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GEOPHYSICS, VOL. 72, NO. 3 �MAY-JUNE 2007�; P. S149–S154, 11 FIGS.10.1190/1.2712113

moothing imaging condition for shot-profile migration

ntoine Guitton1, Alejandro Valenciano2, Dimitri Bevc1, and Jon Claerbout2

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ABSTRACT

Amplitudes in shot-profile migration can be improved if theimaging condition incorporates a division �deconvolution in thetime domain� of the upgoing wavefield by the downgoing wave-field. This division can be enhanced by introducing an optimalWiener filter which assumes that the noise present in the data hasa white spectrum. This assumption requires a damping parame-ter, related to the signal-to-noise ratio, often chosen by trial anderror. In practice, the damping parameter replaces the small val-ues of the spectrum of the downgoing wavefield and avoids divi-sion by zero. The migration results can be quite sensitive to thedamping parameter, and in most applications, the upgoing anddowngoing wavefields are simply multiplied. Alternatively, thedivision can be made stable by filling the small values of the

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pectrum with an average of the neighboring points. This averag-ng is obtained by running a smoothing operator on the spectrumf the downgoing wavefield. This operation called the smoothingmaging condition. Our results show that where the spectrum ofhe downgoing wavefield is high, the imaging condition withamping and smoothing yields similar results, thus correcting forllumination effects. Where the spectrum is low, the smoothingmaging condition tends to be more robust to the noise levelresent in the data, thus giving better images than the imagingondition with damping. In addition, our experiments indicatehat the parameterization of the smoothing imaging condition,.e., choice of window size for the smoothing operator, is easynd repeatable from one data set to another, making it a valuableddition to our imaging toolbox.

INTRODUCTION

Shot-profile migration �SPM� algorithms based on one-way prop-gators generate kinematically accurate images of the subsurface. Inany implementations, amplitudes are generally ignored within the

maging process, and are therefore considered fairly unreliable.owever, obtaining accurate amplitudes is becoming crucial formplitude variation with angle �AVA� analysis or 4D processing.ore important, accurate amplitudes can help unravel important

eologic information that a simpler one-way propagator would not.To obtain better amplitudes, three paths exist. First, the propaga-

ors used during the wavefield extrapolations need to be modified tonclude some amplitude information. This route has been exploredy numerous authors �e.g., Wapenaar et al., 1999; Sava et al., 2001�.or instance, Zhang et al. �2005b� show how the SPM algorithm cane modified to emulate WKBJ amplitudes.

Second, the imaging condition can be improved to yield better re-ection coefficients. To reach this goal, Claerbout �1971� showed

hat the receiver wavefield needs to be divided by the source wave-

Manuscript received by the Editor September 20, 2006; revised manuscrip13DGeo Inc., Santa Clara, California. E-mail: [email protected]; dimitr2Stanford University, Department of Geophysics, Stanford Explorati

tanford.edu.2007 Society of Exploration Geophysicists.All rights reserved.

eld, making sure that division by zero does not occur. The imagingondition can also impact the spatial resolution of the final imageValenciano and Biondi, 2003�, as well as decrease imaging artifactsMuijs et al., 2005�. Often, however, the receiver and source wave-elds are correlated because the parameterization of a stable divisionan be difficult.

Third, the imaging process can be cast as an inverse problem. Fornstance, least-squares migration with regularization has proved ef-ective at improving amplitudes with incomplete surface data �Nem-th et al., 1999� and irregular subsurface illumination because ofomplex structures �e.g., Prucha et al., 1999; Kuehl and Sacchi,002�. Because inversion is an expensive process, some strategiesave been developed to approximate the Hessian �Plessix and Mul-er, 2002; Rickett, 2003; Guitton, 2004� or estimate it on a targetrea only �Valenciano et al., 2006�.

This paper presents a method for improving the imaging condi-ion in SPM by approximating the division of the receiver wavefieldy the source wavefield. This approximation consists of smoothinghe denominator, as opposed to adding a stabilization parameter

ed January 8, 2007; published onlineApril 6, 2007.o.com.ect, Stanford, California. E-mail: [email protected]; jon@sep.

SEG license or copyright; see Terms of Use at http://segdl.org/

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damping�. In 2D, we illustrate this method with the Sigsbee2a andluto 1.5 data sets. In 3D, we present results obtained with the SEG/AGE salt model data set, also known as C3-NA. These exampleshow that the smoothing imaging condition yields stronger ampli-udes in areas of poor illumination and can image steeply dippingvents better than the crosscorrelation imaging condition.An impor-ant property of the smoothing imaging condition is that it yieldsleaner images than the damping imaging condition, where the noiseevel is high and the illumination is low. In addition, from a practicaliewpoint, choosing a window size for the smoothing operator turnsut to be much easier than choosing an adequate stabilization param-ter for the division with damping.

THEORY

For shot-profile migration, an approximation of the reflection co-fficient is given by �Claerbout, 1971�

Irc�x� = �xs

��

U��,x,xs�D��,x,xs�

, �1�

here x = �x,y,z� is each image position, � is the angular frequency,nd xs = �xs,ys,zs� is each source position. Note that the upgoing �U�nd downgoing �D� wavefields are also known as receiver andource wavefields, respectively. Physically, equation 1 states that aeflector exists where U and D coincide in time and space. Equationwill be unstable wherever D equals �or is close to� zero. For this

eason, it is customary to multiply both the numerator and the de-ominator by the complex conjugate of D �i.e. D�� and add a stabili-ation parameter � as follows:

Id�x� = �xs

��

U��,x,xs�D��,x,xs�D��,x,xs�D��,x,xs� + �

. �2�

e call equation 2 the damping imaging condition. Note that equa-ion 2 is equivalent to equation 1 multiplied by an optimal Wiener fil-er, assuming that the spectrum of the noise is white. This assump-ion breaks down when, for instance, residual multiples are presentn the data. In theory, � is related to the S/N. Similarly, the term with-n the two summations in equation 2 is the least-square inverse of thebjective function Q �omitting ��,x,xs� for clarity� or

Q�r� = �Dr − U�2 + ��Ir�2, �3�

here r is the reflection coefficient and I the identity matrix. Therere different ways to improve equation 2. Valenciano and Biondi2003� proposed a 2D deconvolution scheme by summing over fre-uencies and source wavenumbers, thus increasing the resolutionnd decreasing the noise level of the migrated image �Muijs et al.,005�. The summations can be moved inside or outside the numera-or and denominator �Plessix and Mulder, 2002; Zhang et al.,005a�. From an inversion point of view, better images can be ob-ained if regularization operators other than an identity matrix aresed in equation 3.

Equation 2 poses a serious problem to practitioners: how do westimate the damping parameter �? One can envision many strate-ies based on the statistics of D��,x,xs�D��,x,xs�. Alternatively, �ould be selected with a technique known as cross-validation, whereifferent values of � are selected and the residual power is plotted toorm a characteristic L-shaped curve. When tuned correctly, theseechniques can deliver very satisfying results. However, they are


ore difficult to use in a production environment in which robust-ess and reliability are often more important than mathematical ac-uracy. The imaging condition is usually estimated by crosscorrelat-ng the downgoing �D� and upgoing �U� wavefields �Claerbout,971� as follows:

Ic�x� = �xs

��

U��,x,xs�D��,x,xs� . �4�

e call equation 4 the crosscorrelation imaging condition. Jacobs1982� analyzes in detail the differences between equations 4 and 2.

moothing imaging condition

We propose approximating the deconvolution imaging conditionf equation 2. The main goal of this method is to emulate the decon-olution while being practical and robust. The main concept of thisaper is to fill the zero of D��,x,xs�D��,x,xs� with neighboringoints, as opposed to a more arbitrary constant value in equation 2.y “arbitrary,” we mean that in effect, although being theoretically

elated to the S/N, � is often chosen by trial and error. Therefore, weropose the following imaging condition:

Is�x� = �xs

��

U��,x,xs�D��,x,xs��D��,x,xs�D��,x,xs��x,y,z�

, �5�

here � .��x,y,z� stands for smoothing in the image space in the x,y, anddirections. We call equation 5 the smoothing imaging condition.he smoothing operator can be a rectangle, a triangle, a Gaussian

unction, etc. In this paper, a triangle function is used. Although notresented here, it would be worthwhile to study the behavior of theigration with respect to the shape of the smoothing operator. Fill-

ng the zero of a spectrum with neighboring values is not new: inter-olation with prediction-error filters performs similarly. Althoughquation 2 can be derived analytically from least-squares or filteringheory, equation 5 can not. Our formulation merely assumes that

�D��,x,xs�D��,x,xs��x,y,z� D��,x,xs�D��,x,xs� + � .

�6�

he terms in equation 6 behave differently. Where D is strong, equa-ion 2 converges to equation 1; in contrast, equation 5 does not, be-ause all values will be smoothed. Now, when D is small, equation 2onverges to equation 4; in contrast, equation 5 fills the spectraloles with neighboring values. Therefore, for the smoothing imag-ng condition, we might expect better results than the damping imag-ng condition where D is weak, and lesser results where D is strong.ur results seem to indicate that even in the case of strong signal, theethods yield similar results, however.For the smoothing imaging condition, only one parameter is need-

d, i.e., the size of the smoothing window n = �nx,ny,nz�. This pa-ameter has more physical appeal than � in equation 2 because wean choose it based on the frequency we are propagating or on theomplexity of the model. For instance, although not done here, iteems reasonable to increase the size of the smoothing window withncreasing frequency. Note that in all our results, we use a horizontalmoothing n = �n ,n � only.
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look at the denominator

As an illustration of this method for a source at x = 15.0 km in theigsbee2a velocity model �Figure 1, we estimate the illuminationaps D��,x,xs�D��,x,xs� at 5 �Figure 2a� and 45 Hz �Figure 3a�.e also show the smoothed versions �D��,x,xs�D��,x,xs��x in

igures 2b and 3b. We do not show the illumination map for��,x,xs�D��,x,xs� + � because it would look identical to��,x,xs�D��,x,xs� with a constant �. In both Figures 2 and 3, red

olors mean high values, whereas blue colors mean low values.The effect of the smoothing is clear in Figures 2b and 3b. The

moothing retains the main illumination information while fillinghe zero. It also attenuates the strong variations of amplitude in the il-umination paths of Figure 3a at 45 Hz. These patterns come fromhe destructive interference of waves traveling in different directionsnd are not related to the sampling of the velocity model. They mightamage the final image if the imaging condition with dampingequation 2� is not parameterized correctly. Figures 2 and 3 illustratelso that low frequencies need small smoothing windows and thatigh frequencies need bigger ones �the same is true for � in equation�. In practice, however, because it would require more parameters,e opted for a single size for all the frequencies.Better imaging conditions help correct amplitude in areas of poor

llumination. They also help boost the amplitude of the migration op-rator at high dips. Figure 4 displays two impulse responses in a v�z�edium using the crosscorrelation imaging condition �Figure 4a�

nd the smoothing imaging condition �Figure 4b�. Dips close to 90°re stronger after division. This can have an important impact in thenal image where faults and steep reflectors are present.

ize of the smoothing window

In the Sigsbee2a results, we selected a horizontal smoothing win-ow of 4 km. As illustrated in the next sections, we select the sameize for Pluto and a 3.2�3.2 km window for C3-NA. Remarkably,hese numbers are essentially the same. Because we can learn fromrevious models, choosing the right size for n becomes more sys-ematic and repeatable from one migration to the next.

Although not shown here, we conducted a series of tests to ana-yze the sensitivity of our results with respect to the size of themoothing window. These results indicate that the migration result isuite robust to this parameter. Conducting similar tests with theamping imaging condition proved more challenging because wead to iterate several times before finding a satisfying �. We com-ared how the image would evolve by decreasing both � in equationand n in equation 5; we noticed that the image deteriorates much

aster with the damping imaging condition than with the smoothingmaging condition. Both results would eventually look the same forery small values of both � and n.In the next section, we present migration results for three synthet-

c data sets. They illustrate that the smoothing imaging condition in-reases amplitudes in areas of poor illumination and yields similaresults to the damping imaging condition. They also prove that am-litude of steeply dipping reflectors, such as faults, is improved.

EXAMPLES

In this section, we illustrate our imaging condition on two 2D dataets and on one 3D data set.


igration of the Sigsbee2a data set

Figure 5 displays a comparison of three images obtained withhree different imaging conditions. Compared to the crosscorrela-ion imaging condition of Figure 5a, both the damping �Figure 5b�nd smoothing �Figure 5c� imaging conditions show balanced am-litude below the salt; they also look very similar, which indicateshat the smoothing imaging condition is effectively correcting for il-umination effects.

Looking at more details in Figure 6, we notice that the noise levels higher when the damping imaging condition is used. For instance,he strong reflector at z = 4.6 km appears discontinuous in Figure

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Figure 1. Aclose-up of the velocity model for the Sigsbee2a data set.Illumination maps are estimated for this model in Figures 2 and 3.

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b)DD at 5 Hz < DD at 5 Hz>

Figure 2. Illumination maps D�D at 5 Hz �a� without and �b� withsmoothing. Red colors correspond to high values, whereas blue col-ors correspond to low values. The size of the smoothing window is4 km. The zero has been filled after smoothing. The patterns of inter-fering fringes come from the destructive interference of seismicwaves traveling in different directions.


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b. As shown in Figure 6d, this discontinuity corresponds to a lowalue of the illumination map at 5 Hz.Although not shown here, thisow illumination corridor is present for a wide range of frequencies.s anticipated, by using the smoothing imaging condition in Figurec, we obtain a better image. A higher � for the damping imagingondition would mitigate this effect, but would also decrease theverall balancing effect of the division. One can also envision apace-varying �; however; this option would probably require morearameters, something we want to avoid for practical reasons.

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c)U D` U D` / ( +eps) U D` /< >` D D D D

igure 5. Migration results of the Sigsbee2a data set for �a� cross-orrelation imaging condition, �b� damping imaging condition� = 0.0001�, and �c� smoothing imaging condition. Amplitudes be-ow the salt are balanced after division. Note that �b� and �c� are veryimilar.

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d) U D / < D D> ` `

U D ` D D U D ` / ( +eps) `

D D ` at 5 Hz

igure 6. Close-up of the migration results of the Sigsbee2a data setor �a� crosscorrelation imaging condition, �b� damping imagingondition �� = 0.0001� and �c� smoothing imaging condition. Paneld� shows the illumination map at 5 Hz. Red colors correspond toigh values, whereas blue colors correspond to low values. Thetrong reflector in �a� at z = 4.6 km appears discontinuous when us-ng the damping imaging condition. This discontinuity coincidesith a very low value of D�D in Figure 6d.

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b)DD at 45 Hz < DD at 45 Hz>

igure 3. Illumination maps D�D at 45 Hz �a� without and �b� withmoothing. Red colors correspond to high values, whereas blue col-rs correspond to low values. The size of the smoothing window iskm. The patterns of interfering fringes come from the destructive

nterference of seismic waves traveling in different directions.

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x (km)

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igure 4. Impulse responses in a v�z� medium for �a� the crosscorre-ation imaging condition and �b� the smoothing imaging condition.he clipped values for each impulse are shown in red. High dips getigher amplitudes after division.


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igration of the Pluto 1.5 data set

Figures 7 and 8 display a comparison of three images obtainedith three different imaging conditions. Similar to what we ob-

erved for the Sigsbee2a data set, areas of low illumination havetronger amplitudes when the damping and smoothing imaging con-itions are used. In Figure 7, the steep flanks at z = 4.5 km are wellmaged with both smoothing and damping. The flat reflector at z

9 km shows more balanced amplitudes below the salt in both Fig-res 7 and 8.

These two 2D data examples illustrate two important aspects ofhe smoothing imaging condition: First, it yields images similar tohat of the damping imaging condition. Second, it tends to have low-r noise levels because the denominator of equation 5 is smooth atvery frequency. In addition, the size of the smoothing window n isasy to select. In the next section, we illustrate the smoothing imag-ng condition on the C3-NAdata set.

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a) b) c)` DU D / +eps)D( DD` /< >DUU D ```

igure 7. Migration results of the Pluto 1.5 data set for �a� cross-orrelation imaging condition, �b� damping imaging condition� = 0.001�, and �c� smoothing imaging condition. The steep flanksre well imaged in �b� and �c�.

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a) b) c)DD` /< >DU `D/ +eps)D(` DU` DU

igure 8. Migration results of the Pluto 1.5 data set for �a� cross-orrelation imaging condition, �b� damping imaging condition� = 0.001�, and �c� smoothing imaging condition. The steep flanksre well imaged in �b� and �c�. Areas of poor illumination below.0 km have stronger amplitudes in �b� and �c�. Damping andmoothing yield similar results.


igration of the C3-NA data set

Figure 9 shows a window of the velocity model for the migrationesults of Figures 10 and 11. For this example, because we proved inhe 2D examples that the damping and smoothing imaging condi-ions yield similar results �except where the noise is strong and the il-umination is low�, we applied the crosscorrelation and smoothingmaging conditions only. In this example, we notice that the flat re-ectors at z3.5 km have stronger amplitudes when the smoothing

maging condition is used. In addition, the salt flank at x = 6.0 kmhows more balanced amplitudes.

One important effect of the smoothing imaging condition appearsn the fault planes. The faults are stronger in Figure 11 than in Figure0. Similar to Figure 4, steep dips get higher amplitudes, unraveling

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igure 9. Velocity model for the C3 data set used in Figures 10 and1. The top panel is a depth slice at z = 2.0 km, the center panel is aonstant crossline slice at y = 3.6 km, and the right panel is a con-tant inline slice at x = 9.08 km. The lines show where these sectionntersect in the 3D volume.

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igure 10. Migration result obtained with the crosscorrelation imag-ng condition. Both Figures 10 and 11 are scaled to one and dis-layed with the same clip �0.03�.


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S154 Guitton et al.

mportant geologic information. Note that kinematically the accura-y of the one-way propagator is not increased: we witness an ampli-ude effect only.

It is important to notice that having higher amplitudes for steeplyipping reflectors and for regions of low illumination might createroblems if the S/N is small or if the velocity is not accurate. Figure1 illustrates this point very well for y�5.0 km. There, migration ar-ifacts are boosted because of the lack of reflected energy in the sub-alt area. In the inline panel, we also notice weak linear events paral-el to the salt flank appearing after smoothing. Because of a poor illu-

ination, noisy events, such as internal multiples, can affect our im-ge because they are not incorporated into our noise model; in otherords, their spectrum is not white. This important drawback is not

pecific to the smoothing imaging condition, however. It merely re-ects some limitations of the division imaging condition with damp-

ng or smoothing.

CONCLUSION

We presented the smoothing imaging condition. At each frequen-y and for each shot, the image is divided by a smoothed version ofhe source illumination map, as opposed to a damped one. As exem-lified in this paper, this imaging condition is superior to the damp-ng imaging condition in three important aspects. First, the parame-

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igure 11. Migration result obtained with the smoothing imagingondition. Compared to Figure 10, the salt flanks and the flat re-ector at the bottom of the image have stronger amplitudes. Faults

hat were weak in Figure 10 are well imaged. The subsalt areay�5.0 km� becomes noisier owing to a very poor illumination.


erization is very straightforward and physically appealing: we onlyeed to decide on the size of the smoothing window n, which oper-tes in the image space. Second, although not shown here, our exper-ments indicate that the migration result is quite robust to n. For in-tance, this parameter is almost equal for all the results presentedere. Third, the smoothing imaging condition tends to decrease theoise level present in the damping imaging condition. This can bemportant for real data examples where the velocity model is not ac-urate.

ACKNOWLEDGMENTS

We thank Biondo Biondi for many useful suggestions during thisork.Alejandro Valenciano and Jon Claerbout thank the sponsors of

he Stanford Exploration Project for their financial support. Theigsbee2a and Pluto 1.5 data sets were made available by the formerembers of the SMAART J.V.

REFERENCES

laerbout, J. F., 1971, Toward a unified theory of reflector mapping: Geo-physics, 36, 467–481.

uitton, A., 2004, Amplitude and kinematic corrections of migrated imagesfor nonunitary imaging operators: Geophysics, 69, 1017–1024.

acobs, A., 1982, The pre-stack migration of profiles: Ph.D. thesis, StanfordUniversity.

uehl, H., and M. Sacchi, 2002, Robust AVP estimation using least-squareswave-equation migration: 72nd Annual International Meeting, SEG, Ex-pandedAbstracts, 281–284.uijs, R., K. Holliger, and J. O. Robertsson, 2005, Prestack depth migrationof primary and surface-related multiple reflections: 75th Annual Interna-tional Meeting, SEG, ExpandedAbstracts, 2107–2110.

emeth, T., C. Wu, and G. T. Schuster, 1999, Least-squares migration of in-complete reflection data: Geophysics, 64, 208–221.

lessix, R., and W. Mulder, 2002, Amplitude-preserving finite-differencemigration based on a least-square formulation in the frequency domain:72nd Annual International Meeting, SEG, Expanded Abstracts, 1212–1215.

rucha, M., B. Biondi, and W. Symes, 1999, Angle-domain common imagegathers by wave-equation migration: 69th Annual International Meeting,SEG, ExpandedAbstracts, 824–827.

ickett, J., 2003, Illumination-based normalization for wave-equation depthmigration: Geophysics, 68, 1371–1379.

ava, P., B. Biondi, and S. Fomel, 2001, Amplitude-preserved common im-age gathers by wave-equation migration: 71st Annual International Meet-ing, SEG, ExpandedAbstracts, 296–299.

alenciano, A., and B. Biondi, 2003, 2D deconvolution imaging conditionfor shot profile migration: 73rd Annual International Meeting, SEG, Ex-pandedAbstracts, 1059–1062.

alenciano, A., B. Biondi, and A. Guitton, 2006, Target-oriented wave-equa-tion inversion: Geophysics, 71, no. 4, A35–A38.apenaar, K., A. J. V. Wijngaarden, W. van Geloven, and T. van der Leij,1999, ApparentAVAeffects of fine layering: Geophysics, 64, 1939–1948.

hang, Y., J. Sun, C. Notfors, S. H. Gray, L. Chernis, and J. Young, 2005a,Delayed-shot 3D depth migration: Geophysics, 70, no. 5, E21–E28.

hang, Y., G. Zhang, and N. Bleistein, 2005b, Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration: Geo-
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