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GEOPHYSICS, VOL. 72, NO. 3 MAY-JUNE 2007; P. A29A33, 7 FIGS.10.1190/1.2437573

ocal seismic attributes

ergey Fomel1

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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 neighborhoodof each point. I define local attributes with the help of regular-ized inversion and demonstrate their usefulness for measur-ing local frequencies of seismic signals and local similaritybetween different data sets. I use shaping regularization forcontrolling the locality and smoothness of local attributes. Amulticomponent-image-registration example from a nine-component land survey illustrates practical applications oflocal attributes for measuring differences between registeredimages.

INTRODUCTION

Seismic attribute is defined by Sheriff 1991 as a measurementerived from seismic data. Such a broad definition allows for manyses and abuses of the term. Countless attributes have been intro-uced in the practice of seismic exploration, Brown, 1996; Chennd Sidney, 1997, which led Eastwood 2002 to talk about attributexplosion. Many of these attributes play an exceptionally importantole in interpreting and analyzing seismic data Chopra and Marfurt,005.

In this paper, I consider two particular attribute applications:

Measuring local frequency content in a seismic image is impor-tant both for studying the phenomenon of seismic wave attenu-ation and for the processing of attenuated signals.

Measuring local similarity between two seismic images is use-ful for seismic monitoring, registration of multicomponentdata, and analysis of velocities and amplitudes.

Manuscript received by the Editor November 17, 2006; revised manuscrip1University of Texas at Austin, John A. and Katherine G. Jackson Schoo

-mail: sergey.fomel@beg.utexas.edu.2007 Society of Exploration Geophysicists.All rights reserved.

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Some of the best known seismic attributes are instantaneous at-ributes such as instantaneous phase or instantaneous dip Taner etl., 1979; Barnes, 1992, 1993. Such attributes measure seismic fre-uency characteristics as being attached instantaneously to each sig-al point. This measure is notoriously noisy and may lead to un-hysical values such as negative frequencies White, 1991.

In this paper, I introduce a concept of local attributes. Local at-ributes measure signal characteristics not instantaneously, at eachata point, but in a local neighborhood around the point. Accordingo the Fourier uncertainty principle, frequency is essentially an un-ertain characteristic when applied to a local region in the time do-ain. Therefore, local frequency is more physically meaningful than

nstantaneous frequency. The idea of locality extends from local fre-uency to other attributes, such as the correlation coefficient be-ween two different data sets, that are evaluated conventionally inliding windows.

The paper starts with reviewing the definition of instantaneousrequency. I modify this definition to that of local frequency by rec-gnizing it as a form of regularized inversion and by changing regu-arization to constrain the continuity and smoothness of the output.he same idea is extended next to define local correlation. Finally, I

llustrate a practical application of local attributes using an examplerom multicomponent-seismic-image registration in a nine-compo-ent land survey.

MEASURING LOCAL FREQUENCIES

efinition of instantaneous frequency

Let ft represent a seismic trace as a function of time t. The corre-ponding complex trace ct is defined as

ct = ft + iht , 1

here ht is the Hilbert transform of the real trace ft. One can alsoepresent the complex trace in terms of the envelope At and the in-tantaneous phase t, as follows:

d December 1, 2006; published online March 13, 2007.sciences, Bureau of Economic Geology, University Station, Austin, Texas.

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ct = Ateit. 2

By definition, instantaneous frequency is the timederivative of the instantaneous phase Taner etal., 1979

t = t = Im ctct

=ftht ftht

f2t + h2t. 3

Different numerical realizations of equation 3produce slightly different algorithms Barnes,1992.

Note that the definition of instantaneous fre-quency calls for division of two signals. In a linearalgebra notation,

w = D1n , 4

where w represents the vector of instantaneousfrequencies t, n represents the numerator inequation 3, and D is a diagonal operator madefrom the denominator of equation 3. A recipe foravoiding 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 for the identity operator. Stabiliza-tion by does not, however, prevent instanta-neous frequency from being a noisy and unstableattribute. The main reason for that is the extremelocality of the instantaneous-frequency measure-ment, governed only by the phase shift betweenthe signal and its Hilbert transform.

Figure 1 shows three test signals for comparingfrequency attributes. The first signal is a syntheticchirp function with linearly varying frequency.Instantaneous frequency shown in Figure 2 cor-rectly estimates the modeled frequency trend.The second signal is a piece of a synthetic seismictrace obtained by convolving a 40-Hz Rickerwavelet with synthetic reflectivity. The instanta-neous frequency Figure 2b shows many varia-tions and appears to contain detailed information.However, this information is useless for charac-terizing the dominant frequency content of thedata, which remains unchanged because of sta-tionarity of the seismic wavelet. The last test ex-ample Figure 1c is a real trace extracted from aseismic image. The instantaneous frequencyFigure 2c appears to be noisy and even containsphysically unreasonable negative values. Similarbehavior was described by White 1991.

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igure 1. Test signals for comparing frequency attributes. a Synthetic chirinear frequency change. b Synthetic seismic trace from convolution of aectivity with a Ricker wavelet. c Real seismic trace from a marine survey

Chirp instantaneous frequency

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igure 2. Instantaneous frequency of test signals from Figure 1.

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Local seismic attributes A31

efinition of local frequency

The definition of the local frequency attributetarts by recognizing equation 5 as a regularizedorm of linear inversion. Changing regularizationrom simple identity to a more general regulariza-ion operator R provides the definition for localrequency as follows:

wloc = D + R1n. 6

he role of the regularization operator is ensuringontinuity and smoothness of the local frequencyeasure. A different approach to regularization

ollows from the shaping method Fomel, 2005,007. Shaping regularization operates with amoothing shaping operator S by incorporatingt into the inversion scheme as follows:

wloc = 2I + SD 2I1Sn. 7

caling by preserves physical dimensionalitynd enables fast convergence when inversion ismplemented by an iterative method. A naturalhoice for is the least-squares norm of D.

Figure 3 shows the results of measuring localrequency in the test signals from Figure 1. I usedhe shaping-regularization equation 7 with thehaping operator S defined as a triangle smoother.he chirp-signal frequency Figure 3a is correct-

y recovered. The dominant frequency of the syn-hetic signal Figure 3b is correctly estimated toe stationary at 40 Hz. The local frequency of theeal trace Figure 3c appears to vary with timeccording to the general frequency attenuationrend.

This example highlights some advantages ofhe local-attribute method in comparison with theliding-window approach:

Only one parameter the smoothing radiusneeds to be specified, as opposed to severalwindow size, overlap, and taper in the slid-ing-window approach. The smoothing radiusdirectly reflects the locality of the measure-ment.The local-attribute approach continues themeasurement smoothly through the regions ofabsent information, such as the zero-ampli-tude regions in the synthetic example, wherethe signal phase is undefined. This effect is im-possible to achieve in the sliding-window ap-proach unless the window size is always largerthan the information gaps in the signal.

Figure 4 shows seismic images from compres-ional PP and shear SS reflections obtained byrocessing a land nine-component survey. Figureshows local frequencies measured in PP and SS

mages after warping the SS image into PP time.h