Introducing Geophysical Inversion

7/26/2019 Introducing Geophysical Inversion

1/43

Inversion conceptsIntroducing geophysical inversi

This chapter deals with basic concepts underlying geophysical inversion. Four sections provide anoverview of essential ideas without mentioning mathematical details. Extensive use of flowcharts andfigures should quickly introduce what inversion does, what is needed, and what can be expected for

outcomes.

Note that the current chapter !nversion "oncepts# is conceptual in nature while "hapter $ !nversionTheory# covers the same ground in a more rigorous manner. !t is recommended that the conceptualchapter be studied first before tackling the details provided in the theory chapter.

Inversion% Estimating models of physical property distributions based on geophysical survey data.

Overview

&eophysical remote sensing data can be used to help solve practical environmental, engineering orexploration problems. !n some cases, when only limited knowledge about the subsurface is required,inferences drawn directly from the data can be sufficient, as Figure 'a illustrates. This is a familiarlinear framework in which data are gathered, then processed, plotted and interpreted.

(owever, when more detailed information about the subsurface is needed, quantitative models of theearth need to be estimated. This is geophysical inversion, as Figure 'b illustrates. (ere the frameworkis not so simple. )ata are used to constrain possible modelsof the earth, which have been estimatedusing some procedure suitable to the particular problem.

This short package does not explain details, but it should provide a conceptual understanding of whatis involved when trying to solve the geophysical inverse problem.

NOTES about the last page:*ection f. Inverting 2D D datais challenging at this level. !tassumes some understanding of )" resistivity surveying, and some experience with geophysicaldata. (owever, it does illustrate what+s involved in carrying out inversion work. eaders with nomore background than a good understanding of this -!nversion "oncepts- chapter will gain a clearerimpression of geophysical inversion by reading through the page carefully. There is a slightly moreadvanced version of this page at the end of hapter ! "Inversion Theor#".!f "hapter $ is to becovered, you might gain more by deferring this page until then.


2/43

'a. &eophysical information can come directly from data.

'b. !f models are needed, inversion must be applied.


3/43

Inversion conceptOutline of inversi

On this page:

This page uses interactive figures to provide a pictorial introduction to geophysical inversion. ou

will be able to characteri/e 0ust what the geophysical inverse problem is, and what processes anddecisions are involved. 1ey concepts are% observed versus predicted data2 prior information or

knowledge#2 and that a model is acceptable if it can predict data and if it is consistent with priorinformation.

!n a typical geophysical survey, we put energy into the ground and record a response, which we referto as data or observations Figure 'a#. The values of the data depend upon the distribution of physicalproperties in the subsurface. This is illustrated in Figure ' which is a generic slide for all geophysicalsurveys. The goal of the inverse problem is to determine the distribution of the physical property orproperties that gave rise to the data Figure 'b#. 3nfortunately, this is not rigorously possible inpractical surveys because only a limited number of data can ever be recorded and the data are alsoinaccurate. Nevertheless, approximate solutions can be found, and the methodology is designed toinclude other information about the problem so that the calculated solution is more likely to representthe true earth structure.

'a. 4easurements of physical processes are gathered over the earth and a data setis produced.

'a

'b

'b. The purpose is to learn something about what is beneath the surface.5nswering this using the data is inversion.

reating models based upon data

!nversion is a mathematical procedure that can take on several forms. !n order to generateunderstanding about the subsurface without digging or drilling, measurements must be gathered, datamust be generated from these measurements, and some degree of understanding about what is beinginvestigated -prior knowledge-# should exist. Then inversion processing can be carried out, usingthe data and prior knowledge as input. The result will be a set of -models- characteri/ing how therelevant physical property is distributed in the ground. These models will have characteristicsdetermined by the inversion method used, by the data and by prior knowledge. Figures 6a76eillustrate.


4/43

a. 4easurements in the field over 6b. 5 data set is made from these measurements. 4ore or less data procesmay be involved, and plots or graphs must be made to help withinterpretation and data quality assurance.

6c. 8rior information is what you 6d. !nversion is the processing step that uses data and prior knowledge as inputo estimate models of how physical properties are distributed under the gr

he region of interest must first be made.

already know about the problem.

6c

6c


5/43

$e%uirements of inversion

Extracting from the flow chart of Figure 6 we have the basic ingredients of inversion summari/ed inFigure 9%

measured data

prior information

4odels are estimates of a physical property distribution.6e.

the inversion algorithm a physical property distribution within the volume being studied

The section of Figure 9a consists of a model estimation algorithm and two decisions

which must be made before the model is considered satisfactory. These are shown in Figure 9b, andthey can be summari/ed as follows%

'. &odel estimation and predicting data:The inversion algorithm must estimate the values ofparameters that define the model. The first estimate will be based upon a prior understandingof the situation. 8redicting data means calculating the measurements that would be made by asurvey over the model.

6. Decision '( Suitable misfit::ne decision is whether the predicted data caused by thecurrent model# are sufficiently similar to the real survey measurements. !n other words, is themisfit between observedandpredicted data sufficiently small;

9. Decision 2( Optimal model: The second decision is whether the the model itself is -optimal-in a mathematical sense. *uccessful inversion requires careful design of this optimi/ationproblem.


6/43

!n order to make these decisions, both data and prior information are necessary. They both serve tolimit the possibilities for a recovered model, therefore we say that measurements and priorknowledge about likely properties and structures# act as constraints upon the optimi/ation problem.

9a. 5 very generic flow chart with essentialcomponents of inversion

9b. 8rinciple requirements of inversion%

':5 model must be estimated and resultinpredicted data must be calculated.2: 8redicted and measured data must comfavourably.): The model must optimal in some well7defined sense.

5 solution to the inverse problem is obtained byestimating a model, testing it using misfit andacceptibility criteria, and then iteratively perturbing i.e.ad0usting# the model until the two testing criteria aresatisfied. The arrows in Figure 9b show how informationis used and where iteration feedback# occurs.

:f course, once a mathematically acceptible model has been obtained, the geoscience professionalmust asks whether the result is geologically and geophysically plausible. This generally involves

some degree of interaction with people who understand both the fundamentals of inversion and thespecific geologic and geophysical problem.

The next section explains some of the implications of the two decision boxes in Figure9b. *mall icons in the shape of this flow chart like the one shown to the right# will be

used to highlight which section of this chart is being discussed.


7/43

Inversion conceptThe inversion proc

Ob*ectives

This page outlines how inversion works without going into details. 5fter absorbing the concepts, youshould be able to%

'. Explain the difference between observed and predicted data26. 3nderstand why both are needed29. &ive three reasons why we say that the inversion problem is -non7unique-2$. 3nderstand that there will still be significant -prior knowledge,- even though geophysical work

may be done at a site where very little previous work was done.


8/43

inversion does not take a raw data set, manipulate it directly and output some sort of answer. That iswhy the flow chart is not a simple linear path with data at one end and resulting model at the other.!nstead, inversion processing starts with an initial estimate for the earth model, then proceeds with aforward calculation on that model to predict what measurements would be if a survey were carried outover that initial model. The resulting data set is called the predicted data. :nly at this stage is the actualfield data, or -observed data,- brought in. These survey results are compared to predictions generated

for the initial model. This is the first of two decisions in the inversion algorithm, and both are describedin more detail next.

Decision number ': +itting the data

:nce there is a preliminary model, a predicted data set for that model, and an observeddata set collected in the field, the inversion algorithm can go to work on the two decisionsthat have to be made within the inversion process. The important business of how toestimate a modelwill be discussed in detail in a subsequent chapter#. The decisions are notnecessarily made -first- and -second-, but we will start by outlining the misfit decisionusing the following figure%

6a. :nce a model has been estimatedinitially it could be as simple as a halfspace# ...

6b. ... a data set must be calculated byforward modelling. This data set is referred to aspredicted data.

6c. )ata from field surveys are now brought in, ... 6d. ... and compared to the calculated dataobtained by forward modelling from theestimated model.

,etting the process started

3nlike other geophysical -data processing- procedures such as filtering or data reduction,


9/43

This set of figures does not explain how each

step is done, it simply outlines what is donewhen testing to see if the model is capable ofexplaining the observations field data#. 5lso,we have not explained what the criteria are fordeciding whether two data sets are -sufficiently

similar.- That will be covered in the nextchapter.

- crucial concept: nonuni%ueness

The other decision involves a more subtle issue. For most inversion problems that will be discussed inthis work, we will have digiti/ed the Earth with more cells than we have data values. Therefore, it willbe impossible to find a unique value for each cell without additional information. This issue is referredto as the problem of non7uniqueness. 5ll problems like this are called -under7determined,- and it is

6e. !f the two data sets are not sufficientlysimilar, a new model is estimated and theprocess repeated. !f the two data sets aresufficietly similar, we can proceed to the nextstep in the process.

common to say that they have non7unique solutions. That is, all problems with more unknowns thandata will have an infinite number of possible solutions. Furthermore, there will be errors associatedwith every data point, making the problem even more difficult to solve.

/arametric problems:*ituations when there are fewer parameters describing your model than there

are measurements are called -over7determined- problems. They are usually dealt with in a completelydifferent manner, but interpretation of over7determined problems is also non unique. For example, if aparticular geometric shape is assumed in the parametric solution a buried cylinder perhaps# one uniquesolution will be recoverable. (owever a different parameteri/ation could be invoked 7 perhaps a buriedmulti7sided ob0ect 7 so the interpretation is in fact non7unique. *ee *ection $.> for more discussion ofparametric problems. The main point here is that, because we want the flexibility of explaining as muchof the true complexity of the 9) Earth as possible, non7uniqueness is a pervasive characteristic ofsolving the geophysical inverse problem.

The next figures illustrate non7uniqueness for under7determined problems, when the Earth is dividedinto many more cells than there are data values. The data set on top is a synthetic magnetics data set

generated by measuring the fields on the surface above a cube of magnetic material buried between thedepths of


10/43

!f the region where magnetic material can occur is constrainedto only a thin 6


11/43

Decision number 2: Optimal model

(ow does the issue of non7uniqueness relate to decision number two, which relates towhether the model is an optimum one. *ince an infinite number of models are possible,alternative information must be used to constrain the models so that one can be chosen infavour of all the rest. This is where prior information what we already know about the problem# comesin. =hat do we know; !n other words, what constitutes prior knowledge; egardless of how much orhow little work has been done at the field site, a significant amount may be known about thegeoscience problem. (ere are examples%

There will be geological information. For instance, it may be known that background rocks

hosting the target are non7magnetic limestone for example#. !t may be known that there is non7magnetic overburden that is at least a few metres thick. The basic structural situation may bereasonably well understood from surface mapping and other work.

There will be geoph#sical information. For example, we know that magnetic susceptibility

must take on positive values. )ensity contrasts, on the other hand, may be either positive ornegative depending upon whether -target- materials are more or less dense than host materials.

There will be logical information. For example, it is sensible to look for a -simple- solution. !t

will be possible to find an arbitrarily complicated model to explain the data but it will N:T be

possible to find an arbitrarily simple solution if there is any pattern to the data set. Therefore itmakes sense to look for the simplest model that can explain the data. -*imple- may mean aslittle structure as possible, or as little departure from an initial structure as possible, or astructure that is as smooth as possible.

*election of an -optimum- model means constraining the infinite number of solutions to those that areconsistent with what we already know or assume about the situation. Exactly how this priorinformation influences the inversion process is discussed in detail in the next chapter.

0sable results

)isplaying and using 9) results of inversion often takes care. !t is rather easy to give the

wrong impression about the result if the model cannot be examined visually in aconvenient and versatile manner.

The next figures show the model obtained by inverting the data set shown in Figure 9 above, using amethod that allows susceptibility to occur with equal likelihood anywhere in the volume.

$a. The data set is shown above an isosurfaceimage of the susceptible /one recovered.

$b. 5n isosurface is an image with all cells with valuesless than some chosen cutoff value hidden from view. Thchoice of cutoff significantly affects the appearance of thmodel.


12/43

c. 5lternatively, a slice through the volume can beisplayed with the cell value coded by colour. This gives aorrect impression of the range, and distribution pattern, ofalues in the mode.

$d.

$e. $f. !t can be useful to display slices through thevolume in several directions.

$g. 5 combination may be useful, and the direction usedo view the model may contribute to understanding the

model that has been recovered.

!t can be useful to display slices through thevolume in several directions.

!t can be useful to display slices through thevolume in several directions.


13/43

Summar# the fundamental flow chart

,eneral problems

=hat is needed to invert a data set; This flow chartsummari/es the requirements for proceeding with inversionof geophysical data for general problems. The flowchart forparametric problems is shown by rolling the mouse over the

chart, and is described later in this section. The introductorypage for the next chapter is an interactive version of thischart which provides details for each portion. For thepresent, it is sufficient to introduce the chart, and tosummari/e the procedure for implementing inversion asfollows%

,iven% !n order to obtain models of the subsurface

by inversion, it is necessary to start with field data,estimates of errors and noise on those data, aforward modelling calculation procedure, and well

described prior information or assumptions about

the situation. Discreti1e% )escribe the earth by dividing it into cells, each with fixed si/e and unknown but constant

value of the relevant physical property. There are other ways of describing your model of the earth,but this ")7:4 focusses on the use of rectangular cells.#

hoose decision criteria% The choices made for how data predictions and survey measurements arecompared, and how an optimal model is chosen based upon prior information# are crucial forobtaining useful results.

Inversion% Find values for cells which are consistent with both the measured data and the prior

information. The inversion process is implemented using mathematical optimi/ation theory. Evaluate the inversion result% 3se a comparison between predicted data and field

measurements, and take into account what is known or expected about the earth+s properties and

structures. Iteration is inevitable% No initial outcome should be used without exploring a range of equally

possible models. Interpret the result% models of physical properties must be interpreted in terms of useful

geologic or geotechnical parameters. The models must be easy to understand and convenient tomanipulate.

/arametric problems

5s mentioned above, when there are fewer parameters than data values,the problem does not involve choosing from an infinite variety of

reproduce the data as closely as possible is sufficient.The procedures for parametretic problems are described in greaterdetail in subsequent chapters.

5s the theory of inversion is developed in subsequent sections,it will become clear that the primary difficulties are that the

procedures are computationally demanding.

potential models 7 ad0usting parameters until the model can

solution is non7unique, and that the necessary mathematical


14/43

Inversion conceptFlow chart for geophysical inversi

Before inversion can be carried out, the requirements of this flow chart must be addressed.

1.

,iven:

'. Field observations6. Error estimates9. 5bility to forward model$. 8rior knowledge

2.)iscreti/e the Earth

3.

"hoose a *uitable4isfit "riterion and

)esign 4odel :b0ectiveFunction

4.8erform !nversion

3sually ad0ust misfitand or the ob0ective

function

5.Evaluate

esults

!terate

Interpret preferred models

esults


15/43

Notes on iteration

Geophysical inversion is an iterative process. :wing to the nonuniqueness of the problem, severalequally valid solutions should be obtained. &enerally a model obtained from a first successfulinversion should be refined by exploring the importance of misfit and by ad0usting the modelob0ective function. These ad0ustments should be made within the context of as much understandingof the problem as possible. Then the preferred model of the earth can be chosen based upon the rangeof acceptable models, and what is already known about the problem and the geology. !n other words,the person doing inversion must work as a member of a team with professionals who have geologic,geotechnical, geochemical, andAor other relevant expertise.


16/43

The use of geophysics in general is also usually part of an iterative process. &eophysical informationcan build on geologic information already obtained, and it can help guide further investigations as thepro0ect proceeds from preliminary reconnaissance, through follow7up of anomalies in the field,delineation of subsurface details, and further pro0ect development.

Inversion concepSuitable problems for geophysical inversi

For those making use of geophysical inversion for the first time, it is natural to ask -will inversioncontribute towards my problem?- This page provides a ten7point outline of criteria to consider whenanswering this question.

The range of problems re%uiring inversion

!f your geoscience question can be answered withoutknowing the values and distributions ofphysical properties within the ground, then rigorous inversion may not be necessary. :ne example isan ob0ect search question such as locating underground storage tanks# for which a simple map of ageophysical anomaly might provide a clear indication of where the desired ob0ect is located.

!nversion is essentially a processing step that attempts to find the cause for a set of measurements.Therefore inversion can contribute to geoscience problems at any scale. *ee the sidebarfor

examples of inversion being used at all scales of problems, from studying the structure of a wholeplanet, down to characteri/ing features at the scale of only a few cubic meters.

ecall that inversion is essentially a processing step that attempts to find the cause for a set of

measurements. Therefore, inversion can contribute to geoscience problems at any scale. (ere are

examples to show how inversion has been used at all scales of problems, from studying the structure of

a whole planet, down to characteri/ing features at the scale of only a few cubic meters.

1. Crustal thickness of ars Data% Topography, gravitational field

/rior information% )ensities, assumption thatcrustal densities are uniform.

Errors% =ell defined in the original reference.

$esulting model% Cariations in crustal thickness

for the whole planet. Some limitations of the model% True crustal

densities are unlikely to be uniform2 resolution is

approximately 'D?km6

From uber, 4. T. et al.Internal structure and earthermal evolution of Mars from Mars Global urvetopography and gravity, Science 6D@, '@DD7'@96???#.

Topography

crustal thickness

Free7air gravity


17/43

!. "eologic structure under #$$ s%uare kilometers&San 'icolas( e)ico*

Data% '@,>@' measurements of gravity, after

standard data reduction processing. Errors% *tatistically defined for inversion with a

minimum plus a percentage. /rior information% "onstrain to find a -smooth-

model2 positive and negative density contrastspermitted2 equal likelihood of variations at alldepths.

$esulting model% 9) distribution of density

contrast. *hown via mouseover on the ad0acentimage is surface pro0ection of contrasts that aregreater than ?.'gAcc.

Some limitations of the model% *tructures

appear smoothed, cell si/es are large 6


18/43

-. Characterie a hidden utilities trench

Data% 9D values of dipole7dipole )" resistivity

measurements, recorded with four source7receiver spacings.

Errors% *tatistically defined for inversion with a

minimum plus a percentage. /rior information% *mooth model, uniform host

materials, errors defined statistically. $esulting model% 6) cross section# smooth

model with high resistivities blueAgreen#indicating surface and trench material, and lowresistivities redAyellow# indicating surroundingsoils.

Some limitations of the model% Not fully 9),

small data set does not extend far enough to fullycharacteri/e host materials.

*ee "hapter 6, -foundations-, the -&eophysicalsurveys- section for more details about )" resistisurveys.

)" resistivity data. I7axis is line position,7axis is source7receiver spacing

4odel resulting from inversion of data.

/. 0iscriminating between a 2O and scrap metal

m6of data, target is one ob0ect#

Data% )ecaying magnetic fields that were

induced in the target by a source ofelectromagnetic energy, gathered at '?cmspacing along lines 'ft apart.

Errors% 8arametric inveresion does not use

errors in the way that is usual when solvingunder7determined problems.

/rior information% Target shape and metal7type

are known. $esulting model% Thirteen parameters,

including location, orientation, andcharacteristics of two induced dipole -sources-.

Some limitations of the model% Empirical

relations are necessary to discriminate betweenscrap metal and 3I:s.

)ata at two of 6? time intervals.8arameters for the parametric model.

Ten aspects affecting suitabilit# of problems for inversion

5s the needs of exploration, engineering, environmental, and other industries become more sophisticated,

so too do the requirements for inexpensive, non7invasive acquisition of detailed quantitative informationabout subsurface materials. !n the image to the right, the value of density throughout the volume ofinterest has been estimated by inversion of ground7based gravity data set, in order to characteri/e anore deposit as quantitatively as possible.

The question now is, -what aspects of a problem affect its suitability for inversion?!The following ten points below should be considered.


19/43

'.8hysical propertycontrast

2.!lluminationenergy

).8roblem si/e !."onsistent data Jmodel type

.Topography

3.8ermissiblelocations of buriedfeatures

4."onsistencywith priorknowledge

5.5ccurate,clearly understooddata

6.=ell7characteri/eddata errors

'7."onsistencybetween discreti/ationJ data

'. /h#sical propert# contrast% There must be a physical property contrast corresponding to thegeological problem. This is true for all geophysical work, and it is true for inversion. !f the data

contain no response related to the target, inversion will recover nothing.

E8ample% !n the "entury )eposit case history in "hapter #, the model of electrical

conductivity obtained by inversion of )" resistivity data did not show where the ore bodywas, although other structural information was obtained. (owever, the chargeability modeldid include /ones of chargeable material corresponding with economic ore. 5 table ofphysical property values obtained by drilling confirms that the ore body+s electricalconductivity is similar to host rocks, while it+s chargeability is significantly different fromsurrounding geologic materials.

2. Illumination energ#% )ata should be gathered with source energy interacting with the target in asmany different ways as possible. =hen the source energy cannot be moved, some prior knowledgeabout how material is likely to be distributed can be incorporated into the inversion. This is done forpotential fields data 7 see chapters on inverting magnetic and gravity data.

E8ample% :ne variety of )" resistivity survey a so7called +gradient array+ survey# involves

using only a single location for source electrodes. This type of data is hard to invertsuccessfully and techniques similar to inverting magnetic or gravity data may be necessary.4ore discussion can be found in the *an Nicolas case history of "hapter , in section 9,!"egional scale geophysics-, under -#hargeability-.

). /roblem si1e% =hat is meant by problem si/e; This issue is covered in detail throughout the ")7:4, but there are two essential aspects% the number of cells used to discreti/e the Earth referred to

as$#, and the number of data values referred to asM#. The numerical implementation of inversionschemes will involve working with matrix calculations that are as big as$xM.

E8ample% (ow serious is this; !magine a normal airborne survey covering an

area $km by $km, involving survey lines spaced '??m apart and measurementspacing along the lines of


20/43

!. onsistent data and model t#pe% !nversion for ') or 6) models see the model types summarypage in the -Foundations- chapter# can only produce sensible results if the measurements areunaffected by geologic conditions that change in the -missing- direction. !n addition, if the data donot contain information about variations in all 9 dimesions, then full 9) inversions are not likely tobe successful. !n other words, the data set and the choice of inversion methodology must beconsistent.

E8ample% )" resistivity and !8 surveys are commonly gathered along survey lines. To cover

large areas, several lines may be used. These lines are usually rather far apart compared to themeasurement spacing along the lines. !ndividual 6) inversions are recommended for eachdata set gathered along a line. Fully 9) inversions using many lines will likely be successfullonly if the survey line spacing is less than the maximum electrode spacing along the lines. 5nexample of the latter situation is given in the "luny, 4t. !sa case history of chapter .

.Topograph#% 6) and 9) inversions must have good topography data available. 4any types of dataare affected by topography, so these affects will be explained by erroneous structrues if topography isnot correct in the inversion model.

E8ample% The 6) synthetic model shown to the

right has some variations in electrical conductivityunder a mountain and a valley. *ynthetic )"resistivity data generated over this model have been

inverted with and without proper topography. "lickthe following buttons to see the two resulting modelsof the earth+s distribution of electrical conductivity%

True model

First inversion result *econd inversion result

3./ermissible locations of buried features% !f there are geologic featuresaffecting the data which do N:T lie within the volume encompassed by themodel, then the inversion will be forced to place erroneous features withinthe model in order to account for those components of the data. This is veryimportant( especially for magnetic and gravity surveys.

4. onsistenc# with prior 9nowledge% !f a process is designed to generate -smooth- models, youshould expect to interpret the recovered models in terms of smooth variations of the physicalproperty. This is not necessarily a problem if the -smooth- models can be interpreted in terms of

structures expected. The point here is that interpretations can be effective only when the inversionprocess being used is properly understood.

E8ample% The figures to the right show how a discrete block of conductive material may be

revealed by inversion using a process that returns smooth models. 4ove your mouse over the figuresee the inversion result. This issue is much clearer with a good understanding of why inversionprocedures do what they do.


21/43

5. -ccurate clearl# understood data% !t should be obvious that inversion results can be only asgood as the input data. !n addition to having accurate data, it is necessary to know exactly what thephysical measurements were, and how the input data were generated from those measurements.8redicted data cannot be produced properly, and therefore a successfull inversion outcome cannot beexpected if all the relevant details are consistent with the forward modelling procedure used in theinversion.

&ore details% For the outline of this point, see )ecision number '% fitting the data, in "hapter

9, - !nversion "oncepts-.

6. ;ell characteri1ed data errors% !n addition to understanding exactly where the original datacome from, there must be an estimate of the errors that are associated with each data value. !t is notcommon for quantitative statistics to be available, so assumptions often must be made about theerrors associated with data.

'7. onsistenc# between discreti1ationand data% The si/e of cells used in the model should besmaller than the si/e of all features that will affect the measurements. !n other words, data must N:Tcontain information caused by features that are smaller than the cell si/e.

&ore details% To meet this demand with magnetic or gravity data, it may be necessary to

calculate an equivalent data set that would have been gathered some distance from thesurface. The relevant data processing step is called upward continuation.

&ore details% For other types of data it may be necessary to increase the errors assigned to

data if geologic features exist which are very small andAor close to the measurement location.


22/43

Inversion conceptsInverting !0 0C resistivity data

ontents and ob*ectives%n this page&

'. !ntroducing the scenario. )" resistivity data, errors and prior information

. )iscreti/ing the Eartha. &etting started with )"!86) 7 the datab. *pecifying errors*pecifying the desired optimal model>a. First inversion

@. *econd inversionD. )epth of investigation

. 5lternative models a. 5d0usting the reference mo

c. 5d0usting misfit

'?. "onclusions

n this section, fundamental concepts are implemented for a synthetic 6) )" resistivity data set. Treatment is notgorous because the purpose is to illustrate how inversion is applied, and what the effects of various decisions will b

he model recovered by inversion.

his page is challenging at this level. !t assumes some understanding of )" resistivity surveying, and some experienwith geophysical data. (owever, it does illustrate what+s involved in carrying out inversion work. !f you are reading tom the !5& cd7rom, there is a slightly more advanced version of this page at the end of hapter ! "Inversionheor#".!f "hapter $ is to be covered, you might gain more by deferring this page until then.

'. Introducing the scenario

To illustrate the work involved in carrying out inversion, we will use a synthetic )" resistivity survey and the 3B"&!F+s forward modeling J inversion program library called )"!86). The artificial 6) electrical conductivity structof the Earth that is used to generate the data will be revealed later, but for now we will start with the data set and wothrough an inversion sequence as if we were doing a normal 0ob.

&eneric flow chart icons are included to help remind you of where in the inversion methodology we are currentlyworking. There are occasional sidebars to provide clarifaction without interrupting the flow. Luestions to guid

your thinking about inversion are included via the questions icon . "lick hereto see all the questions on one pa

2. D resistivit# data errors and prior information

First all four pre7requisites in the top box of the flow chart must be considered. They are% '. Fieldobservations, 6. Error estimates, 9. 5bility to forward model and $. 8rior knowledge. Each pre7requisite isdescribed next.

The measurementsinclude source current, resulting potentials, and the geometry of electrodes, but recordeddata usually are apparent resistivities. The conventional plotting procedure called pseudosections is not self

b. 5d0usting the model norm

>b. Evaluating results

explanatory, therefore, a sidebaris provided to briefly describe them. There is no other processing of data prio

inversion, except if voltages were not saved during the survey 7 then they must be derived by working the apparentresistivity formula backwards, assuming a source current of ' 5mpere and the known geometry of electrodes. 5lsocourse, the input data must be formatted into a text file with the proper format. The input file used in this exercise cseen here. !n this file, columns are not labelled 7 they are left to right# transmitter electrode positions along the linecolumns#, receiver electrode positions 6 columns#, normali/ed potential measurement, and an estimate of the standdeviation of that measurement. This last column can be added using the inversion program, as explained in part < b


23/43

rrorsin the data arise for various reasons including inaccurate locations of electrodes and electrical noise signals ce in the range of low milli7volts or on the order of microvolts#. 5lso, if the real Earth is very different from being twimensional near any electrodes, then -geologic errors- occur because the features causing data cannot be recoverederfectly with a strictly two7dimensional 6)# method.

orward modelling of )" resistivity data based upon a 6) earth discreti/ed as shown below# is carried out usingquations that are explained in section 9 8rinciples# of -)" resistivity- in the -&eophysical *urveys- section of "ha-Foundations-.

rior 9nowledgeincludes field geology, -educated guesses- about physical property values, expected structures, etchese will be used in the interpretation. !n addition, the methodology makes use of implied prior knowledge. Forxample, the program looks for smooth models that are close to a reference model that can be defined by the user. Theference model is often simply a uniform Earth with fixed electrical conductivity.

D $esistivit# essentialsExtensive details about )" resistivity survey are provided in a complete chapter on )" resistivty and

!8 surveys.

!n )" resistivity surveys, current is put into the ground using two transmitting electrodes that are

planted into the ground, and resulting voltages caused when the currents flow through the buriedgeologic materials are measured elsewhere using a second pair of electrodes. The four7electrode

geometry for )" resistivity surveys varies, but all electrodes are placed in7line for 6) surveys. The

image shows many electrodes planted in a small scale engineering survey. 4ineral exploration surveys

may involve lines up to several kilometers long.

)ata are usually converted to apparent resistivities. These values will be true values of earth materials

only if the Earth is uniform within range of the survey. :therwise, they will represent a complex

average of all materials through which current has flowed. 5pparent resistivities are calculated using

the formula

The value & is a geometric factor which based upon the exact geometry of the four electrodes.There are severalcommon geometric arrangements. !n the equation,Iand delta'are current and voltage differenbetween two receiver electrodes# respectively.

)ata are plotted as a so7called pseudosection. This is not a true section because the vertical axis does not represe

depth. The four figures to the right show how four values are plotted. The animation below shows all data gathe

along a survey line plotted onto a pseudosection.


24/43

)atum ' )atum 6 )atam 9

)atum $


25/43

. Discreti1ing the earth

) models of the Earth are cross7sectional views of structure and materials directly under the survey line. 5ery important assumption is that there are no variations perpendicular to the cross7section. For inversionssing the 3B"7&!F methodology, the cross7section is discreti/ed using rectangular cells.

maller cells are used to fill the /one where current has been in0ected. These determine the spacial resolutionf the model. 4ore cells may seem to be better, but with only limited data, the finest feasible resolution wille obtained using cells with widths equal to half the electrode spacing, and depths equal to a quarter of the electrodepacing.

The mesh used to define a discreti/ed 6) model under the surveyline. The survey extended from ?m to 6??m. ones outside thisregion are -8adding-, explained in the text.

arger cells are used around the main region of interest to allow for mathematically smooth transitions towards the edf the domain. =ithout this buffer or -padding-# /one, it would be impossible to obtain sensible values for the regio

nterest. Marger cells are acceptible since this /one of the model will not be interpreted.

opography must be approximated using rectangular cells. =hen a -default- mesh is used with 3B"7&!F codes, therogram builds a discreti/ed 6) Earth with cells in the /one under the electrodes that are ?.


26/43

a. ,etting started with DI/2D the data

he first task is to inspect the data set itself in order to gain first impressions and to ensure that sensible errors arepplied. For some hints on familiari/ation with the data, click the questions button to the right.

)ata pseudosection for our synthetic data set.

The graphical user interface for )"!86).

=hen first examining a )" resistivity data several questions should be considered. The two main ob0ectives areo gain some appreciation for what is likely to be in the ground prior to carrying out the inversion. The questionutton for some suggestions of applicable things to think about.

b. &isfit criterion: specif#ing errors

=e need to 0udge the reliability of the data set, and to specify standard deviations for the data accordingly.4aking this 0udgement requires some knowlege about the acquisition of data in the field, and for the

kelihood that there were problems or external sources of noise.

rue statistical estimates of error from the field can only be obtained if many versions of the data set werebtain. *ince this is almost never done, it is common to assume there are random &aussian errors, andandard deviations for each datum can be applied using both a percentage of the datum and a minimum value, or offor the data set used in our synthetic example, the added noise is a random &aussian value based upon


27/43

8seudosection display in )"!86), with the properties dialogue for a single datum.

f you have some field experience, some questions you might like to consider regarding errors are in Luestion *et

. Errors can also be ad0usted for individual data points if you suspect any datum is particularly noisy. Forxample, it is not uncommon for all data values recorded at one electrode location to have additional noise, due forxample to a poor electrical contact, a nearby metallic fence, or other reasons. *pecifics for every datum can bexaminied in the data display program by clicking on any data point see the previous figure#.

. Specif#ing the desired optimal model

5 so7called model ob(ective functionis used to define the type ofoptimum- model the inversion algorithm is looking for. Thisunction is a way of quantifying desirable features of a physicalroperty model. The inversion chooses an optimal model byearching for a model which will minimi/e this function sub(ect to

he constraint that the chosen model can generate predicted datahat satisfy the misfit criteria. The model ob0ective function is


28/43

nd it is defined using two components coloured blue and pink above%

'. The algorithm will try to find a model that is as close as possible to a reference model defined either as a halfspace by default a halfspace with a resistivity equal to a weighted average of measured apparent resistivities#as some other, more complicated model defined by the user if there is enough prior knowlege#.

6. The model will be as smooth as possible in the I and directions.

n fact, the significance of each component is controlled using the -5lpha- coefficients s, x, and / in the equabove. Therefore the user can request a model that emphasi/es either component ' or component 6.

)efault values of these coefficients are determined by the program based upon the length scales of the survey and mehe inversion+s task is to find a model that minimi/es this relation2 the result will be an optimal model.

or the program )"!86), the default specifications for these -5lpha- parameters have been found to work well as a ttempt, but experimentation and ad0ustment of the parameters defining the desired model type is expected during theourse of inversion processing. This will be discussed in the -5lternative models- section below.

a. +irst inversion

he first inversion should be run only after learning as much as possible from the raw data, including how toet errors properly. The )"!86) user interface has defaults for all parameters except the input file name.

f our synthetic data set has errors assigned using


29/43

/rogress of DI/2D through iterations

The true model for this example is shown first.:verburden K 9'> :hm7m, a conductive blockK 96 :hm7m, and host geology K '??? :hm7m. )ataproduced from this Earth structure is shown in the


30/43

!teration9< i = Observed data


31/43

!teration>%note 6

< i = Observed data


32/43

b. Evaluating results

he &3! after completing the inversion is shown in the next figure. 8oint your mouse to either model ,

, or buttons to display an image of the corresponding user7interface window for the inversion

hat was 0ust completed.

"hifactKdflt note

Introducing Geophysical Inversion

Documents