local seismic attributes

8/13/2019 Local Seismic Attributes

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Local seismic attributes

Sergey Fomel1

ABSTRACT

Local seismic attributes measure seismic signal character-

istics not instantaneously, at each signal point, and not glo-bally, across a data window, but locally in the neighborhood

of eachpoint.I define local attributes withthe helpof regular-

ized inversion and demonstrate their usefulness for measur-

ing local frequencies of seismic signals and local similarity

between different data sets. I use shaping regularization for

controlling the locality and smoothness of local attributes. A

multicomponent-image-registration example from a nine-

component land survey illustrates practical applications of

local attributes for measuring differences between registered

images.

INTRODUCTION

Seismic attributeis defined by Sheriff1991 as a measurement

derived from seismic data. Such a broad definition allows for many

uses and abuses of the term. Countless attributes have been intro-

duced in the practice of seismic exploration, Brown, 1996; Chen

and Sidney, 1997, which led Eastwood 2002 to talkaboutattribute

explosion. Many of these attributes play an exceptionally important

role in interpreting and analyzing seismic data Chopra and Marfurt,

2005.

In thispaper, I considertwo particular attribute applications:

1 Measuring local frequency contentin a seismic image is impor-

tantboth for studying the phenomenon of seismic wave attenu-

ation and forthe processing of attenuated signals.

2 Measuring local similarity between two seismic images is use-

ful for seismic monitoring, registration of multicomponent

data, andanalysis of velocities and amplitudes.

Some of the best known seismic attributes are instantaneous at-

tributes such as instantaneous phase or instantaneous dip Taner et

al., 1979; Barnes, 1992, 1993. Such attributes measure seismic fre-

quency characteristics as being attachedinstantaneously to eachsig-nal point. This measure is notoriously noisy and may lead to un-

physical values suchas negative frequenciesWhite, 1991.

In this paper, I introduce a concept of local attributes. Local at-

tributes measure signal characteristics not instantaneously, at each

data point, but in a local neighborhood around the point. According

to the Fourier uncertainty principle, frequency is essentially an un-

certain characteristic when applied to a local region in the time do-

main.Therefore, local frequency is more physicallymeaningful than

instantaneous frequency. The idea of localityextendsfrom local fre-

quency to other attributes, such as the correlation coefficient be-

tween two different data sets, that are evaluated conventionally in

slidingwindows.

The paper starts with reviewing the definition of instantaneous

frequency. I modify this definition to that of local frequency by rec-

ognizing it as a form of regularized inversion and by changing regu-

larization to constrain the continuity and smoothness of the output.

The same idea is extended next to define local correlation. Finally, I

illustrate a practical application of local attributes using an example

from multicomponent-seismic-image registration in a nine-compo-

nent land survey.

MEASURING LOCAL FREQUENCIES

Definition of instantaneous frequency

Letft represent a seismictrace as a function of time t. The corre-

sponding complextrace ct isdefined as

ct = ft + iht, 1

where ht is the Hilberttransformof thereal traceft. One can alsorepresentthe complex tracein terms of the envelopeAtand thein-stantaneous phaset, as follows:

Manuscript received by the Editor November17, 2006; revised manuscript received December 1, 2006; publishedonlineMarch 13, 2007.1University of Texas at Austin, John A. and Katherine G. Jackson School of Geosciences, Bureau of Economic Geology, University Station, Austin, Texas.

E-mail: [email protected]. 2007Societyof Exploration Geophysicists.All rights reserved.

GEOPHYSICS,VOL.72, NO.3 MAY-JUNE2007; P. A29A33,7 FIGS.10.1190/1.2437573

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ct = Ateit . 2

By definition, instantaneous frequency isthe time

derivative of the instantaneous phase Taner et

al., 1979

t = t = Im

ct

ct=

ftht ftht

f2t + h 2t . 3

Different numerical realizations of equation 3

produce slightly different algorithms Barnes,

1992.

Note that the definition ofinstantaneous fre-

quency calls fordivision of twosignals.In a linear

algebra notation,

w = D 1n, 4

where w represents the vector of instantaneous

frequencies t, n represents the numerator inequation 3, and D is a diagonal operator made

from the denominator of equation 3. A recipe for

avoiding division by zero is adding a small con-

stant to the denominator Matheney and No-

wack, 1995. Consequently, equation 4 trans-

forms to

winst=D + I1n, 5

where I stands forthe identityoperator. Stabiliza-

tion by does not, however, prevent instanta-

neous frequency from being a noisy and unstable

attribute. The main reason for that is the extreme

locality of the instantaneous-frequency measure-

ment, governed only by the phase shift between

the signal and itsHilbert transform.

Figure 1 shows three testsignals forcomparing

frequency attributes.The first signal is a synthetic

chirp function with linearly varying frequency.

Instantaneous frequency shown in Figure 2 cor-

rectly estimates the modeled frequency trend.

Thesecondsignal isa pieceof a synthetic seismic

trace obtained by convolving a 40-Hz Ricker

wavelet with synthetic reflectivity. The instanta-

neous frequency Figure 2b shows many varia-

tions and appears to containdetailed information.However, this information is useless for charac-

terizing the dominant frequency content of the

data, which remains unchanged because of sta-

tionarity of the seismic wavelet. The last test ex-

ampleFigure 1cis a real trace extracted from a

seismic image. The instantaneous frequency

Figure 2c appears to be noisy and even contains

physically unreasonable negative values. Similar

behaviorwas described by White 1991.

Chirp signal

0 1 2 3 4Time (s)

Synthetic trace

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1

0

0.5

1

0.01

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50

0

50

100

0 0.01 0.02 0.03 0.04Time (s)

Seismic trace

0 1 2 3 4 5Time (s)

Amplitude

Amplitude

Amplitude

a)

b)

c)

Figure 1. Test signals for comparing frequency attributes. a Synthetic chirp signal withlinear frequency change.bSynthetic seismic trace from convolution of a synthetic re-flectivity with a Ricker wavelet.c Realseismic trace from a marinesurvey.

Chirp instantaneous frequency

0 1 2 3 4Time (s)

Synthetic instantaneous frequency

0 0.01 0.02 0.03 0.04Time (s)

Siesmic instantaneous frequency

Time (s)0 1 2 3 4 5

30

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100

80

60

40

20

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Frequ

ency(Hz)

Frequency(Hz)

Frequency(Hz)

a)

b)

c)

Figure2. Instantaneous frequency of test signalsfrom Figure 1.

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Definition of local frequency

The definition of the local frequency attribute

starts by recognizing equation 5 as a regularized

form of linear inversion. Changing regularization

from simple identityto a more general regulariza-

tion operator R provides the definition for local

frequency as follows:

wloc =D + R1n. 6

The role of the regularization operator is ensuring

continuity and smoothnessof the local frequency

measure. A different approach to regularization

follows from the shaping methodFomel, 2005,

2007. Shaping regularization operates with a

smoothingshapingoperator Sby incorporating

it intothe inversion schemeas follows:

wloc =2I + SD 2I1Sn. 7

Scaling by preserves physical dimensionality

and enables fast convergence when inversion is

implemented by an iterative method. A naturalchoice for is the least-squares normofD.

Figure 3 shows the results of measuring local

frequency in the test signals from Figure 1. I used

the shaping-regularization equation 7 with the

shaping operatorS definedas a trianglesmoother.

The chirp-signal frequency Figure 3a is correct-

ly recovered.The dominant frequency of the syn-

thetic signalFigure 3bis correctly estimated to

be stationary at 40 Hz. The localfrequency of the

real traceFigure 3c appears to vary with time

according to the general frequency attenuation

trend.

This example highlights some advantages of

the local-attributemethodin comparison withthe

sliding-window approach:

Only one parameter the smoothing radius

needs to be specified, as opposed to several

window size, overlap, and taperin the slid-

ing-window approach. The smoothing radius

directly reflects the locality of the measure-

ment.

The local-attribute approach continues the

measurement smoothly through the regionsof

absent information, such as the zero-ampli-

tude regions in the synthetic example, where

thesignalphaseis undefined.This effectis im-

possible to achieve in the sliding-window ap-proachunlessthe windowsize is always larger

thanthe information gaps in the signal.

Figure 4 shows seismic images from compres-

sional PP and shear SS reflections obtained by

processing a landnine-component survey. Figure

5 shows localfrequenciesmeasured in PPand SS

images after warping the SS image into PP time.

Theterm image warpingcomes from medicalim-

agingWolberg, 1990and refers, in this case, to

Chirp local frequency

0 1 2 3 4Time (s)

Synthetic local frequency

0 0.01 0.02 0.03 0.04

Time (s)

Seismic local frequency

0 1 2 3 4 5Time (s)

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25

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10

5

0

100

80

60

40

20

0

100

50

0Frequency(Hz)

Frequency(Hz)

Frequency(Hz)

a)

b)

c)

Figure 3. Local frequency of testsignals from Figure1.

0 50 100 150 200 250 300 350 400 450

Trace

0 50 100 150 200 250 300 350 400 450

Trace

0

1

2

3

4

SSPP

0

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Time(s)

Time(s)

a) b)

Figure 4. a PPand b SS imagesfrom a nine-component landsurvey.

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0 100 200 300 400

Trace0 100 200 300 400

Trace

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PP local frequency SS local frequency

54

52

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46

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38

Frequency(Hz)

a) b)

Figure 5. Local frequency attribute ofa PPand b warped SS images.

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squeezing theSS imageto PPreflectiontime to makethe two images

displayin the same coordinate system. We can observe a general de-

cay of frequency with time, caused by seismic attenuation. After

mapping squeezing to PP time, the SS image frequency appears

higherin theshallow partof the image, because ofa relatively lowS-

wave velocity, but lower in the deeper part of the image, because of

the apparently stronger attenuation of shear waves.A low-frequency

anomaly in the PPimage might be indicative of gas presence. Identi-fying and balancing nonstationary frequency variations of multi-

component images is an essential part of the multistep image-regis-

trationtechnique Fomel andBackus, 2003; Fomel et al.,2005.

MEASURING LOCAL SIMILARITY

Consider the task of measuring similarity between two different

signalsatand bt. One can define similarity as a global correla-tion coefficient and then perhaps measure it in sliding windows

across the signal. The local construction from the previous section

suggests approachingthis problemin a more elegant way.

Definition of global correlation

Global correlation coefficient betweena t and bt can be de-fined as the functional

=at,bt

at,atbt,bt, 8

where x t,yt denotes thedot productbetween twosignals

xt,yt = xtytdt. 9

According to equation 8, the correlation coefficient of two

identical signalsis equal to 1, andthe correlation of two signals with

opposite polarity is minus 1. In all the other cases, the correlation

will be less than 1 in magnitude, according to the Cauchy-Schwartz

inequality.

The global equation 8 is inconvenient becauseit supplies only one

number for the whole signal. The goal of local analysis is to turn

the functional into an operator and to produce local correlation as a

variable function t that identifies local changes in the signal

similarity.

Definition of local correlation

In a linear algebra notation, the squared correlation coefficient

fromequation 8 can be represented as a productof two least-squares

inverses

2 = 12, 10

1 =aT

a1

aT

b , 112 =b

Tb1bTa. 12

wherea is a vector notation fora t,b is a vector notation for bt,

and xTy denotesthe dotproduct operation defined in equation 9. Let

A be a diagonal operator composed fromthe elementsofa and B be a

diagonal operator composed from the elements of b. Localizing

equations 11 and 12 amounts to adding regularization to inversion.

Scalars1and 2turn into vectorsc1and c2defined, using shaping

regularization,as

c1 =2I + SATA 2I1SATb, 13

c2 =2I + SBTB 2I1SBTa . 14

To define a local similarity measure, I apply the componentwise

productof vectors c1 and c2. It isinterestingto note that ifone applies

an iterative, conjugate-gradient inversion for computing the inverse

operators in equations 13 and 14, the output of thefirst iterationwill

be the smoothed product of the two signalsc1 = c 2= SATb. This is

equivalent, with an appropriate choice ofS, to the algorithm of fast

local crosscorrelation proposed by Hale 2006.

The local-similarity attribute is useful for solving the problem of

multicomponent image registration. After an initial registration us-

ing interpreters nails DeAngelo et al., 2004 or velocities from

seismic processing, a useful registration indicator is obtained bysqueezing and stretching the warped shear-wave image while mea-

suring its local similarity to the compressional image. Such a tech-

nique was namedresidual scanand was proposed by Fomel et al.

2005. Figure 6 shows a residual scan for registration of multicom-

ponent images from Figure 4. Identifying and picking points of high

local similarity enablesmulticomponent registration with high-reso-

lution accuracy. The registration result is visualized in Figure 7,

which shows interleaved traces from PP and SS images before and

after registration. The alignment of main seismicevents is an indica-

tion of successful registration.

300

1

0.5

7

0

0.5

1

1.5

Time(s)

0.9 1 1.1 0

100

200

300

400

Relative gamma

Inlin

e

Figure6. Residual warpingscan formulticomponent PP/SSregistra-tion computed with the help of the local similarity attribute. Pickingmaximum similarity trendsenables multicomponent registration.

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CONCLUSIONS

I have introduced a concept of local seismic attributes and speci-

fied it for such attributes as local frequency and local similarity. Lo-

cal attributes measure signal characteristics not instantaneously, at

eachsignalpoint, and not globally, across a datawindow, but locally

in theneighborhoodof eachpoint. They findapplications in different

steps of multicomponent-seismic-image registration. One can ex-

tend the ideaof local attributes to other applications.

ACKNOWLEDGMENTS

I would like to thank BobHardage, Milo Backus, and BobGraeb-

ner forintroducing me to the problemof multicomponent dataregis-

trationand formany useful discussions.The dataexample waskind-

ly provided by Vecta. This work was partially supported by the U.S.

Department of Energy through Program DE-PS26-04NT42072 and

is authorized for publication by the director, Bureau of Economic

Geology, the Universityof Texasat Austin.

REFERENCES

Barnes, A. E., 1992, The calculation of instantaneousfrequency and instantaneous bandwidth: Geophys-ics, 57, 15201524.

, 1993, Instantaneous spectral bandwidth anddominant frequency with applications to seismic re-flection data:Geophysics,58, 419428.

Brown, A. R., 1996, Seismic attributes and their classi-

fication:The Leading Edge,15, 1090.Chen, Q., and S. Sidney, 1997, Seismic attribute tech-

nology for reservoir forecasting and monitoring: TheLeading Edge,16, 445456.

Chopra, S., and K. J. Marfurt, 2005, Seismic attributes A historical perspective: Geophysics, 70, no. 5,3SO28SO.

DeAngelo,M. V., R. Remington, P. Murray,B. A. Hard-age, R. Graebner, and K. Fouad, 2004, Multicompo-nent seismic technology for imaging deep gas pros-pects: The Leading Edge,23, 12701281.

Eastwood, J., 2002, The attribute explosion: The Lead-ingEdge, 21, 994.

Fomel, S., 2005, Shapingregularization in geophysical estimation problems:75th Annual International Meeting, SEG, Expanded Abstracts, 16731676.

, 2007, Shaping regularization in geoghysical estimation problems:Geophysics,72, no.2, R29R36.

Fomel, S., and M. Backus, 2003, Multicomponent seismic data registration

by least squares: 73rdAnnual International Meeting, SEG, Expanded Ab-stracts, 781784.

Fomel,S., M. Backus, K. Fouad,B. Hardage, andG. Winters,2005, Amulti-stepapproachto multicomponentseismicimage registrationwith applica-tion to a West Texas carbonate reservoir study: 75th Annual InternationalMeeting,SEG, ExpandedAbstracts,10181021.

Hale,D., 2006, Fast localcross-correlationsof images: 76thAnnual Interna-tional Meeting,SEG, ExpandedAbstracts,31603163.

Matheney, M. P., and R. L. Nowack, 1995, Seismic attenuation values ob-tained from instantaneous frequency matching and spectral ratios: Geo-physical Journal International,123, 115.

Sheriff, R. E., 1991, Encyclopedic Dictionary of Exploration Geophysics:SEG.

Taner,M. T., F. Koehler,and R. E. Sheriff,1979, Complexseismic traceanal-ysis: Geophysics, 44, 10411063.

White, R. E., 1991, Properties of instantaneousseismic attributes: The Lead-ingEdge, 10, 2632.

Wolberg,G., 1990, Digitalimage warping:IEEE ComputerSocietyPress.

0 20 40 60 80 100 120 140 160 180Inline

0

0.5

1

1.5

Time(s)

Interleaved (before)

0 20 40 60 80 100 120 140 160 180

Inline

0

0.5

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Time(s)

Interleaved (after)

a) b)

Figure 7. Interleavedtraces from PPand warped SS images a before and b after multi-component registration. The checkerboard pattern on major seismic events inadisap-pears in b, which is an indicationof successful registration.

Local seismic attributes A33

local seismic attributes

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