-
L tratu ap
C
such as false stratigraphic truncations that are related to
later-
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tmates. Such a process, called spectral inversion, produces
results
GEOPHYSICS, VOL. 73, NO. 2 MARCH-APRIL 2008; P. R37R48, 22
FIGS.10.1190/1.2838274al changes in rock properties, and masking
geologic informa-tion, such as updip limits of thin layers. We
conclude that datathat are inverted spectrally on a trace-by-trace
basis showgreater bedding continuity than do the original seismic
data,suggesting that wavelet side-lobe interference produces
falsebedding discontinuities.
INTRODUCTION
that differ from conventional seismic inversion methods.In this
article, we discuss the basic theory of spectral inversion,
develop a new spectral-inversion algorithm, and show field
exam-ples of improved bed-thickness determination and enhanced
strati-graphic imaging that can be achieved with the process.
Widess modelThe Widess 1973 model for thin-bed reflectivity
teaches that the
fundamental limit of seismic resolution is /8, where is the
wave-ayer-thickness determination and ssing spectral inversion:
Theory and
harles I. Puryear1 and John P. Castagna1
ABSTRACT
Spectral inversion is a seismic method that uses a priori
in-formation and spectral decomposition to improve images ofthin
layers whose thicknesses are below the tuning thickness.We
formulate a method to invert frequency spectra for layerthickness
and apply it to synthetic and real data using com-plex spectral
analysis.Absolute layer thicknesses significant-ly below the
seismic tuning thickness can be determined ro-bustly in this manner
without amplitude calibration. We ex-tend our method to encompass a
generalized reflectivity se-ries represented by a summation of
impulse pairs. Applica-tion of our spectral inversion to seismic
data sets from theGulf of Mexico results in reliable well ties to
seismic data, ac-curate prediction of layer thickness to less than
half the tun-ing thickness, and improved imaging of subtle
stratigraphicfeatures. Comparisons between well ties for spectrally
in-verted data and ties for conventional seismic data illustratethe
superior resolution of the former. Several stratigraphicexamples
illustrate the various destructive effects of thewavelet, including
creating illusory geologic information,According to the Widess 1973
model, seismically thin layers be-ow one-eighth of a wavelength in
thickness cannot be resolved.owever, such thin layers might be
significant reservoirs or impor-
ant flow units within reservoirs. Exploration and development
geo-
la
pe
o
Manuscript received by the Editor 18April 2007; revised
manuscript received 31 O1University of Houston, Department of
Geosciences, Houston, Texas, U.S.A. E-m2008 Society of Exploration
Geophysicists.All rights reserved.
R37igraphic interpretationplication
hysicists frequently are faced with the task of inferring layer
thick-ess for layers such as these where the top and base of the
layer can-ot be mapped distinctly. Consequently, determining layer
proper-ies for such seismically thin beds is of great interest in
explorationnd development applications.
Although tuning-thickness analysis based on the theory of
Widess1973 and Kallweit and Wood 1982 has been the
thickness-map-ing method of choice for several decades, Partyka et
al. 1999, Par-yka 2005, and Marfurt and Kirlin 2001 demonstrate the
effec-iveness of spectral decomposition using the discrete Fourier
trans-orm DFT as a thickness-estimation tool. However, such
methodsave difficulty with thin layers if seismic bandwidth is
insufficient todentify the periodicity of spectral peaks and
notches unambiguous-y. This difficulty motivates the development of
methods that do notequire precise identification of peaks and
troughs within the seismicandwidth.
Partyka 2005, Portniaguine and Castagna 2004, 2005, Puryear2006,
Chopra et al. 2006a, 2006b, and Puryear and Castagna2006 show that
inversion of spectral decompositions for layerroperties can be
improved when reflection coefficients are deter-ined
simultaneously. The result is a sparse-reflectivity inversion
hat can be parameterized to provide robust layer-thickness
esti-ength. Essentially, constructive wavelet interference and
measuredmplitude in the time domain peak at /4. The waveform shape
andeak frequency continue to change somewhat as amplitude decreas-s
to /8, at which point the waveform approximates the derivativef the
seismic wavelet. As the layer thins below /8, the waveform
ctober 2007; published online 27 February 2008.ail:
[email protected]; [email protected].
-
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R38 Puryear and Castagnaoes not change significantly, but
amplitude steadily decreases, asemonstrated in Figure 1. In this
figure, amplitudes are obtainedrom the convolution of a 30-Hz
Ricker wavelet with a wedgeodel.From this point of view, there are
no means to differentiate be-
ween amplitude changes associated with
reflection-coefficienthanges and thickness changes below /8, making
this thickness aard resolution limit for broadband analysis in the
time domain.orse yet, in the presence of noise and wavelet
broadening, the tran-
ition between /4 and /8 is obscured, sometimes making /4
aractical limit of resolution. The key assumptions for the
Widessodel are that the half-spaces above and below the layer of
interest
ave the same acoustic impedance and that the acoustic impedancef
the thin layer is constant.
eneralized reflectivity modelAlthough the theory of Widess 1973
is valid when the assump-
ions are satisfied, nature rarely accommodates such strict
theoreti-al provisions. The theory of spectral inversion is based
on the real-zation that the Widess model for thin-bed reflectivity
presupposes aeflectivity configuration that is actually a
singularity in the continu-m of possible reflection-coefficient
ratios. Any reflection-coeffi-ient pair can be decomposed into even
and odd components, withhe even components having equal magnitude
and sign and the oddomponents having equal magnitude and opposite
sign, as describedy Castagna 2004 and Chopra et al. 2006a, b.
igure 1. Plot of amplitude versus thickness. Note the increase
inmplitude over background as the tuning thickness /4 is
ap-roached. Below tuning, the amplitude rolls off nearly linearly,
andhe waveform approximates the derivative of the wavelet at
/8.
igure 2. Any arbitrary pair of reflection coefficients r1 and r2
can beepresented as the sum of even and odd components. The even
pairas the same magnitude and sign, and the odd pair has the same
mag-
itude and opposite sign. The identity is illustrated in Figure
2. The Widess model assumeshat reflection-coefficient pairs are
perfectly odd, which can be aood approximation for certain target
classes such as a sand layer en-ased in a shale matrix. However,
the assumption of an odd-reflec-ivity pair implies the worst
possible resolution for thin beds. Even amall even component in the
reflection-coefficient pair can increasehe resolvability of a layer
significantly. The improvement in resolu-ion results from the fact
that the even component constructively in-erferes as thickness
approaches zero. In contrast, the odd compo-ent destructively
interferes. Thus, the even component is more ro-ust against noise
as thickness approaches zero see Tirado, 2004.
We calculated peak frequency and peak amplitude from
equationsiven by Chung and Lawton 1995. Figure 3a shows the effect
ofhinning on the peak frequency of a reflection-coefficient pair
withven and odd components. For the model, the total peak
frequencyncreases with decreasing thickness and then returns to the
peak fre-uency of the wavelet rather than that of the derivative of
the wavelets predicted by the Widess model. Interestingly, the
total peak fre-uency shows significant and continuous change down
to zero thick-ess. Likewise, the total peak amplitude Figure 3b
does not ap-roach zero with thickness as predicted by the Widess
model.
The example indicates that the reflection-amplitude trend canhow
significant variation from the Widess curve Figure 1 as theayer
thickness approaches zero when the even component is nonze-o. Thus,
significant information below the Widess resolution limit isot
captured by traditional amplitude-mapping techniques, whichssume
equal and opposite reflection coefficients. Such examples ofnequal
reflection coefficients at the top and base of a layer, whichre the
rule rather than the exception for most real-world seismic
re-ections events, reinforce the need for a more generalized
approach
o thin-bed amplitude analysis.Based on the fact that spacing
between spectral peaks and notches
s a deterministic function of layer thickness, our objective was
toevelop a new algorithm to invert reflectivity using the constant
pe-iodicity in the frequency domain. Our development started with
thexpression for an impulse pair in the time domain, from which
we
a)
b)
igure 3. a Peak frequency and b peak amplitude as a function
ofhickness for the even component, the odd component, and the
total.n a, there is peak-frequency information below the tuning
thick-ess. In b, total peak amplitude approaches the
even-componentmplitude below tuning. Layer model parameters are
r10.2, r20.1, and f0 30 Hz.
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Layer-thickness determination R39ormulated a numerical algorithm
using complex spectral analysisnd then tested the algorithm on
synthetic wedge models. We ex-ended our development to multiple
layers, and we tested our meth-d on 3D seismic data from the Gulf
of Mexico shelf, comparingeismic data, spectrally inverted seismic
data, and well-log data.
METHODS
pectral-inversion methodWe applied windowed Fourier transforms
to several reflectivityodels to generate data for spectral
inversion and used complex
pectral analysis to formulate the inversion algorithm. The
algo-ithm described herein defines the inversion for reflectivity
using theonstant periodicity of the amplitude spectrum for a layer
of a givenhickness, taking advantage of the fact that spacing
between spectraleaks and notches is precisely the inverse of the
layer thickness inhe time domain Partyka et al., 1999; Marfurt and
Kirlin, 2001. Es-entially, layer thickness can be determined
robustly from a narrowand of frequencies with a high
signal-to-noise ratio S/N. To provehis concept, note that the
entire reflectivity spectrum for a singleayer could be
reconstructed from amplitudes at three frequencies inhe absence of
noise.
Beginning with the expression for an impulse pair in the time
do-ain as expressed by Marfurt and Kirlin 2001 Figure 4,
gt r1 t t1 r2 t t1 T , 1
here r1 is the top reflection coefficient, r2 is the base
reflection coef-cient, t is a time sample, t1 is a time sample at
the top reflector, and T
s layer thickness. Locating the analysis point at the center of
the lay-r yields
gt r1 t T2 r2 t T2 . 2aking the Fourier transform of the shifted
expression 2 gives
gt, f r1 expi2 ft T2 r2 expi2 ft T2 , 3
here f is frequency and gf is the complex spectrum. Simplifyingy
using trigonometric identities and taking the real part yields
Regf 2recos fT , 4here re is the even component of the
reflection-coefficient pair.imilarly, the imaginary part of the
complex spectrum is
Imgf 2rosin fT , 5here ro is the odd part of the
reflection-coefficient pair.Figure 5 shows plots for both the even
and odd reflectivity spec-
ra corresponding to equations 4 and 5 for a layer with thickness
10 ms and reflection coefficients r1 0.2 and r2 0.1. Al-
hough both even and odd spectra show the same notch period,
thewo are shifted by half of the frequency spacing. For the
individualeal and imaginary components, the constant period in the
spectrums related to the symmetrical location of the analysis point
at the cen-
er of the layer. This placement effectively divides the
reflection-co- Ifficient pair into perfectly odd and even
components, thereby elimi-ating the phase variation for each. The
effect of violating this condi-ion is discussed inAppendix A.
To maintain constant periodicity in the spectrum while
shiftinghe analysis point away from the layer center, we compute
the modu-us of the real and imaginary components of the spectrum,
which isnsensitive to phase. Beginning with general expressions for
the realnd imaginary time-shifted spectra,
Ime2i ftgf 2ro sin fTcos2 ft 2re cos fTsin2 ft 6
nd
Ree2i ftgf 2re cos fTcos2 ft 2ro sin fTsin2 ft , 7
t can be shown Appendix B that
Ot,k GfdGfdf 2Tk sin2 fT , 8
here Gf is amplitude magnitude as a function of frequency,
kre
2 ro2, and Ot,k is the cost function at each frequency. The
so-
xm (km)
t(s)
t2
t1
T
Reflectivity = r1
Reflectivity = r2
igure 4. Two-layer reflectivity model from Marfurt and
Kirlin,001.
igure 5. Amplitude versus frequency plots for a even and b
oddomponents of the reflection-coefficient pair r1 0.2 and r2
0.1.
n this example, the even component is dominant.
-
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bitple
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R40 Puryear and Castagnaution to equation 8 occurs when the sum
of cost functions Ot,k,valuated at each frequency, is minimized
over the range of frequen-ies within the analysis band. One data
term exists at every samplerequency, so the performance of the
method is determined by the/N over a given analysis band i.e., more
frequencies with a high/N yield a more stable and accurate
inversion.We found the global minimum of equation 8 for a given
analysis
and by searching physically reasonable model parameters k and Tn
two-parameter model space and minimizing the objective func-ion
Figure 6. Although it is costly and impractical for more
com-licated cases, the global search method guarantees the
avoidance ofocal minima for the single-layer case. The remaining
model param-ters then are determined by
roGf24 k cos2 fT , 9re k ro2, 10
igure 6. The difference or error function between the data and
aange of model parameter pairs is calculated. The blue bulls-eye
ishe correct model solution T 10 ms, k 0.02. The black arrowhows a
local minimum.
)
)
igure 7. a Forward-model and b inverse-model schemes usedor the
synthetic. The plots show magnitude of amplitude versus fre-uency.
In a, multiplication of the wavelet with the reflectivitypectrum
and addition of noise in the frequency domain yields theeismic
signal. In b, division of the seismic signal by the wavelet inhe
frequency domain yields the noisy reflectivity band. A smooth-5ng
filter produces the inversion band for input to the model.nd
t11
2i f ln gfr1 r2e2i fT , 11here t1 is the time sample at the top
reflector r1 and gf is the com-lex spectrum for the
reflection-coefficient pair. Equation 11 can beerived by taking the
Fourier transform of equation 1 and solving for
1. The reflection coefficients r1 and r2 can be recovered by
recom-osing the odd and even components of the pair calculated
usingquations 9 and 10, the reverse of the operation illustrated in
Figure. Thus, it is straightforward to compute the remaining
componentsf the layer-reflectivity model from the initial
parameters k and T.ote that although the derivation Appendix B of
the algorithm as-
umes that the even reflectivity component is greater than the
odd re-ectivity component, the solution is the same for the
antithetical as-umption.
odeling resultsWe test the method by convolving a 30-Hz Ricker
wavelet with
eflection-coefficient pairs that have various ratios. We
produceedge models with 4-ms sampling for a predominantly odd
reflec-
ion-coefficient pair, r10.2 and r2 0.1, and a predominantlyven
reflection-coefficient pair, r1 0.2 and r2 0.1. The tuninghickness
of a thin-bed model with a Ricker wavelet is given byhung and
Lawton 1995:
tR6
2 f0, 12
here f0 is the dominant wavelet frequency. For a 30-Hz
Rickeravelet, tR 13 ms. The convolution of the wavelet with the
reflec-
ion-coefficient pair in the time domain is equivalent to
multiplica-ion with the reflectivity spectrum in the frequency
domain. The in-ersion performs perfectly in the absence of noise
for layers of anyhickness.
To achieve a more realistic model, we added noise in the time
do-ain, and we measured and controlled the noise level by
computing
he ratio of the area under the spectrum of the signal to that of
theoise in the frequency domain. We tested the model with 1% and
5%oise levels. The forward-modeling procedure is illustrated in
Fig-re 7a.
The addition of noise causes instability in the inversion for
veryhin layers, partly because the reflectivity spectrum approaches
a flatpectrum as T approaches zero. We mitigated this problem by
apply-ng the arbitrary constraint0.03k0.03 to ensure that the
re-ection-coefficient strength could not exceed what typically is
ob-erved in seismograms.After Fourier-transforming the
time-domainignal, we removed the wavelet overprint by dividing the
magnitudef the amplitude of the seismic signal by that of the
wavelet at eachnalysis frequency.
We tested the inversion at different noise levels while varying
thenalysis band and smoothing filter. These experiments showed
thathe optimal analysis band and the optimal smoothing filter are
deter-
ined by the noise level. We achieved optimal results for the
1%oise case using a 25-Hz bandwidth sampled at 2-Hz frequency
in-rements and centered on the peak signal frequency. A signal is
cor-upted by noise, so it was necessary to narrow the bandwidth.
For the
% noise case, we achieved optimal results with a 20-Hz
bandwidth.
-
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Layer-thickness determination R41More smoothing is required to
stabilize the derivative operator asoise is added. The process
resulting in the smoothed inversion bands illustrated in Figure 7b.
We took derivatives of the magnitude ofhe amplitude with respect to
frequency at each observation frequen-y sample using a first-order
central difference approximation. Fi-ally, we multiplied the
magnitude of the amplitude at each observa-ion frequency by the
derivative of the magnitude of the amplitude athat frequency and
minimized the error between the model definedy equation 8 and the
data for the range of frequencies within thenalysis band Figure
6.
We applied the inversion defined by equation 8 using a
256-ms-indow fast Fourier transform FFT for spectral decomposition
toedge models that have predominantly odd and even
reflection-co-
fficient pairs Figure 8. Figure 9a and b shows the result of the
in-ersion black compared with the true reflectivity model green
anded for 1% and 5% levels of noise, respectively, for the
predomi-antly odd reflection-coefficient pair; Figure 9c and d
shows theseesults for the predominantly even pair. The tuning
thickness is rep-esented by the vertical black line.
We compared the 5% noise results with the corresponding
ampli-ude-mapping results brown using tuning analysis Widess,
1973;allweit and Wood, 1982. The algorithm performed nearly
perfect-
y for both configurations at 1% noise levels as expected,
illustratinghe principle that noiseless data could yield extremely
high resolu-ion.
Although the noise level is unrealistically low for most
seismicata, the results highlight the importance of meticulous
noise sup-ression during acquisition and processing. For thin
layers, the re-ults are useful far below tuning. For the 5% noise
cases, the absoluterror does not increase significantly for very
thin layers because ofhe reflectivity constraint, although the
percent error increases asayer thickness decreases.
Reflection-coefficient estimates wereomparably accurate.As
expected, accuracy deteriorates as noise in-reases beyond 5%. Thus,
for a given wavelet peak frequency andnalysis band, noise level
rather than tuning thickness determineshe limit of resolution.
The tuning-analysis predictions are shifted toward thickerayers
for the even and odd components because of the assumptionf
equal-magnitude reflection coefficients at the top and base.
Thessumption in the relationship between amplitude and thickness
de-ends on the particular mapping scheme. We plotted the results
ofhe amplitude-mapping techniques that generated the least error
inhickness prediction. For the predominantly odd pair, the
optimalmplitude-mapping method was a simple linear regression
fromeak tuning to zero thickness. However, this technique
generatedarge errors and negative thickness predictions for the
predominant-y even pair. So for the predominantly even pair, we
assumed both ofhe reflection coefficients were equal to the peak
background ampli-ude for the thick layer and mapped thickness from
the tuning ampli-ude to zero thickness.
We tested the inversion on a reflector model that violates the
basicssumptions of the method. The model includes two layers
definedy three reflectors, all with reflection coefficients equal
to 0.1. Theop layer has twice the thickness of the lower layer. We
decomposedhe spectrum using an 80-ms DFT with a Gaussian taper
centered onhe thicker layer.
Figure 10 shows the model green and red and the resulting
inver-ion black. The predicted layer thickness is measured from the
topeflector to the black line, yielding a thickness value greater
than that
f the thicker layer but less than that of the two layers
combined. The ebserved predicted thickness results from the
interference of the twoayers in the frequency domain, creating a
period corresponding to aingle layer that is slightly thicker than
the thickest layer. The reflec-ion-coefficient predictions for the
lower reflector are closer to theum of the two base reflection
coefficients r2 r3 0.2 for thinayers and to the single base
reflection-coefficient value r2 r3
0.1 for thicker layers. In practice, additional spikes widely
spacedrom the reflectors of the layer should be considered noise
for theingle-layer model.
Although we value the single-layer model for its ease of
invert-bility, it is necessary to extend the inversion scheme so
that it simul-aneously can invert seismograms containing multiple
interferingayers for most real cases.
xtension of the method to multiple layersRecognizing that a
seismogram can be represented as a superposi-
ion of impulse pairs, the inversion for the properties of a
single layers extended easily to encompass a general
reflectivity-series inver-ion by considering the spectrum versus
time acquired using a mov-ng window as a superposition of
interference patterns originating atifferent times. The inversion
process for reflection coefficients andayer thickness is performed
simultaneously for all impulse pairs af-ecting the local seismic
response.
)
)
igure 8. Original reflectivity wedge models for a a
predominantlydd reflection-coefficient pair and b a predominantly
even reflec-ion-coefficient pair. The vertical black line defines
the tuning thick-ess for the 30-Hz Ricker wavelet convolved with
the reflection-co-
fficient pair blue.
-
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p
w
til
a
c
F eflectiop eflectiop iginala techniw al blact he reflec
R42 Puryear and CastagnaLet us represent the reflectivity series
rt as a summation of evennd odd impulse pairs:
rt
retII t Tt dt
rotII t Tt dt ,13
here ret and rot are the magnitudes of the impulse pairs as
aunction of time, Tt is the time series of layer time thicknesses,
II isn odd impulse pair, and II is an even impulse pair. Assuming a
con-olutional seismogram and known wavelet wt, f, a spectral
decom-osition of a seismic trace st, f is then
st, f wt, f
tw
tw
retcos fTt
irotsin fTtdt , 14here tw is window half-length. The multilayer
case involves more
han two reflectors, so it is necessary to use an objective
function fornversion that properly accounts for interference
between multipleayers.
) b)
) d)
igure 9. Result of the thickness inversion of a predominantly
odd rair with a 1% noise and b 5% noise and a predominantly even
rair, also with c 1% noise and d 5% noise. The plot shows the ornd
red along with the inversion black and the amplitude-mappinghich
uses the incorrect assumption of equal reflectivity. The vertic
he tuning thickness for the 30-Hz Ricker wavelet convolved with
tient pair blue.If the wavelet spectrum is known, we can solve for
rt and Tt byptimizing the objective function Ot,re,ro,T by
Ot,re,ro,T
fL
fH
eRest, f/wt, f
tw
tw
retcos fTtdt oImst, f/wt, f
tw
tw
rotsin fTtdtdf , 15here f l is low-frequency cutoff, fh is
high-frequency cutoff, and e
nd o are weighting functions, the ratio of which can be adjusted
tond an acceptable trade-off between noise and resolution. For
higho/e, the reflectivity approaches the Widess model, and the
resolu-
tion limit becomes /8. We summarize the multi-layer inversion
process in Figure 11.
Multilayer synthetic exampleTo validate the multilayer inversion
technique
in a controlled situation, we generated a modelcontaining
several arbitrary layers Figure 12a,from which we generated a
synthetic trace Fig-ure 12b. The reflectivity spikes were
convolvedwith a 30-Hz Ricker wavelet. The identical tracewith
variable random noise is repeated in Figure12 for clarity. We
tested the inversion using win-dows of different lengths and
centered at differentlocations along the synthetic trace. Assuming
noa priori insight, we set the weighting function re-lationship o e
and minimized the objectivefunction given by equation 15 using a
standardleast-squares conjugate-gradient inversion.
We inverted the data from a 200-ms Gaussian-tapered Fourier
transform, thereby recovering theoriginal model Figure 12c.
Shortening the totalwindow length to 100 ms and maintaining
theGaussian taper, we computed the Fourier trans-form for time
samples between 50 and 150 ms,respectively. The window was shifted
one timesample at a time, with the results of each previouswindow
acting as a constraint for the next inver-sion. If no new
reflectors appeared in the windowor exited the window as it shifted
downward, theresult was the same as in the previous window. Ifthere
were no reflectors in the window, the algo-rithm inverted the
noise, a result that was not par-ticularly problematic because the
noise was notamplified.
n-coefficientn-coefficientmodel greenque brown,k line
definesction-coeffi-o
w
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Layer-thickness determination R43The results for the windows
centered at 50 and 150 ms are shownn Figure 12d and e. As expected,
the shorter windows divide theverall longer series of reflectors
into two isolated sets. By superpo-ition, we can sum the two sets
of reflectors to obtain the longer se-ies. Although the window
length can be varied according to the de-ired result, there are
practical limits to the length of an optimal in-ersion window,
which often are defined by trial and error. If theindow is too
short, frequency resolution suffers; if the window is
oo long, time resolution is lost see Castagna et al., 2003.
REAL DATA RESULTS
omparison to well-log dataWe studied the results of the
application of the multilayer spectral-
nversion process Figure 11 to two seismic data sets from the
Gulff Mexico shelf. Well information for lithologic interpretation,
in-luding P-wave, resistivity, and deep induction, was available
Fig-re 13. We created a synthetic tie between the input seismic
data andhe well-log data, stretching the well-log data to time
without refer-nce to the inverted data for an unbiased,
quantitative layer-thick-ess comparison between the well-log data
and the inverted data.he wavelet extracted from the well for the
synthetic is shown inigure 14. We also visually compared the
well-log data, the invertedata, and the original data to assess the
difference in vertical resolu-ion between the inverted data and the
original data Figures 15 and6.
In the original seismic tie to the well Figure 13, we achieved
aelatively good fit r 0.64 between the reflectivity convolvedith
the extracted wavelet and the seismic trace. However, because
he seismic data are much lower in frequency than the log data,
the fits useful only as an approximation for aligning gross
lithologic pack-ges.Agreat deal of useful information is lost to
the seismic wavelet.
The thickness inversion provides a significantly better
representa-ion of the layering observed in the log data than the
original seismicata does. Figure 15 shows the spectral inversion
for reflectivity, dis-layed with a 90 phase shift to emphasize
relative impedancehanges and a slight time shift from the original
synthetic to providetie to the well with higher fidelity. The need
for this time shift be-
igure 10. Result of the inversion for a two-layer model red,
violat-ng the assumptions of the method. For thicker layers, the
predictedayer thickness from the inversion black, measured from the
top re-ector, falls between the thickness of the thicker layer and
that of the
otal package. As the reflectors converge, the inversion becomes
un-
table. te)
igure 12. a The original model is convolved with a 30-Hz
Rickeravelet to create b a synthetic seismogram. c The
spectral-inver-
ion result using a 200-ms window centered at 100 ms recovers
theriginal reflectivity series. d The spectral-inversion result
using a00-ms window centered at 50 ms and e the spectral-inversion
re-ult using a 100-ms window centered at 150 ms recover the
portionsf the reflectivity series contained within the window. By
superposi-igure 11. Flowchart for the method of the inversion,
where dout ishe output trace, tc is the time sample at the center
of the window, andt is the sampling interval.
) b)ion, c d e.
-
cu
ts
tgv
dthtbyw
c
tbw
t
v
n
C
s
ts
tiw
fdt
FctT
Ff
Fdlcsolved at the tuning thickness.
FcadAw
R44 Puryear and Castagnaomes apparent only after reflectivity
inversion. The lithologic col-mn shows sand and shale packages
interpreted primarily from thehree logs see Figure 13. We initially
distinguished sands andhales using a gamma-ray log, but we used
resistivity, deep induc-ion, and impedance for the bulk of the
interpretation becauseamma-ray data were absent in the well with an
available P-waveelocity log.
Generally, sands correspond to higher resistivity and separation
ofeep induction and shallow-resistivity curves as a result of
mud-fil-rate invasion. In the interval shown in Figure 15, the
sands haveigher impedance than the shales. Comparing the lithologic
columno the inverted data, the thickness inversion clearly shows
layeringelow the tuning thickness the peak frequency of the data is
16 Hz,ielding a one-quarter-wavelength resolution of about 31 ms.
Aell-resolved sand-shale sequence that has a total thickness
very
lose to the tuning thickness is indicated also in Figure 15.
Althoughhe thickness inversion effectively delineates the layering
sequenceelow tuning, it fails to capture gradational impedance
changesithin thin layers, as in the case of shale grading to sand,
shown in
he lithologic column.The following example demonstrates the vast
improvement in
ertical resolution for discrete layers that is achieved by using
thick-ess inversion rather than the original data.
omparison to conventional seismic dataViewing the comparison of
the thickness inversion to the original
eismic data in Figure 16, both with a90 phase rotation, it is
clearhat boundaries between layers are indistinct on the original
seismicection. Layers below the 31-ms tuning thickness are not
resolved inhe original seismic data. Geologic detail is obscured by
the wavelet-nterference patterns, which become more apparent when
comparedith the inversion. A skilled interpreter can decipher
meaningful in-
ormation embedded in the wavelet-interference patterns, but it
isesirable to remove these artifacts altogether to allow direct
accesso the underlying geology.
igure 16. Original seismic data left compared with the
lithologicolumn center and the thickness inverted data right. Both
imagesre phase-rotated by90. Note the failure of the original
seismicata to delineate thin layering as compared with the inverted
data.lso note that the inverted data do not resolve gradational
changesigure 15. Impedance log left compared with spectrally
invertedata right. Sands are of higher impedance in this interval.
Theithologic section interpreted from the impedance log shows a
closeorrespondence to the inverted data. Note the two-layer
sequence re-igure 14. The phase and amplitude spectra of the
wavelet extractedrom the well. The peak frequency is 16 Hz.igure
13. Well-log data, including deep induction, resistivity,
andomputed impedance, long with the synthetic tie blue, the trace
athe well red, and the seismic traces surrounding the well black.he
correlation coefficient r is 0.64.ithin an individual layer.
-
Ttio
ts
c
tb
u
a
t
S
o
p
ls
fiv
ita
a
lTr
ge
g
bw
ta
Fsantl
a
b
Fawb
Fdnvl
Layer-thickness determination R45hickness comparisonWe
quantitatively compared thicknesses from the well-log data to
hose from the spectral reflectivity inversion, interpreting the
layer-ng from the well-log data, which were stretched from depth to
timen the original synthetic, and comparing the results to the
invertedhicknesses as defined by zero crossings on the90
phase-rotatedpectral reflectivity inversion.
We performed the interpretation between 1700 and 2900 ms.
Theomparison is plotted in Figure 17, with excellent correlation
be-ween well-log data thicknesses and the predicted thicknesses
pinkelow the tuning line red. The mean error for the two data sets
is0.5 ms, indicating that the method is generally accurate and
nbiased. The square of the correlation coefficient is R2 0.94
withstandard deviation of 3.10 ms, corresponding to precision
in
hickness estimation on the order of 15 ft 4.5 m for this
example.
tratigraphic interpretationFigures 1820 show stratigraphic
examples in which both the
riginal seismic data and the spectrally inverted data have
a90hase to highlight layer boundaries.
An example of the wavelet overprint effect is observed on
thearge-scale seismic line comparison of Figure 18. On the
originaleismic line, an apparent discontinuity is inferred at about
1315 msFigure 18a. The discontinuity might be interpreted as a
localizedault with minimal offset or as a stratigraphic
discontinuity in layer-ng. However, comparison with the inverted
data Figure 18b re-eals another picture. Although the inverted data
show more detailn general, the apparent discontinuity in layering
does not exist onhe inverted section despite the fact that the
inverted data are gener-ted using a trace-by-trace operation that
makes no assumptionbout lateral continuity.
On the inverted section, the apparent disruption in layering
actual-y represents a lateral change of rock properties within a
given layer.he discrepancy points to the apparent discontinuity in
reflection ar-
ival times seen on the original data; it is not a geologic
feature but aeophysical effect. Specifically, it is a shift in the
wavelet-interfer-nce pattern caused by an impedance change that
resembles a smalleologic layering discontinuity in the seismic
image.
igure 17. Plot of predicted thicknesses from the inversion
pinkquares versus well-log interpreted thickness blue line is a 1:1
di-gonal, showing a strong correlation between the two. The
thick-esses were interpreted between 1700 and 2900 ms. The
tuninghickness is marked by the red line, and accuracy is
maintained be-
ow one-eighth of a wavelength. aThe smaller-scale image in
Figure 19 shows significant lateralreaks in layering that might be
interpreted as discrete sand bodiesith possible erosion of
previously continuous layers. These fea-
ures can be caused by different types of downslope transport
mech-nisms such as channel incision. The horizons on the original
dataFigure 19a are difficult to continue in places black arrows.
How-
)
)
igure 18. a The original seismic data show a small
discontinuity.b The thickness-inverted data reveal a strikingly
continuous layer,strong indication that the geologic discontinuity
seen in a is aavelet effect rather than a real subsurface feature.
The phase foroth images in90, and red indicates higher
impedance.
a)
b)
igure 19. a The original seismic line has significant
stratigraphiciscontinuities black arrows that might be interpreted
as the termi-i of discrete depositional lobes. b The spectrally
inverted data re-eal a laterally continuous layering characteristic
of undisturbedayer-cake geology. The phase on both images is90.
Timing lines
re 20 ms, and red indicates higher impedance.
-
ela
e
l
gitt
s
gc
bc
v
i
Mm
pv
c
c
s
ps
s
tr
Wt
sFritap
a
b
Fwfml
a
b
FctTn
R46 Puryear and Castagnaver, the spectrally inverted data Figure
19b, which assume no re-ationship between neighboring traces, show
striking continuitylong the same horizons. Once again, the complex
wavelet-interfer-nce pattern creates an illusory geologic scenario
that accompaniesimited resolution.
The previous examples demonstrate artifacts that resemble
geolo-y, and Figure 20 shows an example of the same wavelet effect
eras-ng geologic information. In the original seismic data Figure
20a,he apparent pinch-out of a low-impedance layer is observed,
withhe upper and lower events merging below the resolution of the
layer
a)
b)
igure 20. a The original seismic data shows a pinch-out white
ar-ow where the thin layer becomes unresolved. b The inverted
datamages the pinch-out much farther updip.An apparent erosional
fea-ure black circle is resolved on the inversion. The phase on
both im-ges is90. Timing lines are 20 ms, and red indicates higher
im-edance.
)
)
igure 21. Comparison of a the original zero-phase seismic
dataith b the spectrally inverted data, which are
phase-rotated90
rom a. Channels white arrows show the base of channels haveore
relief and more curvature on the original seismic data. Timingines
are 10 ms, and red indicates higher impedance. lwhite arrow.
However, the spectrally inverted data Figure 20bhow the same
low-impedance layer imaged much farther updip, to-ether with the
resolved bounding layers. In addition, an apparent lo-alized
broadening or bulge in the wavelet in the original data justelow
the first two timing lines is resolved as a possible erosional
in-ision on the inverted data. Such improved detection of
stratigraphicariation has significant implications for better
reservoir character-zation and delineation.
We tested the method on a line of data from a shallow Gulf
ofexico data set with known large incision features,
previouslyapped using the coherence attribute. Figures 21 and 22
show zero-
hase original seismic images and90 phase-rotated
spectral-in-ersion images. Typically, seismic images of channels
show signifi-ant relief from the levy to the thalweg, which appeals
to the intuitiveoncept of a curved-channel geometry.
Figure 21a is an example of a pair of adjacent channels, showing
atrongly curved geometry on the original seismic data.
The90hase-rotated spectrally inverted section of the data Figure
21bhows an alternative image of the channels in which the
curvatureeen in the channel profile is less prominent, hinting at
the possibilityhat some component of the curvature can be
attributed to the rapidock-property changes known to occur across
the strike of a channel.
e believe further investigation of this phenomenon using well
con-rol is warranted.
Figure 22 shows another large channel imaged on the
originaleismic data and on the90 phase-rotated spectral inversion,
re-
)
)
igure 22. A large channel white arrows show the edges of
thehannel imaged on a the original zero-phase seismic data and bhe
spectrally inverted data, which are phase-rotated90 from a.he
thin-bed layering of the channel edges and overall vertical chan-el
extent are imaged more precisely on the inverted data. Timing
ines are 10 ms, and red indicates higher impedance.
-
stis
dtIpz
a
f
c
pv
n
s
m
tt
r
o
ifltfm
ffb
c
r
tpm
s
iit
s
w
tpe
a
c
v
X
gMs
a
e
ttt
A
T
R
w
o
o
tpo
s
p
Layer-thickness determination R47pectively. The tuning effect in
the original data confounds channel-hickness interpretation as the
channel thins toward the levies. Theres also an ambiguity in the
placement of the top and base boundingurfaces of the channel
related to the wavelet phase. The invertedata show a clearer
picture of the channel geometry, with constanthinning of the
channel-fill wedges toward the edges of the channel.n addition, the
top and base bounding surfaces of the channel can beicked more
precisely on the inverted data at the sharply definedero crossings
with less guesswork in the placement of horizons.
Thus, in the workflow of seismic interpretation, spectral
inversiondds visual information that can contribute to delineating
geologiceatures of interest such as channels.
CONCLUSIONS
Beginning with a generalized theory of reflectivity, spectral
de-omposition is used as a tool to unravel the complex
interferenceatterns created by thin-bed reflectivity. These
patterns can be in-erted to obtain the original reflectivity. We
developed and studiedew analytical methods for spectral inversion
based on complexpectral analysis. Spectral inversion yields
accurate thickness deter-inations below tuning, using the inverse
relationship between
hickness and the constant periodicity of spectral interference
pat-erns.
Representing the seismogram as a superposition of simple
layeresponses constitutes a means of imposing on the inversion the
a pri-ri assumption that sedimentary rocks occur as layers with
discretenterfaces at the top and base and can be represented as
such in a re-ectivity series. When this assumption is valid, the
consequence is
hat on the inverted reflectivity trace, there is geologically
meaning-ul information at frequencies outside the band of the
original seis-ic data. When this assumption is false, the recovered
frequency in-
ormation outside the band of the original seismic data will also
bealse. For example, smooth impedance transitions will be inverted
aslocky steps in impedance.
Spectral shape information obtained from spectral
decompositionan be used to drive an inversion with significantly
greater verticalesolution than that of the original seismic data,
thereby improvinghickness estimation, correlation to well logs, and
stratigraphic inter-retation. These results are achieved without
using well-log infor-ation in the inversion as a starting model or
as a constraint. The re-
ulting inversion therefore is unbiased by preconceived ideas.As
ev-denced by the results of applying the method to real data,
spectralnversion has great potential as a practical tool for
seismic explora-ion.
The spectral-inversion methods described in this work
demon-trate improvement in vertical resolution; however, we did not
useell-log information after the wavelet-removal step. It is
desirable
o investigate the effectiveness of using well-log data to
further im-rove vertical resolution of interbedded layers or
gradational chang-s within layers that are not revealed by seismic
spectral inversionlone. In addition, thickness constraints from
spectral inversionould be used as input for more accurate
model-based impedance in-ersion.
ACKNOWLEDGMENTS
The authors would like to thank Gene Sparkman, Carlos
Moreno,
ianhui Zhu, and Oleg Portniaguine of Fusion Petroleum Technolo-
aies for their help and support. Thanks also to Kurt Marfurt and
Scottorton for assistance, suggestions, and contributions.
Financial
upport was provided by ExxonMobil and Shell.
APPENDIX A
THE SHIFT EFFECT
When examining the real and imaginary components
separately,phase shift occurs if the analysis window is not
centered on the lay-r. To study this effect in more detail, we
revisited the original equa-ions. The shift theorem says that a
time sample shift t away fromhe layer center tc in the time domain
is equivalent to a phase ramp inhe frequency domain:
gtc t e2i ftgf . A-1pplying this equivalency,
e2i ftgf cos2 ft i sin2 ft2re cos fT i2ro sin fT . A-2
aking the real component of equation A-2,
Ree2i ftgf 2re cos fTcos2 ft 2ro sin fTsin2 ft .
A-3
earranging yields
Ree2i ftgf 2rocos fTcos2 ft sin fTsin2 ft 2re rocos2 ftcos
fT,
2ro cos2 ft T/2 2re rocos2 ftcos fT , A-4
hich has the form of a modulation and represents the spectral
plotsf time-shifted models. A similar expression can be derived for
thedd component. The phase shift corresponds to a sinusoidal
modula-ion of the signal, which can be viewed as an interference
pattern su-erimposed on another interference pattern. Furthermore,
the periodf the interference pattern is determined by the magnitude
of thehift.
APPENDIX B
INVERSION-MODEL DERIVATION
Applying the shift theorem equation A-1 and taking general
ex-ressions for the real and imaginary spectra,
Ime2i ftgf 2ro sin fTcos2 ft 2re cos fTsin2 ft
B-1nd
-
Ree2i ftgf 2re cos fTcos2 ft 2ro sin fTsin2 ft .
B-2
Er
R
G
T
M
w
V
t
gr
r
t
t
Tt
gRIr
r
G
Variable Definition
dGf / df Derivative of magnitude of amplitude with respectto
frequency
k Even component of reflectivity squared minus odd
t
r
IIs
w
t
OO
C
C
C
C
K
M
P
P
P
P
P
T
W
R48 Puryear and Castagnaxpressing the amplitude spectrum and
settingt 0 as a constanteference,
Gf Ree2i ftgf2 Ime2i ftgf2
4re2 cos2 fT 4r02 sin2 fT . B-3earranging terms,
f 2re2 r02cos2 fT r02 cos2 fT r02 sin2 fT .
B-4aking the derivative,
dGfdf
2Tre2 ro
2cos fTsin fTre2 ro2cos2 fT r02
. B-5
ultiplying and simplifying using trigonometric identities
yields
GfdGfdf 4Tre
2 r02cos fTsin fT
2Tk sin2 fT , B-6here k re2 ro2.
NOMENCLATURE
ariable Definition
Seismic record timet Time-domain impulse pair1 Top reflector in
a two-reflector model2 Base reflector in a two-reflector model1
Time at top reflector in a two-reflector model2 Time at base
reflector in a two-reflector model
Layer two-way traveltime thicknessc Time at layer center in a
two-reflector modelt Time shift
f Frequencyf Frequency-domain impulse responsee Real component
of a function
m Imaginary component of a functione Even component of the
reflection coefficiento Odd component of the reflection
coefficientf Magnitude of amplitude as a function of
frequencycomponent squaredR Tuning thicknessfo Wavelet peak
frequencyt Reflection-coefficient series as a function of time
Convolutional placeholderI Even impulse pairI Odd impulse pairt,
f Time- and frequency-varying seismic tracet, f Time- and
frequency-varying seismic wavelet
w Window half-lengthk,T Frequency-varying objective
functiont,re,ro,T Time- and frequency-varying objective functione
Even-component weighting functiono Odd-component weighting
function
fL Low-frequency cutofffH High-frequency cutoff
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