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GEOPHYSICS, VOL. 72, NO. 2 �MARCH-APRIL 2007�; P. R29–R36, 13 FIGS., 1 TABLE.10.1190/1.2433716

haping regularization in geophysical-estimation problems

ergey Fomel1

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ABSTRACT

Regularization is a required component of geophysical-es-timation problems that operate with insufficient data. Thegoal of regularization is to impose additional constraints onthe estimated model. I introduce shaping regularization, ageneral method for imposing constraints by explicit mappingof the estimated model to the space of admissible models.Shaping regularization is integrated in a conjugate-gradientalgorithm for iterative least-squares estimation. It providesthe advantage of better control on the estimated model incomparison with traditional regularization methods and, insome cases, leads to a faster iterative convergence. Simpledata interpolation and seismic-velocity estimation examplesillustrate the concept.

INTRODUCTION

A great number of geophysical-estimation problems are mathe-atically ill-posed because they operate with insufficient data �Jack-

on, 1972�. Regularization is a technique for making the estimationroblems well posed by adding indirect constraints on the estimatedodel �Engl et al., 1996; Zhdanov, 2002�. Developed originally byikhonov �1963� and others, the method of regularization has be-ome an indispensable part of the inverse-problem theory and hasound many applications in geophysical problems: traveltime to-ography �Bube and Langan, 1999; Clapp et al., 2004�, migration

elocity analysis �Woodward et al., 1998; Zhou et al., 2003�, high-esolution Radon transform �Trad et al., 2003�, spectral decomposi-ion �Portniaguine and Castagna, 2004�, etc.

While the main goal of inversion is to fit the observed data,ikhonov’s regularization adds another goal of fitting the estimatedodel to a priorly assumed behavior. The contradiction between the

wo goals often leads to a slow convergence of iterative-estimationlgorithms �Harlan, 1995�. The speed can be improved considerably

Presented at the 75th Annual International Meeting, SEG. Manuscript recublished online February 15, 2007.

1University of Texas at Austin, John A. and Katherine G. Jackson Sergey.fomel@beg.utexas.edu.

2007 Society of Exploration Geophysicists. All rights reserved.

R29

y an appropriate model reparameterization or preconditioning �Fo-el and Claerbout, 2003�. However, the difficult situation of trying

o satisfy two contradictory goals simultaneously leads sometimeso an undesirable behavior of the solution at the early iterations of anterative-optimization scheme.

In this paper, I introduce shaping regularization, a new generalethod of imposing regularization constraints. A shaping operator

rovides an explicit mapping of the model to the space of acceptableodels. The operator is embedded in an iterative-optimization

cheme �the conjugate-gradient algorithm� and allows for betterontrol on the estimation result. Shaping into the space of smoothunctions can be accomplished with efficient low-pass filtering. De-ending on the desirable result, it is also possible to shape the modelnto a piecewise-smooth function, a function following geologicaltructure, or a function representable in a predefined basis. I illus-rate the shaping concept with simple numerical experiments of datanterpolation and seismic-velocity estimation.

REVIEW OF TIKHONOV’S REGULARIZATION

If the data are represented by vector d, model parameters by vec-or m, and their functional relationship is defined by the forward-

odeling operator L, the least-squares optimization amounts toinimizing the least-squares norm of the residual difference Lm −. In Tikhonov’s regularization approach, one additionally attempts

o minimize the norm of Dm, where D is the regularization operator.hus, we are looking for the model m that minimizes the least-quares norm of the compound vector �Lm − d �Dm�T, where � is acalar scaling parameter. The formal solution has the well-knownorm

m̂ = �LTL + �2DTD�−1LTd , �1�

here m̂ denotes the least-squares estimate of m, and LT denotes thedjoint operator. One can carry out the optimization iteratively withhe help of the conjugate-gradient method �Hestenes and Steifel,952� or its analogs. Iterative methods have computational advan-ages in large-scale problems when forward and adjoint operators

the Editor April 7, 2006; revised manuscript received September 13, 2006;

of Geosciences, Bureau of Economic Geology, Austin, Texas. E-mail:

a�

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re represented by sparse matrices and can be computed efficientlySaad, 2003; van der Vorst, 2003�.

In an alternative approach, one obtains the regularized estimate byinimizing the least-squares norm of the compound vector �p r�T

nder the constraint

�r = d − Lm = d − LPp. �2�

ere, P is the model-reparameterization operator that translates vec-or p into the model vector m, r is the scaled-residual vector, and �as the same meaning as before. The formal solution of the precondi-ioned problem is given by

m̂ = Pp̂ = PPTLT�LPPTLT + �2I�−1d , �3�

here I is the identity operator in the data space. Estimate 3 is mathe-atically equivalent to estimate 1 if DTD is invertible and

�DTD�−1 = PPT = C . �4�

tatistical theory of least-squares estimation connects C with theodel covariance operator �Tarantola, 2004�. In a more general case

f reparameterization, the size of p may be different from the size of, and C may not have the full rank. In iterative methods, the pre-

onditioned formulation often leads to faster convergence. Fomelnd Claerbout �2003� suggest constructing preconditioning opera-ors in multidimensional problems by recursive helical filtering.

SMOOTHING BY REGULARIZATION

Let us consider an application of Tikhonov’s regularization to onef the simplest possible estimation problems: smoothing. The task ofmoothing is to find a model m that fits the observed data d, but is in aertain sense smoother. In this case, the forward operator L is simplyhe identity operator, and the formal solutions 1 and 3 take the form

m̂ = �I + �2DTD�−1d = C�C + �2I�−1d . �5�

1

2

3

4

5

0 5 10 15 20

Time (sample)

25 30 35 40

1

2

3

4

5

–0.4–0.3–0.2–0.1 0 0.1 0.2 0.3 0.4

Frequency (cycle)

a) b)

igure 1. �a� Impulse response of regularized smoothing. Repeatedmoothing converges to a Gaussian bell shape. �b� Frequency spec-rum of regularized smoothing. The spectrum also converges to aaussian.

moothness is controlled by the choice of the regularization opera-or D and the scaling parameter �.

Figure 1 shows the impulse response of the regularized smoothingperator in the 1D case when D is the first difference operator. Thempulse response has exponentially decaying tails. Repeated appli-ation of smoothing in this case is equivalent to applying an implicituler finite-difference scheme to the solution of the diffusion equa-

ion

� m

� t= − DTDm. �6�

he impulse response converges to a Gaussian bell-shape curve inhe physical domain, while its spectrum converges to a Gaussian inhe frequency domain.

As far as the smoothing problem is concerned, there are betterays to smooth signals than applying equation 5. One example is tri-

ngle smoothing �Claerbout, 1992�. To define triangle smoothing ofD signals, start with box smoothing, which, in the Z-transform no-ation, is a convolution with the filter

Bk�Z� =1

k�1 + Z + Z2 + ¯ + Zk� =

1

k

1 − Zk+1

1 − Z, �7�

here k is the filter length. Form a triangle smoother by correlationf two boxes:

Tk�Z� = Bk�1/Z�Bk�Z� . �8�

riangle smoothing is more efficient than regularized smoothing,ecause it requires twice less floating-point multiplications. It alsorovides smoother results, while having a compactly supported im-ulse response �Figure 2�. Repeated application of triangle smooth-ng also makes the impulse response converge to a Gaussian shape,ut at a significantly faster rate. One can also implement smoothingy Gaussian filtering in the frequency domain or by applying otherypes of band-pass filters.

a)

1

2

3

4

5

0 5 10 15 20

Time (sample)

25 30 35 40

1

2

3

4

5

–0.4 –0.3–0.2–0.1 0 0.1 0.2 0.3 0.4

Frequency (cycle)

b)

igure 2. �a� Impulse response of triangle smoothing. Repeatedmoothing converges to a Gaussian bell shape. �b� Frequency spec-rum of triangle smoothing. Convergence to a Gaussian is faster thann the case of regularized smoothing. Compare to Figure 1.

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Shaping regularization R31

SHAPING REGULARIZATION IN THEORY

The idea of shaping regularization starts with recognizingmoothing as a fundamental operation. In a more general sense,moothing implies mapping of the input model to the space of ad-issible functions. I call the mapping operator shaping. Shaping op-

rators do not necessarily smooth the input, but they translate it inton acceptable model.

Taking equation 5 and using it as the definition of the regulariza-ion operator D, we can write

S = �I + �2DTD�−1 �9�

r

�2DTD = S−1 − I . �10�

ubstituting equation 10 into equation 1 yields a formal solution ofhe estimation problem regularized by shaping:

m̂ = �LTL + S−1 − I�−1LTd = �I + S�LTL − I��−1SLTd .

�11�

he meaning of equation 11 is easy to interpret in some specialases:

If S = I �no shaping applied�, we obtain the solution of unregu-larized problem.If LTL = I �L is a unitary operator�, the solution is simply SLTdand does not require any inversion.If S = �I �shaping by scaling�, the solution approaches �LTd as �goes to zero.

The operator L may have physical units that require scaling. Intro-ucing scaling of L by 1/� in equation 11, we can rewrite it as

m̂ = ��2I + S�LTL − �2I��−1SLTd . �12�

he � scaling in equation 12 controls the relative scaling of the for-ard operator L, but not the shape of the estimated model, which is

ontrolled by the shaping operator S.Iterative inversion with the conjugate-gradient algorithm requires

ymmetric positive-definite operators �Hestenes and Steifel, 1952�.he inverse operator in equation 12 can be symmetrized when thehaping operator is symmetric and representable in the form S

HHT with a square and invertible H. The symmetric form of equa-ion 12 is

m̂ = H��2I + HT�LTL − �2I�H�−1HTLTd . �13�

hen the inverted matrix is positive-definite, equation 13 is suitableor an iterative inversion with the conjugate-gradient algorithm. Ta-le 1 contains a complete algorithm description.

FROM TRIANGLE SMOOTHINGTO TRIANGLE SHAPING

The idea of triangle smoothing can be generalized to produce dif-erent shaping operators for different applications. Let us assumehat the estimated model is organized in a sequence of records, as fol-ows: m = �m1 m2 . . . mn�T. Depending on the application, theecords can be samples, traces, shot profiles, etc. Let us further as-ume that, for each pair of neighboring records, we can design a pre-

iction operator Zk→k+1, which predicts record k + 1 from record k.global prediction operator is then

Z = �0 0 0 ¯ 0 0

Z1→2 0 0 ¯ 0 0

0 Z2→3 0 ¯ 0 0

0 0 Z3→4 ¯ 0 0

¯ ¯ ¯ ¯ ¯ ¯

0 0 0 ¯ Zn−1→n 0

� . �14�

he operator Z effectively shifts each record to the next one. Whenocal prediction is done with identity operators, this operation isompletely analogous to the Z operator used in the theory of digital-ignal processing. The Z operator can be squared, as follows:

able 1. Algorithm for conjugate-gradient iterative inversionith shaping regularization. The algorithm follows directly

rom combining equation 13 with the classic conjugate-radient algorithm of Hestenes and Steifel (1952).

ONJUGATE GRADIENTS WITH SHAPING �L,H,d,�,tol,N�1 p←0

2 m←0

3 r←−d

4 for n←1,2, . . . ,N

5 do

6 gm←LTr − �m

7 gp←HTgm + �p

8 gm←Hgp

9 gr←Lgm

10 �←gpTgp

11 if n = 1

12 then �←0

13 �0←�

14 else �←�/�̂

15 if �� tol or �/�0 � tol

16 then return m

17

�sp

sm

sr� ← �gp

gm

gr� + ��sp

sm

sr�

18 � ← �/�srTsr + ��sp

Tsp − smT sm��

19

�p

m

r� ← �p

m

r� − ��sp

sm

sr�

20 �̂←�

21 return m

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R32 Fomel

Z2 = �0 0 ¯ 0 0 0

0 0 ¯ 0 0 0

Z2→3Z1→2 0 ¯ 0 0 0

0 Z3→4Z2→3 ¯ 0 0 0

¯ ¯ ¯ ¯ ¯ ¯

0 0 ¯ Zn−1→nZn−2→n−1 0 0

� .

�15�

n a shorter notation, we can denote prediction of record j fromecord i by Zi→j and write

Z2 = �0 0 ¯ 0 0 0

0 0 ¯ 0 0 0

Z1→3 0 ¯ 0 0 0

0 Z2→4 ¯ 0 0 0

¯ ¯ ¯ ¯ ¯ ¯

0 0 ¯ Zn−2→n 0 0

� . �16�

ubsequently, the prediction operator Z can be taken to higher pow-rs. This leads immediately to an idea on how to generalize boxmoothing: predict each record from the record immediately preced-ng it, the record two steps away, etc., and average all of those predic-ions and the actual records. In mathematical notation, a box shaperf length k is then simply

Lateral (km)

0

0.2

0.4

0.6

Tim

e (s

)

1.2 1.41

) b)0.80.60.40.20

Later

0

0.2

0.4

0.6

Tim

e (s

)

0.60.40.20

Lateral (km)

0

0.2

0.4

0.6

Tim

e (s

)

1.2 1.41

) d)0.80.60.40.20

Distan

0

0.2

0.4

0.6

Tim

e (s

)

0.60.40.20

igure 3. Shaping by smoothing along local dip directions accordingquation 20. �a� An example image, �b� local dip estimation, �c� smooers along local dips, and �d� impulse responses of oriented smoothiocations in the image space.

Bk =1

k�I + Z + Z2 + ¯ + Zk� , �17�

hich is completely analogous to equation 7. Implementing equa-ion 17 directly requires many computational operations. Noting that

�I − Z�Bk =1

k�I − Zk+1� , �18�

e can rewrite equation 17 in the compact form

Bk =1

k�I − Z�−1�I − Zk+1� , �19�

hich can be implemented economically using recursive inversionf the lower triangular operator I − Z. Finally, combining two gen-ralized box smoothers creates a symmetric, generalized trianglehaper

Tk = BkTBk, �20�

hich is analogous to equation 8. A triangle shaper uses local predic-ions from both the left and the right neighbors of a record and aver-ges them using triangle weights.

Figure 3 illustrates generalized triangle shaping by constructing aonstationary smoothing operator that follows local structural dips.igure 3a shows a synthetic image from Claerbout �2006�. Figure 3b

s a local dip estimate obtained with plane-wave destruction �Fomel,002�. Figure 3c is the result of applying triangle smoothing orientedlong local dip directions to a field of random numbers. Orientedmoothing generates a pattern reflecting the structural compositionf the original image. This construction resembles the method of

Claerbout and Brown �1999�. Figure 3d showsthe impulse responses of oriented smoothing forseveral distinct locations in the image space. Asillustrated later in this paper, oriented smoothingcan be applied for generating geophysical Earthmodels that are compliant with the local geologi-cal structure �Sinoquet, 1993; Versteeg andSymes, 1993; Clapp et al., 2004�.

Appendix A describes general rules for com-bining elementary shaping operators.

EXAMPLES

Two simple examples in data regularizationand seismic-velocity estimation illustrate themethod of shaping.

1D inverse data interpolation

I start with a simple 1D example: a benchmarkdata-regularization test used previously by Fomeland Claerbout �2003�.

The input synthetic data are irregular samplesfrom a sinusoidal signal �Figure 4�. The task ofdata regularization is to reconstruct the data on aregular grid. The forward operator L in this caseis forward interpolation from a regular grid usinglinear �two-point� interpolation.

2

1

0

–1

–2

1.2 1.4

)1.2 1.4

rator Tk fromandom num-ine different

al (km)10.8

ce (km10.8

to opething r

ng for n

i1tctfict

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mnsdtAp

V

sfwsSfit

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F

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Shaping regularization R33

Figure 5 shows some of the first iterations and the final results ofnverse interpolation with Tikhonov’s regularization using equationand with model preconditioning using equation 3. The regulariza-

ion operator D in equation 1 is the first finite difference, and the pre-onditioning operator P in equation 3 is the inverse of D or causal in-egration. The preconditioned iteration converges faster, but its veryrst iterations produce unreasonable results. This type of behavioran be dangerous in real large-scale problems, when only few itera-ions are affordable.

Figure 6a shows some of the first iterations and the final result ofnverse interpolation with shaping regularization, where the shapingperator S was chosen to be Gaussian smoothing with the impulse-esponse width of about 10 samples. The final result is smoother, andhe iteration is both fast-converging and producing reasonable re-ults at the very first iterations. Thanks to the fact that the smoothingperation is applied at each iteration, the estimated model is guaran-eed to have the prescribed shape.

Examining the spectrum of the final result �Figure 7�, one can im-ediately notice the peak at the dominant frequency of the initial si-

usoid. Fitting a Gaussian shape to the peak defines a data-adaptivehaping operator as a band-pass filter implemented in the frequencyomain �dashed curve in Figure 7�. Inverse interpolation with the es-imated shaping operator recovers the original sinusoid �Figure 6b�.nalogous ideas in the model-preconditioning context were pro-osed by Liu and Sacchi �2001�.

a) b)

igure 4. The input data �b� are irregular samples from a sinusoid �a�.

iter=1

iter=3

iter=5

iter=7

iter=300

iter=1

iter=3

iter=5

iter=7

iter=300

50 100 150 200

Sample

50 100 150 200

Sample

Regularization Preconditioninga) b)

igure 5. �a� The first iterations and the final result of inverse interpo-ation with Tikhonov’s regularization using equation 1 and �b� with

odel preconditioning using equation 3. The regularization opera-or D is the first finite difference. The preconditioning operator P

D−1 is causal integration. The number of iterations is indicated inhe plot labels.

elocity estimation

The second example is an application of shaping regularization toeismic-velocity estimation. Figure 8 shows a time-migrated imagerom a historic Gulf of Mexico data set �Claerbout, 2006�. The imageas obtained by velocity continuation �Fomel, 2003�. The corre-

ponding migration velocity is shown in the right plot of Figure 8.haping regularization was used for picking a smooth velocity pro-le from semblance gathers obtained in the process of velocity con-

inuation.The task of this example is to convert the time-migration velocity

o the interval velocity. I use the simple approach of Dix inversionDix, 1955�, formulated as a regularized inverse problem �Valen-

iter=1

iter=3

iter=5

iter=7

iter=300

iter=1

iter=3

iter=5

iter=7

iter=300

50 100 150 200

Sample

50 100 150 200

Sample

Shaping 1 Shaping 2a) b)

igure 6. The first iterations and the final result of inverse interpola-ion with shaping regularization using equation 13. �a� The shapingperator H is low-pass filtering with a Gaussian smoother. �b� Thehaping operator H is band-pass filtering with a shifted Gaussian.haping by band-pass filtering recovers the sinusoidal shape of thestimated model. The number of iterations is indicated in the plot la-els.

Frequency (cycle)

Spectrum shaping 2

1.2

1

0.8

0.6

0.4

0.2

0

Rel

ativ

e m

agni

tude

0.30.250.20.150.10.050

igure 7. Spectrum of the estimated model �solid curve� fitted to ahifted Gaussian �dashed curve�. The Gaussian band-limited filterefines a shaping operator for recovering a band-limited signal.

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usor

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Flocity. Shaping by triangle smoothing.

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R34 Fomel

iano et al., 2004�. In this case, the forward operator L in equation 11s a weighted time integration. There is a choice in choosing thehaping operator H.

Figure 9 shows the result of inversion with shaping by trianglemoothing. While the interval-velocity model yields a good predic-ion of the measured velocity, it may not appear geologically plausi-le, because the velocity structure does not follow the structure ofeismic reflectors as seen in the migrated image.

Following the ideas of steering filters �Clapp et al., 1998; Clapp etl., 2004 � and plane-wave construction �Fomel and Guitton, 2006�, Istimate local slopes in the migration image using the method oflane-wave destruction �Fomel, 2002� and define a triangle plane-ave shaping operator H using the method of the previous section.he result of inversion, shown in Figures 10 and 11, makes the esti-ated interval velocity follow the geological structure and thus ap-

ear more plausible for direct interpretation. Similar results were ob-ained by Fomel and Guitton �2006�, using model parameterizationy plane-wave construction, but at a higher computational cost. Inhe case of shaping regularization, about 25 efficient iterations wereufficient to converge to the machine-precision accuracy.

CONCLUSIONS

Shaping regularization is a new general method for imposing reg-larization constraints in estimation problems. The main idea ofhaping regularization is to recognize shaping �mapping to the spacef acceptable functions� as a fundamental operation and to incorpo-ate it into iterative inversion.

There is an important difference between shaping regularizationnd conventional �Tikhonov’s� regularization from the user per-pective. Instead of trying to find and specify an appropriate regular-zation operator, the user of the shaping-regularization algorithmpecifies a shaping operator, which is often easier to design. Shapingperators can be defined following a triangle construction from localredictions or by combining elementary shapers.

I have shown two simple illustrations of shaping applications. Thexamples demonstrate a typical behavior of the method: enforcedodel compliance to the specified shape at each iteration and, inany cases, fast iterative convergence of the conjugate-gradient it-

0

1

2

3

Tim

e (s

)

1.5

2

2.5

3

Est

imat

ed in

terv

al v

eloc

ity (

km/s

)

Lateral position (km)8 9 10 11 12 13 14 15 16

igure 11. Seismic image from Figure 8 overlaid on the interval-ve-ocity model estimated with triangle plane-wave shaping regulariza-ion.

a) 0

1

2

3

Lateral position (km)

Time-migrated image

Tim

e (s

)

1.4

1.6

1.8

2

2.2

Vel

ocity

(km

/s)

8 9 10 11 12 13 14 15 16b)

0

1

2

3

Lateral position (km)

Picked migration velocity

Tim

e (s

)

8 9 10 11 12 13 14 15 16

igure 8. �a� Time-migrated image. �b� The corresponding migra-ion velocity from automatic picking.

a) 0

1

2

3

Interval velocity

Tim

e (s

)

1.6

1.8

2

2.2

Vel

ocity

(km

/s)

1.6

1.8

2

2.2

2.4

2.6

2.8

Vel

ocity

(km

/s)

b) 0

1

2

3

Lateral position (km)

Predicted migration velocity

Tim

e (s

)

8 9 10 11 12 13 14 15 16

Lateral position (km)8 9 10 11 12 13 14 15 16

igure 9. �a� Estimated interval velocity. �b� Predicted migration ve-

a) 0

1

2

3

Interval velocity

Tim

e (s

)

1.4

1.6

1.8

2

2.2

Vel

ocity

(km

/s)

1.5

2

2.5

3

Vel

ocity

(km

/s)

b) 0

1

2

3

Lateral position (km)

Predicted migration velocity

Tim

e (s

)

8 9 10 11 12 13 14 15 16

Lateral position (km)8 9 10 11 12 13 14 15 16

igure 10. �a� Estimated interval velocity. �b� Predicted migrationelocity. Shaping by triangle local plane-wave smoothing creates aelocity model consistent with the reflector structure.

esp

itta

ot

cf

wtrs

b

w

twpa

cfil

B

C

—

C

C

C

D

E

F

—

F

F

F

H

H

J

L

P

SS

Fe

Fi

Shaping regularization R35

ration. The model compliance behavior follows from the fact thathaping enters directly into the iterative process and provides an ex-licit control on the shape of the estimated model.

ACKNOWLEDGMENTS

I would like to thank Pierre Hardy and Mauricio Sacchi for inspir-ng discussions, TOTAL for partially supporting this research, andhree reviewers for helpful suggestions. Publication is authorized byhe director, Bureau of Economic Geology, The University of Texast Austin.

APPENDIX A

COMBINING SHAPING OPERATORS

General rules can be developed to combine two or more shapingperators for the cases when there are several features in the modelhat need to be characterized simultaneously. A general rule for

X sample

0

5

10

15

20

25

30

35

40

Y s

ampl

e

30 35 352520151050

igure A-1. Impulse response for a combination of two shaping op-rators smoothing in two different directions.

X sample

0

10

20

30

40

Y s

ampl

e

30 40

20

20

20100

30

40

20

20

10

0

z sam

ple

igure A-2. 3D impulse response for a combination of two 2D shap-ng operators smoothing in inline and crossline directions.

ombining two different shaping operators S1 and S2 can have theorm

S12 = S1 + S2 − S1S2, �A-1�

here one adds the responses of the two shapers and then subtractsheir overlap. An example is shown in Figure A-1, where an impulseesponse for oriented smoothing in two different directions is con-tructing from smoothing in each of the two directions separately.

Combining two operators that work in orthogonal directions cane accomplished with a simple tensor product, as follows:

Sxy = SxSy , �A-2�

here Sx and Sy are shaping operators that apply in orthogonal x- andy-directions, and Sxy is a combined operator that works in both direc-ions. An example is shown in Figure A-2, where two 2D shapersorking in orthogonal directions are combined to produce an im-ulse response of 3D shaping operator that applies smoothing along3D plane.

Constructing multidimensional recursive filters for helical pre-onditioning �Fomel and Claerbout, 2003� is significantly more dif-cult. It involves helical spectral factorization, which may create

ong, inefficient filters �Fomel et al., 2003�.

REFERENCES

ube, K., and R. Langan, 1999, On a continuation approach to regularizationfor crosswell tomography: 69th Annual International Meeting, SEG, Ex-panded Abstracts, 1295–1298.

laerbout, J. F., 1992, Earth soundings analysis: Processing versus inver-sion: Blackwell Scientific Publications, Co., Inc.—–, 2006, Basic earth imaging: Stanford Exploration Project, http://sepw-ww.stanford.edu/sep/prof/.

laerbout, J., and M. Brown, 1999, Two-dimensional textures and predic-tion-error filters: 61st Annual Conference and Exhibition, EAGE, Extend-ed Abstracts, Session 1009.

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