minor repetition of some material in order to achieve self-

contained manuscripts.

Chapter 2 reviews the requirements of the inversion model and algorithms for the

inversion of shallow seismic refraction data. All methods for inverting shallow

seismic refraction data require reversed and redundant data in order to resolve

wavespeeds and structure within each refractor, and to identify the wavespeed

stratification above the target refractor. However, there are still fundamental

limitations in accurately determining the wavespeed stratification from even the

most complete sets of data. These difficulties indicate that as much of the data

processing as possible should be carried out in the time domain, rather than in

the depth domain. I conclude that the wavespeed analysis and the time-depth

algorithms of the group of processing techniques known as the reciprocal

methods, satisfy these requirements. I also conclude that the variable migration

of the GRM provides a useful approach to the treatment of undetected layers,

wavespeed reversals, variable wavespeed media, anisotropy and non-

uniqueness.

Chapter 3 describes the generation of the refraction time section, which is similar

to the familiar reflection time cross section, through the convolution of pairs of

forward and reverse traces. The convolution section shows the same time

structure on the refracting interface as is obtained with many standard analytical

methods for the inversion of refraction data. In addition, there is good

20

compensation for the effects of geometric spreading and dipping interfaces, with

the result that the amplitudes are essentially a function of the head coefficient,

which is the expression relating the refraction amplitudes to the wavespeeds and

densities in the upper layer and the refractor.

Chapter 4 describes the ambiguities in resolving wavespeeds in the refractor

where there are significant changes in depth. In many cases, these ambiguities

are not resolved with model-based methods of inversion, such as tomography,

because many of the starting models are compatible with the original traveltime

data.

Chapter 5 describes the relationship between the convolved amplitudes and the

specific acoustic impedances. It is shown that the head coefficient is

approximately the ratio of the specific acoustic impedances (the product of the

density and wavespeed). The convolved amplitudes are the square of this ratio.

The amplitudes are then employed to resolve an ambiguity in the wavespeed of a

very irregular refractor.

Chapter 6 describes a 3D refraction survey over a shear zone. This case history

is a compelling demonstration that even simple 3D refraction methods can

provide far more useful geological models than even the most detailed 2D

results. It also demonstrates the use of amplitudes in obtaining qualitative

measures of anisotropy and therefore rock fabric.

Chapter 7 describes the effects of near surface variations on refraction

amplitudes and it provides a valuable insight into the ubiquitous concerns about

the effects of geophone coupling with the ground on the reliability of the

measurement of seismic amplitudes with single detectors.

Chapter 8 describes an elementary approach to signal processing of the

convolution section with dip filtering. Dip filtering in the f-k domain can remove

21

“cross-convolution” artifacts generated by the convolution process, thereby

highlighting later events from other, usually shallower, refractors. A number of

attempts were made to improve the vertical resolution with signature and

predictive deconvolution but without much success.

Chapter 9 describes stacking of refraction data, using a CMP-like approach. A

major conclusion is that methods of data acquisition which are suitable for

stacking are also efficient and suitable for routine field operations

Chapter 10 is a personal perspective of the possible future use of the RCS as

well as the conclusions for this study.

1.5 - References

Aldridge, D. F., and Oldenburg, D. W., 1992, Refractor imaging using an

automated wavefront reconstruction method: Geophysics, 57, 378-385.

Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edition:

McGraw-Hill Inc.

Gunn, P., ed, 1997, Thematic issue: airborne magnetic and radiometric surveys,

AGSO Journal of Australian Geology and Geophysics, 17(2), 1-216.

Hagedoorn, J. G., 1959, The plus-minus method of interpreting seismic refraction

sections: Geophys. Prosp., 7, 158-182.

Hagiwara, T., and Omote, S., 1939, Land creep at Mt Tyausa-Yama

(Determination of slip plane by seismic prospecting): Tokyo Univ. Earthquake

Res. Inst. Bull., 17, 118-137.

22

Hales, F. W., 1958, An accurate graphical method for interpreting seismic

refraction lines: Geophys. Prosp., 6, 285-294.

Hawkins, L. V., 1961, The reciprocal method of routine shallow seismic refraction

investigations: Geophysics, 26, 806-819.

Lanz, E., Maurer H., and Green, A. G., 1998, Refraction tomography over a

buried waste disposal site: Geophysics, 63, 1414-1433.

Mayne, W. H., 1962, Common-reflection-point horizontal data-stacking

techniques: Geophysics, 27, 927-938.

Nestvold, E. O., 1992, 3-D seismic: is the promise fulfilled?: The Leading Edge,

11, 12-19.

Nettleton, L. L., 1940, Geophysical prospecting for oil: McGraw-Hill Book

Company Inc.

Palmer, D., 1980, The generalized reciprocal method of seismic refraction

interpretation: Society of Exploration Geophysicists.

Palmer, D., 1986, Refraction seismics: the lateral resolution of structure and

seismic velocity: Geophysical Press.

Palmer, D, 2000, Can new acquisition methods improve signal-to-noise ratios

with seismic refraction techniques?: Explor. Geophys., 31, 275-300.

Rockwell, D. W., 1967, A general wavefront method, in Musgrave, A .W., Ed.,

Seismic Refraction Prospecting: Society of Exploration Geophysicists, 363-415.

23

Sjogren, B., 1979, Refractor velocity determination - cause and nature of some

errors: Geophys. Prosp., 27, 507-538.

Sjogren, B., 1984, Shallow refraction seismics: Chapman and Hall.

Sjogren, B., 2000, A brief study of applications of the generalized reciprocal

method and some of the limitations of the method: Geophys. Prosp., 48, 815-

834.

Thornburg, H. R., 1930, Wavefront diagrams in seismic interpretation: AAPG

Bulletin, 14, 185-200.

Walker, C. S., and Win, M. A., 1997, A new standard in the practice of

engineering seismic refraction, in McCann, D. M., Eddleston, M., Fleming, P. J.,

and Reeves, G. M., eds., Modern geophysics in engineering geology: The

Geological Society, 391-398.

Weimar. P., and Davis, T. L., 1996, Applications of 3-D seismic data to

exploration and production, Geophysical Developments Series, no. 5: Society of

Exploration Geophysicists.

Whiteley, R. J., 1992, Comment on the ‘The resolution of narrow low-velocity

zones with the generalized reciprocal method’ by Derecke Palmer: Geophys.

Prosp., 40, 925-931.

Yilmaz, O, 1988, Seismic data processing: Society of Exploration Geophysicists.

Zhang, J., and Toksoz, M. N., 1998, Nonlinear refraction traveltime tomography:

Geophysics, 63, 1726-1737.

24

Chapter 2

Inversion of Shallow SeismicRefraction Data – A Review

2.1 - Summary

All methods for inverting shallow seismic refraction data require reversed and

redundant data in order to resolve wavespeeds and structure within each

refractor, and to identify the wavespeed stratification above the target refractor.

However, there are fundamental limitations in accurately determining the

wavespeed stratification from even the most complete sets of data. Not all layers

are necessarily detected in the traveltime data, because some layers are either

too thin, or the wavespeeds are less than that in the overlying layer.

Furthermore, the wavespeed stratification cannot be determined with high

precision within those layers which are detected, because the refracted rays do

not penetrate deeply enough, or because the horizontal rather than the vertical

wavespeed is measured.

The difficulties in accurately determining the inversion model indicate that as

much of the data processing as possible should be carried out in the time

domain, rather than in the depth domain. The wavespeed analysis and the time-

depth algorithms of the group of processing techniques known as the reciprocal

methods, satisfy these requirements.

25

In addition, there is another fundamental issue of non-uniqueness in determining

lateral variations in wavespeeds in the refractor. This requires the use of

refraction migration in order to accommodate the offset distance. However,

incorrect migration distances which would result from the use of incorrect

wavespeeds in the layers above the target refractor, can still generate results

which satisfy the traveltime data. This problem can be overcome with the use of

multiple migration distances with the generalized reciprocal method (GRM) and

the use of the minimum variance criterion.

The GRM is a logical advancement of pre-existing refraction inversion methods.

It combines the horizontal layer approximations of the intercept time method, the

wavespeed analysis and time-depth algorithms of the traditional reciprocal

methods, and the accommodation of the offset distance with refraction migration

of the delay time and Hales’ methods. The variable migration of the GRM

provides a useful approach to the treatment of undetected layers, wavespeed

reversals, variable wavespeed media, anisotropy and non-uniqueness.

2.2 - Introduction

The refraction method was the first seismic technique to be used in petroleum

exploration, and in the 1920’s, it achieved spectacular success in Iran and the

Gulf Coast of the USA. Although refraction methods were soon superseded by

reflection methods, they were still commonly used in many areas where single

fold reflection methods were not effective. However, with the development of

common midpoint methods in the 1950’s, the use of refraction methods in

petroleum exploration decreased even further.

Today most seismic refraction surveys are carried out to map targets in the near

surface region for geotechnical, groundwater and environmental applications,

26

and for statics corrections for seismic reflection surveys. On a line kilometre

basis, statics corrections clearly constitute the greatest use of the method.

The 1950’s represent a significant period in the development of refraction

techniques. Almost all of the major issues had been identified and many

advances had been achieved in the years prior to that date. They include the

mapping of irregular refractors, complex wavespeed functions in the layers above

the target refractor, undetected layers, wavespeed reversals, anisotropy, and

refraction migration.

In the last fifty years, the development of the refraction method has been virtually

stagnant and most research has tended to focus on the various methods for

inverting traveltime data. However, in many cases, it is apparent that the models

used for inversion are not cognizant of the realities of the near surface

environment and that implausible assumptions are often made.

This study reviews the major issues associated with the inversion of seismic

refraction traveltime data, especially that acquired in the near surface

environment, where geological conditions can change rapidly. I conclude that

the generalized reciprocal method (GRM) (Palmer 1980, 1986) is a logical

evolution of the major inversion methods, which can usefully address the issues

of resolution, ambiguity and non-uniqueness.

2.3 - Field Data Requirements

The first stage of the inversion of the traveltime data is the determination of an

appropriate model. Generally, this is a qualitative stage in which an assessment

is made of the number of layers that can be recognized confidently in the

traveltime data, and in which each arrival is assigned to a particular refractor. It

requires reversed traveltime data for which there are shot points in both the

27

forward and reverse directions, in order to resolve lateral variations in depths to

and wavespeeds within each refractor. In addition, redundant data in which there

are several shot points on either side of the array of detectors, are also essential.

Hinge points or changes in slope which shift horizontally with each graph indicate

new layers, while hinge points which shift vertically indicate changes in depth or

wavespeed within the same layer. These requirements are routinely satisfied

with shallow refraction operations which employ a high density of shot points

(Walker and Win, 1997), and they are described in more detail in Palmer (1986),

Palmer (1990), and Lankston (1990).

2.4 - Undetected Layers

However, this process is only effective if there is a monotonic increase in

wavespeeds from layer to layer with increasing depth and if the thickness of each

layer is greater than a minimum value. Layers, which are thin in relation to the

thicknesses and wavespeeds of the surrounding layers, can escape detection

(Maillet and Bazerque, 1931; Soske, 1959). Furthermore, even layers which are

thick are not detected if there is a reversal in wavespeed from the layer above

(Domzalski, 1956; Knox, 1967). These are the well-known undetected layer

problems and various methods for determining maximum errors have been

described by many authors (Merrick et al, 1978; Whiteley and Greenhalgh,

1979).

2.5 - Incomplete Sampling of Each Layer

The difficulties in accurately specifying the inversion model extend to the

determination of the wavespeed within each layer. In Hagedoorn (1955),

traveltimes are computed for a simple two layer model, in which the wavespeed

28

in the upper layer varies linearly with depth. A variety of other wavespeed

functions are then fitted to the traveltime graphs with an accuracy of better than

0.5%, but nevertheless the errors in the computed depths to the refractor are

between 10% and 29%.

Hagedoorn’s (1955) study is of fundamental significance to the inversion of all

refraction data using any approach. It demonstrates that even in the absence of

undetected layers, the wavespeed model in the each layer and therefore its

thickness, cannot be accurately determined with the traveltimes from that layer

alone. It also demonstrates that the selection of the correct wavespeed model is

essential for accurate depth determinations.

The difficulties in accurately determining the parameters of each layer are related

to the inherent errors of extrapolation. The parameters of the wavespeed

function are computed from arrivals which rarely penetrate more than 30% of the

thickness for realistic wavespeed functions. These parameters are then

extrapolated to the remainder of the layer where each wavespeed function can

behave quite differently.

2.6 - Implications for Model-Based Methods of Inversion

Hagedoorn’s (1955) study is especially applicable to model-based inversion or

tomography (Zhu et al., 1992). With these methods, the parameters of a model

of the subsurface are refined by comparing the traveltimes of the model with the

field data. When the differences between the computed and field traveltimes are

a minimum, the model and parameters are taken as an accurate representation

of the wavespeeds in the subsurface.

The performance of refraction tomography has been continually improved

through more efficient inversion and forward modeling routines, (see Zhang and

29

Toksoz, 1998 for an overview of these advances). However, the choice of the

model has yet to receive widespread attention, since the role of model-based

inversion is to provide information about the unknown numerical parameters

which go into the model, not to provide the model itself (Menke, 1989, p3).

Perhaps the most common model has been the linear increase of wavespeed

with depth (Zhu et al.,1992; Stefani, 1995; Miller et al., 1998; Lanz et al., 1998),

possibly because of mathematical convenience. However, this model is of

questionable validity as most theoretical (Iida, 1939; Gassman, 1951, 1953;

Brandt, 1955; Paterson, 1956; Berry, 1959), laboratory (Birch, 1960; Wyllie et al.,

1956, 1958), and field studies (Faust, 1951, 1953; White and Sengbush, 1953;

Acheson, 1963, 1981; Hall, 1970; Hamilton, 1970, 1971; Jankowsky, 1970),

suggest a more gentle increase for clastic sediments, such as a one sixth power

of depth function.

Furthermore, the gradients obtained range from 0.342 and 2.5 m/s per metre

(Stefani, 1995), and 2.68 and 4.67 m/s per metre (Zhu et al., 1992), to as high as

40 m/s per metre (Lanz et al, 1998). These values are generally much larger

than those applicable to the compaction of clastic sediments (Dobrin, 1976), but

they are rarely justified on geological or petrophysical grounds.

The combination of the linear increase of wavespeed with depth and the high

gradients probably contributes to instability in the inversion process. The

example of the somewhat paradoxical situation of the poor determination of

wavespeeds in the refractor, despite the fact that over 90% of traveltimes are

from that layer (Lanz et al., 1998, Figure 8), is at variance with the experiences of

most seismologists using more traditional methods of refraction processing.

Furthermore, the use of linear wavespeed functions where constant wavespeed

layering is applicable can result in large gradients which in turn can result in the

ubiquitous ray path diagrams demonstrating almost complete coverage of the

30

subsurface. These diagrams are misleading when the inversion model does not

accurately represent the subsurface, because the shortcomings of extrapolation

are not properly addressed.

2.7 - Anisotropy

Another factor which affects the determination of the inversion model is

anisotropy. Seismic anisotropy, in which the wavespeed in the horizontal

direction is different from that in the vertical direction, has been recognized from

the earliest days of seismic exploration (McCollum and Snell, 1932), and

refraction examples have been described by Hagedoorn (1954) and others. The

significance of anisotropy is that the wavespeeds measured on the traveltime

graphs are horizontal components, whereas vertical components are required for

depth conversion.

2.8 - The Need to Employ Realistic Models for RefractionInversion

Accordingly, the determination of an appropriate inversion model from seismic

refraction traveltime data is not necessarily a straightforward task. It requires an

adequate set of reversed and redundant data, in order to assign each arrival to a

refractor. However, even with such data, there is still no guarantee that all layers

can be detected, either because of thin layers or because of wavespeed

reversals. In these cases, the traveltime data do not provide a complete model

of the layering. Furthermore, the wavespeeds in those layers which are detected

may not be accurate because of the difficulties in deriving the appropriate

wavespeed versus depth function, and because the wrong component is

obtained in the presence of seismic anisotropy. The fact that the traveltime data

31

are neither a complete, an accurate nor a representative indication of the

inversion model should be viewed as a fundamental geophysical reality which

must be accommodated in any approach to refraction inversion.

2.9 - The Large Number of Refraction Inversion Methods

In view of the many applications over the last eight decades, it is not surprising

that the refraction method is characterized by the existence of numerous

approaches for inverting the field data. Standard texts such as Musgrave (1967),

Dobrin (1976), and Sheriff and Geldart (1995), describe almost a score of

techniques which have been used at some time in the past. Each method

represents a compromise between the desire for mathematical exactness and

the realities of geophysical robustness and computational convenience.

Most of these methods have not seen regular use and are more of curiosity

value, rather than being practical inversion methods. The more commonly used

methods have been wavefront reconstruction, the intercept time, the reciprocal

method and the group which employ refraction migration, viz. the delay time.

Hales’ and the generalized reciprocal methods.

2.10 - Wavefront Reconstruction Methods

Perhaps the earliest techniques to be used were the wavefront reconstruction

methods (Thornburg, 1930; Rockwell, 1967; Aldridge and Oldenburg, 1992).

These methods retrace the emerging forward and reverse wavefronts down into

the subsurface. The refractor interface is located at the positions where the sum

of the forward and reverse wavefronts is equal to the reciprocal time. Wavefront

reconstruction methods are generally considered to be the most precise because

32

they make few assumptions or approximate Snell’s law. However, they operate

in the depth domain and therefore require a detailed and accurate knowledge of

the wavespeeds above the target refractor. As discussed above, this is probably

one of the most difficult requirements to satisfy.

2.11 - The Intercept Time Method

Another longstanding technique is the intercept time method (ITM), (Ewing et al,

1939). This method is essentially a ray tracing approach applied to a subsurface

model consisting of homogeneous layers with uniform wavespeeds separated by

plane dipping interfaces. The angle of emergence of each ray is readily

determined from the travelime graphs, and its trajectory in the subsurface is then

computed with the simple application of Snell’s law.

Although the ITM is mathematically precise, it is not geophysically robust.

Discordant dips produce large changes in slope on the traveltime graphs and as

a result, there can be difficulties in recognizing individual layers. Furthermore,

dipping interfaces eventually intersect, thereby resulting in layers which do not

register in the traveltime graphs below a minimum thickness.

Under most circumstances, the horizontal layer approximations are of sufficient

accuracy (Palmer, 1986). These approximations are (i) the use of the law of

parallelism to obtain intercept times (Sjogren, 1980), which are a measure of the

depth to the refracting interface in units of time, (ii) the horizontal layer value of

the depth conversion factor which relates intercept times and layer thicknesses

and (iii) the harmonic mean of the forward and reverse apparent wavespeeds to

obtain a measure of the refractor wavespeeds.

33

2.12 - The Reciprocal Methods

The approximations of the ITM are identical to those which are integral to the

group of techniques known as the reciprocal methods (Hawkins, 1961). This

group had its origins in the 1930’s when it was known as the method of

differences (Edge and Laby, 1931, p.339-340; Heiland, 1963, p.548-549). These

methods are also known as the ABC method in the Americas, (Nettleton, 1940;

Dobrin, 1976), Hagiwara's method in Japan, (Hagiwara and Omote, 1939), and

the plus-minus method in Europe, (Hagedoorn, 1959). There are no fundamental

mathematical differences between each of these methods, and usually the

choice of a particular version is a function of geography. Mathematically, the

reciprocal methods can be demonstrated to be simple extensions of the ITM

whereby depths and wavespeeds, which are determined at the shot points with

the ITM, are also computed at each detector position between the shot points

(Palmer, 1986).

2.13 - Data Processing in the Time Domain

The reciprocal methods employ two fundamental algorithms. The first, the

wavespeed analysis function tV, employs the subtraction of forward and reverse

traveltimes at each detector position. There can be other operations, such as the

addition of the reciprocal time, which is the traveltime from one shot point to the

other, and the halving of the result. However, the essential feature is the

subtraction operation, which effectively removes the effects of any variations in

the thicknesses of the layers above the refractor. The gradient of this function

with respect to distance is the reciprocal of the wavespeed in the refractor, Vn.

tV = (tforward – treverse + treciprocal)/2 (2.1)

d/dx tV = 1 / Vn (2.2)

34

The second algorithm employs the addition of the forward and reverse

traveltimes at each detector position, in order to obtain a measure of the depth to

the refracting interface in units of time. This function, known as the time-depth tG,

can also include other operations, such as the subtraction of the reciprocal time,

and the halving of the result.

tG = (tforward + treverse - treciprocal)/2 (2.3)

The two algorithms of the reciprocal methods represent major advances in the

processing of shallow seismic refraction data. The processing is carried out in

the time domain and therefore it does not require an accurate knowledge of the

wavespeeds in the layers above the target refractor. Although accurate

wavespeeds are necessary for the final conversion to a depth cross-section,

nevertheless, many useful processing operations can be conveniently carried out

in the time domain prior to that step. This advantage is not shared with methods

which operate in the depth domain, such as the wavefront reconstruction

methods and tomography.

The depth zG, is computed from the time-depth and the wavespeeds in the

refractor and the layer(s) above with equation 4, viz.

zG = tG DCF (2.4)

where the DCF, the depth conversion factor relating the time-depth and the

depth, is given by:

DCF = V Vn / (Vn2 - V2)½ (2.5)

or

DCF = V / cos i (2.6)

where

35

sin i = V / Vn (2.7)

and where V is the average wavespeed above the refractor.

2.14 - Accommodation of the Offset Distance with RefractionMigration

The offset distance is the horizontal separation between the point of emergence

of the ray on the refractor interface and the point of detection at the surface. The

offset distance is implicitly accommodated in all refraction techniques which use

a depth conversion factor similar to the horizontal layer approximations of the

ITM in equation 2.5.

In addition, there are several inversion techniques which explicitly accommodate

the offset distance. These methods seek to employ the process known as

refraction migration whereby any traveltime anomalies are laterally shifted by the

offset distance so that they are positioned above their source on the refractor.

They include the delay time method (Gardner, 1939; Barthelmes, 1946; Barry,

1967), Hales’ method (Hales, 1958; Sjogren, 1979, 1984) and the generalized

reciprocal method (GRM) (Palmer, 1980, 1986).

These methods represent a systematic evolution of the refraction migration

concept. In the delay time method, refraction migration is applied individually to

the forward and reverse traveltime graphs, and after a series of adjustments and

corrections, an averaged delay time profile is generated. Hales’ method

essentially achieves the same results more readily with a graphical approach

using reversed traveltime data. In addition, the use of the reversed traveltime

data within a single operation reduces the effects of dip on the offset distance (as

well as the time-depths) to the horizontal layer value.

36

However, both of these methods ideally require an accurate knowledge of the

wavespeeds in the layers above the target refractor, in order to compute the

offset distance. This problem is addressed with the GRM through the use of a

series of offset distances (known as XY distances), and then selecting the

optimum value with a minimum variance criterion (Palmer, 1991). This is a

unique and useful feature of the GRM because under certain conditions, it can

permit the computation of the gross or average wavespeed model above the

refractor for a wide range of models using the optimum XY value. These models

include the single layer with a constant average wavespeed, two layers one of

which may be undetected, variable wavespeed media, and simple transverse

isotropy (Palmer 1981, 1992, 2000b, 2001a).

2.15 - Using Refraction Migration to Recognize Artifacts

The use of refraction migration was once an important part of refraction inversion

when the method was applied to deep targets in petroleum exploration. In those

applications, the offset distances could be hundreds or even thousands of

metres, and refraction migration was essential to ensure that any boreholes were

accurately sited with respect to the target.

However, with the restriction of refraction methods to predominantly shallow

targets in the last fifty years, the use of refraction migration has not always been

considered necessary because the offset distances are commonly only a few

metres or a few tens of metres at most. Furthermore, any improvements in the

resolution of the depths to the refractor were often quite subtle, especially with

large detector intervals, and so it was usually considered difficult to justify the

extra effort in using refraction migration.

The major benefit of using refraction migration in shallow investigations is in the

determination of wavespeeds in the refractor where they are commonly used as

37

a measure of rock strength. It is especially important to detect narrow zones with

low wavespeeds which can be representative of shear zones. However, the

wavespeed analysis function of the reciprocal methods generates narrow zones

with high and low wavespeeds, which are artifacts of inversion algorithm, where

there are changes in depth to the refracting interface.

The use of the GRM to separate genuine lateral variations in the refractor from

artifacts which are a product of the inversion algorithm is described in Palmer

(1991) and Palmer (2001b).

2.16 - Non-uniqueness in Determining Refractor Wavespeeds

The presentations of the wavespeed analysis function and the time-depths for a

range of XY or offset distances, represent families of geologically acceptable

starting models (Palmer, 2000c; 2000c) which satisfy the original traveltime data

(Palmer, 1980, p.49-52; 1986, p.106-107) to better than a millisecond. This is

simply another statement of the fundamental problem of non-uniqueness

common to all inversion processes (Oldenburg, 1984; Treitel and Lines, 1988),

but it is rarely if ever, addressed satisfactorily with refraction methods.

The problems of non-uniqueness are important to all refraction inversion

methods but especially so with model-based methods or tomography. The family

of starting models generated with the GRM can be useful for examining the

extent of the non-uniqueness problem with data obtained during routine surveys.

In many cases, the minimum variance criterion of the generalized reciprocal

method (GRM) can resolve whether lateral variations in the refractor wavespeeds

are genuine or if they are artifacts. However, this approach usually requires

good quality data and small detector intervals in relation to the depth of the

refractor. Commonly, detector intervals of less than about one quarter of the

38

target depth are recommended. In those cases where the effective application of

the GRM is not possible, the use the amplitudes (Palmer, 2001c) is proposed.

2.17 - Fundamental Requirements for Refraction Inversion

In summary, the performance of all methods for inverting shallow seismic

refraction data depends upon the quality of the field data, and the applicability of

the inversion model to the geological realities. Good quality redundant data are

essential for resolving many basic ambiguities. However, there are fundamental

limitations in accurately determining the wavespeed stratification from even the

most complete sets of data. Not all layers are necessarily detected in the

traveltime data, because some layers are either too thin, or the wavespeeds are

less than that in the overlying layer. Furthermore, the wavespeed stratification

cannot be determined with high precision within those layers which are detected,

because the refracted rays do not penetrate deeply enough, or because the

horizontal rather than the vertical wavespeed is measured.

The difficulties in accurately determining the inversion model indicate that as

much of the data processing as possible should be carried out in the time

domain, rather than in the depth domain. The wavespeed analysis and the time-

depth algorithms of the group of processing techniques known as the reciprocal

methods, satisfy these requirements.

In addition, there is another fundamental issue of non-uniqueness in determining

lateral variations in wavespeeds in the refractor. This requires the use of

refraction migration in order to accommodate the offset distance. However,

incorrect migration distances which would result from the use of incorrect

wavespeeds in the layers above the target refractor, can still generate results

which satisfy the traveltime data. This problem can be overcome with the use of

39

multiple migration distances with the GRM and the use of the minimum variance

criterion.

References

Acheson, C. H., 1963, Time-depth and velocity-depth relations in Western

Canada: Geophysics, 28, 894-909.

Acheson, C. H., 1981, Time-depth and velocity-depth relations in sedimentary

basins - a study based on current investigations in the Arctic Islands and an

interpretation of experience elsewhere: Geophysics, 46, 707-716.


automated wavefront reconstruction method: Geophysics 57, 378-385.

Bamford, D., and Nunn, K. R., 1979, In-situ seismic measurements of crack

anisotropy in the Carboniferous limestone of North-west England: Geophys.

Prosp., 27, 322-338.

Barker, J. A., 1991, Transport in fractured rock, in Downing, R. A., and Wilkinson,

W. B., eds., Applied groundwater hydrology: Clarendon Press, 199-216.

Barthelmes, A. J., 1946, Application of continuous profiling to refraction shooting:

Geophysics 11, 24-42.

Barry, K. M., 1967, Delay time and its application to refraction profile

interpretation: in Seismic refraction prospecting, A. W. Musgrave, ed., Society of

Exploration Geophysicists, p. 348-361.

40

Berry, J. E., 1959, Acoustic velocity in porous media: Petroleum Trans. AIME,

216, 262-270.

Birch, F., 1960, The velocity of compressional waves in rocks at 10 kilobars: J.

Geophys. Res., 65, 1083-1102.

Brandt, H., 1955, A study of the speed of sound in porous granular media: J.

Appl. Mech., 22, 479-486

Crampin, S., McGonigle, R., and Bamford, D., 1980, Estimating crack

parameters from observations of P-wave velocity anisotropy: Geophysics 45,

345-360.


McGraw-Hill Inc.

Domzalski, W., 1956, Some problems of shallow seismic refraction

investigations: Geophysical Prospecting 4, 140-166.

Edge, A. G., and Laby, T. H., 1931, The principles and practice of geophysical

prospecting: Cambridge University Press.

Ewing, M., Woollard, G. P., and Vine, A. C., 1939, Geophysical investigations in

the emerged and submerged Atlantic coastal plain, Part 3, Barnegat Bay, New

Jersey section: GSA Bulletin 50, 257-296.

Faust, L. Y., 1951, Seismic velocity as a function of depth and geologic time:

Geophysics, 16, 192-206.

Faust, L. Y., 1953, A velocity function including lithologic variation: Geophysics,

18, 271-288.

41

Gardner, L. W., 1939, An areal plan for mapping subsurface structure by

refraction shooting: Geophysics 4, 247-259.

Gassman, F., 1951, Elastic waves through a packing of spheres: Geophysics,

16, 673-685.

Gassman, F., 1953, Note on Elastic waves through a packing of spheres:

Geophysics, 16, 269.

Gunn, P., ed, 1997, Thematic issue: airborne magnetic and radiometric surveys,

AGSO Journal of Australian Geology and Geophysics, 17(2), 1-216.

Hagedoorn, J. G., 1954, A practical example of an anisotropic velocity layer:

Geophysical Prospecting 2, 52-60.

Hagedoorn, J. G., 1955, Templates for fitting smooth velocity functions to seismic

refraction and reflection data: Geophysical Prospecting 3, 325-338.





Res. Inst. Bull., 17, 118-137.



Hall, J., 1970, The correlation of seismic velocities with formations in the

southwest of Scotland: Geophys. Prosp., 18, 134-156.

42

Hamilton, E. L., 1970, Sound velocity and related properties of marine sediments,

North Pacific: J. Geophys. Res., 75, 4423-4446.

Hamilton, E. L., 1971, Elastic properties of marine sediments: J. Geophys. Res.,

76, 579-604.



Heiland, C. A., 1963, Geophysical exploration; Prentice Hall.

Iida, K., 1939, Velocity of elastic waves in granular substances: Tokyo Univ.

Earthquake Res. Inst. Bull., 17, 783-897.

Jankowsky, W., 1970, Empirical investigation of some factors affecting elastic

velocities in carbonate rocks: Geophys. Prosp., 18 103-118.

Knox, W. A., 1967, Multilayer near-surface refraction computations: in Seismic

refraction prospecting, A. W. Musgrave, ed., Society of Exploration

Geophysicists, p. 197-216.

Lankston, R. W., 1990, High-resolution refraction seismic data acquisition and

interpretation: in Geotechnical and environmental geophysics, S. H. Ward, ed.,

Investigations in geophysics no. 5, vol. 1, 45-74, Society of Exploration

Geophysicists.



43

Maillet, R., and Bazerque, J., 1931, La prospection sismique du sous-sol:

Annales des Mines 20, 314

Mayne, W. H., 1962, Common-reflection-point horizontal data-stacking

techniques: Geophysics, 27, 927-938.

McCollum, B., and Snell, F. A., 1932, Asymmetry of sound velocity in stratified

formations: in Early geophysical papers, Society of Exploration Geophysicists,

Tulsa, 216-227.

Menke, W., 1989, Geophysical data analysis: discrete inverse theory: Academic

Press, Inc.

Merrick, N. P., Odins, J. A., and Greenhalgh, S. A., 1978, A blind zone solution to

the problem of hidden layers within a sequence of horizontal or dipping

refractors: Geophysical Prospecting 26, 703-721.

Miller, K. C., Harder, S. H., and Adams, D. C., and O'Donnell, T., 1998,

Integrating high-resolution refraction data into near-surface seismic reflection

data processing and interpretation: Geophysics 63, 1339-1347.

Musgrave, A. W., 1967, Seismic refraction prospecting: Society of Exploration

Geophysicists, Tulsa.


11, 12-19.


Company.

44

Oldenburg, D. W., 1984, An introduction to linear inverse theory: Trans IEEE

Geoscience and Remote Sensing, GE-22(6), 666.



Palmer, D., 1981, An introduction to the generalized reciprocal method of seismic

refraction interpretation: Geophysics 46, 1508-1518.



Palmer, D, 1990, The generalized reciprocal method – an integrated approach to

shallow refraction seismology: Exploration Geophysics 21, 33-44.

Palmer, D., 1991, The resolution of narrow low-velocity zones with the

generalized reciprocal method: Geophysical Prospecting 39, 1031-1060.

Palmer, D., 1992, Is forward modelling as efficacious as minimum variance for

refraction inversion?: Exploration Geophysics 23, 261-266, 521.

Palmer, D, 2000a, Can new acquisition methods improve signal-to-noise ratios

with seismic refraction techniques?: Exploration Geophysics, 31, 275-300.

Palmer, D., 2000b, The measurement of weak anisotropy with the generalized

reciprocal method: Geophysics, 65, 1583-1591.

Palmer, D., 2000c, Can amplitudes resolve ambiguities in refraction inversion?:

Exploration Geophysics 31, 304-309.

Palmer, D., 2000d, Starting models for refraction inversion: submitted.

45

Palmer, D., 2001a, Model determination for refraction inversion: submitted.

Palmer, D, 2001b, Comments on “A brief study of the generalized reciprocal

method and some of the limitations of the method” by Bengt Sjögren,

Geophysical. Prospecting, submitted.

Palmer, D, 2001c, Resolving refractor ambiguities with amplitudes, Geophysics66, 1590-1593.

Paterson, N. R., 1956, Seismic wave propagation in porous granular media:




Sheriff, R. E., and Geldart, L. P., 1995, Exploration Seismology, 2nd edition:

Cambridge University Press.



Sjogren, B., 1980, The law of parallelism in refraction shooting: Geophysical

Prospecting 28, 716-743.


Soske, J. L., 1959, The blind zone problem in engineering geophysics:


Stefani, J. P., 1995, Turning-ray tomography: Geophysics 60, 1917-1929.

46


Bulletin, 14, 185-200.

Treitel, S., and Lines, L., 1988, Geophysical examples of inversion (with a grain

of salt): The Leading Edge 7, 32-35.

Walker, C. S., and Win, M. A., 1997, A new standard in the practice of

engineering seismic refraction, in McCann, D. M., Eddleston, M., Fleming, P. J.,

and Reeves, G. M., eds., Modern geophysics in engineering geology: The

Geological Society, 391-398.

Whiteley, R. J., and Greenhalgh, S. A., 1979, Velocity inversion and the shallow

seismic refraction method: Geoexploration 17, 125-141.

White, J. E., and Sengbush, R. L.,, 1953, Velocity measurements in near surface

formations: Geophysics, 18, 54-69.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1956, Elastic wave velocities in

heterogeneous and porous media: Geophysics, 21, 41-70.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1958, An experimental

investigation affecting elastic wave velocities in porous media: Geophysics, 23,

459-593.



Zhu, X., Sixta, D. P., and Andstman, B. G., 1992, Tomostatics: turning-ray

tomography + static corrections: The Leading Edge 11, 15-23.

47

Chapter 3

Imaging Refractors with theConvolution Section

3.1 - Summary

Seismic refraction data are characterized by large moveouts between adjacent

traces and large amplitude variations across the refraction spread. The

moveouts are the result of the predominantly horizontally traveling trajectories of

refraction signals, while the amplitude variations are the result of the rapid

geometric spreading factor, which is at least the reciprocal of the distance

squared.

The large range of refraction amplitudes produces considerable variation in

signal-to-noise (S/N) ratios. Inversion methods which use traveltimes only,

employ data with a wide range of accuracies, which are related to the variations

in the S/N ratios.

The time section, generated by convolving forward and reverse seismic traces,

addresses both issues of large moveouts and large amplitude variations.

The addition of the phase spectra with convolution effectively adds the forward

and reverse traveltimes. The convolution section shows the structural features of

the refractor, without the moveouts related to the source to detector distances.

48

Unlike the application of a linear moveout correction or reduction, a measure of

the refractor wavespeed is not required beforehand.

The multiplication of the amplitude spectra with convolution, compensates for the

effects of geometric spreading and dipping interfaces to a good first

approximation, and it is sufficient to facilitate recognition of amplitude variations

related to geological causes. These amplitude effects are not as easily

recognized in the shot records.

The convolution section can be generated very rapidly from shot records without

a detailed knowledge of the wavespeeds in either the refractor or the overburden.

3.2 - Introduction

In this study, I propose the application of full trace processing as one method of

addressing the fundamental issue of the large variations in signal-to-noise (S/N)

ratios with seismic refraction data.

I begin with a discussion of the effects of geometric spreading on two shot

records from a shallow seismic refraction survey. The data demonstrate that the

spreading is large, it is not adequately described with the reciprocal of the

distance squared expression and it dominates any geological effects. These

large variations in amplitudes result in large variations in S/N ratios and in turn, in

large variations in the accuracies of the measured traveltimes.

Next, I briefly review various methods of full trace processing and then propose

the generation of a refraction time cross-section by the convolution of forward

and reverse traces. I demonstrate that convolution provides very good

compensation for geometric spreading and for the variations in amplitudes

caused by changes in the dip of the refracting interface.

49

Figure 3.1: Field record for shot point at station 1, presented at constant gain.

The large drop in amplitudes from about station 51 can be clearly seen.

Finally, I present a convolution section across a complex refractor in which there

are large variations in depths and wavespeeds. The image presents the same

50

time structure that would be obtained with the standard methods of processing

traveltime data, while the amplitudes are a function of the head coefficient, which

is the expression relating the refraction amplitudes to the petrophysical

parameters of the upper layer and the refractor.

3.3 - The Large Variations in Signal-to-Noise Ratios withRefraction Data

A long standing problem with the acquisition of seismic refraction data is the

relatively high source energy requirements, which are necessary to compensate

for the rapid decrease of signal amplitudes with distance. For signals which have

traveled several wavelengths within a thick refractor with a plane horizontal

interface, the geometrical spreading factor is approximately the reciprocal of the

distance squared (Grant and West, 1965), and it is much more rapid than the

equivalent function for reflected signals which is the reciprocal of the distance

traveled.

Figures 3.1 and 3.2 are two shot records presented at a constant gain, and

illustrate the large variations in S/N ratios. The shot points are offset

approximately 120 m from each end of a line of 48 detectors, which are 5m apart.

Qualitatively, each shot record exhibits high amplitudes close to the shot point,

followed by greatly reduced amplitudes from about station 51 onwards. Figure

3.3 shows the amplitudes of the first troughs of the forward shot data, normalized

to the value at station 50. As expected, the amplitudes show the rapid fall with

distance from the shot, with the variation between the near and far traces being a

factor of 20, or 26 decibels. The reduction with distance is much more rapid than

the reciprocal of the distance squared spreading function, which is also shown in

Figure 3.3, and the reciprocal of the cube of the distance appears to be a much

closer approximation.

51


The large drop in amplitudes from about station 51 is even more pronounced

than on the previous record.

52

Figure 3.3: Amplitudes of the first trough measured on the forward shot record,

together with the reciprocals of the distance squared and distance cubed

geometric effects.

A similar result occurs with the reverse shot data in Figure 3.4. The amplitudes

decrease much more rapidly than a reciprocal of the distance squared function,

53

and in this case, the variation between the near and far traces is a factor of 60, or

36 decibels. Again, a reciprocal of the distance cubed function is a better

approximation, although the fit with the low amplitude values is not particularly

close.

Figures 3.3 and 3.4 demonstrate that the reduction in amplitude with distance is

large, and that it dominates any secondary effect caused by geological

variations. An interpretation of the traveltime data derived from these shot

records is presented in Chapter 5 (Palmer, 2000c), and it shows rapid changes in

the depth to the main refractor, which in this case is the base of the weathering,

as well as large variations in the wavespeed of the refractor. Accordingly, the

challenge is to effectively separate the amplitude variations related to geological

factors from those caused by geometrical spreading.

In addition, Figures 3.3 and 3.4 demonstrate the difficulties in employing

corrections for geometrical spreading based on widely accepted theoretical

treatments. The reciprocal of the distance squared function only applies to

homogeneous media separated by plane horizontal interfaces, and only after the

signal has traveled 5-6 times the predominate wavelength of the pulse (Donato,

1964). These latter results are in keeping with model studies (Hatherly, 1982),

and are the norm, rather than the exception in most shallow refraction surveys.

Furthermore, this example highlights the very large variations in S/N ratios at

each detector for the usual ensemble of shot points and in turn, the considerable

range of accuracies in the measured traveltime data for most refraction surveys.

At any given location, a detector will be close to a source, and the measured

traveltimes will be comparatively accurate, because of the high S/N ratio.

However for the traveltime in the reverse direction, the source-to-receiver

distance will be much larger, and the accuracy will be greatly reduced, because

of the lower S/N ratio. Such large variations in accuracies adversely affect the

quality of data processing with any method.

54

Figure 3.4: Amplitudes of the first trough measured on the reverse shot record,

together with the reciprocals of the distance squared and distance cubed

geometric effects.

Most methods for the processing of seismic refraction data use simple scalar first

arrival traveltimes, and the problem is normally perceived as achieving

55

satisfactory, rather than uniform S/N ratios. Commonly, a simple gain function is

applied to adjust amplitudes to a convenient level, but this still does not alter the

large variations in S/N ratios. With statics corrections for reflection surveys,

typically a limited source-to-detector interval over which the refraction data are of

sufficient quality, is selected. For geotechnical, groundwater and environmental

studies, the source energy levels are usually increased as far as environmental

and cultural factors permit, or vertical stacking with repetitive sources is

employed.

The following section reviews full trace processing and the issue of the large

variations in S/N ratios.

3.4 - Full Trace Processing Of Refraction Data

Perhaps the simplest approach to full trace processing, is the application of a

linear moveout (LMO) correction to each shot record. With this approach, which

is also known as reduction, each refraction trace is shifted or reduced by a time

equal to the source-to-detector distance, divided by a velocity, which is usually

the known or estimated wavespeed in the target of interest, (Sheriff and Geldart,

1995, Fig. 11.10). The result is normally presented as a set of traces for which

the first arrivals occur at the sum of the source point and detector delay times.

One benefit of this presentation is that it maps any variations in the target depth

in terms of the delay times.

However, this process does not address the basic issue of the large variation in

S/N ratios across the refraction recording spread. The degradation of the arrivals

at the more distant detectors is usually very significant, particularly with crustal

and earthquake studies. Furthermore, it is usually inconvenient to include any

reverse shot records within the same presentation, and therefore to readily

accommodate any lateral variations in wavespeed with irregular refractors.

56

Other approaches are the broadside and fan shooting methods, in which the

source is usually located at an offset point, orthogonal to the center of a linear or

circular array of detectors. Since the source-to-detector distances are essentially

constant, the geometric spreading effects are also constant, and there are much

smaller variations in the S/N ratios from trace to trace. Furthermore, corrections

for the source-to-detector distances, such as with an LMO, in order to emphasize

any structural anomalies in the target refractor, are not essential because such

time shifts are virtually constant also. Examples of the imaging or migration of

broadside data (Mcquillan et al, 1979, Figure 7/15), indicate some of the

possibilities of full trace processing of refraction data.

These methods represent the first true 3D seismic methods for exploration and

pre-date the current reflection 3D methods by many decades (Sheriff and

Geldart, 1995). As such, they will eventually be incorporated into the routine

refraction methods of the future. However, the methods described above do

have two major limitations. They do not determine wavespeeds in the refractor,

nor are they able to separate source and receiver delay times without additional

information, such as borehole control, or the simultaneous recording of a

conventional in-line profile orthogonal to the broadside pattern.

A recent method of imaging refractors with forward and reverse data, is

downward continuation using the tau-p transform (Hill, 1987). It can achieve

good resolution by accommodating diffraction and shadow zone effects. Like all

wavefront methods, it requires an accurate knowledge of the wavespeed of the

upper layer, but this is probably one of the least reliable parameters determined

in most refraction surveys (Chapter 2; Palmer, 1992; Appendix 2).

In this study, I describe the generation of a refraction time section through the

convolution of forward and reverse traces as an effective method of addressing

the fundamental issues of large S/N variations and large moveouts with refraction

57

data. The result, the refraction convolution section (RCS), is similar in

appearance to the familiar reflection time cross section, in which the results are

displayed for example, as a series of wiggle traces.

There are several benefits to processing with this approach. The first is that it is

extremely rapid, avoiding in particular the familiar time consuming tasks of

determining first arrival traveltimes. The second is that little, if any, a priori

information on overburden or refractor wavespeeds is required, although of

course such information is essential for the generation of final depth cross

sections. Accordingly, the convolution section is a very convenient presentation

for an assessment of the quality of processing using other detailed methods,

such as tomography.

In addition, the approximate compensation for large variations in the S/N ratios

facilitates the vertically stacking of refraction data, in a manner analogous to the

common midpoint method with reflection data. This in turn, suggests more

efficient methods of data acquisition with lower environmental impact, particularly

for geotechnical investigations (Palmer, 2000a).

The benefits to interpretation are that the amplitudes obtained through

convolution are essentially a function of the refractor wavespeeds and/or

densities, rather than the source to detector separation. In general, high

wavespeeds and/or densities in the refractor produce low amplitudes. This

relationship between amplitudes and contrasts in the parameters of the refractor

and the overburden provides an additional valuable method for resolving

ambiguities, especially with model-based methods of refraction inversion

(Palmer, 2000c).

The concept of the convolution section was first proposed by Palmer (1976), but

initial tests with Vibroseis data were not especially encouraging, because of

correlation noise before the first breaks (K B S Burke, pers. comm., circa 1982).

58

However, the method was later successfully applied to synthetic data (Taner et

al, 1992).

3.5 - Imaging The Refractor Interface Through The Addition ofForward And Reverse Traveltimes

The unambiguous resolution of dip with plane interfaces or structure with

irregular interfaces, and variable wavespeed within the refractor, usually requires

forward and reverse traveltime data, or off-end data with a high density of source

points, from which the equivalent reversed traveltime data can be generated.

Accordingly, the majority of refraction processing methods explicitly identify and

use forward and reverse traveltimes within their algorithms. These methods

include the wavefront construction methods (Thornburg, 1930; Rockwell, 1967;

Aldridge and Oldenburg, 1992), the conventional reciprocal method (CRM),

(Hawkins, 1961), which is also known as the ABC method in the Americas,

(Nettleton, 1940; Dobrin, 1976), Hagiwara's method in Japan, (Hagiwara and

Omote, 1939), and the plus-minus method in Europe, (Hagedoorn, 1959), Hales'

method, (Hales, 1958; Sjogren, 1979; Sjogren, 1984), and the generalized

reciprocal method (GRM), (Palmer, 1980; Palmer, 1986).

There are minor differences in detail between the algorithms for each of these

methods. These differences include whether the reciprocal time, the time from

the forward shot point to the reverse shot point, is used, the inclusion of the

factor of a half, or whether the offset distance, which is the horizontal separation

between the point of refraction on the interface and the detector position on the

surface, is accommodated through the operation known as refraction migration

(Palmer, 1986, p.74-80).

59

Figure 3.5: Traveltime data for a line crossing a major shear zone in

southeastern Australia. The station interval is 5 m. The traveltimes for the offset

shots which are offset 120 m from either end at stations 1 and 97, are shown in

bold.

Nevertheless, each of these methods includes an algorithm in which the forward

and reverse traveltimes are added, in order to obtain a measure of the depth to

the refractor in units of time. This process of addition averages most of the dip

effects to the horizontal layer approximations and replaces the moveout with a

constant value for all detectors between the forward and reverse source points.

With the CRM and GRM, this constant is then removed by subtracting the

reciprocal time. Finally, the result is halved to derive a parameter which is

60

essentially the mean of the forward and reverse delay times. The result is known

as the time-depth, where

time-depth = (tforward + treverse - treciprocal)/2. (3.1)

Figure 3.5 presents the traveltime data recorded across a major shear zone in

southeastern Australia with a set of collinear shots and receivers. The station

interval is 5 m, and the shot points are at stations 1 which is offset 120 m to the

left, 25, 49, 73 and 97 which is offset 120 m to the right. The traveltimes indicate

a three layer model consisting of a thin surface layer of friable soil with a

wavespeed of about 400 m/s, a thicker layer of weathered material with a

wavespeed of approximately 700 m/s, and a main refractor with an irregular

interface.

Figure 3.6: Time-depths computed from traveltime data with shot points offset

120 m from each end of the geophone array at stations 1 and 97.

An example of the application of equation 3.1 is shown in Figure 3.6, using the

traveltime data measured from the shot records shown in Figures 3.1 and 3.2,

61

and summarized in bold in Figure 3.5. The time-depths have been computed

with a reciprocal time of 147 ms, (Palmer, 1980, equation 33), and an optimum

XY value of 5 meters.

The XY value is the separation between the pairs of forward and reverse

traveltimes used in equation 3.1, and it is usually a multiple of the detector

spacing. The optimum XY value is obtained with the minimum variance criterion

described elsewhere (Palmer, 1980, p.31-35) and it is the sum of the forward and

reverse offset distances. This sum is essentially independent of the dip angles,

unlike the individual forward and reverse components. At the optimum XY value,

the forward and reverse rays are refracted from near the same point on the

refractor and the smoothing effects of other XY values are minimized.

3.6 - The Addition of Traveltimes With Convolution

The traditional methods for the inversion of refraction data, can be categorized by

how the addition of the forward and reverse traveltimes is implemented. The

wavefront construction and Hales' methods achieve it graphically, while the CRM

and GRM achieve it with the simple addition of two numbers.

In this study, I demonstrate the use of convolution of forward and reverse traces

to effectively achieve the addition.

The convolution process has usually been associated with filtering. Its effect can

be described in the frequency domain, as the multiplication of the amplitude

spectra and the addition of the phase spectra of the two functions.

A similar result occurs with the convolution of two seismic refraction traces. The

amplitude spectra are multiplied, and the arrival times, which are contained within

the phase spectra, are added.

62

Alternatively, the addition of first arrival times with convolution can be

demonstrated with the z transform notation (Sheriff and Geldart, 1995). The

digitized seismic trace can be represented as a polynomial in z, in which the

exponent represents the sample number. The forward trace F(z) is given by

F(z) = fm zm + fm+1 zm+1 + fm+2 zm+2 + .... (3.2)

where fj = 0 for j < m.

The forward traveltime is m, since fm is the first non-zero amplitude for the

forward trace and therefore represents the onset of seismic energy. Similarly,

the reverse trace R(z) is given by

R(z) = rn zn + rn+1 zn+1 + rn+2 zn+2 + .... (3.3)

where rj =0 for j < n. In this case, the reverse traveltime is n, since rn is the first

non-zero amplitude.

Convolution in the z domain is achieved by polynomial multiplication, ie.

F(z) * R(z) = fm rn zm + n + (fm rn+1 + fm+1 rn) zm + n + 1

+ (fm rn+2+ fm+1 rn+1 + fm+2 rn) zm+n+2 + .... (3.4)

It can be seen that the first non-zero coefficient is fm rn and it occurs at the time m

+ n, which is at the sum of the forward and reverse traveltimes.

63

Figure 3.7: Convolution section generated by convolving forward and reverse

shot records. The traces are presented at constant gain with no trace

equalization.

64

The convolution section generated with the shot records in Figures 3.1 and 3.2

and an XY separation of 5 m, is shown in Figure 3.7. Each trace in fact

represents the time-depth, as both the subtraction of the reciprocal time and the

halving of the time scale have been carried out. (These operations were readily

achieved with software for processing seismic reflection data, by treating the

reciprocal time as a static correction and by halving the sampling interval in the

trace headers.)

It is immediately apparent that the moveout has been removed by the

convolution process. The convolution section shows the same structure on the

refractor interface as that obtained in Figure 3.6 with the traveltime data.

In addition, perhaps the other striking effect of the convolution section is the

convenient presentation of the amplitude information. It is clear that convolution

has compensated for the very large amplitude variations related to geometrical

spreading and other factors with the shot records, and that the signal-to-noise

ratios of the convolved traces are very similar. Although the compensation is not

exact, as will be shown below, it is still sufficient to permit the recognition of

amplitude variations related to geological factors.

However, the interface computed using traveltimes in Figure 3.6 is about 10 ms

shallower than that recognizable from the convolution section in Figure 3.7. This

discrepancy arises from the various gain functions used with each approach.

The time-depths in Figure 3.6 were computed with traveltimes at which the first

onset of seismic energy was detected on the shot records, using as high a gain

as was possible without the background noise causing any detectable deflections

before the first breaks. This gain is usually sufficient to cause clipping of most of

the seismic data after the first arrivals. On the other hand, the presentation gain

in Figure 3.7 is much lower, and it has been selected to permit the examination of

the first few cycles after the computed time-depth.

65

3.7 - The Effects of Geometrical Spreading on the ConvolutionSection Amplitudes

The shot record amplitudes shown in Figures 3.3 and 3.4 demonstrate the very

large variations due to geometrical spreading, as well as the difficulties in

selecting an appropriate mathematical description. Figure 3.8 shows normalized

theoretical amplitudes for reciprocal distance squared and reciprocal distance

cubed functions for a shot at station 1. The values are normalized to that at

station 72, which is the most distant detector from the shot at station 1. The

variation in amplitude between the first and last detectors is about 19 db for

reciprocal distance squared spreading, while it is 28.6 db for the reciprocal

distance cubed case, with an average of about 24 db.

Figure 3.8 also shows the geometrical effects for the convolved traces, obtained

with equation 3.5, viz.

Geometric factor convolved trace = 1 / (Xn (L-X)n) (3.5)

where, X is the distance from one shot point to the detector, L is the shot point to

shot point distance, which in this case is 480 m, and n is 2 for the reciprocal

distance squared and 3 for the reciprocal distance cubed cases. The convolved

amplitudes have been normalized to the minimum values which are at station 49,

the midpoint of the shot point to shot point distance. The maximum variation in

the convolved amplitudes is between the ends and the midpoint of the detector

array, and is 5 db for n equal to 2 and 7.5 db for n equal to 3, with an average of

about 6 db.

It is clear that convolution has reduced the effects of geometrical spreading by

approximately 18 db, but that a residual geometric effect of about 6 db still

remains. However, the reduction is sufficient to be able to recognize amplitude

variations related to geological effects. This is shown in Figure 3.9, with the

66

convolved amplitudes as well as the convolved amplitudes which have been

corrected for the residual geometric spreading with equation 2.5 for n equal to

both 2 and 3 and normalized to the value midway between the two shot points.

The first positive amplitudes are low and erratic, and so the absolute values of

the following first negative which are much larger and more consistent, are used.

Figure 3.8: Geometric spreading factors for shot records with the shot point at

station 1, and the convolution section for shot points at stations 1 and 97, for

reciprocal distance squared and cubed functions.

67

Figure 3.9: First positive and negative normalized amplitudes measured on the

convolution section. The first negative amplitudes are also shown with inverse

distance squared and inverse distance cubed geometric corrections.

68

Figure 3.10: The product of the forward and reverse amplitudes of the first

trough measured on the shot records, together with the product corrected for

inverse distance squared and inverse distance cubed geometric effects.

Figure 3.10 shows the product of the forward and reverse amplitudes presented

in Figures 3.3 and 3.4, together with the values corrected for the geometric effect

with equation 3.5. The pattern of amplitude variations is similar to that in Figure

69

3.9, confirming that convolution has in fact multiplied the amplitudes, and that the

product has greatly reduced the geometrical effect.

In both Figures 3.9 and 3.10, it is possible to separate the convolved and

multiplied amplitudes into four regions which correlate well with those recognized

in chapter 5, (Palmer, 2001), using wavespeed and depth. Correction of the

convolved and multiplied amplitude products with the theoretical geometrical

effects improves the ease in recognizing the four regions, but does not alter the

general features of the amplitudes.

3.8 - Effects Of Refractor Dip On Convolution Amplitudes

The convolution of forward and reverse traces provides an approximate

correction for the effects of a dipping interface on the amplitudes measured with

vertical component geophones. Suppose the angle from the vertical at which a

critically refracted ray approaches the surface is α for a horizontal refractor. The

vertical component measured with the standard geophone will be the forward or

reverse amplitude multiplied by cosα. Therefore, the convolved amplitude will be

multiplied by cos2α, ie.

Convolved Amphorizontal refractor = cos2α Ampforward Ampreverse (3.6)

Next, suppose the refractor has a dip of θ. The vertical component measured will

be the shot amplitude multiplied by cos(α+θ) in one direction, cos(α-θ) in the

reverse direction.

Vertical Shot Amp dipping refractor = cos(α±θ) Amp (3.7)

The vertical component of the convolved amplitude is given by equation 3.8, viz.

70

Convolved Ampdipping refractor =(cos2α cos2θ - sin2α sin2θ) Ampforward Ampreverse

(3.8)

For small dip angles, say less than about fifteen degrees, the second order terms

in sinθ can be neglected, while the cos2θ term is approximately one. Therefore,

to sufficient accuracy the product of the forward and reverse amplitudes achieved

with convolution is given by

Convolved Ampdipping refractor = cos2α Ampforward Ampreverse (3.9)

Accordingly, amplitudes computed for plane horizontal refractors (Heelan, 1953;

Werth, 1967) can still be usefully applied to dipping layers when convolution is

employed.

3.9 - Conclusions

Seismic refraction acquisition techniques are characterised by large source to

receiver distances. Commonly, these distances are greater than about four

times the depth of the target, whereas for reflection methods, the equivalent

distances are less than the target depth. The large distances produce

commensurately large moveouts between adjacent traces and large amplitude

variations between the near and far traces.

The wide range of refraction amplitudes is the result of the rapid geometric

spreading factor, which is at least the reciprocal of the distance squared, and it

produces considerable variation in S/N ratios. Accordingly, most refraction

inversion methods use traveltime data with widely varying accuracies, which are

related to the large variations in signal-to-noise ratios.

71

The time section, generated by convolving forward and reverse seismic traces

together with a static shift equal to the reciprocal time, addresses both issues of

large moveouts between adjacent traces and large amplitude variations.

The addition of the phase spectra with convolution effectively adds the forward

and reverse traveltimes. This process of addition is common to most of the

standard techniques for the inversion of refraction data. The convolution section

after shifting by the reciprocal time, shows the same structural features of the

refractor in units of time, as is obtained with the standard approaches.

Furthermore, the convolution section can be generated without a prior knowledge

of the wavespeeds in either the upper layer, as is required with the downward

continuation methods, or in the refractor, as is required with the application of a

linear moveout correction or reduction. This latter is especially important where

there are significant lateral variations in the wavespeed of the refractor.

The multiplication of the amplitude spectra with convolution, to a good first

approximation, effectively compensates for the effects of geometric spreading,

which can be significantly larger than the commonly assumed reciprocal of the

distance squared function. This compensation is generally sufficient to be able to

recognize amplitude variations related to geological causes, which are not as

easily detected in the shot records. The correlation of any amplitude variations

with the structural variations on the interface of the refractor can be more

conveniently and more rapidly carried out using the convolution section, than for

example by multiplying amplitudes measured on the shot records.

If necessary, a geometric correction based on the product of a reciprocal of the

distance power function in the forward and reverse directions, can be applied to

the convolution section. This correction exhibits a much reduced variation

72

compared with those for the individual shot records, and it is most useful near the

shot points where it can have a value of up to a factor of about 2, or 6 decibels.

The ease and convenience of generating the convolution section facilitate its

inclusion in the routine processing of seismic refraction data using any method.

3.10 - References


automated wavefront reconstruction method: Geophysics, 57, 378-385.


McGraw-Hill Inc.

Donato, R. J., 1964, Amplitude of P head waves: J. Acoust. Soc. Am., 36, 19-25.

Grant, F. S., and West, G. F., 1965, Interpretation theory in applied geophysics:

McGraw-Hill Inc.


sections: Geophys. Prosp, 7, 158-182.



Res. Inst. Bull., 17, 118-137.



73

Hatherly, P. J., 1982, Wave equation modelling for the shallow seismic refraction

method: Expl. Geophys., 13, 26-34.



Heelan, P. A., 1953, On the theory of head waves: Geophysics, 18, 871-893.

Hill, N. R., 1987, Downward continuation of refracted arrivals to determine

shallow structure: Geophysics, 52, 1188-1198.

McQuillan, R., Bacon, M., and Barclay, W., 1979, An introduction to seismic

interpretation: Graham & Trotman Limited.


Company Inc.

Palmer, D., 1976, An application of the time section in shallow seismic

refraction studies: Master's thesis, The University of Sydney.





Palmer, D., 1992, Is forward modeling as efficacious as minimum variance for

refraction inversion?: Explor. Geophys. 23, 261-266, 521.

Palmer, D., 2000a, Can new acquisition methods improve signal-to-noise ratios

with seismic refraction techniques?: Explor. Geophys., 31, 275-300.

74

Palmer, D., 2000b, Can amplitudes resolve ambiguities in refraction inversion?:

Explor. Geophys., 31, 304-309.

Palmer, D., 2001, Resolving Refractor Ambiguities With Amplitudes: Geophysics

66, 1590-1593.








Taner, M. T., Matsuoka, M., Baysal, E., Lu, L., and Yilmaz, O., 1992, Imaging

with refractive waves: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys.


Bulletin, 14, 185-200.

Werth, G. A., 1967, Method for calculating the amplitude of the refraction arrival,

in Musgrave, A. W., Ed., Seismic refraction prospecting: Society of Exploration

Geophysicists, 119-137.

75

Chapter 4

Starting Models For RefractionInversion

4.1 - Summary

The algorithms of the generalized reciprocal method (GRM) are applied to a set

of reversed traveltime data for a two layer model with a synclinal refractor

interface, in order to generate a family of starting models. Each starting model

shows much the same depths as the original model, but each has a narrow zone

in the refractor with an anomalous wavespeed. The traveltimes through each of

the starting models differ from those for the original model by less than a

millisecond. If any were used as starting models for tomographic or model-based

inversion, then the final result would show only minor differences. This example

demonstrates the non-uniqueness of model-based inversion.

In order to address the issues of non-uniqueness with model-based inversion, it

is recommended that a range of starting models, such as those which can be

generated with the GRM, be used.

Alternatively, other approaches, which aim to resolve these ambiguities, can be

employed. In many cases, the minimum variance criterion of the GRM can

resolve whether lateral variations in the refractor wavespeeds are genuine, or

whether they are artifacts of the inversion algorithm. In addition, it is proposed

76

that the amplitudes of the refraction convolution section can indicate where there

are genuine changes in the wavespeed of the refractor, because the amplitudes

are a function of the contrasts in wavespeeds between the refractor and the layer

above.

4.2 - Introduction

The inversion of seismic refraction data with model-based or tomographic

methods consists of deriving a starting model of the subsurface with standard

algorithms, and then testing it by comparing the computed traveltimes of the

model with the observed data. If there are differences, then the model is

adjusted until an acceptable agreement is achieved. Commonly, several

iterations may be required.

While most geophysicists are satisfied to generate a model which reproduces the

observations, there is the fundamental theoretical reality that an infinite number

of solutions can reproduce the data (Oldenburg, 1984; Treitel and Lines, 1988),

although not all of these solutions will be geologically plausible. This non-

uniqueness becomes more significant where the data are inaccurate and

incomplete as is often the case with field data, and where models, which do not

fit the data precisely, are accepted.

The issues of non-uniqueness are not usually considered with shallow refraction

tomography. The non-uniqueness includes the wavespeeds in both the

overburden and the refractor and they are often inter-related. One compelling

example is the somewhat paradoxical situation of the poor determination of


from that layer (Lanz et al, 1998, Figure 8). This situation is at variance with the

experiences of most seismologists using more traditional methods of refraction

77

processing, and it is probably related to the use of a linear wavespeed function

with a very high gradient in the upper layer.

Previous studies (Hagedoorn, 1955) have demonstrated the ambiguities in

determining the wavespeed stratification within a single layer above the refractor.

Even in the absence of undetected layers, generally known as hidden layers

within the blind zone and reversals in wavespeed, it is not possible to accurately

specify the mathematical function which describes the wavespeed in the

overlying layer. As a result, there is a large range in the depths to the refractor

computed with the various mathematical functions which can be fitted to the first

arrival traveltime data with acceptable accuracy.

Palmer (1992, Appendix 2) has demonstrated, that when the refractor interface is

sufficiently irregular in relation to its depth, the generalized reciprocal method

(GRM) (Palmer, 1980; 1986), can significantly improve the accuracy of the depth

computations for a wide range of mathematical functions in the upper layer. The

mathematical functions include wavespeed reversals and transverse isotropy,

which are not adequately addressed with other approaches.

This study examines the non-uniqueness in the determination of the wavespeeds

in the refractor. In these cases, the non-uniqueness is usually related to the

starting model for the inversion process and in turn, to the selection of the

inversion algorithm used to generate that model.

I demonstrate that a range of geologically plausible starting models can be

readily generated from the one set of reversed traveltime data with the algorithms

of the GRM, and that each of these models fits the data to an acceptable

accuracy of a few milliseconds. I conclude that the selection of the initial starting

model is critical with model-based methods of refraction inversion. I further

conclude that the issues of non-uniqueness, which currently are not adequately

examined with most model-based methods for inverting refracting data, can be

78

addressed by testing a family of starting models which can be generated for

example, with the GRM. Finally, I propose the use of two methods for resolving

ambiguities, namely, the minimum variance criterion of the GRM and the use of

amplitudes in the refraction convolution section.

4.3 - Inversion Of A Two Layer Model With The GRM Algorithms

Figure 4.1 shows a simple two layer model with isotropic homogeneous seismic

wavespeeds separated by a synclinal interface. It represents an obvious step for

increasing the complexity of the interpretation model over the simple two layer

case with plane interfaces. The dips of the sloping interfaces are ± 9.2°, which

are relatively large. This model was used to generate the traveltime data shown

in Figure 4.2, which in turn were processed or inverted using the two algorithms

of the GRM for computing time-depths and refractor wavespeeds.

Figure 4.1: Two layer model with a synclinal interface.

The time-depth tG, at G is given by equation 4.1, viz.

tG = (tAY + tBX - tAB - XY/Vn)/2 (4.1)

79

where A, X, G, Y, and B are collinear, A and B are source points, X and Y are

detectors and G is midway between X and Y, tAY is the traveltime from A to Y, tBX

is the traveltime from B to X, tAB is the reciprocal time, the traveltime from the

source at A to the source at B, and Vn is the wavespeed in the refractor.

Figure 4.2: Traveltimes generated for two layer model with a synclinal interface

shown in Figure 4.1. The station spacing is 5 m.

Figure 4.3 shows the time-depths computed for XY values from zero to 30 m in

increments of 5 m, which is the detector spacing. Each set of time-depths shows

the synclinal structure of the refractor, although there are minor differences in

detail around the hinge point at station 12.

80

Figure 4.3: Time-depths computed for the synclinal model in Figure 4.1 for a

range of XY values. The reciprocal times have been systematically decreased

with increasing XY value, in order to separate each set of graphs for clarity.

The second function computed with the GRM is the refractor wavespeed analysis

function tV, given by equation 4.2, viz.

tV = (tAY - tBX + tAB)/ 2 (4.2)

Two parameters can be derived from this function. The first is the wavespeed in

the refractor Vn, from the reciprocal of the gradient, ie

d/dx tV = 1 / Vn (4.3)

81

Figure 4.4: Wavespeed analysis function computed for the synclinal model in

Figure 4.1 for a range of XY values.

Figure 4.4 shows the wavespeed analysis function for the same range of XY

values used in Figure 4.3. Each set of graphs for a given XY value shows the

same wavespeed in the refractor of 2820 m/s, except for a short interval around

the hinge point at station 12. Here the wavespeed ranges from as low as 2000

m/s to as high as 4800 m/s.

The second parameter determined from the wavespeed analysis function is the

intercept of tV at the source point, which is the time-depth tA at a distance of ½XY

from the source point, ie

82

tA = tV |x=0 (4.4)

For this two layer model, the time-depths presented in Figure 4.3 can be

converted into depths zG, with equation 4.5, viz.

zG = tG / DCF (4.5)


depth, is given by:

DCF = V Vn / (Vn2 - V2)½ (4.6)

or

DCF = V / cos i (4.7)

V is the average wavespeed above the refractor and

sin i = V / Vn (4.8)

Figure 4.5: A summary of the starting models which can be generated from the

traveltime data for the synclinal model in Figure 4.1. The region with the variable

wavespeeds near the hinge point of the interface is an artifact.

83

Figure 4.5 is a summary of the range of depth models which can be generated

with the XY values from zero to 30 m. Although the depth sections reproduce the

synclinal structure of the original model, there is an additional segment in the

second layer with wavespeeds from 2000 m/s to 4800 m/s which is not present in

the original model. This additional segment could represent a weathered dyke or

a shear zone for the low wavespeed cases or an unweathered dyke or a silicified

shear zone for the high wavespeed cases. Therefore, all models are geologically

both plausible and significant. Nevertheless, they are artifacts generated by

equation 4.2, the refractor wavespeed analysis algorithm.

4.4 - Time Differences Between Starting Models

Figure 4.6 shows the time-depths for the range of XY values from zero to 30 m

plotted without the vertical separation obtained by changing the reciprocal time

tAB, in equation 4.1. This presentation, which emphasizes the subtle variations

between different XY values, shows that the time-depth values are identical for

the planar sloping surfaces, but diverge by less than 2 ms in the vicinity of the

hinge point.

The smaller values are associated with the XY values which are less than the

optimum of 15 m and in turn are associated with the zone of lower wavespeeds

in the refractor. Although there is a slightly higher DCF computed with equation

4.6 in this narrow region, there is still a reduction in depth at the hinge point, and

in turn a small decrease in traveltimes in the upper layer. This decrease

approximately compensates for the slight increase in traveltimes in the region of

lower wavespeeds in the refractor.

The larger time-depth values are associated with the XY values which are

greater than the optimum of 15 m and in turn are associated with the zone of

84

higher wavespeeds in the refractor. Although there is a slightly lower DCF

computed with equation 4.6 in this narrow region, there is still an increase in

depth at the hinge point, and in turn a small increase in traveltimes in the upper

layer. This increase approximately compensates for the slight decrease in

traveltimes in the region of higher wavespeeds in the refractor.

Figure 4.6: Time-depths computed for the synclinal model in Figure 4.1 for a

range of XY values. The reciprocal times are identical for each XY value, and it

results in an emphasis of the subtle variations between different XY values.

Therefore, while there are small differences in depths at the hinge point, they are

matched with compensating changes in the wavespeeds in the refractor. The

85

final result is that there are few significant differences in the traveltimes in final

depth models.

Figure 4.7: Refractor wavespeed analysis function computed for the synclinal

model in Figure 4.1 for a range of XY values. The reciprocal times are identical

for each XY value.

Figure 4.7 is a similar presentation in which the wavespeed analysis function in

equation 4.2 is presented with identical reciprocal times for all XY values. Again

the aim is to emphasize the subtle variations between each set of values. It can

be seen that the maximum difference between the values computed for XY

86

values of zero and 30 m is 2.2 ms and that maximum difference between the any

set and those computed with optimum XY value of 15 m is less than 1.1 ms.

Figure 4.7 also demonstrates a fundamental problem in determining wavespeeds

in narrow intervals of the refractor with seismic refraction methods. The variation

in wavespeed in the refractor of 2000 m/s to 4800 m/s is very large and

geologically significant. However, these variations in wavespeed do not result in

commensurately large changes in traveltimes. An inspection of Figure 4.4 shows

that a single wavespeed can be fitted to each set of points with an accuracy of

better than a millisecond.

The significance of Figures 4.6 and 4.7 is that the time differences between each

model of the refractor computed with the selected range of XY values are subtle

and are generally within 1 ms of that computed with the optimum XY value of 15

m. These differences are typical of the acceptable residuals for most model-

based or tomographic methods of inversion.

4.5 - Agreement Between Starting Models And Traveltime Data

The small time differences between the various models as shown in Figures 4.6

and 4.7 suggest that each model should closely honor the original traveltime

data. Such a result is in fact the norm with the GRM, because the algorithms

seek to separate or analyze the traveltimes into the source point and detector

time-depths, together with the traveltime in the refractor, while still preserving the

original traveltime data. This is demonstrated by the simple addition of equations

4.1 and 4.2, viz.

tAY = tG + tV + ½ XY/Vn (4.9)

From equations 4.3 and 4.4, it is readily shown that

87

tV = tA + AG/Vn (4.10)

Equations 4.9 and 4.10 can be combined to obtain

tAY = tG + tA + AY / Vn (4.11)

4.6 - Discussion

This study illustrates some of the inherent problems of non-uniqueness with

determining wavespeeds in the refractor. Using a simple model and the GRM

algorithms, it is possible to generate a family of starting models, each of which

has much the same depths to the refractor as the original model but each of

which includes a narrow zone in the refractor with an anomalous wavespeed.

Even with the noise-free model data used in this study, the time differences are

generally less than one millisecond, which is the error commonly assigned to the

measurement of traveltimes from field data, and which is within the range of

acceptable residuals for tomography. Therefore, if any were used as starting

models for tomography, then there would be minimal differences with the final

result of the inversion process. Furthermore, all of these models are geologically

meaningful and hence cannot be readily discarded.

(As an aside, geologically meaningless models can also be generated with the

GRM, simply by increasing the XY value in the wavespeed analysis function in

equation 4.2, until negative wavespeeds are obtained with equation 4.3.)

Frequently, the algorithms of the standard reciprocal method (SRM) (Hawkins,

1961), which is a special case of the GRM with a zero XY value, are used to

generate starting models. These algorithms are probably the most commonly

used throughout the world for shallow seismic refraction investigations, because

88

of their simplicity and robustness. These algorithms, which are also known as

the ABC method in the Americas (Nettleton, 1940; Dobrin, 1976), Hagiwara's

method in Japan (Hagiwara and Omote, 1939), and the plus-minus method in

Europe (Hagedoorn, 1959), can be viewed as simple extensions of the

slope/intercept method (Ewing et al, 1939), whereby computations are extended

from the source points to each detector location (Palmer, 1986).

However, this study demonstrates that the starting model generated by any

single method, such as an SRM analysis of the traveltime data need not

necessarily converge to the correct model. Therefore, in order to address the

issues of non-uniqueness, it is recommended that model-based methods of

inversion test a family of starting models such as those which can be readily

derived with the GRM.

The results over the Elura orebody (Hawkins and Whiteley, 1980) demonstrate

the significance of artifacts. The claim, that the massive sulphide orebody was

characterized by a low wavespeed, attracted considerable debate (Emerson,

1980), and was at variance with laboratory tests on hand specimens (MacMahon,

1980). An alternative analysis with the GRM indicated that the low wavespeed

was probably an artifact which coincided with an increase in the depth of the

regolith over the orebody (Palmer, 1980b). Many of the qualitative aspects of the

model study above can be recognized in the Elura case history.

The inability of model-based inversion methods to recognize artifacts can also

have important legal implications. There are instances where the combination of

the SRM and ray tracing is a contractual requirement of major geotechnical

investigations, in order to obviate claims for compensation by construction

companies for unexpected variations in site conditions.

Numerous model studies and case histories (Palmer, 1980; Palmer, 1986,

Palmer, 1991) demonstrate that the minimum variance criterion of the GRM is

89

frequently able to resolve whether lateral variations in wavespeeds in the

refractor are genuine or are artifacts. For the model study above, the wavespeed

analysis function in Figure 4.4, shows that the optimum XY value of 15 m, that

the measured wavespeed is same as the model and that no artifacts are

generated.

However, the effective application of the GRM is not always possible, often

because the detector interval is too large. In these cases, alternative methods

are required. Other studies demonstrate that the amplitudes of the refraction

convolution section (Palmer, 2001a; Chapter 5) can indicate where there are

genuine changes in the wavespeed of the refractor. These amplitudes are a

function of the contrasts in wavespeeds between the refractor and the layer

above, and therefore provide another approach which is independent of the

traveltime data.

4.7 - Conclusions

The inversion of seismic refraction data with model-based methods or

tomography consists of deriving a starting model of the subsurface with standard

algorithms, and then testing it by comparing the computed traveltimes of the

model with the observed data. If there are differences, then the model is

adjusted until an acceptable agreement is achieved. Commonly, several

iterations may be required.

However, a simple model study illustrates the inherent problems of non-

uniqueness with this approach. The GRM is able to generate a family of starting

models, all of which are geologically meaningful and all of which are compatible

with the original traveltime data. If any were used as starting models for

tomography, then there would be minimal differences with the final result of

inversion.

90

In view of the significance of the starting model, it is recommended that model-

based methods of inversion test a range of starting models such as those which

can be readily generated with the GRM. In general, the models which can be

derived with the GRM tend to be compatible with the original traveltime data.

In many cases, the minimum variance criterion of the generalized reciprocal

method (GRM) can resolve whether lateral variations in the refractor wavespeeds

are genuine, or whether they are artifacts of the inversion algorithm.

In those cases where the effective application of the GRM is not possible, then

alternative methods are required. It is proposed that the amplitudes of the

refraction convolution section (Palmer, 2001a; Chapter 5) frequently can indicate

where there are genuine changes in the wavespeed of the refractor.

4.8 - References

Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edn.: McGraw-

Hill Inc.

Emerson, D. W., 1980, The geophysics of the Elura orebody, Cobar, NSW: Bull.

Aust. Soc. Explor. Geophys., 11, 347.

Ewing, M., Woollard, G. P., and Vine, A. C., 1939, Geophysical investigations in

the emerged and submerged Atlantic Coastal Plain, Part 3, Barnegat Bay, New

Jersey section: Bull. GSA, 50, 257-296.

Hagiwara, T., and Omote, S., 1939, Land creep at {Mt} {Tyausa-Yama}


Res. Inst. Bull., 17,118-137.

91

Hagedoorn, J. G., 1955, Templates for fitting smooth velocity functions to seismic

refraction and reflection data: Geophys. Prosp., 3, 325-338.





Hawkins, L. V., and Whiteley, R. J., 1980, The seismic signature of the Elura

orebody: Bull. Aust. Soc. Explor. Geophys., 11, 325-329.



MacMahon, B. K., 1980, Discussion in Emerson, D. W., ed., The geophysics of

the Elura orebody, Cobar, NSW: Bull. Aust. Soc. Explor. Geophys., 11, 346.


Company Inc.

Oldenburg, D. W., 1984, An introduction to linear inverse theory: Trans IEEE

Geoscience and Remote Sensing, GE-22(6), 666.

Palmer, D., 1980a, The generalized reciprocal method of seismic refraction


Palmer, D., 1980b, Comments on "The seismic signature of the Elura orebody":

Bull. Aust. Soc. Explor. Geophys., 11, 347.

92

Palmer, D., 1986, Refraction seismics - the lateral resolution of structure and

seismic velocity: Geophysical Press


generalized reciprocal method: Geophys. Prosp., 39, 1031-1060.


refraction inversion?: Explor. Geophys., 23, 261-266, 521.

Palmer, D., 2001a, Resolving refractor ambiguities with amplitudes: Geophysics

66, 1590-1593.

Palmer, D., 2001b, Model determination for refraction inversion: Geophysics,

submitted.


of salt): The Leading Edge, 7, 32-35.

93

Chapter 5

Resolving Refractor Ambiguities WithAmplitudes

5.1 - Summary

Amplitudes are used to constrain refraction models. This study demonstrates

that the refraction time section generated through the convolution of forward and

reverse refraction traces together with a static shift, facilitates the convenient

recognition of amplitude variations related to changes in refractor wavespeed.

For large contrasts in wavespeeds between the upper layer and the refractor, the

head coefficient is approximately proportional to the ratio of the specific acoustic

impedances. Since the convolution operation effectively multiplies the

amplitudes of the forward and reverse arrivals, the convolved amplitudes are

proportional to the square of this ratio. In general, the higher the contrast in the

refractor wavespeed and/or density, the lower the amplitude. The regions

recognized in the wavespeed analysis function correlate with those determined

with amplitudes, thereby providing an additional constraint on inversion with

model-based approaches.

94

5.2 - Introduction

The inversion of seismic refraction data with model-based methods is inherently

ambiguous, and artifacts, which are geologically plausible and significant, can be

introduced by the algorithms commonly used to generate starting models. In

many cases, the minimum variance criterion of the generalized reciprocal method

(GRM) (Palmer, 1980; Palmer, 1986; Palmer, 1991) can resolve whether lateral

variations in the refractor wavespeeds are genuine, or whether they are artifacts

of the inversion algorithm. As an additional constraint, this study demonstrates

that any genuine lateral changes in refractor wavespeed should also have an

associated amplitude expression.

Amplitudes are not commonly used in seismic refraction studies, mainly because

the very large geometric spreading component dominates any variations related

to wavespeed in the refractor. For near surface investigations, the source-to-

detector distances are generally less than five or six times the dominant

wavelength. As a result, the geometric spreading can be very rapid and it is not

satisfactorily described with the commonly used reciprocal distance squared

expression. However, the multiplication of amplitudes through the convolution of

forward and reverse traces effectively compensates for geometric spreading

(Palmer, 2001). The resultant amplitudes are then described with the head

coefficient.

This study shows that the head coefficient is a function of the contrasts in

wavespeeds and/or densities between the upper layer and the refractor, and that

there are changes in the convolved amplitudes where there are genuine changes

in the wavespeed of the refractor. Since the convolution operation effectively

multiplies the amplitudes of the forward and reverse arrivals, the convolved

amplitudes are proportional to the square of the head coefficient.

95

The paper begins with a discussion of refraction amplitudes with plane horizontal

interfaces and proposes a simplification of the head coefficient for large contrasts

in wavespeeds. The paper then applies this simplification to a case history study

from southeastern Australia where the refractor model is quite complex with both

large variations in depths to and wavespeeds in the refractor.

5.3 - Amplitude and Wavespeed Relationships

The expression for the amplitude of the head wave for a thick refractor with a

plane horizontal interface has been derived by Heelan (1953) and Zvolinskii,

(Werth, 1967), who showed that:

Amplitude = K F(t) / (rL3)½ (5.1)

where K is the head coefficient, which depends on the elastic properties of the

upper and lower layers, F(t) is the displacement potential of the incident pulse, r

is the source to detector distance, and L is the distance the wave has traveled

within the refractor.

The expression for K given by Werth (1967) is

K = 2 ρ χ [λ1 (1 + 2 m γ2) + λ2 (ρ - 2 m γ2)]2 (5.2)

[γ2(1 + 2 m γ2 - ρ)2 + ρ χ λ2 χ λ1 (1 + 2 m γ2)2]2

where γ = VP1 / VP2

ρ = ρ1 / ρ2

m =ρ (VS12 / VP1

2) - VS22 / VP1

2

χ = (1 - γ2)½

λ1 = (VP12 / VS1

2 - γ2) ½

96

λ2 = (VP12 / VS2

2 - γ2) ½

VP1 = compressional wavespeed in upper medium

VS1 = shear wavespeed in upper medium

ρ1 = density in upper medium, and similarly for the lower medium 2.

The evaluation of K for a selected set of elastic parameters (O’Brien, 1967;

Cerveny and Ravindra, 1971) shows that the amplitude decreases as the

contrast in the wavespeed between the two media increases. This result is in

keeping with the observations of field data (O’Brien, 1967), and is confirmed by

the results presented in this study.

Intuitively this result seems unexpected: high wavespeeds and densities are

usually associated with more competent rocks and therefore with better energy

transmission properties. However, these results have parallels with the Zoeppritz

equations used in reflection seismology, wherein high transmission coefficients

occur with low contrasts in the specific acoustic impedance, while low

transmission coefficients occur with high contrasts.

For strong contrasts in wavespeeds, ie for γ → 0,

K ∝ ρ γ = ρ1 VP1 / ρ2 VP2 (5.3)

Equation 5.3 is probably valid for γ as large as 0.7 (Cerveny and Ravindra, 1971,

Figure 3.11), which would constitute a major proportion of shallow seismic

refraction applications. For larger values of γ, there is a rapid increase in

amplitude, which is given by (Cerveny and Ravindra, 1971, p139):

K → 1 / (1 – (VP1 / VP2)2) ½ (5.4)

In the generation of the convolution section, the amplitudes of the forward and

reverse traces are multiplied, and to a reasonable approximation, the effects of

97

dipping refractors are minimized (Palmer, 2001). Therefore, the approximation of

equation 5.3 can be applied directly, and so the convolution amplitude varies

approximately as the square of the contrast in the specific acoustic impedances

between the overburden and the refractor.

The simplified relationship between amplitudes and the specific acoustic

impedances in equation 5.3 takes no account of variations in shear wavespeeds,

attenuation or diffractions which constitute a large proportion of the refracted

signal with irregular interfaces. However, the approximation suggested in

equation 5.3, may be a practical approach to relating refraction amplitudes to

petrophysical parameters for routine applications.

5.4 - Mt Bulga Case History

Refraction data were recorded across a major sub-vertical shear zone which

occurs near the contact between Ordovician volcanics and meta-sediments at Mt

Bulga, in southeastern Australia. Figure 5.1 presents the traveltime data for five

shots from the original ensemble of nine, representing every other shot. They

indicate a three layer model consisting of a thin surface layer of friable soil with a

wavespeed of about 400 m/s, a thicker layer of weathered material with a

wavespeed of approximately 700 m/s, and a main refractor with an irregular

interface.

The traveltime data were processed or inverted using the two algorithms of the

generalized reciprocal method (GRM) (Palmer, 1980; Palmer,1986) for

computing time-depths and refractor wavespeeds.

The time-depth tG, were computed with equation 5.5, viz.

tG = (tAY + tBX - tAB - XY/Vn)/2 (5.5)

98

where A, X, G, Y, and B are colinear, A and B are source points, X and Y are

detectors and G is midway between X and Y, tAY is the traveltime from A to Y, tBX

is the traveltime from B to X, tAB is the reciprocal time, the traveltime from the

source at A to the source at B, and Vn is the wavespeed in the refractor. Figure

5.2 shows the time-depths computed with an XY value of 5 meters and a

reciprocal time of 147 ms, (Palmer, 1980, equation 33), and they detail the quite

irregular shape of the refractor interface.

Figure 5.1: Traveltime data for a line crossing a major shear zone at Mt Bulga.

Offset shot points are 120 m from either end. Station spacing is 5 m.

99

Figure 5.2: Time-depths computed from traveltime data with shot points at

stations 1 and 97.

The refractor wavespeeds are obtained from the reciprocal of the gradient of the

wavespeed analysis function tV, given by equation 5.6, viz.

tV = (tAY - tBX + tAB)/ 2 (5.6)

d / dx tV = 1 / Vn (5.7)

Figure 5.3 shows the generalized wavespeed analysis function for an XY value of

5 meters. The refractor can be separated into four main regions with

wavespeeds of 5,000 m/s, 2,200 m/s, 5,000 m/s, and 2,600 m/s. The scatter of

points about the line between stations 25 and 54 has been interpreted as errors

in picking first arrivals, rather than lateral variations in the wavespeed of the

refractor.

The time-depths presented in Figure 5.2 are converted into depths zG, with

equation 5.8, viz.

100

Figure 5.3: The generalized wavespeeds analysis function for a 5 meter XY

value.

zG = tG / DCF (5.8)


depth, is given by:

DCF = V Vn / \(Vn2 - V2)½ (5.9)

or

DCF = V / cos i (5.10)

101

where V is the average wavespeed above the refractor and

sin i = V / Vn (5.11)

Figure 5.4 shows the depth section in which the upper two layers have been

approximated with a single wavespeed using an average wavespeed

approximation (Palmer, 1980, equation 27) of 700 m/s. This approximation

effectively ignores the thin surface layer but introduces only negligible errors in

depth computations.

Figure 5.4: Edited depth section computed with the time-depths shown in Figure

5.2, using an average first layer wavespeed of 700 m/s and the four refractor

wavespeeds of 5000 m/s, 2200 m/s, 5000 m/s and 2600 m/s.

The first region is between stations 24 and 54, and has a wavespeed of 5,000

m/s and depths ranging from 15 meters to 30 meters, with an average of about

102

25 meters. This region could be further divided with the zone between stations

42 and 54 having a slightly higher wavespeed.

Figure 5.5: Convolution section generated with shot records with source points

at stations 1 and 97.

103

The second region is between stations 54 and 62. It has a wavespeed of 2,200

m/s and an average depth of over 30 meters. It corresponds with the inferred

location of the major shear zone.

The third region is between stations 62 and 67. It has a wavespeed of 5000 m/s

and depths range from 28 to 15 meters. This region is relatively narrow, and the

wavespeed is not well determined, possibly because it is associated with a major

change in depths. A re-interpretation of Figure 5.3 indicates that a lower value of

less than 4000 m/s could be assigned to this region.

The fourth region is between stations 67 and 73. It has a wavespeed of 2600

m/s and an average depth of about 15 meters.

The convolution section presented in Figure 5.5, shows the same structure as

the time-depths in Figure 5.2, and it is possible to recognize four regions with the

relative amplitudes of approximately 1, 5, 2, and 4 (Palmer, 2001; chapter 3,

Figure 3.9). These regions correspond with those determined on the basis of

refractor wavespeeds in Figure 5.3. The ratios of the wavespeeds in each region

to the average wavespeed in the overburden are 0.14, 0.32, 0.14, and 0.27. The

square of these ratios normalized to the lowest value are 1, 5.2, 1, 3.7. They are

similar to the ratios of the convolved amplitudes, except for the third region,

where a refractor wavespeed of 3540 m/s would be compatible with the observed

amplitude.

This case history provides a compelling demonstration of the correlation between

amplitudes and wavespeeds with a complex refractor exhibiting large changes in

both depth and wavespeed. Not only do major changes in wavespeed result in

marked amplitude variations, such as the contrast between the regions with

wavespeeds of 5000 m/s and 2200 m/s, but subtle changes within each region

can also be recognized. For example, the region between stations 42 and 54

104

has a slightly higher wavespeed and lower convolved amplitudes than is the case

between stations 25 and 42.

This example also demonstrates the ability of amplitudes to help resolve any

ambiguities in the determination of refractor wavespeeds. The third region of the

refractor between stations 62 and 67 is relatively narrow and has a large change

in depth. Both of these factors may affect the amplitudes and the accuracy of the

measurement of the wavespeed. A re-interpretation of the wavespeed analysis

function together with the amplitudes suggest that a lower wavespeed of

between 3540 m/s and 4000 m/s would be more appropriate for this interval.

5.5 - Conclusions

The inversion of refraction data can be ambiguous. Artifacts, such as narrow

zones with higher and lower wavespeeds can be produced where there are

changes in the depth to the refractor. In general, forward modeling does not

recognize or correct these artifacts.

The amplitudes of the refracted signals provide another means of recognizing

genuine lateral variations in wavespeed within the refractor, once the large

effects of geometric spreading are removed. This study uses the convolution of

forward and reverse seismic traces to compensate for geometrical spreading.

The refraction time section obtained in this way facilitates the correlation of

structure on the refractor interface with amplitudes, and in turn with wavespeeds

within the refractor.

The case history has large variations in depths to and wavespeeds within the

refractor and provides a searching test of the method. The regions in the

refractor recognized with the wavespeed analysis function correlate closely with

the regions recognized with the convolved amplitudes or the amplitude products.

105

Low contrasts in the wavespeeds between the refractor and the overlying layer

produce higher amplitudes than is the case with high contrasts.

This feature alone is very useful in constraining the generation of artifacts with

model-based inversion. The gross features of the interpretation model

recognizable in the convolution section should also have corresponding

expressions in the results obtained with other approaches.

For large contrasts in the wavespeeds and/or densities between the upper layer

and the refractor, the head coefficient is approximately proportional to the ratio of

the specific acoustic impedances. In turn, the amplitudes in the convolution

section are proportional to the square of this ratio because the forward and

reverse amplitudes are multiplied with convolution. This approximation is

satisfactory for three of the regions examined in the case history. Furthermore, it

supports a revision of the wavespeeds in a narrow region of the refractor to a

lower value than was initially inferred.

The convolution section is a very effective single presentation for combining the

information depicting the geometry of the refractor which is obtained with the

time-depth algorithm, and the information depicting the wavespeed in the

refractor which is usually obtained with the wavespeed analysis algorithm. It is

very rapid to generate, avoiding in particular the familiar time consuming tasks of

determining first arrival traveltimes and amplitudes. In addition, little, if any, a

priori information on upper layer or refractor wavespeeds is required, although of

course such information is essential for the generation of final depth cross

sections.

Accordingly, the convolution section is an extremely useful and convenient

presentation for inclusion in the routine processing of seismic refraction data

using any method.

106

5.6 - References

CervenY, V., and Ravindra, R., 1971, Theory of seismic head waves: University

of Toronto Press.

Heelan, P. A., 1953, On the theory of head waves: Geophysics, 18, 871-893.

O’Brien, P. N. S., 1967, The use of amplitudes in seismic refraction survey, in

Musgrave, A. W., ed., Seismic refraction prospecting: Society of Exploration



Interpretation: Society of Exploration Geophysicists.





Palmer, D., 2001, Imaging refractors with the convolution section: Geophysics

66, 1582-1589.

Werth, G. A., 1967, Method for calculating the amplitude of the refraction arrival,

in Musgrave, A. W., Ed., Seismic refraction prospecting: Society of Exploration


107

Chapter 6

Efficient Mapping Of Structure AndAzimuthal Anisotropy With Three

Dimensional Shallow SeismicRefraction Methods

6.1 - Summary

A three dimensional (3D) seismic refraction survey was carried out across a

shear zone.

The data were processed with the generalized reciprocal method (GRM) rather

than with tomographic inversion because of the relatively small volume of data,

the occurrence of large variations in depth to and wavespeeds within the main

refractor and the presence of azimuthal anisotropy.

The results show that there is an increase in the depth of weathering and a

decrease in wavespeed in the sub-weathering associated with the shear zone.

Although the shear zone is generally considered to be a two dimensional (2D)

feature, the significant lateral variations in both depths to and wavespeeds within

the refractor in the cross-line direction indicate that it is best treated as a 3D

target. These variations are not predictable on the basis of a 2D profile recorded

earlier.

108

The amplitudes of the refracted signals are approximately proportional to the

ratio of the specific acoustic impedances between the upper layer and the

refractor and they provide a convenient and detailed measure of apparent

azimuthal anisotropy or rock fabric. The amplitudes also contain additional

useful geological information, although some of the cross-line amplitudes could

not be completely explained.

Qualitative measures of azimuthal anisotropy are obtained from the wavespeeds

and the time-depths computed from the traveltime data with the GRM algorithms

and from the amplitudes. These three methods give similar consistent results,

with the direction of the greater wavespeed being approximately parallel to the

direction of the dominant geological strike. Furthermore, all three methods show

that the direction of the greater wavespeed is approximately orthogonal to the

direction of the dominant geological strike in one region adjacent to the shear

zone.

The in-line results show that both accurate refractor depths and wavespeeds can

be computed with moderate cross-line offsets, say less than 20 m, of shot points.

These results demonstrate that swath shooting with a number of parallel

recording lines would be adequate for 3D surveys over targets such as highways,

damsites and pipelines. Only a modest increase in shot points over the

requirements for the normal 2D program would be required in the cross-line

direction for measuring azimuthal anisotropy and rock fabric with amplitudes.

6.2 - Introduction

In the last two decades, three dimensional (3D) seismic reflection methods have

revolutionized the exploration for, and production of petroleum resources. The

improved images of the subsurface geology are a result of the recognition that

109

most geological targets are in fact three dimensional, and that it is essential to

employ spatial sampling densities and processing methods which recognize and

accommodate this reality. It is now generally accepted that in many cases, two

dimensional (2D) seismic reflection methods give an incorrect rather than an

incomplete picture of the sub-surface (Nestvold, 1992).

By contrast, 3D refraction methods (Zelt, 1998; Bennett, 1999; Deen et al, 2000)

are not very common. However, there are compelling reasons for the expedient

development of 3D shallow refraction methods for routine use in geotechnical,

environmental and groundwater applications.

Geological structures and the corresponding depths to and wavespeeds within

bedrock, can show as much variation in the cross-line direction as in the in-line

direction. In the vast majority of near-surface studies, such variations are

significant.

There is a need to address azimuthal anisotropy of wavespeeds. Anisotropy can

be caused by lamination, foliation or by the preferred orientation of joints and

cracks within the refractor, and it is another important parameter for assessing

rock strength for rippability and foundation design. However, its most important

near-surface application may be in the determination of fracture porosity in

crystalline rocks for the development of groundwater supplies for domestic and

irrigation purposes, in studies of contaminant transport especially of radioactive

wastes (Barker, 1991), the stability of rock slopes and seepage from dams, the

construction of underground rock cavities for storing water, gas, etc, and the

construction of tunnels.

The relationship between anisotropy and crack parameters has been the subject

of considerable research in the past (Crampin et al, 1980; Thomsen, 1995).

Nevertheless, there are no established approaches for the routine mapping of

these parameters with shallow geotechnical or environmental targets, although

110

radial surveys to measure azimuthal anisotropy (Bamford and Nunn, 1979; Leslie

and Lawton, 1999) represent the first steps in that direction.

6.3 - Data Processing With The GRM

This study describes the results of a 3D shallow seismic refraction survey

recorded some time ago across a shear zone at Mt Bulga in southeastern

Australia. The data are processed with a traditional approach using the

generalized reciprocal method (GRM) (Palmer, 1980; Palmer1986), rather than

with tomographic inversion for the following reasons.

The wavespeeds in the refractor range from less than 2000 m/s in the shear

zone to more than 5000 m/s in the adjacent rocks. Recent case histories (Lanz

et al, 1998), demonstrate that current tomographic inversion methods cannot yet

reliably resolve wavespeeds in the main refractor, even though over 90% of the

traveltimes originated from the refractor. In those cases where stable inversion

has been achieved, the variations in wavespeeds are generally less than about

5% (Zelt, 1998).

The volume of data is low in contrast to that generally considered desirable for

effective tomographic inversion. As a comparison, the approximately 2000

traveltimes for 120 detector positions used in this study are much less than the

more than 50,000 traveltimes for 29 detector positions used in the tomographic

analysis of Zelt and Barton, (1998). For most routine shallow refraction

investigations, the costs of recording at least an order of magnitude of additional

shot points can be prohibitive.

Model studies and case histories (Palmer, 1980; Palmer, 1991) demonstrate that

the GRM can resolve large variations in the depths to and wavespeeds within

111

refractors using considerably smaller data volumes than is the case with most

tomography programs.

Azimuthal anisotropy is rarely accommodated with most tomography programs.

Isotropy is normally assumed in order to employ as many traveltimes from as

many directions as possible in the inversion process. In addition, the traveltime

differences due to anisotropy are quite small, and are often within the accepted

range for the residuals of inversion.

In this study, the amplitudes of the refracted head waves are used to map

anisotropy. Previous studies have shown that the head coefficient, the

parameter which controls the amplitude of the refracted signal, is approximately

proportional to the ratio of the specific acoustic impedances of the overburden

and the refractor (Palmer, 2001b; chapter 5). However, the head wave

amplitudes are generally dominated by the rapid variation due to geometric

spreading. Another study (Palmer, 2001a; chapter 3), demonstrates that the

effects of geometrical spreading and dipping interfaces can be accommodated

with either the multiplication of the amplitudes of the forward and reverse traces,

or by the convolution of those traces. In this study, the ratios of the amplitude

products for pairs of shot points with varying azimuths are used as a qualitative

measure of azimuthal anisotropy.

6.4 - Survey Details

The data used in this study were acquired in approximately the same location as

a 2D set of data described previously (Palmer, 2001a; chapter 3). The survey

was carried out shortly after the area had undergone complete clearing of the

native vegetation and subsequent planting of tube stock for a pine plantation. As

a result, the survey pegs which marked out the exploration grid, had been

removed, and so the precise relationship between the two surveys is not known.

112

However, the cross-line numbers in this study correlate approximately with the

station numbers on the 2D profile.

Figure 6.1: Plan of in-line and cross-line geophones and shot points. Shots 1 to

15 are shown as bold symbols and were recorded with in-lines 17 and 21. Shots

16 to 42 are shown as open symbols and were recorded with cross-lines 45 to

69.

The data were recorded with a 48 trace seismic system using a roll switch and

single 40 Hz detectors. Shot holes were drilled to depths of between 1 and 2.5 m

with a small trailer mounted drill rig. Charge sizes were between 1 and 3 kg of a

high velocity seismic explosive.

113

Initially, two parallel lines 20 m apart, with each consisting of 24 geophones at a

5 m spacing were set out. These in-lines were located approximately either side

of the earlier 2D profile. Five shot points, nominally 60 m apart, were located

along each line, while another four oblique shot points offset 60 m from the end

of each line of geophones in the in-line direction and offset 60 m in the cross-line

direction were also recorded, making a total of fourteen shots.

A second series of seven parallel cross-lines which were 20 m apart, and each of

which consisted of twelve geophones at a 5 m separation were then set out.

There were four shot points on each cross-line and the shots were nominally 60

m apart. These lines were recorded in groups of four by simply rolling through

from one end to the other. A total of twenty seven shots were recorded in the

cross-line directions.

Figure 6.1 is a plan of the two geophone arrangements and shot point locations.

6.5 - Analysis of the In-line Traveltime Data

The traveltimes were hand picked from the field monitors, and standard

corrections for the uphole time and the system delay in the analogue

components were applied. The previous 2D study (chapter 3; Palmer, 2001a),

showed that a three layer model was applicable. It consists of a thin surface

layer of friable soil with a wavespeed of about 400 m/s, a thicker layer of

weathered material with a wavespeed of approximately 700 m/s, and a main

refractor with an irregular interface with wavespeeds between approximately

2000 m/s and 5000 m/s.

The traveltime data for in-line 21 for all fourteen shots are shown in Figure 6.2.

The graphs for the shot points which are offset by 60 m from the geophone

spreads in the in-line direction and which are located along cross-lines 33 and

114

81, namely shots 1 to 4 and 8 to 11, all show arrivals which originate from the

main refractor. The graphs in the forward and reverse directions appear to be

essentially parallel, but in fact gradually converge. Also, there is an unresolved

inconsistency in traveltimes between stations 45 and 49, which is related to very

low amplitude arrivals on the shot records.

Figure 6.2: Traveltime data recorded on in-line 21 with in-line, adjacent and

oblique shot points. In general, the graphs gradually converge in each direction

of recording. The inconsistencies in the reverse traveltimes can be seen

between cross-lines 45 and 49.

Figure 6.3 shows the refractor wavespeed analysis function tV, computed with

equation 6.1, using a 5 m XY value, for four shot pairs. They are shots 2 and 9

115

which are collinear with the detectors, shots 3 and 10 which are collinear with the

adjacent parallel line of detectors on in-line 17, shots 1 and 11 which form a

northwest-southeast shooting orientation, and shots 4 and 8 which form a

northeast-southwest shooting orientation.

Figure 6.3: Refractor wavespeed analysis function computed for the in-line,

adjacent and oblique shot pairs. The wavespeeds for the oblique shot pairs have

been corrected with the cosine of 30 degrees which is the angle between in-line

21 and the line joining the shot points.

tV = (tforward - treverse + treciprocal)/ 2 (6.1)

116

where treciprocal is the traveltime from the forward shot point to the reverse shot

point, and it is a constant for a given shot pair and a set of collinear detectors.

The wavespeed in the refractor along in-line 21, is obtained from the reciprocal of

the gradient of tV for the shot pairs which are collinear with the detectors, namely

shots 2 and 9. Between cross-lines 45 and 50, the wavespeed is not well

determined because of the unresolved inconsistency in the traveltimes

mentioned previously, but it appears to be greater than 4000 m/s. The value of

5000 m/s shown in Figure 6.3 is taken from the earlier adjacent 2D results

previously referenced.

The wavespeed is 1850 m/s between cross-lines 50 and 60.

Between cross-lines 60 and 69, the wavespeed is 3930 m/s. However, this

region can be further separated into an interval between cross-lines 60 and 64

with a wavespeed of approximately 5000 m/s followed by an interval between

cross-lines 64 and 69 with a wavespeed of approximately 3000 m/s. This

separation is consistent with the results of the earlier 2D survey.

The refractor wavespeeds computed with shots 3 and 10 which are located along

the adjacent in-line 17, are essentially the same as those determined above.

However, the wavespeeds computed with the northeast-southwest and

northwest-southeast oblique shot points are higher mainly because of the angle

of about 30 degrees between the line of the detectors and the line joining the two

shot points. The wavespeeds between cross-lines 50 and 60 of the oblique

shots shown in Figure 6.3 are the product of the measured values and the cosine

of 30 degrees. They show that the corrected wavespeeds are higher in the

northeast-southwest direction than in the northwest-southeast direction.

There is some question about the validity of the wavespeeds derived from the

oblique shot pairs, because no account has been taken of the fact that most of

117

the detectors are not collinear with the two shot points, as is assumed with

equation 6.1. For these shot pairs, the reciprocal time increases as the offset of

the geophone from the line joining the shot points increases. Nevertheless,

these results have been included, because they are consistent with other results

to be described below.

Figure 6.4: Time-depths computed for the in-line, adjacent and oblique shot

pairs. The reciprocal times for the oblique shots have been adjusted so that the

time-depths are the same between cross-lines 45 and 49, in order to emphasize

the systematic divergence from the in-line values.

Figure 6.4 shows the time-depths computed with equation 6.2 using a 5 m XY

value, for the four shot pairs used in Figure 6.3. The reciprocal time for the in-

line shots 2 and 9 was computed with equation 33 of Palmer (1980). The

118

reciprocal times for the other shot pairs could not be derived as conveniently, and

they have been adjusted until the differences in the time-depths between cross-

lines 45 and 49 were minimized. This facilitates the recognition of the systematic

divergence of the time-depths for the oblique shot pairs from the collinear values.

time-depth = (tforward + treverse - treciprocal)/2. (6.2)

The increase in the time-depths between cross-lines 50 and 60 corresponds to

the region in the refractor with the low wavespeed.

The systematic divergence of the time-depths computed with the oblique shot

pairs from cross-line 49 to cross-line 69, can be employed as a qualitative

measure of azimuthal anisotropy in the following way.

The time-depths tG, are related to the depths zG, with equation 6.3,

zG = tG / DCF (6.3)


depth, is given by:

DCF = V Vn / (Vn2 - V2)½ (6.4)

or

DCF = V / cos i (6.5)


sin i = V / Vn (6.6)

119

It is reasonable to assume that the point of critical refraction below each station is

much the same whether the energy propagating in the refractor is traveling in the

northeast-southwest direction or the northwest-southeast direction. This implies

that the depth to the refractor is the same irrespective of the direction of

measurement. Therefore, any variations in the time-depth at each station will be

related to variations in the DCF through equation 6.3.

Figure 6.5: The ratio of the time-depths computed with shots 4 and 8 in the

northeast-southwest direction and shots 1 and 11 in the northwest-southeast

direction.

The time-depths for the oblique shots were then re-adjusted in the following

manner. In-line 21 intersects the line joining shots 1 and 11 at cross-line 53 and

the line joining shots 4 and 8 at cross-line 61. The time-depths for the oblique

shots at these points were computed from the in-line depths with equation 3

using the refractor wavespeeds appropriate to each direction as shown in Figure

120

6.3. The reciprocal times for the oblique shots were then adjusted until the

depths at the intersections were the same as those just computed. Finally, the

ratios of the time-depths in the two oblique directions were computed and they

are presented in Figure 6.5.

As with the wavespeed analysis function in Figure 6.3, this method of detecting

azimuthal anisotropy makes no allowances for the variations in reciprocal time for

the detectors which are not collinear with the shot points. The reciprocal time

subtracted in equation 6.2 should be increased for the detectors offset from the

line joining the shot points, in order to take into account the extra path length in

the refractor. The use of a constant reciprocal time should increase the

computed time-depths at the offset detectors and as a result, the time-depth

profile should appear to be flattened. However, no such flattening is obvious in

Figure 6.4.

Despite these reservations, the results are presented because they are

consistent with those determined with other approaches. In particular, the region

between cross-lines 45 and 49 shows values less than one, while the remainder

shows values greater than one. These results are qualitatively similar to those

derived from amplitude ratios in Figure 6.8 below and from a comparison of the

in-line and cross-line wavespeeds.

These results also demonstrate the benefits of including an analysis of the

residuals as a function of azimuth with tomographic methods. The differences in

traveltimes between the offset shots in Figure 6.2 show little variation about the

mean and as a result, the time-depths also show the same small variations. For

example, the variations about a zero mean difference in the time-depths in Figure

6.4 are less than a few milliseconds. Although such variations are within the

acceptable ranges of residuals for most tomographic approaches, nevertheless,

there may still be a systematic correlation with azimuth and therefore an

indication of azimuthal anisotropy.

121

Figure 6.6 shows the approximate depths to the refractor obtained with equation

6.3.

Figure 6.6: Depths to the main refractor computed with an average wavespeed

of 700 m/s in the upper two layers.

6.6 - Analysis of the In-line Amplitude Data

The amplitudes of the first arrivals were hand picked from the trace values with a

utility in Visual_SUNT, a seismic reflection processing software package. A

correction was applied for geometric spreading using a reciprocal of the distance

cubed expression, which previous studies had indicated was appropriate for this

site (Palmer, 2001a; chapter 3). The corrected amplitudes for the four pairs of

shots described above, were then multiplied.

122

Figure 6.7 shows the amplitude products for shots 2 and 9 which are colinear on

in-line 21. They show low values between cross-lines 45 and 49, which

correspond with the wavespeed of 5000 m/s, higher values between cross-lines

50 and 62, which correspond with the wavespeed of 1850 m/s, and lower values

between cross-lines 63 and 69 which correspond with the wavespeed of 3930

m/s. The amplitudes in this last region gradually increase towards cross-line 69,

and correspond with the decrease in wavespeed when the region is further sub-

divided into two regions.

Figure 6.7: Amplitude products corrected for geometric spreading for shots 2

and 9.

These results are consistent with previous studies which demonstrate that the

amplitude product is approximately proportional to the square of the ratio of the

123

specific acoustic impedances of the overburden and the refractor (chapter 5;

Palmer, 2001b). Since the wavespeeds in the layers above the main refractor

exhibit little lateral variation in the in-line direction, the amplitudes are essentially

a function of the wavespeeds and densities in the refractor.

The amplitudes of the other shot pairs show the same general pattern, as well as

the detailed features such as the higher values at cross-lines 54, 56, 59 and 61

on line 17. These variations can be attributed to changes in the coupling of the

detectors, or near-surface changes in the wavespeeds.

Figure 6.8: An apparent anisotropy factor obtained from the square root of the

ratio of the corrected amplitudes for the two pairs of oblique shots.

Figure 6.8 shows the square root of the ratio of the amplitudes obtained with

shots 4 and 8 in the northeast-southwest direction to the amplitudes obtained

124

with shots 1 and 11 in the northwest-southeast direction. This parameter should

reflect a relative anisotropy factor, since it is not possible to provide an absolute

scale, because as yet, there is no method for compensating for the different

energy levels and coupling of each shot. However, an approximate scaling factor

was obtained from the ratio of the wavespeeds in the different directions for the

region with the low wavespeeds between cross-lines 50 and 62 in Figure 6.3.

An examination of the cross-line data described below, shows that in general the

wavespeeds are higher in the cross-line direction, that is along the dominant

geological strike, than in the in-line direction. However, the exception is the

region between cross-lines 45 and 49 where the reverse applies. The fact that

Figure 6.8 is consistent with this model provides confidence in the validity of the

relative anisotropy factor.

6.7 - Analysis of the Cross-line Traveltime Data

The traveltime data recorded in the cross-line direction show that the same three

layer model is applicable in the cross-line direction as for the in-line direction.

However the wavespeeds in the second layer show more variation and range

from 540 m/s on cross-line 57 to more than 1000 m/s on cross-lines 45 and 69.

Figure 6.9 summarizes the traveltime data for the shot points at the ends of each

cross-line.

The refractor wavespeed analysis function for each cross-line is shown in Figure

6.10. In general the pattern is similar to that determined for the in-line directions,

namely a zone of low wavespeeds between cross-lines 49 and 61, and zones of

higher wavespeeds on cross-lines 45, 65 and 69.

125

Figure 6.9: Stacked traveltimes for the shot points at each end of the cross-

lines.

On cross-line 45, the wavespeed is 3380 m/s, while on cross-line 49, there is a

lateral change from that value to 2000 m/s. Furthermore, there is a

corresponding change in the wavespeeds of the second layer shown in Figure

6.9. The 3380 m/s wavespeed correlates with the second layer values of 1020

m/s to 1200 m/s, while the 2000 m/s on cross-line 49 correlates with a value of

810 m/s for the second layer. This correlation between the wavespeeds in

second layer and main refractor on cross-line 49 together with the change in

126

refractor wavespeeds on the two in-line profiles at cross-line 50 are consistent

with a lateral change in wavespeed occurring on cross-line 49. The significance

of this result is that a major change in refractor wavespeed has been resolved

along cross-line 49, even though the contact between the two zones is probably

not orthogonal to cross-line 49. Theoretical studies (Sjogren, 1984, p168-173)

have predicted that there should be errors in the determination of accurate

refractor wavespeeds.

The lateral change in the wavespeed on cross-line 49 also provides an

explanation for the inconsistent traveltimes obtained on the in-line profiles with

the shot points on cross-line 81. The seismic trace consists of a low amplitude

early arrival from the high wavespeed zone, followed by the high amplitude later

arrival from the adjacent low wavespeed zone. It is possible that one of these

arrivals may be a side swipe.

The wavespeed of 3380 m/s determined in the cross-line direction is significantly

less than the value of about 5000 m/s determined previously with the 2D profile

in the in-line direction. It contrasts with the remainder of the survey area, in

which the wavespeeds are greater in the cross-line direction. However it is

consistent with qualitative measures of azimuthal anisotropy obtained with time-

depth ratios in Figure 6.5 and with amplitude ratios in Figure 6.8.

The wavespeed of 2000 m/s between cross-lines 53 and 61 is 7.5% larger than

the value of 1850 m/s measured in the in-line direction.

On cross-lines 65 and 69, the wavespeed is 6500 m/s between in-lines 19 and

21 and 2000 m/s to 2100 m/s elsewhere. Although the accuracy of the

wavespeed in this center interval is not high because it is measured over a

limited number of points, it is still higher than the in-line value of 3930 m/s

determined between cross-lines 60 and 69.

127

Figure 6.10: Stacked wavespeed analysis function for the offset shots for cross-

lines 45 to 69.

These results are a compelling demonstration that there can be important 3D

effects even with a nominally 2D geological structure. The lateral change in

wavespeeds on cross-line 49 generates inconsistent arrivals on the in-line data

which are more readily explained with the cross-line data. In addition, the rock

fabric in the region between cross-lines 45 and 49, as measured with the

apparent anisotropy factor, is approximately orthogonal to the dominant

128

geological strike direction and to the fabric in the remaining regions of the

refractor. One geological interpretation is that this region has undergone rotation

during the formation of the shear zone.

Figure 6.11: Isometric view of the cross-line time-depths.

Figure 6.11 is an isometric view of the cross-line time-depths and shows that the

variations in the cross-line direction can be considerable even for a nominally 2D

structure with a line orientation which attempted to parallel the dominant strike

direction.

6.8 - The Cross-line Amplitude Data

The cross-lines were recorded in groups of four with the shots being collinear

with either the second or third line in the group. The amplitudes for the seven

129

cross-lines shown in Figure 6.12, were obtained by combining the corrected

amplitude products for each line using the offset shot pairs for that line. As there

was usually some variation in the energy levels from each shot due to shot hole

depth or local geological conditions affecting coupling of the energy, it was

necessary to scale each set of amplitudes to a common level. This was

achieved by determining an average scaling factor between adjacent lines using

the two shot pairs collinear with those two lines.

Figure 6.12: Isometric view of the cross-line amplitude products corrected for

geometric spreading.

The amplitude products corrected for geometric spreading are shown in Figure

6.12. In general, the amplitudes reflect the wavespeeds in the refractor. The low

wavespeeds between cross-lines 53 and 61 produce an increase in the

amplitudes, while the higher wavespeeds between cross-lines 45 and 49 and on

130

cross-line 65 produce lower amplitudes. However, there are a number of

departures from this trend where a detailed correlation is made.

On cross-line 45, there is a gradual increase in amplitude from in-line 13 to in-line

25, which is consistent with an increase in the wavespeed in the overlaying layer

from 1020 m/s to 1200 m/s. On cross-line 49, the decrease in wavespeed in the

refractor from 3380 m/s to 2000 m/s is matched with a decrease in the

wavespeed in the overlaying layer from 1020 m/s to 810 m/s, resulting in only a

minor increase in amplitudes.

There is no obvious explanation for the decrease in amplitudes at each end of

cross-lines 53 to 61. While the higher amplitudes in the center of each line

correlate with the low refractor wavespeeds, there is little evidence for any

significant variation in wavespeeds in the cross-line direction.

Variations in topography and density might provide an explanation. The

topography along the in-lines 17 and 21 is lower than that of the surrounding

survey area and there is an ephemeral creek located across the southeastern

corner. Accordingly, the edges of the survey area may have lower moisture

levels and therefore lower densities in the second layer. Furthermore, no

account has been taken of shear wavespeeds, which also affect the head

coefficient.

The gradual increase in amplitudes on cross-lines 53 to 61 along in-line 13

correlates with an overall increase of wavespeeds in the second layer from about

600 m/s to about 800 m/s. However, these values are much the same as the in-

line values of about 700 m/s, and so do not provide a complete explanation for

the decrease in amplitudes at each end of cross-lines 53 to 61.

131

The locally high values on cross-lines 55 and 61 correlate with similar peaks on

the in-line results, suggesting a geological source, rather than variations in

geophone coupling.

It is also difficult to fully explain the amplitudes on cross-lines 65 and 69 because

they do not readily correlate with wavespeeds or the in-line results. The high

amplitudes between in-lines 21 and 24 correlate with low refractor wavespeeds

shown in Figure 6.9, but they are not matched with similar amplitudes for the low

wavespeeds between in-lines 13 and 17. Furthermore, the high wavespeed

region between in-lines 17 and 21, exhibits both low but more commonly high

amplitudes.

Figure 6.13: Summary of wavespeeds and interpreted faults plotted over the

contours of the time-depths in milliseconds. The bold arrows indicate the

directions of the higher wavespeeds.

Despite these apparent inconsistencies, it is possible that the amplitudes are still

providing a viable model of the wavespeeds in the refractor. The low

wavespeeds which are implied by the high amplitudes, correlate with the

132

separation of the region between cross-lines 60 and 69 into two intervals with

wavespeed of approximately 5000 m/s and 3000 m/s, and with a wavespeed of

2600 m/s detected on the earlier 2D data between cross-lines 67 and 73.

Figure 6.13 is a summary of the wavespeeds in the in-line and cross-line

directions plotted over a contour map of the time-depths. The boundaries of the

regions with different wavespeeds are interpreted as faults. An additional fault

along in-line 17 might also be inferred on the basis of the cross-line amplitudes.

Although this discussion has focused on the variations in amplitudes due to

variations in wavespeeds, it is recognized that other factors, such as inelastic

attenuation, can affect amplitudes. The inelastic attenuation in the refractor at

each detector for a given shot pair, is reduced to a constant amount with the

amplitude product or convolution and therefore, it is not a significant factor. The

inelastic attenuation in the overburden is not compensated with the amplitude

product, and it may be an important factor in lossy media. However, in this case

history, the travel path in the overburden is less than two times the dominant

wavelength of the seismic energy (700 m/s / 35 Hz), and is considered to be a

second order effect.

6.9 - Discussion and Conclusions

The results of this study are a convincing demonstration of the benefits of 3D

shallow refraction methods. Although the shear zone at Mt Bulga is considered

to be a 2D structure, the significant spatial variations in depths, wavespeeds and

azimuthal anisotropy demonstrate that it is best viewed as a 3D target.

The depths to the refractor show considerable variation in the cross-line direction

as well as in the in-line direction. There is a general increase in depths which is

associated with the lower wavespeeds of 1850 m/s to 2000 m/s between cross-

133

lines 49 and 61. However, the increase in depths between in-lines 20 and 25 on

cross-line 49 could not be confidently predicted on the basis of the results from

either the earlier 2D profile or the two in-line profiles in this study. Similarly, the

lateral variations in wavespeeds on cross-lines 65 and 69 also require the

additional coverage in the cross-line direction to be detected and resolved.

In general, the amplitudes correlate with the ratio of the wavespeeds between the

refractor and the layer above. However, there are a number of anomalies in the

cross-line results for which as yet there are no obvious explanations. The

geology of the survey area is quite complex and it is probable that drilling or

excavation would be required to obtain a complete explanation of the observed

amplitudes.

The amplitude ratios provide a convenient approach to determining azimuthal

anisotropy. The qualitative correlation between the measures of azimuthal

anisotropy obtained with wavespeeds, time-depths and amplitude ratios provides

confidence in the validity of the results. This is especially the case with the

region between cross-line 45 and 49 where the direction of the maximum

wavespeed is approximately orthogonal to that for the remainder of the survey

area and to the dominant geological strike direction.

A major benefit of using amplitudes as a measure of the wavespeeds and

therefore anisotropy, is that a value can be determined at each detector, whereas

several collinear detectors are usually required if traveltimes are employed. In

addition, it is probable that the amplitudes may be a more sensitive measure of

anisotropy than traveltimes.

There were a number of difficulties in combining some of the in-line and cross-

line amplitude results. This suggests that data be recorded in a single pass

using several parallel lines of detectors, rather than with a number separate

recording setups.

134

The results of this study also show that shots laterally offset by up to 20 m still

produce results similar to the in-line shots. Therefore, a single line of shot points

together with a number of parallel recording lines, would be efficacious for

recording 3D refraction data along narrow swaths. A minimum of three parallel

lines is suggested, while five or more would give better cross-line determinations

of wavespeeds with the traveltime data.

Such a recording program would be suitable for many types of geotechnical

investigations as for example with highways and damsites, which require only

relatively limited coverage in the cross-line direction. Accordingly, the benefits of

the additional sampling in the cross-line direction can be achieved without a

commensurate increase in the number of shot points. Typically, an increase of

about 100% over an equivalent 2D program may be sufficient.

The refractor mapped in this study has large spatial variations in depths,

wavespeeds and azimuthal anisotropy and therefore it provides a searching test

of any approach seeking to resolve each of these parameters. The results of this

study demonstrate that simple and efficient 3D refraction methods using the

GRM can provide more useful geological interpretations than would be the case

with even detailed 2D approaches.

6.10 - References

Bamford, D., and Nunn, K. R., 1979, In-situ seismic measurements of crack

anisotropy in the Carboniferous limestone of North-west England: Geophys.

Prosp., 27, 322-338.

Barker, J. A., 1991, Transport in fractured rock, in Downing, R. A., and Wilkinson,

W. B., eds., Applied groundwater hydrology: Clarendon Press, 199-216.

135

Bennett, G., 1999, 3-D seismic refraction for deep exploration targets: The

Leading Edge, 18, 186-191.

Crampin, S., McGonigle, R., and Bamford, D., 1980, Estimating crack

parameters from observations of P-wave velocity anisotropy: Geophysics, 45,

345-360.

Deen, T. J., Gohl, K., Leslie, C., Papp, E., and Wake-Dyster, K., 2000, Seismic

refraction inversion of a palaeochannel system in the Lachlan Fold Belt, Central

New South Wales: Explor. Geophys., 31, 389-393.

Lanz, E., Maurer, H., and Green, A. G., 1998, Refraction tomography over a


Leslie, J. M., and Lawton, D. C., 1999, A refraction-seismic field study to

determine the anisotropic parameters of shales: Geophysics, 64, 1247-1252.


11, 12-19.







136

Palmer, D., 2001a, Imaging refractors with convolution: Geophysics 66, 1582-

1589.

Palmer, D., 2001b, Resolving refractor ambiguities with amplitudes: Geophysics

66, 1590-1593.


Thomsen, L., 1995, Elastic anisotropy due to aligned cracks in porous rock:

Geophys. Prosp., 43, 805-829.

Zelt, C. A., and Barton, P. J., 1998, 3D seismic refraction tomography: a

comparison of two methods applied to data from the Faeroe Basin: J. Geophy.

Res., 103, 7187-7210.

Zelt, C. A., 1998, Lateral velocity resolution from 3-D seismic refraction data:

Geophysical Journal International, 135, 1101-1112.

137

Chapter 7

Effects Of Near-Surface LateralVariations On Refraction Amplitudes

7.1 - Summary

Increases in refracted amplitudes not related to changes in the head coefficient

are usually associated with increases in traveltimes in the near-surface layers,

while decreases in amplitudes are associated with decreases in traveltimes.

These correlations demonstrate that the amplitude variations are related to

variations in the near surface geology, rather than to variations in the coupling of

the detectors with the ground.

The change in amplitude can be described with the transmission coefficient of

the Zoeppritz equations. Correction factors can be applied for those surface

conditions which are sufficiently extensive to permit the measurement of the

wavespeed. Where this is not possible, then the lowest amplitude or amplitude

product is representative of the head coefficient for the main refractor.

138

7.2 - Introduction

The generation of a refraction time section through the convolution of forward

and reverse seismic traces (Palmer, 2001a), provides a powerful and convenient

approach to resolving some of the ambiguities in the inversion of shallow seismic

refraction data (Palmer, 2001b). 2D and 3D case histories demonstrate that the

approach is efficacious with refractors exhibiting large variations in depths and

wavespeeds. The head coefficient is approximately the ratio of the specific

acoustic impedance in the upper layer to that in the refractor, while amplitudes in

the convolution section are the square of that ratio.

However, there can be geological situations where the refraction amplitudes are

not predicted by the head coefficient or its approximations. These situations

include lateral variations in the near surface layers and/or variations in the

coupling of the geophones with the ground.

The coupling of geophones, especially with the standard single geophone per

trace of most shallow seismic refraction operations, is a ubiquitous concern with

quantitative analysis of refraction amplitudes. Pieuchot (1984) reviewed earlier

work (Bycroft, 1956; Fail et al, 1962; Lamer, 1970), in which the effects of the

weight and diameter of geophones on coupling were considered. He concluded

that the size of modern geophones was adequate to produce satisfactory

coupling, and that the common geophone spike lengths of 50 mm to 100 mm

further guaranteed satisfactory coupling. Field trials (E J Polak, 1969) in which

the amplitudes of bunched geophones were measured, demonstrate that

variations in amplitudes related to planting are minor.

These conclusions are supported by a seismic refraction profile across a narrow

massive sulfide orebody at Mt Bulga in southeastern Australia. Originally, this

profile was recorded to observe whether there are any variations in refraction

amplitudes related to an unambiguous increase in density associated with the

139

mineralization. However, it is a complex case history which combines lateral

changes in the wavespeeds in both the refractor and the weathered layer above,

as well the density changes associated with the mineralization. Furthermore, the

results provide a valuable insight into the relative importance of the effects of

near surface lateral variations and geophone coupling with the ground on the

measurement of seismic amplitudes with single detectors. In particular, there is

a consistent correlation between amplitude variations of the refracted signal and

minor traveltime variations in the near surface layers. These results indicate that

near-surface geology rather than geophone coupling is the dominant cause of

seismic amplitude “statics”.

7.3 - Traveltime Results

The Mt Bulga massive sulfide orebody is narrow with a width generally less than

about 10 m. Nine shots, each consisting of small explosive charges in shallow

hand augered shot holes, and nominally 30 m apart, were recorded with a 48

trace seismic system using single geophones which were 2.5 m apart.

The centre of the seismic line at station 49 was located on the crest of a small

ridge, which also marked the location of the sulfide orebody. The rocks on either

side of the mineralization are Ordovician meta-sediments. Between stations 25

and 49, these sediments crop out, and there was some difficulty in auguring the

shot holes to a satisfactory depth and in planting the geophones. Between

stations 49 and 73, there is no outcrop, and the production of the shot holes was

much easier, as was the planting of the geophones.

The traveltime graphs are shown in Figure 7.1. They show that a two layer

model of the wavespeeds is generally satisfactory, and that there is a significant

lateral change in the wavespeeds of the first layer. Between stations 25 and 48

where the Ordovician meta-sediments sediments crop out, the wavespeed of the

140

first layer is 1500 m/s. Between stations 48 and 53 there is no outcrop, due

partly to the mining of the enriched supergene zone over a century ago and to

recent restoration of the site for a pine tree plantation. Here, the wavespeed of

the first layer is 900 m/s. On the other side of the orebody, between stations 53

and 72, the wavespeed of the first layer is 1000 m/s.

Figure 7.1: Traveltime data for a line crossing a narrow massive sulfide orebody

at Mt Bulga. The shot point interval is nominally 30 m.

Between stations 26 and 28, the traveltimes increase in both the forward and

reverse directions. This increase is inferred to be the result of an increase in the

thickness of surface layer of soil, because there is no lateral offset between the

increases in the forward and reverse traveltime graphs. A wavespeed of

141

approximately 500 m/s can be obtained from the graph with the shot point at

station 25.

Between stations 69 and 71, the traveltimes decrease in both the forward and

reverse directions. As with the previous case, there is no lateral offset, and so

this decrease is inferred to be the result of a reduction in the thickness of the

surface layer of soil.

Figure 7.2: Time-depths computed from traveltime data with shot points at

stations 1 and 97. The shading highlights a distinctive pattern of time-depth

anomalies which are centred on station 56 and which have their origin in the very

near-surface soil layer.

142

By contrast, the increases in the traveltimes in the forward and reverse directions

on either side of station 49 are offset by about two detector intervals, indicating

that the corresponding increases in depth occur in the main refractor. This

corresponds with an optimum XY value of 5 m, which is obtained from both the

wavespeed analysis function in Figure 7.3 and the offset in amplitudes in Figure

7.7.

The time-depths computed with the traveltimes for the shots at stations 1 and 97

and using a reciprocal time of 120 ms, and XY values from 0 to 10 m in

increments of 2.5 m which is the trace spacing, are presented in Figure 7.2. The

increase in the time-depths over the orebody is readily apparent.

The wavespeed analysis function is shown in Figure 7.3, using the traveltimes for

the shots at stations 1 and 97, and XY values from 0 to 15 m in increments of 2.5

m, the trace spacing. The optimum XY value has been taken as 5m, although it

may be a little less, possibly about 4m, because the graphs for the XY values of

2.5 m and 5 m are symmetrical about their average. The wavespeed in the

refractor is 5000 m/s between stations 25 and 48, 3430 m/s between stations 48

and 62, 2400 m/s between stations 62 and 68, and possibly about 5000 m/s

between stations 68 and 72. The region with the wavespeed of 2400 m/s is

along strike from the shear zone detected in another study (Palmer, 2001a;

chapter 3), and it is probably a continuation of that feature.

The depth section computed with these wavespeeds is shown in Figure 7.4. The

depths have been plotted vertically below the surface reference point and require

an additional operation (Palmer, 1986), which is equivalent to reflection migration

or imaging. The increase in the depth of weathering in the vicinity of station 50 is

probably caused by the more rapid breakdown of the sulfides or the removal of

ore and rock during mining.

143

Figure 7.3: The generalized wavespeed analysis function for a range of XY

values from zero to 15 m in increments of 2.5 m, which is the station separation.

The values for a 5 m XY value show the least variation related to the increased

depth of weathering over the orebody, and at least four zones with different

wavespeeds can be recognized.

144

Figure 7.4: Depth section computed with the time-depths shown in Figure 7.2.

The vertical to horizontal exaggeration is approximately 4:1.

7.4 - Effects of Near-surface Lateral Variations on Amplitudes

The two shot records with shot points at stations 1 and 97, are shown in Figures

7.5 and 7.6. They both show the large decrease in amplitude with increasing

shot-to-detector distance. In addition, there is a very obvious reduction in

amplitudes on the few traces centered on station 50, which is the location of the

massive sulfide orebody.

145


The large drop in amplitudes at stations 51 and 52 occurs near the location of the


146


The large drop in amplitudes at stations 49 and 50 occurs near the location of the


147

Figure 7.7: Amplitudes of the first cycles of the field records with shot points at

stations 1 and 97.

The amplitudes of the first cycles are shown in Figure 7.7, and show the usual

decrease with distance from the shot point, together with the sudden decrease at

around station 50. The amplitudes are somewhat erratic for the forward shot

between stations 26 and 49, which is the region of outcrop and where some

difficulties were experienced in planting the geophones. In addition, there is

good correlation between similar features on the reverse shot point, such as at

stations 32 and 43.

148

Figure 7.8: Uncorrected amplitude products of the first cycles of the field

records with shot points at stations 1 and97.

The product of the shot amplitudes, shown in Figure 7.8 with an XY separation of

zero, exhibits the same erratic nature as the shot amplitudes between stations 26

and 49. The amplitude products at stations 32 and 43 for example, are higher

than those at the adjacent stations.

A similar correlation is possible at station 56, where there is an increase in

amplitudes on both forward and reverse shots, and a corresponding increase in

the amplitude product. Furthermore, there is an increase in traveltime at this

station, and in turn, an increase in time-depths computed with a zero XY value as

149

shown in Figure 7.2. An examination of Figure 7.1, shows that the increase in

the traveltime occurs at the same geophone location for both the forward and

reverse shots, that is, there is no lateral displacement. Therefore, the increase in

depth occurs in the surface soil layer, rather than in the main refractor. The near-

surface origin is also supported by the characteristic pattern produced the time-

depths with in Figure 7.2 (Palmer, 1986, p107-111).

These results indicate that the occurrence of the soil surface layer produces

increases in seismic amplitudes. This is clearly indicated at station 56, as well as

at stations 32, 35, 41 and 43. In the latter cases, the time-depth anomalies are

not as large, but nevertheless, they are consistent with the hypothesis.

The hypothesis is further supported by the large increase in the surface soil layer

at stations 26 and 27 which corresponds with an increase in amplitudes, as well

as the decrease in the surface layer at stations 70 and 71, which correlates with

a decrease in amplitudes.

The variations in amplitudes with varying surface layers can be explained with

the transmission coefficients of the Zoeppritz equations, viz.:

Trans Coeff = 2 v lower ρ lower / (v lower ρ lower + v upper ρ upper) (7.1)

where

v upper is the wavespeed in the upper or surface layer,

ρ upper is the density in the upper or surface layer,

v lower is the wavespeed in the lower layer, and

ρ lower is the density in the lower layer.

This form of the equation is a little different from the standard, because the signal

is travelling upwards from the refractor.

150

In general, v upper < v lower and ρ upper < ρ lower. Therefore, the transmission

coefficient in equation 7.1 will vary from one, that is, there is no surface soil layer,

to two, that is, the surface soil layer wavespeed and density are much less than

those of the layer below. In those cases where there is a surface soil layer, there

will be an increase in amplitudes, because the transmission coefficient will

usually be greater than one.

The wavespeeds and densities in the upper soil layer can vary over quite large

ranges. Furthermore, it can be difficult to accurately map any rapid lateral

variations with for example, seismic methods. Therefore, it may not always be

possible to conveniently derive correction factors based on the Zoeppritz

equation.

In such situations where there is significant variation in the surface soil layer, it is

suggested that the minimum values, rather than the average values, be taken as

representative of that region. For example, the amplitude product for the region

between stations 30 and 48 will be taken as about 1.3, rather than the average of

about 2 or the maximum of about 2.5.

However, a wavespeed of approximately 500 m/s can be recovered for the

surface soil layer between stations 26 and 28. If the densities are ignored, then

the transmission coefficient computed with equation 7.1, is 1.5. Since the

amplitudes are multiplied in Figure 7.8, the transmission coefficient must be

squared, prior to application. The squared correction factor of 2.25 satisfactorily

accounts for the increase in amplitude, as shown in Table 1.

151

7.5 - Relationships Between Amplitudes and RefractorWavespeeds

The normalized amplitude products are shown in Figure 7.9 for XY values from

zero to 10 m. The values shown include the addition of a constant, namely, 1 for

an XY value of 2.5 m, 2 for an XY value of 5 m, and so on, in order to separate

the graphs.

The anomalous amplitude product at station 56 is readily seen on the graph for

the zero XY value. As the XY value is systematically increased, the forward and

reverse amplitude anomalies are separated, with the result that the anomalous

product separates into two, which correspond with the forward and reverse shot

amplitude values. The forward amplitude systematically moves to the right, while

the reverse value moves to the left. The pattern is similar to that produced by

traveltime anomalies which originate in the near-surface (Palmer, 1986, p107-

111), as shown in Figure 7.2 for station 56.

This pattern with the amplitude products which can be seen clearly at station 56,

can also be recognized with some difficulty between stations 27 and 48.

Figure 7.7 shows that the very low amplitudes associated with the massive

sulfides, occur at stations 51 and 52 on the forward shot and 49 and 50 on the

reverse shot. The amplitude products in Figure 7.9, show that this interval is a

minimum for an XY value of 5 m, and that it occurs at stations 50 and 51. For

other XY values, this zone is wider.

The accompanying table summarizes the amplitude products and the correlation

with wavespeeds. In general, the agreement is good.

152

Figure 7.9: Uncorrected products of the amplitudes of the first cycles of the field

records with shot points at stations 1 and 97 for a range of XY values from zero

to 10 m.

The normalized squared ratios of the wavespeeds in the final row have been

corrected for the additional near-surface layer of soil between stations 26 and 29,

and for an inferred density factor of 2.8 for the mineralized region in the centre

153

between stations 50 and 51. If a density of 2.4 tonnes/m3 is assumed for the

meta-sediments, then the resulting density for the mineralization is 6.6

tonnes/m3. This density is a little high, but it is possible to reduce it using a

higher wavespeed in the mineralization. This is reasonable because the

measured wavespeeds may not be especially accurate over such a narrow

interval.

Station 26 – 29 29 – 50 50 – 51 51 – 62 62 – 68 68 -72NormalizedAmplitudeProduct

2.9 1.3 0.13 1.0 2.2 1.5

V1 (m/s) 1500 1500 900 1000 1000 ? 1600

V2 (m/s) 5000 5000 3430 3430 2400 ? 5000

(V1/ V2)2 0.09 0.09 0.07 0.07 0.17 ? 0.10

Normalized(V1/ V2)2

1.3 1.3 1 1 2.4 ? 1.5

Corrected(V1/ V2)2

2.9

(soil

layer)

1.3 0.13

(orebody

density)

1 2.4 ? 1.5

Table 1: Summary of amplitude products and wavesppeds.


This case history provides another good example of the correlation between

head wave amplitude products and the ratios of the wavespeeds. It is complex

with many large variations in depths as well as wavespeeds in both the refractor

and the layer above. Furthermore, it qualitatively confirms the importance of

densities on head wave amplitudes.

154

The case history also provides a valuable insight into the importance of near-

surface variations and geophone coupling on the measured refraction

amplitudes.

Between stations 50 and 72 where there is no outcrop, the amplitudes and the

amplitude products are essentially a function of the wavespeeds in the refractor

and the layer above. However, at station 56, there is an increase in amplitudes

which correlates with an increase in traveltimes and time-depths. These results

indicate that the amplitude variation is related to the near-surface layering, rather

than to the coupling of the geophones with the ground.

The results for the region between stations 26 and 50, where there was

extensive outcrop, support this interpretation, because all of the amplitude

anomalies can be associated with traveltime anomalies.

The presentation of both amplitude products and time-depths for a range of XY

values from zero to more than the optimum value, provides a convenient and

effective method for recognizing near-surface anomalous zones of limited lateral

extent.

The increases in amplitudes are compatible with the transmission coefficients of

the Zoeppritz equations. As the seismic signal approaches the surface from the

refractor, there is an increase in seismic amplitude where there is another layer

with lower wavespeed and or density.

In general, this change in amplitude can be ignored when there are several

continuous layers above the refractor, because the same increase in amplitudes

occurs at each detector. In these situations, the amplitudes are adequately

described with the head coefficients, together with a geometric spreading factor.

155

Where there are lateral changes in the surface layers, such as the irregular

development of a surface soil layer, there can be large variations in amplitudes

by a factor of between 1 and 2. If these layers have sufficient lateral extent so

that they can be mapped, such as the region between stations 26 and 28, then

an approximate correction factor can be computed with the transmission

coefficients of the Zoeppritz equations.

However, this is not always possible. Under these circumstances, the minimum

amplitudes are probably the most representative.

7.7 - References

Bycroft, G. N., 1956, Forced vibrations of a rigid plate on a semi-infinite elastic

space: Roy. Soc. London, 248, 327-368.

Fail, J. P., Grau, G., and Lavergne, M., 1962, Couplage des sismographes avec

le sol: Geophys. Prosp., 10, 128-147.

Lamer, A., 1970, Couplage sol-geophone: Geophys. Prosp., 18, 300-319.




seismic: Geophysical Press.

Palmer, D., 2001a, Imaging refractors with the convolution section: Geophysics

66, 1582-1589.

156

Palmer, D., 2001b, Resolving refractor ambiguities with amplitudes: Geophysics

66, 1590-1593.

Pieuchot, M., 1984, Seismic instrumentation: Geophysical Press.

Polak, E. J., 1969, Attenuation of seismic energy and its relation to the properties

of rocks: Ph D thesis, University of Melbourne, p4.7-4.9.

157

Chapter 8

Enhancement of Later Events in theRCS with Dip Filtering

8.1 - Summary

Later events, which occur in the shot records, are also treated in the same

manner as first events with the convolution process. Both the addition of the

traveltimes and the multiplication of amplitudes take place. However, there can

be additional features in which cross-convolution artifacts are also generated.

These artifacts which are formed by the convolution of events from different

refractors, occur as relatively steeply dipping events in the refraction convolution

section (RCS) and therefore, they can be removed by dip filtering. The filtered

RCS shows better continuity of events than is the case with the unfiltered

section.

For events which have traveled through the surface layer, the filtered RCS shows

a series of events which occur at a time which is a function of the distance

between the two shot points and the wavespeed in the surface layer. The time of

this event can be used to improve the estimates of the wavespeed in the surface

layer.

158

8.2 - Introduction

The generation of the refraction convolution section (RCS) (Palmer, 2000)

produces a set of traces with the superficial appearance of a seismic reflection

section. It has been demonstrated that the RCS reproduces the time structure of

the refractor interface with first arrivals, while the amplitudes which are largely

corrected for geometric spreading, are essentially a function of the square of the

head coefficient. It has also been demonstrated that the head coefficient is given

approximately by the ratio of the wavespeeds in the upper layer and refractor.

The RCS amplitudes can be employed to image the refractor, to resolve some of

the ambiguities in the determination of wavespeeds in the refractor, and to obtain

a measure of azimuthal anisotropy with three dimensional methods.

To date, research has focused primarily on the portion of the RCS which

corresponds with the first arrivals, and little attention has been directed at later

events. However, the convolution process performs the same operations on later

arrivals as it does with the first events. These operations are the addition of the

traveltimes in the forward and reverse directions, which replaces of moveout from

trace to trace with a constant amount equal to the reciprocal time, the time from

the forward shot point to the reverse shot point, and the multiplication of the

amplitudes. The addition of the traveltimes produces the relative time structure

on the refracting interface, while the amplitude product effectively compensates

for the large geometric spreading which is characteristic of refraction data. The

true time structure on the interface can be obtained by subtracting the reciprocal

time. As the reciprocal time generally decreases with deeper layers, the

shallower layers occur at later times in the RCS.

One feature of the RCS is the generation of what will be termed cross-

convolution events with later arrivals. In this case, the convolution operation

adds arrivals from different refractors, and therefore generates artifacts which

have no geophysical significance. For example, it is possible to produce an

159

event which is the addition of the traveltimes from the refractor in the forward

direction, with the traveltimes from the surface layer in the reverse direction. In

practice, these artifacts occur as pairs, that is there is also an event produced by

the addition of the traveltimes from the surface layer in the forward direction, with

the traveltimes from the refractor in the reverse direction. Because of the

different moveouts or wavespeeds, these events appear as relatively steeply

dipping features in the RCS.

This study describes the use of dip filtering in the f-k domain (Sheriff and Geldart,

1995), to remove the cross-convolution events, with the aim of enhancing those

later events which may have geological significance.

8.3 - Generation of Useful Events and Artifacts in the RCS

The generation of useful later events in the RCS can best be demonstrated with

the ground-coupled air wave. While it is recognized that the imaging of the air

wave has minimal geological significance, it is employed in this study because its

high amplitude improves its clarity in the RCS.

Figure 8.1 is a shot record from a shallow seismic refraction survey at Mt Bulga,

which has been described previously (Palmer, 2001). The record shows the first

arrivals between about 70 ms and 130 ms as very low amplitude signals, and a

very high amplitude event between 350 ms and 1000 ms. The first arrivals are

refracted from the base of the weathering, while the second arrivals are the

ground-coupled air wave.

Figure 8.2 shows the shot record in the reverse direction. The same two events

can be clearly identified, but in this case, the relative amplitude of the ground-

coupled air wave is lower.

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Figure 8.1: A shot record showing low amplitude first arrivals between about 70

ms and 130 ms refracted from the base of the weathering, and the high

amplitude ground-coupled air wave between 350 ms and 1000 ms.

161

Figure 8.2: Reverse shot record in the reverse direction. The relative amplitude

of the ground-coupled air wave between 350 ms and 1000 ms is lower.

162

Figure 8.3: The RCS generated with the two shots shown in Figures 8.1 and

8.2. The sampling interval has been halved but there has been no subtraction of

a reciprocal time.

163

The RCS generated with these two shots is shown in Figure 8.3. The sampling

interval has been halved (Palmer, 2001), but there has been no subtraction of a

reciprocal time. The presentation gain is low so that the portion of the RCS

which corresponds with the convolved events from the refractor around 150 ms,

becomes essentially featureless. However, the gain facilitates the recognition of

the strong event at approximately 700 ms between stations 29 and 68. The

limited lateral extent of this event occurs because the recording time of one

second was insufficient to record the air wave at the distant detectors.

In addition, the presentation gain highlights the cross-convolution events which

start a few traces from the left side of the section at 300 ms and continue to

about 550 ms near the right side of the section. The recognition of the

companion artifact which starts on the right hand side and finishes on the left is

not as clearly evident in Figure 8.3 and requires more careful inspection.

8.4 - Removal of Cross-convolution Artifacts with Dip Filtering

The transformation of the in RCS in Figure 8.3 from the time-distance domain to

the frequency-wavenumber (fk) domain with the double Fourier transform, is

shown in Figure 8.4. It shows signal centered on the frequency axis, which

corresponds with the horizontal events, and signal spread out parallel to the

wavenumber axis, which is inferred to correspond with the cross-convolution

events.

Figure 8.6 shows the fk domain after the application of a filter to remove all signal

other than that centered on the frequency axis, while Figure 8.5 shows the RCS

after the application of the filter. The event which corresponds with the time-

depth of the ground-coupled air wave can be clearly seen at about 0.710 s.

164

Figure 8.4: The transformation of the in RCS in Figure 8.3 from the time-distancedomain to the frequency-wavenumber (fk) domain with the double Fouriertransform.

Figure 8.6: The transformation of the in RCS in Figure 8.5 from the time-distance domain to the frequency-wavenumber (fk) domain with the doubleFourier transform.

165

Figure 8.5: Refraction convolution section in Figure 8.3 after dip filtering to

remove cross-convolution events.

166

8.5 - Times for Near-surface Events in the Uncorrected RCS

The RCS in Figure 8.5 which has not been corrected by the subtraction of a

reciprocal time, facilitates the computation of wavespeeds for the near surface

layers.

The traveltime in the forward direction t forward, of a seismic signal travelling

through a near-surface layer, that is, for which the depth can be ignored is

tforward = x / V1 (8.1)

Similarly, the traveltime in the reverse direction treverse, at the same detector is

treverse = (d – x) / V1 (8.2)

where x is the forward shot-to-detector distance and d is the separation between

the forward and reverse shot points, and V1 is the wavespeed in the near-surface

layer.

In the RCS in Figure 8.5, these times are firstly summed, then halved, and they

occur at a time tRCS, where:

tRCS = d / 2 V1 (8.3)

It can be readily shown that the ground-coupled airwaves in Figures 1 and 2 has

wavespeeds of about 335 m/s. Using a value of d, the shot point to shot point

distance, of 480 m, the value of tRCS computed with equation is 0.716 s. This

value is similar to that measured above in Figure 8.5.

167

Figure 8.7: Shot record with shot point at station 26.

168

8.6 - Near-surface Wavespeeds from the Uncorrected RCS

Figures 8.7 and 8.8 are two shot records with shot points at the ends of the

geophone spread at stations 26 and 73. The presentation gains are low to

facilitate recognition of a series of events with a wavespeed of approximately 400

m/s, that is, they arrive at the geophones most distant from the shot points after

about 0.600 s. These events occur over an interval of about 0.15 s, and they are

interpreted to be arrivals from the near-surface layer, rather than the ground-

coupled airwave, because of their lower frequency and inferior continuity

compared to the ground-coupled airwaves in Figures 8.1 and 8.2.

A comparison of the unfiltered and filtered RCS in Figures 8.9 and 8.10, shows

that the dip filtering has removed the cross-convolution events, and that the

horizontal and near-horizontal events are emphasized.

Using equation 8.3, it is readily demonstrated that the group of events with the

wavespeeds of 400 m/s should occur at a time tRCS, of 0.3 s. Figure 8.10 shows

a series of events from about 0.33 s to about 0.47 s with higher amplitudes than

the adjacent events. These events are interpreted to represent signals which

have traveled in the surface soil layer. Using the minimum time of 0.33 s and

equation 8.3, the revised wavespeed for this layer is 360 m/s.

It is also possible to recognize a series of events from about 0.18 s with higher

amplitudes than the adjacent events. These events may correspond with arrivals

which travel through the second layer with a wavespeed of approximately 700

m/s.

While the event associated with the ground-coupled air wave convincingly

demonstrates the generation of meaningful later events in the RCS, the

application to events from the near-surface layers is not as clear. It is likely that

further processing, such as deconvolution may be useful. However initial

169

attempts at simple deconvolution methods were not successful, suggesting that

considerably more research may be required to develop suitable techniques.

Figure 8.8: Reverse shot record with shot point at station 73.

170

Figure 8.9: Unfiltered convolution section.

171

Figure 8.10: Dip filtered convolution section.

172

8.7 - Conclusions

Later events, which occur in the shot records, are also treated in the same

manner as first events with the convolution process. Both the addition of the

traveltimes and the multiplication of amplitudes take place. However, there is an

additional feature in which cross-convolution artifacts are also generated. These

artifacts occur as relatively steeply dipping events in the RCS and therefore, they

can be removed by dip filtering. The filtered RCS shows better continuity of

events than is the case with the unfiltered section.

For events which have traveled through the surface layer, the filtered RCS shows

a series of events which occur at a time which is a function of the distance

between the two shot points and the wavespeed in the surface layer. The time of

this event can be used to improve the estimates of the wavespeed in the surface

layer.

8.8 - References

Palmer, D., 2001, Imaging refractors with the convolution section: Geophysics

66, 1582-1589.



173

Chapter 9

Stacking Seismic Refraction Data inthe Convolution Section

9.1 - Summary

The refraction convolution section (RCS) is an effective domain to vertically stack

shallow seismic refraction data, in order to improve signal-to-noise ratios (S/N).

The convolution operation essentially compensates for the effects of geometrical

spreading, and generates traces with much the same S/N ratios. Such traces

are optimum for stacking, unlike the traces on the original shot records.

A major benefit of stacking in the RCS domain is that it takes places before the

measurement of times or amplitudes. With other approaches which do not

routinely employ stacking, such as tomography, any variations in data quality are

addressed with the application of statistical methods to the traveltimes

determined on the original field data.

An essential requirement for stacking in the RCS domain are data which have

been acquired with a continuous roll along approach typical of reflection

methods, rather than with the more common static spread. Such operations are

more efficient and produce more data from the critical near-surface layers, but

they would require significant re-capitalization of most shallow seismic field

operations.

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9.2 - Introduction

It is well known that the source energy requirements for seismic refraction

surveys are considerably greater than those for seismic reflection surveys for the

same target. The maximum source-to-detector distance for seismic reflection

surveys is generally less than the target depth, whereas the minimum source-to-

detector distance with refraction surveys is usually greater than four times the

target depth. In addition, the geometric spreading component is the reciprocal of

the distance traveled for reflected signals, while the corresponding function for

refracted signals is the reciprocal of the distance squared. Both the longer path

lengths and the more rapid spreading factors result in low refraction amplitudes

and therefore higher source energy requirements. Commonly, the refraction

source is more than ten times the size of the reflection source for the same

target.

Explosives are the standard energy source in most shallow seismic refraction

surveys, and adequate signal-to-noise (S/N) ratios are readily achieved by

increasing the size of the charge. However, this is not always practical in many

environmentally sensitive or urban areas, and it normally results in either poor

quality data due to insufficient charge sizes, or more commonly, no acquisition of

data at all.

In some cases, it is possible to use vertical signal stacking with repetitive

sources, such as hammers and dropping weights. Nevertheless, this approach

can be of limited usefulness, because many repetitions can be required to obtain

reasonable S/N ratios, especially where urbanization is the major source of

noise.

In addition, vertical stacking can result in slow rates of progress where there are

many source points. Walker et al, (1991) demonstrate that one of the most

important factors in improving the reliability of shallow seismic refraction

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interpretation is a detailed mapping of the wavespeeds in the layers above the

target refractor using a high density of source points. It is not uncommon to

employ a source point between every other pair of geophones.

This study demonstrates the use of vertical stacking with a CMP-like method

using the refraction convolution section (RCS). Redundant or multi-fold

refraction data are acquired with a continuous roll along approach, which is

standard with reflection acquisition. Multiple overlapping RCS are generated with

pairs of shots with the same shot point to shot point separation. The ensemble of

RCS are then sorted and gathered, in much the same way as reflection shot

records, and the gathers are then stacked.

The RCS is a suitable domain in which to stack refraction data when the shot

size, depth and separation are uniform, because the events have approximately

the same S/N ratios. With effective vertical stacking, the S/N ratio improves as

the square root of the number of traces in the stack, but only when the S/N ratios

of the original traces are much the same. Excessively noisy traces, that is traces

with an anomalously low S/N ratio, can significantly degrade the stack and

reduce the benefits of stacking. This situation occurs with stacking refraction

shot records, because there can be large variations in S/N ratios related to the

effects of geometric spreading. Traces at a given station with nearby source

points will have high S/N ratios while traces at the same station with more distant

source points will have lower S/N ratios. The large range in S/N ratios with

refraction shot records significantly reduces the effectiveness of stacking traces

from various shot records with a surface consistent approach.

Shearer (1991) demonstrates stacking shot records in which the shot-receiver

distance is preserved, but not the individual station locations, in order to improve

S/N ratios with earthquake data (see also Lay and Wallace, 1995, p215-216).

However, this approach is not a viable option with shallow refraction data,

176

because it does not accommodate lateral variations in either the depths to or

wavespeeds in the target refractor.

A major advantage of stacking in the RCS domain is that S/N ratios can be

enhanced prior to the measurement of parameters, such as times or amplitudes.

By contrast, tomographic methods measure traveltimes from the shot records

which have varying S/N ratios, and then seek to minimize any errors with a

statistical approach.

9.3 – The Cobar Stacked RCS Section

High resolution data were recorded with a 48 trace recorder, and single 40 Hz

geophones with a 10 m spacing, as part of a regional seismic reflection survey

(Drummond et al, 1992) across the Cobar Basin (Glen et al, 1994), in the central

west of NSW, Australia. The aim of these high resolution lines was to image the

near surface layers, which in these areas were dipping predominantly in the

vertical direction. The seismic source was a 10 kg charge of a high velocity

seismic explosives at a depth of 40 m, and the shot point interval was 30 m.

The data were recorded with off-end shots in both the forward and reverse

directions in order the obtain large shot-to-detector distances for the vertically

dipping reflection targets. For example, the first shot was at station 96, and the

geophones were from station 95 to station 48. The next shot was at station 90

and the geophones from station 89 to 42, a shift of 60 m. Subsequent shots

continued through to station 48 with geophones between stations 47 and 0. The

geophone array then remained static while the shot points at 60 m intervals

within the array at stations 42, 36, etc., were recorded. The recording process

was then reversed. The shot point at station 3 was recorded with geophones

between stations 4 and 51. Subsequent shots were at 60 m intervals (stations 9,

15, etc.), and the geophone array was moved up by the same amount in each

177

case (stations 10 to 57, 16 to 63, etc.). The resulting maximum refraction fold is

six, which is comparatively low.

Figure 9.1: Forward shot record number 65. Shot point is at station 33

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Figure 9.2: Reverse shot record number 43. Shot point is at station 90.

The first six to twelve traces near the shot point were arrivals from the surface

layer. In order to generate as many useful convolution traces as possible, pairs

179

of shots spaced 63 stations apart were used. This reduced the fold to between

one and four.

Figures 9.1 and 9.2 are two shot records which show the familiar rapid decay of

refracted energy with distance, and in turn, the large variation in S/N ratios with

offset.

Figures 9.3 to 9.8 are a series of RCS over intervals of approximately 30

stations. The structure of the refractor can be readily seen in the RCS. A major

feature of the five RCS is the approximately uniform S/N ratios.

Figure 9.9 is the stacked section, obtained from the five sorted and gathered

sections. While the structure of the refracting interface can be recognized, there

is only a modest improvement in the S/N, due mainly to the low fold of between

180

one and three. Nevertheless, it demonstrates that stacking is efficacious.

Clearly, a much higher fold is necessary to obtain the full benefits of stacking.

181

Figure 9.9: Stacked convolution section.

182

9.4 - The Static Geophone Spread

For the effective use of stacking, it is necessary to use RCS with relatively

uniform S/N ratios, and therefore to employ uniform acquisition parameters. The

relevant parameters are consistent charge size and depth, as well as a uniform

separation between the forward and reverse sources. Although these field

parameters are the norm with the refraction data acquired with routine reflection

surveys for petroleum and coal exploration and for regional reflection surveys in

fold belts, they are uncommon with most shallow seismic refraction surveys

carried out for geotechnical, groundwater and environmental applications.

The majority of shallow seismic refraction surveys carried out for geotechnical

and other shallow applications acquire data in discrete units or spreads. With

these surveys, a static spread of detectors is used with a multiplicity of source

points located at several offset positions on one side, through the spread and to

offset positions on the other side. The number of shot points recorded for each

spread has increased substantially in recent years in order to improve the

determination of the wavespeed stratification above the target refractor, and it is

now common to record more than eleven shots for a spread of 12 detectors

(Walker et al, 1991). For the typical survey length of 400 m for a road cutting,

approximately eight spreads of 12 detectors with a 2 detector overlap are

required (Walker et al, 1991), making a total of 88 shot points.

However, it is questionable whether even this considerable number of shot points

achieves the stated objectives of defining the wavespeed stratification within the

weathered layer. An inspection of published data (Walker et al, 1991, Fig. 6),

shows that the vast majority of the traveltime data (~ 90%) are arrivals refracted

from the base of the weathering. Although it is essential to ensure some

redundancy in the traveltime data in order to resolve the fundamental ambiguity

of determining the number of layers detected (Palmer, 1986, p21-29; Lankston,

1992), the majority of the data from the main refractor are generally not used in

183

the subsequent data processing stages. It questions the fundamental

effectiveness of the static spread approach to acquiring shallow seismic

refraction data.

There are also concerns about the efficiency of field operations with static

spreads. There can be a comparatively large number of shot points per unit

distance because of the occupation of many shot point locations on two and

often three occasions, as well as the common practice of using an overlap of

several detectors. The repeated occupation of shot points can be

environmentally damaging as well as time consuming. Furthermore, field

operations do not progress smoothly, because the acquisition of data ceases

while the spread of detectors is retrieved and then re-deployed for the next

adjacent spread.

The static spread approach also results in a wide range of source energy

requirements for the different offsets and the different layers above the target.

Relatively low source energies are required for signals propagating in the shallow

near-surface layers, while considerably greater source energies are required for

the deeper target refractors. Since the majority of traveltimes are from the main

refractor, there can be a large source energy requirement. As mentioned

previously, many of these times are not used in the data processing, which

suggests that more efficient approaches may be possible.

This study proposes the continuous acquisition of shallow seismic refraction data

be employed routinely for geotechnical, groundwater and environmental

applications.

9.5- Continuous Acquisition of Shallow Seismic Refraction Data

Continuous acquisition of redundant or multi-fold data is the norm with seismic

reflection methods. With this technique, the source point maintains a fixed

184

position in relation to the detector spread, which for land operations is usually a

split spread with the source in the centre of the detectors. The constant

geometry is obtained by laying out more detectors than there are channels in the

recording instruments and then selecting the required channels with a roll along

switch. Continuous and efficient operations are achieved with a single pass of

the seismic source along the line in conjunction with the continual removal of

detectors from the start of the line after they are no longer required, and their

placement at the other end to which the source is progressing.

Better and more uniform coverage of all refractors commonly occurs.

Roll along acquisition methods can provide better data for either the conventional

or convolution approaches where two or more adjacent static spreads would

normally be employed. Comparisons of field operations show that in fact, there

can be a reduction in the number of shot points per unit distance of coverage.

For example, for a 400 m long survey for a road cutting using a 15 m shot

spacing and a 5 m detector interval, a total of only about 55 shots would be

required. A 12 channel seismic recorder would not be suitable for roll along

operations, because the maximum shot to detector distance of 30 m would

generally be insufficient to record enough arrivals from the base of the

weathering. However, a 24 channel seismic recorder, which is widely used in

shallow refraction surveys, would be suitable, and in many cases might even

permit a reduction in the trace spacing to 3m to further improve the resolution of

the wavespeed stratification in the weathered layer. A 48 channel system would

provide further improvements in data quality through additional reductions in

trace spacing, as well as enhanced capabilities with swath or partial three

dimensional profiling.

The comparatively large number of shot points per unit distance with adjacent

static spreads is a result of the occupation of shot point locations on two and

often three occasions, as well as the common practice of using an overlap of

185

several detectors. In contrast, the continuous recording of refraction data with

the roll along method involves the once-only occupation of each source point,

and incorporates a uniform overlap as an integral part of the method.

Accordingly, it represents a more efficient use of equipment and field personnel

with a lower environmental impact.

For surveys in which only limited coverage is required, it is still desirable to

replace the single static spread with a quasi roll along approach. However, the

number of source points may in fact increase with this approach, in order to

obtain sufficient data redundancy for stacking.

9.6 – Determination of Fold with RCS Data

The maximum fold obtained with continuous refraction acquisition using a split

spread shooting method is similar to that obtained with reflection data, viz.

Maximum fold = Number of detectors / (Shot spacing x 2) (9.1)

Note that the shot spacing is given as the number of detector intervals.

The validity of equation 9.1 can be demonstrated with a simple example.

Suppose that the recording system has 48 channels, the shot is at station 25 and

that the live geophones are from stations 1 to 24 and 26 to 49. For the same

split spread recording pattern, reversed shots at stations 1 and 49, which

represent a shot spacing of 24 stations, are the minimum necessary to compute

a time-depth at each detector, provided all arrivals are from the target refractor.

The maximum resulting fold is therefore one.

186

At the other extreme, if the shot spacing is reduced to a single detector interval,

then the maximum fold is 24. For the more common shot spacing of two detector

intervals, the maximum fold is 12.

In general, not all detectors will record arrivals from the target refractor. Suppose

that the first six arrivals on either side of the shot point are from layers other than

the target. For a shot spacing of two detector intervals, the fold will be 9. This

represents a substantial improvement in efficiency over the static spread

approach. The fold or redundancy of nine would be adequate to resolve most

ambiguities in layer recognition, as well as providing moderate improvements in

S/N through stacking. In addition, 25% of the arrivals are from the shallow

surface layers, which represents a significant improvement over the 10% for

typical static spreads, while the overall shot density per unit distance has been

decreased by as much as 40%. Further increases in the proportion of arrivals

from the near surface layers could be achieved by reducing the station spacing.

This analysis has used shot points at the stations themselves, rather than

midway between, which is also common. The benefit of shots at the detectors is

that reciprocal times, the times between the forward shot point and the reverse

shot point can be readily measured in both directions and then averaged. Any

traveltime delays caused by disturbed ground caused by previous shots, can be

avoided by offsetting the shot points by a few metres at right angles to the line.


The refraction convolution section (RCS) is an effective domain to vertically stack

shallow seismic refraction data, in order to improve signal-to-noise ratios (S/N).

The convolution operation essentially compensates for the effects of geometrical

spreading, and generates traces with much the same S/N ratios. Such traces

are optimum for stacking, unlike the traces on the original shot records.

187

A major benefit of stacking in the RCS domain is that it takes place before the

measurement of times or amplitudes. With other approaches which do not

routinely employ stacking, such as tomography, any variations in data quality are

addressed with the application of statistical methods to the traveltimes

determined on the original field data.

Stacking data in the RCS requires data with uniform acquisition parameters, such

as are typical of seismic reflection surveys for petroleum exploration on land

using split spread and CMP methods.

This would require a major change in field methods with most shallow seismic

refraction operations. In particular, it would require upgrades in acquisition

systems from 12 or 24 channels to at least 48 and preferably 60 channels for 2D

operations, together with roll along and radio shot firing system capabilities.

However, the costs of re-capitalization would be quickly recovered through

improved operational efficiencies.

The use of a roll along acquisition program would result in a reduction in shot

points by up to 40% and in turn a reduction in field time by at least the same

amount. The daily costs for the average three man field crew with 24 trace

equipment are about $A3000. Accordingly, the cost of a new 60 trace field

system at $A90,000 would be equivalent to thirty days of saved field time.

One obvious application of stacking with the RCS is with the computation of

statics, the corrections for variations in the elevations of source and detectors

and for the weathered layer, for regional seismic reflection surveys in fold belts

(Palmer et al., 2000). Accurate weathering corrections are especially important

with regional reflection studies, because continuous reflectors, which are

common in seismic reflection data in sedimentary basins, are very rare in data

recorded across fold belts. As a result, residual statics routines are not effective,

188

and therefore detailed refraction statics analyses are necessary. Frequently, the

arrivals from the base of the weathering can be poor quality and some form of

signal enhancement prior to the measurement of traveltimes might be beneficial.

9.8 - References

Drummond, B, Goleby, B, Wake-Dyster, K, Glen, R and Palmer, D, 1992, New

tectonic model for the Cobar Basin, NSW points to new exploration models for

targets in the Lachlan Fold Belt: BMR Research Newsletter 16, 16-17.

Glen, R.A., Drummond, B.J., Goleby, B.R., Palmer, D. and Wake-Dyster, K.D.,

1994. Structure of the Cobar Basin New South Wales based on seismic

reflection profiling: Australian Journal of Earth Sciences 41, 341-352.

Lankston, R. W., 1990, High-resolution refraction seismic data acquisition and

processing, in Ward, S. H., ed. Geotechnical and environmental geophysics, vol.

1, Investigations in geophysics no. 5: Society of Exploration Geophysicists, 45-

74.

Lay, T., and Wallace, T. C., 1995, Modern global seismology: Academic Press.



Palmer, D., Goleby, B., and Drummond, B., 2000, The effects of spatial sampling

on refraction statics: Explor. Geophys., 31, 270-274.

Shearer, P., 1991, Imaging global body wave phases by stacking long-period

seismograms: J. Geophys. Res., 96, 20,353-20,364.

189

Walker, C., Leung, T. M., Win, M. A., and Whiteley, R. J., 1991, Engineering

seismic refraction: an improved field practice and a new interpretation program,

REFRACT: Explor. Geophys., 22, 423-428.

190

Chapter 10

Discussion and Conclusions

10.1 - Shallow Refraction Seismology for the New Millennium: APersonal Perspective

The point of departure for this study was that most current shallow seismic

refraction operations have not taken advantage of advances in technology for

acquisition, processing or interpretation, they are under-capitalized, they are

relatively inefficient, and that the current seismic reflection technology provides

compelling models for the advancement of shallow refraction seismology. Based

on the results of this work, what then are the major features of seismic refraction

operations which might be appropriate to the requirements and the technology of

the new millennium?

A major achievement of this study is a demonstration of the superiority of 3D

results over 2D. There is simply no substitute for the improved quality and

quantity of information which can be obtained from even simple cost-effective 3D

surveys such as that described in this study. It is essential that 3D refraction

methods be adopted as a matter of some priority.

It is likely that the acceptance of 3D shallow refraction methods will parallel the

acceptance of 3D reflection methods by the petroleum exploration and

production industries and the acceptance of high resolution airborne magnetic

and radiometric data by the mineral exploration industries. Initially, cost was

191

considered to be the major reason for the relatively low levels of acceptance of

these methods. However, this situation changed rapidly when it was widely

demonstrated that high spatial sampling densities in all directions, is one, if not

the most important factor, in reducing risk through improved geological

interpretations. This conclusion is supported by the 3D results described in this

study.

The development of a 3D oriented approach implies the use of specialist seismic

contractors for acquisition in order to employ field systems with greatly increased

capabilities, as well as to promote efficient field operations. It is difficult to justify

the use of relatively expensive professional expertise to carry out routine

unskilled field duties with under-capitalized systems and inefficient operations.

Increasing channel capacity to at least 150 and doubling the number of shot

points could achieve efficient 3D field operations. This would result in an

increase of at least an order of magnitude in the amount of data, and in turn it

would dictate the use of efficient methods of data processing and interpretation.

Full trace processing with the RCS is a simple and efficient approach for

processing any volume of seismic refraction data.

It is likely that the increased quantity and quality of data obtained with 3D surveys

might stimulate a change in the roles of the geophysicist from acquisition and

processing towards interpretation. It also implies inclusion of other geoscientists

at earlier stages of the interpretation process, in order to generate more complex

and more geologically meaningful interpretation models.

The format of data processed with the RCS facilitates the convenient application

of current reflection processing and interpretation technology to shallow seismic

refraction data. Although the existing software developed specifically for

refraction seismology represents many man-years of effort, it is relatively

insignificant when compared with the software developed for reflection

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seismology. Just as the use of imaging processing software, which was

developed originally for remotely sensed data, has increased the detail of the

geological interpretation of magnetic, radiometric and gravity data, so seismic

reflection software is a vast resource which has the potential to extract even

greater information from refraction data. In particular, the data processed with

the RCS is suitable for analysis with software used for the interpretation of

processed seismic reflection data. Such software includes basic functions for

picking times and amplitudes of horizons, as well as post-processing functions,

such as attribute analysis. Attribute processing of RCS data may have as large

an impact on increasing the detail of the interpreted geological model as it has

with reflection data.

The author’s preference for an approach which is essentially an extension of the

GRM, is hardly surprising. However, other approaches, such as tomography are

currently not viable alternatives. The major shortcoming of tomography is that

the large increase in the number of shot points, commonly by at least an order of

magnitude over a simple GRM approach suggested here, would result in high

and possibly prohibitive costs of acquisition. Furthermore, tomography has yet to

satisfactorily address either the issues of non-uniqueness, large variations in

wavespeeds in the refractor, or anisotropy.

The RCS offers a new approach to generating more complex geological models

from shallow seismic refraction data through the use of the complete seismic

refraction trace and therefore, the use of amplitudes as well as traveltimes. In

time, it may stimulate the development of routine refraction methods which are

comparable in sophistication to current 3D reflection methods.

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10.2 - Conclusions

This study demonstrates that the refraction convolution section (RCS), generated

by the convolution of forward and reverse shot records, is an efficacious

approach to full trace processing of shallow seismic refraction data.

The convolution operation effectively adds the first arrival traveltimes of each pair

of forward and reverse traces. Like the many standard methods for processing

refraction data which use addition to obtain a measure of the depth to the

refracting interface in units of time, the RCS also produces a similar time image

of the refractor. In this study, the reciprocal time, the traveltime from the forward

shot point to the reverse shot point, is subtracted and the result is then halved

(by halving the sample interval of the trace headers) to form the equivalent of the

time-depth function of the generalized reciprocal method (GRM).

The generation of the RCS requires no estimates of, or assumptions about the

wavespeeds in either the refractor or the overlying layer. Any lateral changes in

refractor wavespeeds are accommodated through the use of forward and reverse

data.

The convolution operation also multiplies the amplitudes of first arrival signals.

This operation compensates for the large effects of geometric spreading to a very

good first approximation, with the result that the convolved amplitude is

essentially proportional to the square of the head coefficient. The signal-to-noise

(S/N) ratio of the RCS shows much less variation than those on the original shot

records.

A significant achievement of this study is the demonstration that the head

coefficient is approximately proportional to the ratio of the specific acoustic

impedances in the upper layer and in the refractor, under the conditions

encountered in most shallow seismic refraction surveys. These conditions are

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that there is a reasonable contrast between the specific acoustic impedances in

the layers. Although the original theoretical formulations of the head coefficient

were published almost half a century ago, the very limited use of refraction

amplitudes since that time has not resulted in convenient approximations which

might facilitate practical quantitative analysis with routine surveys. It is likely that

the simplification proposed in this work will promote greater use of amplitudes in

routine shallow seismic refraction surveys.

A major part of this study has been the demonstration of the usefulness of either

the convolved amplitudes or the equivalent shot amplitude products. The two 2D

case histories at Mt Bulga demonstrate both the correlation between amplitudes

and wavespeeds, and the use of amplitudes in addressing any ambiguities in the

determination of wavespeeds.

Non-uniqueness in determining wavespeeds in the refractor is an important

issue. Although most geophysicists tacitly accept that the inversion of seismic

refraction data need not necessarily produces a unique solution, the results of

most inversion routines still do not adequately reflect this reality. The non-

uniqueness can occur in the determinations of wavespeeds in both the upper

layer and the refractor and often, they are inter-related. This study proposes

several solutions to non-uniqueness in the refractor wavespeeds. Firstly, the

GRM can be used to generate a family of acceptable starting models for model-

based inversion or tomography. Secondly, the minimum variance criterion of the

GRM can be employed to determine a most likely starting model. Finally,

amplitudes can provide additional valuable information to constrain any starting

models.

The RCS can also include a separation between each pair of forward and

reverse traces in order to accommodate the offset distance in a manner similar to

the XY spacing of the GRM and to improve lateral resolution. The offset distance

is the horizontal separation between the point of refraction on the interface and

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the point of detection at the surface. Although the differences between the updip

and downdip offset distances can be large, their sum which is obtained with the

optimum XY value, is relatively insensitive to the dip angle. It facilitates the

application of the RCS to deeper refractors where the offset distances are

significant, as well as to very shallow refractors.

The refraction profile across the Mt Bulga massive sulfide orebody demonstrates

that there can be lateral separations of the amplitudes on the forward and

reverse shot records. In this case it was the distinctive low amplitudes

associated with the mineralization. This separation is similar to the optimum XY

value determined from the refractor wavespeed analysis function. In addition,

this case history demonstrates that the accommodation of the offset distance

with finite XY values is efficacious for improving lateral resolution with shallow

refractors and with detector separations as small as 2.5 m.

Another important achievement of this study is the examination of the effects of

variations in the near-surface soil layers on amplitudes or “amplitude statics”.

The profile across the Mt Bulga orebody demonstrates that the increases in the

thickness of the surface soil layer correlate with increases in refraction

amplitudes, and that these increases are adequately described with the

transmission coefficients of the Zoeppritz equations. Where these surface layers

are laterally continuous, the same increases in amplitudes occur at each

detector, and therefore the relative amplitudes are preserved. However, where

the surface layers are laterally discontinuous, the amplitudes can be quite

variable. If the wavespeeds in these zones can be measured, then corrections

can be applied with the Zoeppritz equations. Where this is not possible, then the

minimum amplitudes, rather than an average should be used.

Perhaps the most exciting aspects of this study are the results of the 3D survey.

Even with the nominally 2D structure of the shear zone, there are important

lateral variations in both refractor depth and wavespeed, which could not be

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predicted on the basis of the earlier 2D survey. In addition there are important

variations in the direction of the rock fabric as inferred from the qualitative

measures of azimuthal anisotropy. These results are a compelling

demonstration that more useful geological interpretations are possible with

simple 3D sets of data with complete spatial coverage in all directions, rather

than with the most detailed inversion of 2D sets of data.

Just as 3D reflection methods have revolutionized petroleum exploration and

production, so it is anticipated that shallow 3D seismic refraction methods will

eventually be recognized as a cost-effective approach to minimizing risk,

especially with geotechnical and environmental investigations. The results of the

3D survey raise the question of whether the 2D model of the subsurface is a

satisfactory approximation for most seismic refraction targets.

Another significant advantage of the use of 3D amplitudes, is that they provide a

measure of refractor wavespeeds at each detector, whereas the analysis of

traveltimes provides a measure over several detectors, commonly a minimum of

six. Therefore, amplitudes effectively improve the spatial resolution by almost an

order of magnitude. It is likely that amplitudes will facilitate the extraction of even

more detail with, for example, the attribute processing methods currently being

used with the interpretation of 3D seismic reflection data.

The RCS provides another approach to the use of later events. “Cross-

convolution” artifacts can be easily removed with simple dip filtering methods,

thereby highlighting those events from other, generally shallower, layers. It is

likely that the application of standard seismic reflection processing steps such as

dip filtering, deconvolution and migration or imaging, will result in the extraction of

further information from the RCS.

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The RCS also provides an effective approach to the high source energy

requirements of refraction seismology through stacking in a manner similar to the

CMP methods of reflection seismology.

The RCS is a simple and efficient method for full trace processing of shallow

seismic refraction 2D and 3D data. The convolution process is very quick and

not particularly demanding of computing facilities.

The RCS can be viewed as a simple extension of the GRM. It facilitates

improved interpretation of shallow refraction seismic data through the convenient

use of amplitudes as well as traveltimes.

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Appendix 1

Comments on “A brief study of thegeneralized reciprocal method and

some of the limitations of themethod” by Bengt Sjögren.

A.1 - Introduction

Despite the implications of the title, Sjögren (2000) is essentially a rejection of the

generalized reciprocal method (GRM). It is clear that there are very few aspects

of the GRM, which Sjögren finds acceptable, if in fact there are any at all.

I consider that most of the substance of his critique of the GRM is either wrong or

ill informed, while other aspects are simply matters of opinion. A thorough

response to his paper would be quite lengthy and somewhat technical, and so I

will restrict my response to three main issues only. They are the following:

1. Do we always need to define all layers above the target refractor, or are there

situations where the use of an average wavespeed is more appropriate? I

accept that Sjogren’s re-interpreted depth sections for the two case histories

provide a better estimate of total depths across the complete profiles. However,

the original depth sections in Palmer (1991) generated with the average

wavespeeds are appropriate to the objectives of each case history and to those

of the paper as discussed in the third issue.

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2. There are no fundamental mathematical differences between the GRM and

the collection of methods used by Sjögren. From both an assessment of his

descriptions of the various methods and an examination of some of his figures, it

is clear that both approaches generate essentially the same processed data, and

that his mean-minus-T method is identical to the wavespeed analysis function of

the GRM. The main differences are found in how the processed data are

interpreted. The GRM provides a systematic and objective framework, whereas

in my opinion, Sjögren's approach is neither consistent nor objective.

3. Sjögren has not addressed the fundamental aims of Palmer (1991), namely

the demonstration of an objective approach for recognizing and defining narrow

zones with low wavespeeds in the refractor. Sjogren’s approach is not

systematic and it frequently relies on personal judgment, and as a consequence,

it can result in the generation of artifacts, such as those shown in Sjögren (2000,

Fig. 5(a)).

A.2 - The Use of Average Wavespeeds

Sjogren’s assertion of the need to define all layers in geotechnical investigations

rather than use an average wavespeed, is not cognizant of the objectives of each

field study in Palmer (1991). In the case of the collapsed doline, the objective

was to determine its depth. Drilling had not been completely successful because

the loss of circulation in the rubble had stopped progress at depths of about 50 m

and before solid rock was encountered. Therefore, the issue is not whether

Sjogren’s detailed layer by layer approach is more useful than the average

wavespeed approach, but rather why both approaches, which give much the

same maximum depths of about 15 m, are clearly at variance with the drilling.

Irrespective of which interpretation approach is used, it is obvious that the

original objectives have not been achieved, that the refraction method is

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inappropriate for solving the problem, and that the differences in depth

computations between stations 56 and 71 are peripheral to the survey objectives.

Furthermore, Sjogren’s reasoning for his rejection of my explanation that the

refracted energy propagates around rather than under the doline is convoluted,

not convincing and ignores the implications of a genuine three-dimensional

structure.

The second case history was across a fault. An earlier high-resolution reflection

survey had been carried out in order to test whether the method was efficacious

in detecting known faulting in the underlying coal seams. These results were

poor, possible due to the proximity of the line to a busy road and the use of small

explosive charges. The refraction survey was then carried out in order to

generate a more accurate set of statics corrections. The component of the

statics corrections, which effectively replaces the weathered layer with

unweathered material prior to the application of the elevation component, was

generated with the approach described in Dobrin (1976, p.215), Palmer (1995),

and Palmer et al (2000). This method simply scales the time-depths by a factor

which is a function of the average wavespeed in the weathering and the

wavespeed in the refractor.

Therefore, while I accept that Sjogren’s depth computations may be more

appropriate to many types of geotechnical investigations, the use of average

wavespeeds in Palmer (1991) was entirely compatible with the aims of both the

field studies and the paper. Furthermore, the differences in total depths between

Palmer (1991) and Sjogren (2000) do not alter the major conclusions of the

paper with respect to the study of narrow zones with low wavespeeds.

In Palmer (1981, 1992, 2000a, 2000b) I demonstrate the use of the average

wavespeed in accommodating undetected layers, wavespeed reversals and

transverse isotropy. At present, there are no other published approaches to

solving these problems which are commonly encountered in many parts of the

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world, especially those with deep regoliths. Furthermore, Sjogren appears to

have overlooked other case studies (Palmer, 1980) in which the GRM has

effectively defined all layers above the target refractor.

A.3 - The Similarities Between The GRM and Sjogren’s Approach

Sjogren's indication that he does not accept the usefulness of this average

wavespeed is surprising, especially since he also uses an average overburden

wavespeed with Hales' method where there are multiple layers (Sjogren, 2000,

p.819). In that same paragraph, he also describes varying the XY distances, in a

manner analogous to the GRM. It is clear that there are many similarities

between the two average wavespeeds, but that the GRM has extended the

concept to include several important benefits such as the ability to accommodate

undetected layers, as well as minor differences in the interpretation of the

traveltime graphs. Even though Sjogren has emphasized such differences, there

are only minor differences in depth computations between Palmer (1991) and

Sjogren (2000) at the points where the average wavespeeds were determined.

While, it is acknowledged that the accuracy of the average wavespeed can be

reduced with distance from the point of computation, this is not necessarily a

problem, as was the case with the two field studies.

Furthermore, the similarities extend beyond the average wavespeeds. In Palmer

(1986), I describe Hales’ method and conclude that fundamentally, it is very

similar to the GRM. The similarities are that both methods obtain a measure of

the depth to the refracting interface in units of time through the addition of

forward and reverse traveltimes, and a measure of the wavespeeds in the

refractor from the differencing of the same forward and reverse traveltimes. The

equations describing these two operations for each method are virtually identical.

Furthermore, both methods employ refraction migration in order to accommodate

the offset distance, which is the horizontal distance between the point of

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refraction on the interface and the point of detection at the surface. The major

difference between the two methods is that Hales’ method achieves the addition,

subtraction and migration with a graphical approach, while the GRM performs

these same operations arithmetically with the scalar traveltimes prior to their

graphical presentation.

Sjogren’s determination of refractor wavespeed in the main refractor begins with

a version of the ABC method, which is a special case of the GRM with a zero XY

spacing. In the first case history but not the second, he then applies what is

clearly another version of the GRM wavespeed analysis algorithms with finite XY

values. Sjogren (2000, p.825) refers to curves 1 and 2 in his Fig. 4 as having

been computed with migrations of 5 m and 7.5 m. These curves bear a

remarkable resemblance to those computed with the GRM with similar XY values

in Palmer (1991, Fig.16). The minor differences are due to Sjogren’s editing of

the traveltime data. He then applies yet another method, namely Hales’ method,

to further refine his wavespeed determinations.

Accordingly, Sjogren’s approach with a number of methods and my approach

with the GRM are essentially generating the same set of computations for the

determination of refractor wavespeeds. Where Sjogren uses a succession of

different techniques, all of which employ addition and subtraction of forward and

reverse traveltimes, together with accommodation of the offset distance with

migration, the GRM achieves the same results within a single presentation, such

as in Palmer (1991, Fig.16).

Are the differences between Sjogren’s approach and the GRM important? An

examination of the wavespeed analysis function in Palmer (1991, Fig.16), reveals

that the graph for an XY value of 5 m has intervals of steeper gradient which

would correspond with lower wavespeeds, and which occur over the same

intervals with low wavespeeds shown in Sjogren (2000, Fig 5(a)). Therefore,

where Sjogren interprets virtually every change in slope in a single graph as

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evidence of a lateral change in wavespeed, I have concluded that there is

insufficient evidence for the existence of those intervals. This conclusion is

based on an assessment of all of the graphs in Palmer (1991, Fig. 16), and the

recognition of systematic changes in patterns, rather than the detailed

interpretation of a single graph.

The issue then, is whether the approach of Palmer (1991) under-interprets the

data and therefore overlooks intervals with low wavespeeds, or whether Sjogren

(2000) over-interprets the data and generates artifacts which do not really exist.

The issue is important because such features can be very significant in most

geotechnical, groundwater and environmental applications.

A.4 - Recognizing And Defining Narrow Zones With LowWavespeeds In Refractors

This difference between the two approaches introduces the fundamental

question which Palmer (1991) seeks to address. Is there an objective and

systematic approach, which is independent of individual interpretation styles, for

recognizing and defining narrow zones with low wavespeeds in refractors?

At the present time, I am still largely of the opinion that most narrow zones with

low wavespeeds are simply artifacts of the inversion algorithms and individual

interpretation styles. I have several reasons for holding this view.

1. Numerous model studies have shown that the algorithms which seek to

determine the wavespeeds in the refractor through the differencing of forward

and reverse traveltimes, can readily produce narrow zones with alternating high

and low wavespeeds, where there are significant changes in depths to the

refractor. This pattern can be seen for example, in the vicinity of the doline in

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Sjogren (2000, Fig. 5(a)), and it immediately raises doubts about the veracity of

the lateral changes in wavespeed.

2. The traveltime differences through these zones are small, and frequently they

are within the acceptable errors of the data, commonly plus or minus one

millisecond. For example, in Palmer (1991, Fig. 16), the differences between the

computed points and the line representing the fitted wavespeed function are

generally less than a millisecond for the optimum XY value of 5m. (Also see Fig.

2 to be discussed later.) I question whether any minor changes in slope are

statistically significant.

3. The issue is not resolved with forward modeling with either ray tracing or the

eikonal equation which are employed, for example, with tomographic and other

model-based methods of inversion. The GRM is able to generate a family of

geologically acceptable starting models in which the wavespeeds range from low

to high in narrow zones (Palmer, 2000c; 2000d) and which essentially satisfy the

original traveltime data (Palmer, 1980, p.49-52; 1986, p.106-107) to better than a

millisecond. This is simply another statement of the fundamental problem of non-

uniqueness with all inversion processes (Oldenburg, 1984; Treitel and Lines,

1988), but it is rarely if ever, addressed satisfactorily with refraction methods.

Therefore all of the refractor wavespeed models generated with different XY

spacings in Palmer (1991, Fig. 16), satisfy the traveltime data. Although some of

these can be rejected on simple geological grounds, such as those with negative

wavespeeds which obviously are not geologically realizable, there still remains a

range of models which fit the data to an acceptable accuracy.

Therefore, while I accept that Sjogren has a methodology for determining

wavespeeds in his Figures 1, 2, 4 and 7, I do not accept that he has addressed

the fundamental issues of non-uniqueness, that is of recognizing the artifact from

the real. I consider his approach relies heavily on his familiarity with Hales’

method and modified versions of the ABC method which include a migration

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process identical to the GRM, and that there is a significant subjective

component. Furthermore, his approach is poorly explained and the use of

different interpretation methods in a non-systematic manner is confusing.

Accordingly, I do not accept that he has demonstrated he has an objective

approach, or that there are narrow zones with low wavespeeds in the refractor in

Figure 5(a).

Sjogren's non-systematic approach can also be demonstrated with the model

data. Sjogren (2000, Fig. 2) changes the inclinations of the slope lines in the

Hales time loop on the basis of the wavespeeds derived from curve 1 for which

the XY value is zero, in order to improve the resolution of the narrow zone with

the low wavespeed. If that approach were to be employed with the first model

(Palmer, 1991, Fig. 2), using the wavespeed analyses for zero XY shown in

Palmer (1991, Fig. 5), then artifacts with both high and low wavespeeds at the

sloping interface, would be generated.

A.5 - Use Of Alternative Presentations And Amplitudes ForDetermining Wavespeeds In Refractors

In Palmer (1991), I present a systematic and objective criterion, generally known

as minimum variance. It is clear that an important aspect of this approach is to

determine a gross model of the refractor wavespeeds, and then to systematically

fit this model as has been done in Palmer (1991, Fig. 16). This is usually an

iterative process, simply because it is difficult to obtain the correct wavespeeds at

the first attempt. It can be somewhat challenging because of the need to

recognize the pattern of the departure of the computed points from the fitted line

as shown in Palmer (1991, Fig. 5), while at the same time accommodating the

normal errors in field data.

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Recently, I have been experimenting with averaging the wavespeed analysis

function for a range of XY values which range from less than to greater than the

optimum value, with the range of XY values being symmetrical about the

optimum, in order to derive a gross wavespeed model. This process minimizes

many of the apparent changes in wavespeeds due to structure where there are

no narrow lateral changes in wavespeed such as is shown in Palmer (1991, Fig.

5), and it averages many of the errors in picking traveltimes.

Figure 1. Refractor wavespeed analysis function, averaged for XY values from

zero to 10 m.

Using the traveltime data for the doline field study, Figure 1 shows a graph in

which the points computed with the wavespeed analysis function for XY values

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between zero and 10 m have been averaged. While Figure 1 risks introducing

yet another model of the refractor wavespeeds, namely about 2150 m/s between

stations 39 and 58, and about 3240 m/s elsewhere, there is no indication of any

narrow zones with low wavespeeds.

Figure 2. Differences between the averaged refractor wavespeed analysis

function in Figure 1, and the individual refractor wavespeed analysis functions for

XY values from zero to 10 m.

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Figure 2 shows the differences between the average and the computed

wavespeed analysis function for XY values between zero and 10 m. The

patterns of these differences are also consistent with there being no narrow

zones with low wavespeeds. The minimum differences occur for an XY value of

5 m, they are essentially random, and they are generally less than a millisecond.

Figure 3. Amplitudes of the forward and reverse offset shots.

I have also been investigating the use of amplitudes as an additional approach to

addressing this fundamental problem of non-uniqueness. In Palmer (2000e,

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2000f), I demonstrate that the amplitude of the refracted head wave, after

correction for geometrical spreading and refractor dip with either convolution or

multiplication, is essentially a function of the head coefficient. I further

demonstrate that the head coefficient is approximately proportional to the ratio of

the specific acoustic impedances (which is the product of the wavespeed and

density) in the overburden to that in the refractor. Therefore, arrivals from zones

in the refractor with low wavespeeds should exhibit high amplitudes, and vice

versa.

The amplitudes for the shot records are shown in Figure 3. For the shot at

station 1, the amplitudes shown a strong decay between stations 24 and 40,

which is interpreted as the geometric effect, together with an interval with

extremely low amplitudes between stations 46 and 59. The amplitudes for the

reverse shot at station 97 show a much less pronounced geometric effect, but

again there is an interval with very low amplitudes between stations 38 and 44.

These very low amplitudes were a major limitation on the measurement of

accurate and consistent traveltimes.

There are very few model studies on the effects of structure on the refraction

amplitudes. Nevertheless, it is unlikely that the very low amplitudes are

compatible with the simple refraction of energy from under the survey profile, but

rather with some form of scattering. They are compatible with energy

propagating around the doline and being scattered back to the surface through

the highly attenuating medium of the rubble in the collapsed doline, as has been

proposed in Palmer (1991).

Figure 4 shows the product of the amplitudes computed with an XY separation of

5 m. The results can be broadly separated into three regions which correspond

approximately with those recognized in Figure 1 with the wavespeed analysis

function. They include high values between stations 24 and 37, very low values

between stations 37 and 59, and higher values between stations 59 and 71. The

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lower values between stations 37 and 59 suggest higher rather than lower

wavespeeds. The slightly higher values between stations 37 and 42 suggest

lower wavespeeds, but do not correlate with those determined by Sjogren (2000,

Fig5(a)). However, these results should be used with considerable caution

because of the severe attenuation of seismic energy within the rubble of the

collapsed structure, and because the feature is three-dimensional.

Figure 4. Product of the forward and reverse shot amplitudes shown in Figure 3

with an XY value of 5m in arbitrary units.

Therefore, it seems probable that there are no narrow zones with low

wavespeeds associated with the collapsed doline and that Sjogren has

generated artifacts through an over-interpretation of the processed data.

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A.6 - A Systematic Approach With The GRM

In summary, Sjogren concludes that “the GRM must be regarded as being of

limited use for detailed and accurate interpretations of refraction seismics for

engineering purposes.” This conclusion is surprising given the very close

similarities between the GRM, Hales’ and the mean-minus-T methods, the latter

two of which he clearly favours. The fact that key features of his figures bear a

remarkable resemblance to parts of the GRM presentations emphasizes the

essential similarities between the two approaches. Instead, he has sought

differences where none really exist, and as a corollary, he has not recognized

similarities where in reality there are many. On the basis of the fundamental

similarities between the GRM and Sjogren's processed data, I conclude that

Sjogren (2000) is more a demonstration of his interpretation style and experience

using a collection of methods rather than a cohesive assessment of the GRM.

However, his approach is neither systematic nor entirely objective, and as a

result it is prone to the generation of artifacts.

It seems that the aim of his paper is to emphasize minor differences, mainly in

the assignment of layers to the traveltime graphs, and then to illogically imply

fundamental shortcomings of the GRM. As such, his paper lacks balance and

objectivity, and it is more in keeping with seeking a conviction in an adversarial

court system than with a scientific journal.

Sjogren (2000) has produced nothing of substance which requires any

fundamental re-assessment of the main features of the GRM in general, or of

Palmer (1991), in particular. He has not satisfactorily addressed the aims of

Palmer (1991), namely an objective method for the recognition and definition of

narrow zones with low wavespeeds. His conclusion that the GRM is unsuitable

for geotechnical applications is not substantiated, and it is based on an

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incomplete understanding of the GRM and of Palmer (1991), rather than any

genuine shortcomings with the method.

A.7 - The Need To Promote Innovation In Shallow RefractionSeismology

Sjogren’s rejection of the GRM as a useful method for processing and

interpreting shallow seismic refraction data, does little to encourage others to

present new approaches through fear of biased criticism. It is irresponsible and

does not advance the science through balanced and objective debate. In the last

fifty years, innovation in shallow refraction seismology has been rather modest at

best and it has focused predominantly on the various competing methods for

inverting field data. Until there is widespread consensus through a recognition of

fundamental similarities between these inversion methods, there will be little

advancement in other equally important aspects of the science. By comparison,

reflection seismology has achieved major advances through the development of

common midpoint methods, digital signal processing, three-dimensional methods

and sophisticated computer interpretation programs, over the same period of

time. It is now time to move on to the refraction techniques which will be

appropriate to the requirements and the technology of the new millennium.

In Palmer (2000g), I demonstrate the generation of the refraction convolution

section (RCS) through the convolution of forward and reverse traces. The

addition of the traveltimes with convolution is equivalent to that achieved

graphically with Hales’ method and arithmetically with the GRM. The RCS

facilitates full trace processing of seismic refraction data and in turn, the

examination of many important issues such as signal-to-noise ratios, “amplitude

statics”, 3D refraction methods and azimuthal anisotropy, signal processing to

enhance second and later events and stacking data in a manner similar to CMP

reflection methods. A major advantage of the RCS is that it incorporates

213

amplitudes and time structure within a single presentation, facilitating the

resolution of many of the non-uniqueness issues discussed here. It is extremely

rapid and suitable for use with any volume of data, and therefore it can be readily

included in the processing of refraction data with any method.

The RCS is a new approach to obtaining more and better information from

shallow seismic refraction data, and in time it may even supercede the GRM as

well as Hales’ method. However, that possibility can only occur if there is a shift

in culture from one of conflict and an emphasis on minor differences to one of

consensus and an emphasis on fundamental similarities which traditionally, has

characterized the scientific method.

Sjogren’s critique of the GRM does not seek the consensus essential for the

advancement of the science of shallow refraction seismology. Regrettably, it is

neither balanced nor objective, it shows minimal insight into the fundamental

similarities of most methods of refraction inversion, it does not provide an

alternative systematic approach to refraction interpretation, it does not address

the important issues of non-uniqueness, and it does not provide a vision for

future innovation.

A.8 - References


McGraw-Hill Inc.

Oldenburg, D. W., 1984, An introduction to linear inverse theory: Transactions

IEEE Geoscience and Remote Sensing, GE-22(6), 666.



214

Palmer, D., 1981, An introduction to the generalized reciprocal method of seismic

refraction interpretation: Geophysics 46, 1508-1518




generalized reciprocal method: Geophysical Prospecting 39, 1031-1060.

Palmer, D, 1992, Is forward modeling as efficacious as minimum variance for

refraction inversion?: Exploration Geophysics, 23, 261-266, 521.

Palmer, D., 1995, Can linear inversion achieve detailed refraction statics?:

Exploration Geophysics, 26, 506-511.

Palmer, D., Goleby, B., and Drummond, B., 2000a, The effects of spatial

sampling on refraction statics: Exploration Geophysics, 31, 270-274.

Palmer, D., 2000a, Model determination for refraction inversion: submitted.

Palmer, D., 2000b, The measurement of weak anisotropy with the generalized

reciprocal method: Geophysics, 65, 1583-1591.

Palmer, D., 2000c, Can amplitudes resolve ambiguities in refraction inversion?:

Exploration Geophysics 31, 304-309.

Palmer, D., 2000d, Starting models for refraction inversion: submitted.

Palmer, D., 2000e, Imaging refractors with the convolution section: Geophysics,

in press.

215

Palmer, D., 2000f, Resolving refractor ambiguities with amplitudes: Geophysics,

in press.

Palmer, D, 2000g, Digital processing of shallow seismic refraction data with the

refraction convolution section, PhD thesis, UNSW, submitted.

Sjogren, B., 2000, A brief study of the generalized reciprocal method and of

some limitations of the method: Geophysical Prospecting 48, 815-834.


of salt): The Leading Edge 7, 32-35.

216

Appendix 2

Model Determination For RefractionInversion

A.1 - Summary

In recent years, tomography or model-based inversion, has been used to

construct a model of the subsurface from seismic refraction data, mainly to

determine statics corrections for reflection data. With these methods, the

parameters of a model of the subsurface are refined by comparing the

traveltimes of the model with the field data. When the differences between the

computed and field traveltimes are a minimum, the model and parameters are

taken as an accurate representation of the wavespeeds in the subsurface.

In this study I demonstrate the sensitivity of model-based inversion to the

selection of the inversion model, using eleven geological models. In general, the

residuals between the original and modeled traveltimes are better than 0.4%,

indicating that virtually any model can be fitted to the data with high accuracy.

However, the errors in depth computations are between 5% and 10% for simple

monotonic increases of wavespeed with depth. For other models such as a

reversal in wavespeed, the errors are indeterminate, but much larger. The depth

errors are least when the inversion model and the original geological model are

similar, and greatest when the models are different.

217

I also determine the parameters for the same eleven inversion models with the

generalized reciprocal method (GRM). With the GRM, the parameters of each

model are constrained with the optimum XY value. The errors in depth

computations are generally about one third of those for the equivalent model-

based case. Furthermore, the GRM is able to produce reasonable results with

the wavespeed reversal and transverse isotropy models, unlike the model-based

methods. However, the residuals between the direct traveltimes for the inversion

model and the original geological model are up to a factor of four greater than

with the equivalent model-based case. The residuals are least when the models

are similar, and greatest when the models are different.

The GRM determines model parameters by interpolation with the optimum XY

value. This parameter is determined from the refracted traveltimes from the

underlying layer, and therefore it is a function of all wavespeeds and thicknesses

in the overlying layer(s). By contrast, the model-based methods determine the

model parameters from the direct traveltimes from the upper part of the layer(s),

and then extrapolate those parameters throughout the remainder of the layer(s)

I conclude that suitable starting models for tomographic or model-based

inversion have parameters which are determined with optimum XY values, and

which have minimum residuals between the field and modeled traveltimes of the

direct arrivals.

A.2 - Introduction

In recent years, model-based inversion or tomography, has been used to process

seismic refraction data, often to determine statics corrections for reflection data

(Zhu et al., 1992). With these methods, the parameters of a model of the

subsurface are refined by comparing the traveltimes of the model with the field

data. When the differences between the computed and field traveltimes are a

218

minimum, the model and parameters are taken as an accurate representation of

the wavespeeds in the subsurface.

While the performance of refraction tomography has been continually improved

through more efficient inversion and forward modeling routines, (see Zhang and

Toksoz, 1998 for an overview of these advances), one issue, which has yet to

receive widespread attention, is the choice of the model for the inversion

process. This situation is not surprising, since the role of model-based inversion

is to provide information about the unknown numerical parameters which go into

the model, not to provide the model itself (Menke, 1989, p3).

Perhaps the most common model has been the linear increase of wavespeed

with depth (Zhu et al.,1992; Stefani, 1995; Miller et al., 1998; Lanz et al., 1998),

possibly because of mathematical convenience. However, this model is of

questionable validity as most theoretical (Iida, 1939; Gassman, 1951, 1953;

Brandt, 1955; Paterson, 1956; Berry1959), laboratory (Birch, 1960; Wyllie et al.,

1956, 1958), and field studies (Faust, 1951, 1953; White and Sengbush, 1953;

Acheson, 1963, 1981; Hall, 1970; Hamilton, 1970, 1971; Jankowsky, 1970),

suggest a more gentle increase for clastic sediments, such as a one sixth power

of depth function.

Furthermore, the gradients obtained range from 0.342 and 2.5 m/s per metre

(Stefani, 1995), and 2.68 and 4.67 m/s per metre (Zhu et al., 1992), to as high as

40 m/s per metre (Lanz et al, 1998). These values are generally larger than

those applicable to the compaction of clastic sediments (Dobrin, 1976), but they

are rarely justified on geological or petrophysical grounds.

The combination of the linear increase of wavespeed with depth and the high

gradients probably contributes to instability in the inversion process. The

example of the somewhat paradoxical situation of the poor determination of


219

from that layer (Lanz et al., 1998), is at variance with the experiences of most

seismologists using more traditional methods of refraction processing.

In this study, I demonstrate the effects of the choice of the model on refraction

inversion using both the model-based approach and the generalized reciprocal

method (GRM), (Palmer, 1980, 1986). With the model-based approach, I show

that a range of models can be fitted to noise-free traveltime data with acceptable

accuracy, but that there is a wide range in the depths computed to the main

refractor. However, the accuracy of depth computations improves as the

inversion model approaches the subsurface or geological model. In addition,

only a limited number of models can be examined with model-based methods of

inversion.

With the GRM approach, I show that a wider range of models can be addressed,

with generally greater accuracies in depth computations. However, in contrast

with the methods of model-based inversion, the agreement with the traveltime

data is usually much poorer.

I conclude that the appropriate starting model for tomographic inversion is one for

which the depths are similar to those obtained with the GRM, and for which the

field and model traveltimes agree.

A.3 - Model and Inversion Strategies

The model of the subsurface used in this study is shown in Figure A.1, and

consists of two layers with the upper layer having a parabolic variation in

wavespeed with depth, viz.

V(z) = 1750(1 + 0.001 z)½ (A.1)

220

where V(z) is the wavespeed and z is the depth. The rate of increase of the

wavespeed is 0.875 m/s per metre at the surface, while the average is about

0.73 m/s per metre over a depth of 1000m. These values are comparable with

those caused by compaction of clastic sediments (Dobrin, 1976). The parabolic

wavespeed function is a compromise between the more commonly used linear

wavespeed function and the theoretically derived function which is a one sixth

power of depth.

Figure A.1: Two layer model for which the traveltime data shown in Figure A.2

were computed. The seismic wavespeed in the upper layer is a parabolic

function of depth.

The depth to the refractor changes from 750m to 1000m over a horizontal

distance of 500m, and the wavespeed in the refractor is 5000 m/s.

The traveltimes are shown in Figure 2, and are taken from Palmer (1986).

In this study, I examine the fundamental issue of what is the appropriate model

for the wavespeed of the first layer, given that the traveltime data indicate that

there are two layers with a single interface. I approximate the wavespeed in the

first layer with eleven inversion models. They are the constant wavespeed model,

221

the two layer model, including a reversal in wavespeed, the Evjen function, and

transverse isotropy.

Figure A.2: Traveltime data for the model shown in Figure A.1. The direct

arrivals from the shot points at 2400m and 4800m penetrate 85m for a range of

1200m and 300m for a range of 2400m.

In the model-based approach, I determine the wavespeed stratification in the

layers above the refractor from the direct arrivals from those layers. I derive the

parameters for each model from the traveltime graphs for the one and two layer

models, by iteratively adjusting for the Evjen models until the traveltimes for the

222

data and the model agree, or simply by arbitrarily selecting wavespeeds for the

wavespeed reversal and the transverse isotropy models.

With the GRM approach, I determine the wavespeeds in the layers above the

refractor using the optimum XY value, which I derive with the traveltimes from the

refractor. The optimum XY value is determined in the following way.

The refractor wavespeed analysis function tV, is given by equation A.2, viz.

tV = (tforward - treverse}+ treciprocal)/ 2 (A.2)

Equation A.2 was evaluated for a range of XY values, which is the separation

between the detectors at which the forward and reverse traveltimes are recorded,

from 500 m to 1150 m.

I describe the method of determining optimum XY values in some detail

elsewhere (Palmer, 1991). Essentially, the procedure involves the determination

of the overall trends of the wavespeed analysis function, from which the refractor

wavespeed is derived, and then the deviations from those trends, from which the

optimum XY value is obtained. In this study, I use averaging to obtain the

wavespeed in the refractor, and then differencing to obtain the optimum XY

value.

Figure A.3 shows the average of the refractor wavespeed analysis function

computed at each location for the full range of XY values from 500 m to 1150 m.

The reciprocal of the gradient of this function is the refractor wavespeed Vn, viz.

5046 m/s. This value is higher than the true value for the model of 5000 m/s, and

is the result of the non-planar refractor interface. A correction for dip (Palmer,

1980, equation 9), when the refractor interface is approximated with a planar

interface with an average dip of about six degrees (tan-1(250/2400)), improves

the wavespeed estimate to 5018 m/s.

223

Figure A.3: The averaged wavespeed analysis function, computed for the range

of XY values from 500 m to 1150 m in increments of 50 m, the station spacing.

The reciprocal of the gradient of this function is the wavespeed in the refractor,

namely 5046 m/s.

Figure A.4 presents the differences between the computed values of tV for a

selected range of XY values and the averaged values. A visual inspection

indicates that the average optimum XY value is 800 m, because the differences

are the closest to zero. This value is consistent with the values of 700 m and 950

m for each side of the model obtained with a more detailed analysis, (Palmer,

1986).

224

Figure A.4: The differences between the wavespeed analysis function

computed for the range of XY values from 650 m to 1000 m, and the averaged

values shown in Figure A.3. The values for an XY value of 800 m are assessed

as being closest to zero, thereby indicating that the optimum XY value is 800m.

225

Depths for both the model-based inversion and GRM approaches are computed

from the time-depths, tG, where

tG = (tforward + treverse - treciprocal – XY / Vn)/2 (A.3)

The reciprocal time is the traveltime between the two shot points.

As the term implies, the time-depth is a measure of the depth to the refracting

interface in units of time. It is analogous to the one-way reflection time, and it is

often, but incorrectly identified with the delay time. The time-depth is, in fact, an

average of the forward and reverse delay times.

The time-depths range from 342 ms to 429 ms, which correspond to the

horizontal sections of the interface. The average value used in the GRM

calculations is 384 ms.

The time-depth is related to the depth zG, by:

zG = tG DCF (A.4)


depth, is given by:

DCF = V Vn / (Vn2 - V2)½ (A.5)

or

DCF = V / cos i (A.6)


sin i = V / Vn (A.7)

226

Figure A.5: The difference in traveltimes between the original data and the

modeled response for the single layer with a constant wavespeed inversion

model, for the model-based and GRM approaches.

A.4 - Single Layer Constant Wavespeed Inversion Model

The simplest model has a single layer with a constant wavespeed above the

refractor. As the traveltime graphs with the shot points at distances 2400 m and

4800 m show some curvature, the fitting of one straight line is easily recognized

as an approximation. However, for completeness with the model-based

approach, I use a line which passes through the origin and through the graph at a

traveltime of 1103ms and at a distance of 2000m from the shot point, to obtain an

227

average wavespeed of 1813 m/s. The differences in the traveltimes for the data

and the constant wavespeed model range from -16 ms to 19 ms, and are shown

in Figure A.5. These residuals are comparable to those obtained by Stefani

(1995) with the Tibalier Trench data which is at approximately the same depth.

Accordingly, I view the fit between the computed and original data as acceptable.

Figure A.6: Unmigrated depth sections for the single layer with a constant

wavespeed inversion model, for the model-based and GRM approaches.

The depths computed with this wavespeed are shown in Figure A.6, and they

have been plotted vertically, rather than orthogonally to the refractor interface.

228

An additional step, equivalent to the migration operation of reflection data, is

required to accurately image the refractor interface (Palmer, 1986, p.196-198).

The depths computed over the horizontal sections of the refractor are 659.4m

(for an error of -12.1%), and 833.4m (for an error of -16.7%), and they are listed

in Table A.1.

The GRM approach uses the average velocity formula (Palmer, 1980, p.42, eqn.

27; 1986, p.147, eqn. 10.4), for the single layer, constant wavespeed model,

viz.

V = [(XY Vn2) / (XY + 2 tG Vn)]½ (A.8)

The resultant average wavespeed is 2107 m/s, which in turn produces depths

over the horizontal sections of 770.2 m (for an error of 2.7%) and 973.6 m (for an

error of -2.6%). The differences in the traveltimes for the data and the constant

wavespeed model range from zero to -140 ms, and are shown in Figure A.5.

This inversion model emphasizes the appropriateness of the selection of the

starting model with model-based inversion. The constant wavespeed model is

applicable to a wide range a subsurface targets especially those in the near

surface, and it has proven to be efficacious since the earliest days of refraction

seismology. Furthermore, it is often used for field data, where the curvature of

the traveltime graph is not obvious (Palmer, 1983; Zhu et al., 1992). This

situation is not uncommon because the amplitudes at the larger offsets are

usually much weaker than those on the near offsets, thereby resulting in the

arrival times being measured later than is the case with noise-free data.

Even though the differences in the traveltimes between field and the modeled

data are an order of magnitude smaller with the model-based inversion approach

than with the GRM, there is not a commensurate improvement in the depth

computations.

229


modeled response for the two layers with constant wavespeeds inversion model,

for the model-based and GRM approaches.

A.5 - Two Layer Constant Wavespeed Inversion Model

Clearly, the single layer constant wavespeed model is not a completely

satisfactory approximation, and the next step is to consider a two segment

approximation to the traveltime graphs. This model is especially realistic when

the curvature of the traveltime graphs is not obvious with field data, and as a

result, the graphs are approximated with two straight line segments. In this case

230

the two wavespeeds are 1775 m/s and 1932 m/s, and the intercept time is 65ms

for the second layer. Figure A.7 shows that the modeled traveltimes are within

±4 ms of the original data, a result which would normally be considered an

excellent approximation. Figure A.8 shows the computed depths for the

horizontal sections are 695.2 m (for an error of -7.3%), and 882.7 m (for an error

of -11.7%).

Figure A.8: Unmigrated depth sections for the two layers with constant

wavespeeds inversion model, for the model-based and GRM approaches.

For the GRM approach to the two layer approximation, I solve two simultaneous

equations which relate layer thicknesses and wavespeeds to the time-depth, and

231

to the optimum XY value, (Palmer, 1980, p.47: 1986, p.136-137). The results are

the layer thicknesses z1 and z2, after values for the wavespeeds have been set.

The equations are:

tG = z1 cos i1n / V1 + z2 cos i2n / V2 (A.9)

XY = 2 z1 tan i1n + 2 z2 tan i2n (A.10)

where

ijn = sin-1 (Vj / Vn) (A.11)

The values used for the wavespeeds are 1800 m/s for V1, which is suggested by

the traveltime graphs, and 2400 m/s for V2. The computed depths are 735.7 m

(for an error of -1.9%), and 981.2 m (for an error of -1.9%).

The traveltime differences are shown in Figure A.7 and range from -10 ms to +25

ms. As with the single layer inversion model, these differences are much larger

than those for the model-based inversion approach, but the depth computations

are still more accurate.

A.6 - Two Layer Wavespeed Reversal Inversion Model

Suppose that there were sufficient reasons for considering the existence of a

surface layer with a wavespeed of 3600 m/s. In this case, the traveltime graphs

with the shot points at 2400m and 4800m, would be ignored because of

additional, more compelling reasons, such as the existence of a local surface

lens of limestone or permafrost with a high wavespeed, or even extrapolation

from an adjacent area. The model-based inversion approach can only assume

232

that this wavespeed occurs throughout the upper layer, resulting in computed

depths of 1758.6 m (for an error of 137%) and 2222.8 m (for an error of 123%).

The GRM approach to the wavespeed reversal model is similar to the two layer

constant wavespeed model described previously. In that case, the selection of

the second layer wavespeed of 2400 m/s might have appeared to be rather

arbitrary. In fact, almost any wavespeed will suffice, provided it is above a

certain minimum, because the thickness of the corresponding layer is

automatically adjusted to produce an effective average wavespeed similar to that

obtained with equation A.4. Furthermore, it will be noted that there is no

requirement for the wavespeeds to increase monotonically with depth with this

approach, and therefore it can be applied equally validly to the wavespeed

reversal model. Using wavespeeds of 3600 m/s and 1800 m/s, solution of

equations A.9 to A.11 produces depths of 731.81 m (for an error of -2.4%), and

904.7m (for an error of -9.5%).

The wavespeed reversal model is generally acknowledged as one of the most

difficult if not impossible to address satisfactorily with refraction tomography

(Lanz et al., 1998). In the absence of other geological or geophysical data, the

GRM approach provides a measure of constraint for a normally intractable

problem.

A.7 - The Evjen Inversion Model

If in fact, it was recognised that the traveltime data represented a variable

wavespeed medium, then the wavespeed function in the overburden can be

approximated with the Evjen function, viz.

V(z) = V0 (1 + q z)1/m (A.12)

233

For the model-based inversion approach, I determine the variable q, by iteratively

matching the traveltime graphs, once the value of V0 is set at 1750 m/s, and m is

set at 1,2,3,4 or 6.


modeled response for the single layer with a linear function of the depth


The GRM approach to the variable wavespeed model is described in detail in

Palmer (1986, p.175-181), and is outlined in the Appendix.

234

Figure A.9 shows the results for the linear wavespeed function. As with other

inversion models, the model-based inversion is able to achieve excellent

agreement with the traveltime data (-2 ms to + 0.5 ms), while depth computations

are 766.7 m (2.2% error) and 1026.2 m (2.6% error). With the GRM approach,

the differences in traveltimes are up to 12 ms, while the depths are 751.0 m

(0.1% error) and 996.7 m (-0.3% error).


modeled response for the single layer with a parabolic function of the depth


235

Figure A.10 shows the results for the parabolic wavespeed function. The errors

in the traveltimes with model-based methods are zero, and the depth estimates

are 757.2 m (1% error) and 1002.5 m (0.25% error). The GRM depth

computations are of comparable accuracy being 749.5 m (-0.4% error) and 989.6

m (-1% error). The differences in the traveltimes are up to 6 ms.


modeled response for the single layer with a one third power of the depth


The one third power of the depth function produces excellent agreement with the

traveltimes for both the model-based inversion and the GRM approaches. As is

shown in Figure A.11, traveltimes differences are generally less than 1 ms.

236

Depth estimates are also quite accurate with 749.9 m(-0.01% error) and 985.7 m

(-1.5% error) for the model-based inversion, and 751.2 m (0.16% error) and

987.9 m (-1.2% error) for the GRM.


modeled response for the single layer with a one fourth power of the depth


For the one fourth power of the depth function, traveltimes are generally within ±1

ms (see Figure A.12), while the depths are 745.0 m (-0.7% error) and 966.5 m (-

3.4% error) for the model-based inversion. For the GRM approach, the

237

traveltime differences are as much as -5 ms, while the depths are 752.7 m (0.4%

error) and 986.4 m (-1.4% error).


modeled response for the single layer with a one sixth power of the depth


The results for the one sixth power of the depth function are similar to those

described above. For the model-based inversion approach, the traveltime

differences are between -2 ms and +4 ms (see Figure A.13) while the depths are

735.0 m (-2% error) and 954.8 m (-4.5% error). For the GRM approach, the

238

traveltime differences are up to -15 ms, while depths are 755.2 m (0.7% error)

and 984.4 m (-1.6% error).

A.8 - Transverse Isotropy Inversion Model

Finally, there is the case of seismic anisotropy, in which the horizontal

wavespeed is different from the vertical value. This is one area where the

geophysical idealization of the earth can differ substantially from the

petrophysical reality. While there is overwhelming evidence that wavespeeds are

more likely to be anisotropic than isotropic, because of intrinsic anisotropy, cyclic

layering, etc, it is rare for the basic model for most refraction inversion routines to

include this property. No doubt the complexities of the mathematical treatment,

together with the lack of accepted methods for determining anisotropy

parameters from surface measurements, contribute to this situation.

In most cases, the horizontal wavespeed is greater than the vertical value. This

is usually the case with undisturbed sedimentary rocks, because of the effects of

compaction and cyclic layering. However, the reverse condition may apply, such

as with steeply dipping sedimentary or metamorphic rocks, or with non-

hydrostatic stress. Accordingly, simply assuming an anisotropy factor with the

vertical wavespeed being less than the horizontal can sometimes be

inappropriate.

Unless there is additional information, such as bore hole control, seismic

anisotropy is not easily recognised or accommodated with the standard refraction

methods, and any depth and wavespeed computations can have indeterminate

errors. However, for completeness of the comparison, I use the commonly

assumed anisotropy factor for P waves in sedimentary rocks of 1.10, together

with the horizontal wavespeeds of 1800 m/s, 2100 m/s and 2300 m/s, for the

239

model-based inversion approach. The vertical wavespeeds used for depth

conversion are then 1636 m/s, 1909 m/s and 2091 m/s.

The GRM approach to anisotropic overburdens is described in detail in Palmer

(Palmer, 1986). With this approach, I seek to determine the anisotropy factor,

which is the horizontal wavespeed divided by the vertical wavespeed, for which

the wavespeed given by the Crampin equation, viz.

V2(φ) = A + B cos 2φ + C cos 4φ (A.9)

(where φ is the angle from the vertical), is equal to the average wavespeed

modified for anisotropy, viz.

V = [(XY Vn2) / (XY + 2 c tG Vn)]½ (A.10)

where

c = (A - B + C - 8 C cos4 φ) / (A + B + C - 8 C sin4 φ) (A.11)

The value of φ used in equations A.13 and A.15, is that for the critical angle for

an horizontal refractor. I examine three models of anisotropy.

The first model uses a horizontal wavespeed of 1800 m/s, which can be

recovered from the traveltime graphs. For the model-based inversion approach

using an anisotropy factor of 1.1, the computed depths are 664.3 m (-11.4%

error) and 839.7 m (-16% error).

The GRM approach produces an anisotropy factor of 0.855, and depths of 789.7

(5.3% error) and 998.1 m (-0.2% error). The computed value of the anisotropy

factor is unusual in that it is less than unity. Nevertheless, the computed depths

are reasonably close to the true values.

240

The second model uses a horizontal wavespeed of 2100 m/s, which is

approximately the average wavespeed in the first layer. For the model-based

inversion approach using an anisotropy factor of 1.1, the computed depths are

700.0 m (-6.7% error) and 885.1 m (-11.5% error).


m (1.7% error) and 965.5 m (-3.5% error). In this case, the anisotropy factor is

close to unity, which is compatible with the selection of a horizontal wavespeed

similar to the average computed with equation A.4. Therefore, the GRM can

produce an isotropic result if indeed that is appropriate.

The third model uses a horizontal wavespeed of 2400 m/s. For the model-based

inversion approach, the computed depths are 780.4 m (4.1% error) and 986.4 m

(-1.4% error). This is the best result for this approach and it is the result of a

fortuitous combination of the horizontal wavespeed and the assumed anisotropy

factor producing a vertical wavespeed which is comparable with the average

computed with equation A.4.


m (-5.8% error) and 893.6 m (-10.6% error).

Seismic anisotropy, like wavespeed reversals, emphasizes the fact the model-

based inversion is only efficacious when the traveltime data are an accurate and

complete reflection of the subsurface wavespeed stratification, and therefore

when an inversion model can be determined.

241

A.9 - Errors Related to the Optimum XY Value

In general, the errors in the depth computations related to the choice of the

choice of the inversion model, are between 2% to 3% with the GRM approach.

For this study, a single or average optimum XY value is used. I demonstrate

further improvements in accuracy when the optimum XY values, which are

applicable to each side of the sloping segment of the refractor, are used (Palmer,

1986, 1992).

The errors in the depth computations which are related to the errors in the

measurement of the XY value, can be estimated in the following way.

The errors in the optimum XY value are one half of the detector spacing, ie ± 25

m. Therefore the possible range of optimum XY values is 800 m ± 25 m, ie 775

m to 825 m. (An inspection of Figure A.4 suggests that the latter value of 825 m

is probably the best estimate of a single average value. It is also the mean of the

values applicable to either side of the sloping interface (Palmer, 1986).)

The single layer constant wavespeed will be used for convenience. The average

wavespeed computed with equation 4 using optimum XY values of 775 m and

825 m, are 2079 m/s and 2133 m/s. They differ from the value of 2107 m/s used

in the single layer inversion model above, by an average of ± 27 m/s, or about

1.3%. The resulting error in depth calculations is approximately ± 11.4 m or

1.3%.

Alternatively, it can be demonstrated by differentiation of equation A.4 that:

∆V / V ≈ ½ ∆XY / XY (A.16)

Therefore, an error of 3% (25/800) in the optimum XY value, results in an error of

1.5% in the average wavespeed.

242

Figure A.14: Summary of the errors in depth computations for all inversion

models, for the model-based and GRM approaches. The errors for the

wavespeed reversal model using the model-based approach have been

arbitrarily set at 22% and 25%.

243

INVERSION MODEL MODEL-BASED GRM

InversionModel

TrueDepth

Wavespeed Depth Error Wavespeed Depth Error

One Layer 750 m

1000 m

1813 m/s

1813 m/s

659 m

834 m

-12%

-17%

2107 m/s

2107 m/s

770 m

974 m

2.7%

-2.6%

Two Layers 750 m

1000 m

1775 m/s

1932 m/s

1175 m/s

1932 m/s

146 m

695 m

146 m

883 m

-7%

-12%

1800 m/s

2400 m/s

1800 m/s

2400m/s

382 m

736 m

429 m

981 m

-1.9%

-1.9%

Wavespeed

Reversal

750 m

1000 m

3600 m/s

3600 m/s

1759 m

2223 m

137%

123%

3600 m/s

1800 m/s

3600 m/s

1800 m/s

119 m

732 m

174 m

906 m

-2.4%

-9.5%

Evjen, m=1 750 m

1000 m

q=.00048

q=.00048

767 m

1026 m

2.2%

2.6%

q=.000422

q=.000422

751 m

997 m

0.1%

-0.3%

Evjen, m=2 750 m

1000 m

q=.001

q=.001

757 m

1026 m

1 %

0.3%

q=.000941

q=.000941

750 m

990 m

0%

-1.0%

Evjen, m=3 750 m

1000 m

q=.001577

q-.001577

750 m

986 m

0%

-1.5%

q=.00156

q=.00156

751 m

988 m

0.2%

-1.2%

Evjen, m=4 750 m

1000 m

q=.00218

q=.00218

745 m

967 m

-0.7%

-3.4%

q=.00235

q=.00235

753 m

986 m

0.4%

-1.4%

Evjen, m=6 750 m

1000 m

q=.0035

q=.0035

735 m

955 m

-2%

-4.5%

q=.004413

q=.004413

755 m

984 m

0.7%

-1.6%

Anisotropy

VH=1800 m/s

750 m

1000 m

k=1.10

k=1.10

664 m

840 m

-11.4%

-16%

k=0.855

k=0.855

790 m

998 m

-5.3%

-0.2%

Anisotropy

VH=2100 m/s

750 m

1000 m

k=1.10

k=1.10

700 m

885 m

-6.7%

-11.5%

k=1.025

k=1.025

763 m

966 m

1.7%

-3.5%

Anisotropy

VH=2400 m/s

750 m

1000 m

k=1.10

k=1.10

780 m

986 m

4.1%

-1.4%

k=1.21

k=1.21

707 m

894 m

-5.8%

-10.6%

Table A.1: Comparison of depth estimates using model-based and GRM

approaches.

244

A.10 - Discussion and Conclusions

This study demonstrates that virtually any inversion model can be fitted to

traveltime data to a high accuracy. With the noise-free data used in this study,

the accuracy is generally better than ±5 ms for traveltimes of up to 1300 ms or

0.4%. The errors are about four times larger for the single layer constant

wavespeed inversion model, but they are still acceptable.

However, the superior agreement with the traveltime data is not associated with

more accurate depth estimates. The average errors in depth computations are

about 5% to 10% and they are approximately three times larger than those

obtained through a GRM analysis. Figure A.14 summarizes the results.

In those cases where the inversion models are similar to the test model as with

the Evjen functions, the accuracy of the depth computations is quite high. The

accuracy is highest with the Evjen function with exponents of 2 and 3, which are

close to the test model, and it is least for exponents of 1 and 6, which represent

less similar models.

The lower accuracy of depths computed with the one and two layer

approximations, which are valid and widely applicable inversion models,

indicates the significance of the inversion model rather than the superiority of

variable wavespeed models, such as the linear function. As a corollary, it

indicates that the use of linear functions where constant wavespeed models are

applicable will also result in comparable significant errors in depth computations.

The cases of a reversal in the wavespeed and transverse isotropy illustrate a

fundamental shortcoming of model-based inversion. With a wavespeed reversal,

no traveltimes are recorded from the layer, and therefore it is simply not possible

to achieve agreement between the observed data and the computed traveltimes.

With seismic anisotropy, the traveltimes are obtained in the wrong direction. The

245

field data are a measure of the horizontal wavespeed, while the vertical or near-

vertical wavespeed is required for depth conversion. These models demonstrate

that layers or wavespeeds not represented in the traveltime data can only be

modeled after an empirical assignment of parameters, or only if other data are

available, such as borehole control.

By contrast, the differences between the original traveltime data and the

computed traveltimes for models whose parameters are determined with the

GRM, are much larger. In several cases, these differences can be many tens of

milliseconds or up to 11%. However, the computed depths are generally more

accurate than those computed with the model-based approach. Furthermore, the

GRM is able to address the valid and important inversion models of wavespeed

reversal and transverse isotropy, unlike the model-based methods.

The consistent depth computations with the GRM are related to the constraint of

the wavespeed models with the optimum XY value. This parameter is a function

of both the thicknesses and wavespeeds of all layers above the refractor.

Therefore, the accuracy of the GRM approach is due to the inherent accuracy of

interpolation.

By contrast, the maximum depth of penetration of the direct arrivals used in the

model-based approach is less than 300 m (Palmer, 1986, equation 13.18). As a

result, wavespeeds throughout the first layer are determined by extrapolation

from the upper part of that layer. The various inversion models represent

different extrapolation functions and the errors are due to the inherent instability

of extrapolation. This instability is aggravated where there are undetected layers.

Furthermore, the application of linear wavespeed functions to constant

wavespeed layering can result in large gradients (Lanz et al., 1998), which in turn

can result in the ubiquitous ray path diagrams demonstrating almost complete

coverage of the subsurface. These diagrams are misleading when the inversion

246

model does not accurately represent the subsurface, because the shortcomings

of extrapolation are not overcome.

In general, the direct traveltimes through a layer are unable to provide a reliable

wavespeed model of that layer, except in the trivial cases of isotropic constant

wavespeeds. This casts doubts on the benefits of very high precision of the

traveltimes. Although the data used in this study are noise-free and accurate to

0.1 ms, there is not a corresponding increased accuracy in the depth

computations with the model-based approach. Accordingly, it is questionable

whether it is necessary to measure traveltimes to an accuracy of 0.1 ms (Lanz et

al., 1998), especially when the acceptable residuals appear to be between 2 ms

and 5 ms for most near surface targets.

A more reliable measure of the accuracy of depth computations is the optimum

XY value, because it constrains the parameters for each inversion model. This in

turn, is a function of the detector interval. Therefore, trace spacing is at least as

important as the accuracy of the traveltime data in assessing of the accuracy of

refraction inversion.

While it is clear that the quality of the field data will have some effect on the

measurement of the optimum XY value, it is not as critical as with the model-

based methods. This single parameter is obtained through the recognition of a

distinctive pattern of minimum residuals between the computed and averaged

wavespeed analysis function, as shown in Figure A.4. This pattern is less

sensitive to errors in the individual traveltimes.

The major conclusion to be drawn from this study is that the choice of the

inversion model is an important, if not the most important factor in the

performance of model-based inversion. However, the wavespeed model for

each layer cannot be determined uniquely from the traveltime data from that

layer, because the data generally do not provide a complete, an accurate nor a

247

representative sample of that layer. By contrast, the optimum XY value does

provide a complete sample, because it is determined from the traveltimes for the

underlying refractor, and it is able constrain an almost complete range of

inversion models. Therefore, the appropriate model for model-based inversion

has parameters determined with a GRM analysis, together with minimum

residuals between observed and modeled traveltimes.

The results of this study suggests a two stage inversion strategy. Firstly, the

GRM is employed where the refractor is sufficiently irregular, in order to

determine an appropriate model for inversion. Experience indicates that this

situation occurs in about 30% of the data. Secondly, model-based inversion

methods such as tomography are then employed to process the full set of data,

using the wavespeed model determined with the GRM analysis.

Alternatively, if depth estimates obtained with the GRM using the single layer

constant wavespeed model are similar to those obtained with model-based

inversion, then the inversion model can be taken as appropriate.

A.11 - References

Acheson, C. H., 1963, Time-depth and velocity-depth relations in Western

Canada: Geophysics, 28, 894-909.

Acheson, C. H., 1981, Time-depth and velocity-depth relations in sedimentary

basins - a study based on current investigations in the Arctic Islands and an

interpretation of experience elsewhere: Geophysics, 46, 707-716.

Berry, J. E., 1959, Acoustic velocity in porous media: Petroleum Trans. AIME,

216, 262-270.

248

Birch, F., 1960, The velocity of compressional waves in rocks at 10 kilobars: J.

Geophys. Res., 65, 1083-1102.

Brandt, H., 1955, A study of the speed of sound in porous granular media: J.

Appl. Mech., 22, 479-486


McGraw-Hill Inc.

Faust, L. Y., 1951, Seismic velocity as a function of depth and geologic time:


Faust, L. Y., 1953, A velocity function including lithologic variation: Geophysics,

18, 271-288.

Gassman, F., 1951, Elastic waves through a packing of spheres: Geophysics,

16, 673-685.

Gassman, F., 1953, Note on Elastic waves through a packing of spheres:

Geophysics, 16, 269.

Hall, J., 1970, The correlation of seismic velocities with formations in the

southwest of Scotland: Geophys. Prosp., 18, 134-156.

Hamilton, E. L., 1970, Sound velocity and related properties of marine sediments,

North Pacific: J. Geophys. Res., 75, 4423-4446.

Hamilton, E. L., 1971, Elastic properties of marine sediments: J. Geophys. Res.,

76, 579-604.

249

Iida, K., 1939, Velocity of elastic waves in granular substances: Tokyo Univ.

Earthquake Res. Inst. Bull., 17, 783-897.

Jankowsky, W., 1970, Empirical investigation of some factors affecting elastic

velocities in carbonate rocks: Geophys. Prosp., 18 103-118.

Lanz, E., Maurer, H., and Green, A. G., 1998, Refraction tomography over a


Menke, W., 1989, Geophysical data analysis: discrete inverse theory: Academic

Press, Inc.

Miller, K. C., Harder, S. H., and Adams, D. C., and O'Donnell, T., 1998,

Integrating high-resolution refraction data into near-surface seismic reflection

data processing and interpretation: Geophysics, 63, 1339-1347.



Palmer, D., 1983, Comment on "Curved raypath interpretation of seismic

refraction data" by S.A. Greenhalgh and D.W. King: Geophys. Prosp., 31, 542-

543.






refraction inversion?: Explor. Geophys., 23, 261-266, 521.

250

Paterson, N. R., 1956, Seismic wave propagation in porous granular media:


Stefani, J. P., 1995, Turning-ray tomography: Geophysics, 60, 1917-1929.

White, J. E., and Sengbush, R. L.,, 1953, Velocity measurements in near surface

formations: Geophysics, 18, 54-69.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1956, Elastic wave velocities in

heterogeneous and porous media: Geophysics, 21, 41-70.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1958, An experimental

investigation affecting elastic wave velocities in porous media: Geophysics, 23,

459-593.


Geophysics, 63, 1726-1737.

Zhu, X., Sixta, D. P., and Andstman, B. G., 1992, Tomostatics: turning-ray

tomography + static corrections: The Leading Edge, 11, 15-23.

A.12 - Appendix: Definition of Variable Wavespeed Media withthe GRM

The GRM approach to the variable wavespeed model is described in detail in

(Palmer, 1986, p.175-181). Essentially the approach assumes that the Evjen

function in equation A-1a, is applicable, ie.

V(z) = V0 (1 + q z)1/m (A.1a)

251

The wavespeed of the upper layer at the surface, V0, is obtained from the

traveltime graphs. The refractor wavespeed, Vn, the time-depth, tG, and the

optimum XY value are obtained from the application of the GRM wavespeed

analysis and time-depth algorithms.

These parameters are substituted into the left hand side of equation A-2a, which

is a standard integral for integer values of m, the exponent in equation A-1a.

tG Vn / XY = i0∫i1 (sinm-2 i - sin2 i) di / i0∫i1 sinm i (A.2a)

sin i0 = V0 / Vn (A.3a)

sin i1 = V1 / Vn (A.4a)

and V1 is the wavespeed in the variable wavespeed medium, immediately above

the refractor.

The sine of the critical angle, i1, is obtained from equation A-2a, once a value of

m has been selected, either from a cross-plot of V0 / Vn and tG Vn / XY, or by

iteration.

The critical angle is then used to evaluate the relationship between the time-

depth tG and the angles i0 and i1, in equation A-5, to obtain the parameter q (see

equation A-1a), viz.

tG Vn = (m / q V0 sinm-1 i0) i0∫i1 (sinm-2 i - sin2 i) di (A.5a)

Finally the depth, zG, is obtained from

sin i1 = V0(1 + q zG)1/m / Vn (A.6a)

252

Appendix 3

Surefcon.c

/* Copyright (c) Colorado School of Mines, 1999.*/

“

/* All rights reserved. */

/*SUCONV: $Revision: 1.12 $ ; $Date: 1996/09/05 19:24:26 $ */

"

"

#include "su.h"

"

#include "segy.h"

"

#include "header.h"

"

"

/*********************** self documentation **************************/

char *sdoc[] = {

" ",

" SUREFCON - Convolution of user-supplied Forward and Reverse ",

" refraction shots using XY trace offset in reverse shot ",

"

", surefcon <forshot sufile=revshot xy={trace offseted} >stdout ",

" ",

" Required parameters: ",

" sufile= file containing SU trace to use as reverse shot ",

253

" xy = Number of traces offseted from the 1st trace in sufile ",

" ",

" Optional parameters: ",

" none ",

" ",

" Trace header fields accessed: ns ",

" Trace header fields modified: ns ",

" ",

" Notes: It is quietly assumed that the time sampling interval on the",

" output traces is the same as that on the traces in the input files.",

" ",

" Examples: ",

" suconv<DATA sufile=DATA xy=1 | ...

","

",

" Here, the su data file, \"DATA\", convolved the nth trace by ",

"

" (n+xy)th trace in the same file ",

" ",

" ",

NULL};

/* Credits:

* CWP: Jack K. Cohen, Michel Dietrich

* UNSW: D. Palmer, K.T. LEE

*

* CAVEATS: no space-variable or time-variable capacity.

* The more than one trace allowed in sufile is the

* beginning of a hook to handle the spatially variant case.

*

* Trace header fields accessed: ns

* Trace header fields modified: ns

*/

/**************** end self doc ********************************/

segy intrace, outtrace, sutrace;

int

254

main(int argc, char **argv)

{

int nt; /* number of points on input traces */

int ntout; /* number of points on output traces */

int xy; /* the offset number for GRM */

float *forshot; /* forward shot */

int nforshot; /* length of input wavelet in samples */

cwp_String sufile; /* name of file of forward SU traces */

FILE *fp; /* ... its file pointer */

int itr; /* trace counter */

/* Initialize */

initargs(argc, argv);

requestdoc(1);

/* Get info from first trace */

if (!gettr(&intrace) ) err("can't get 1st reverse shot trace");

nt = intrace.ns;

/* Default parameters; User-defined overrides */

if (!getparint("xy", &xy) ) xy = 0;

/* Check xy values */

if (xy < 0) err("xy=%d should be positive", xy);

if (!getparstring("sufile", &sufile)) {

err("must specify sufile= desired forward shot");

} else {

/* Get parameters and set up forshot array */

fp = efopen(sufile, "r");

for (itr = 0; itr <= xy; ++itr) {

if (!fgettr(fp, &sutrace) ) {

err("can't get 1st requested forward trace");

};

};

nforshot = sutrace.ns;

255

forshot = ealloc1float(nforshot);

/* Set output trace length */

ntout = nt + nforshot - 1;

/* Main loop over reverse shot traces */

do {

fprintf(stderr,"rev==%d\t , for=%d\t", intrace.tracf,

sutrace.tracf);

memcpy((void *) forshot,

(const void *) sutrace.data, nforshot*FSIZE);

/* Convolve forshot with revshot trace */

conv(nforshot, 0, forshot,

nt, 0, intrace.data,

ntout, 0, outtrace.data);

/* Output convolveed trace */

memcpy((void *) &outtrace, (const void *) &intrace,

HDRBYTES);

outtrace.ns = ntout;

outtrace.dt = outtrace.dt/2;

/*outtrace.cdp = 2*intrace.tracf + xy;*/

fprintf(stderr,"out_cdp=%d\n", 2*intrace.tracf + xy);

puttr(&outtrace);

} while ( gettr(&intrace) && fgettr(fp, &sutrace) );

} ;

return EXIT_SUCCESS;

}

256

Appendix 4

The Effects of Spatial Sampling onRefraction Statics

Palmer, D., Goleby, B., and Drummond, B., 2000a, The effects of spatial

sampling on refraction statics: Exploration Geophysics, 31, 270-274.

Digital processing of shallow seismic refraction data with the refraction convolution section

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