Migration velocity analysis in factorized VTI media · and considerable structural complexity, velocity model building is conveniently implemented in the prestack depth-migrated domain

CWP-451

Migration velocity analysis in factorized VTI media

Debashish Sarkar & Ilya TsvankinCenter for Wave Phenomena, Dept. of Geophysics, Colorado School of Mines

ABSTRACT

One of the main challenges in anisotropic velocity analysis and imaging is reli-able estimation of both velocity gradients and anisotropic parameters from re-flection data. Approximating the subsurface by a factorized VTI (transverselyisotropic with a vertical symmetry axis) medium provides a convenient way ofbuilding vertically and laterally heterogeneous anisotropic models for prestackmigration. The algorithm for P-wave migration velocity analysis (MVA) intro-duced here is designed for 2-D models composed of factorized VTI layers orblocks with constant vertical and lateral gradients in the vertical velocity VP0.The anisotropic MVA method is implemented as an iterative two-step procedurethat includes prestack depth migration (imaging step) followed by an update ofthe medium parameters (velocity-analysis step). The iterations for a particularblock continue until the corresponding reflection events in image gathers aresufficiently flat. The residual moveout of the migrated events, which is needed tocompute parameter updates, is described by a nonhyperbolic equation governedby two moveout parameters determined from semblance analysis.For piecewise-factorized VTI media treated here, the residual moveout of P-waveevents in image gathers is governed by four effective quantities in each block: (1)the normal-moveout (NMO) velocity Vnmo at a certain point within the block,

(2) the vertical velocity gradient kz, (3) the combination kx = kx

√

1 + 2δ ofthe lateral velocity gradient kx and the anisotropic parameter δ, and (4) theanellipticity parameter η. We show that all four parameters can be estimatedfrom the residual moveout for at least two reflectors within a block and establishthe minimum depth separation between the reflectors and the minimum lateraldistance to be covered by the image gathers. Stable inversion for the parameter ηalso requires using either long-spread data (with the maximum offset-to-depthratio no less than two) from horizontal interfaces or reflections from dippinginterfaces.To find the depth scale of the section and build a model for prestack depthmigration using the MVA results, the vertical velocity VP0 needs to be specifiedfor atleast a single point in each block. When no borehole information aboutVP0 is available, a well-focused image can often be obtained by assuming thatthe vertical-velocity field is continuous across layer boundaries. A synthetic testfor a three-layer model with a syncline structure confirms the accuracy of our

MVA algorithm in estimating the interval parameters Vnmo, kz , kx, and η andillustrates the influence of errors in the vertical velocity on the image quality.

1 INTRODUCTION

Most existing velocity-analysis methods for VTI (trans-versely isotropic with a vertical symmetry axis) mediaapproximate the subsurface with homogeneous or verti-cally heterogeneous layers or blocks (e.g., Alkhalifah andTsvankin, 1995; Le Stunff and Jeannot, 1998; Tsvankin,2001; Grechka et al., 2002). Anisotropic layers, however,

are often characterized by non-negligible lateral veloc-ity gradients that may distort the shape of underlyingreflectors and cause errors in the anisotropic parame-ters. Since lateral homogeneity is an inherent assump-tion in time imaging, whether isotropic or anisotropic, itis justified to ignore lateral gradients in the time-domainvelocity analysis of P-waves in VTI media (e.g., Alkhal-ifah, 1997; Han et al., 2000). In contrast, anisotropic

120 D. Sarkar & I. Tsvankin

depth imaging has to account properly for both verticaland lateral variations of the velocity field.

An analytic correction of normal-moveout (NMO)ellipses for lateral velocity variation in anisotropic mediawas developed by Grechka and Tsvankin (1999). Theirmethod, however, is limited to horizontal layers, smalllateral velocity gradients, and the hyperbolic portion ofreflection moveout. Also, for purposes of depth imaging,we are interested in estimating lateral velocity variationrather than just removing its influence on anisotropic in-version. The main problem in reconstructing a spatiallyvarying anisotropic velocity field is caused by the trade-offs between the velocity gradients, anisotropic param-eters, and the shapes of the reflecting interfaces. Evenin isotropic media, some trade-offs between the veloc-ity field and reflector shapes cannot be resolved even inisotropic media without a priori information. A practi-cal way to incorporate vertical and lateral velocity vari-ations into anisotropic velocity analysis without exces-sively compromising the uniqueness of the solution is toadopt the so-called factorized anisotropic model in whichthe ratios of the stiffness coefficients (and, therefore, theanisotropic parameters) are constant.

Here, we consider a model composed of factorizedVTI blocks, where each block is bounded by plane or ir-regular interfaces. The problem is treated in two dimen-sions, which implies that the vertical incidence planethat contains sources and receivers should coincide withthe dip plane of the subsurface structure. The verticalP-wave velocity VP0 is assumed to vary linearly withineach block, so the vertical (kz) and lateral (kx) gra-dients in VP0 are constant. The kinematics of P-wavepropagation in each block can be described by five pa-rameters: the velocity VP0 defined at a certain spatiallocation, the gradients kz and kx, and Thomsen (1986)anisotropic parameters ε and δ. Although it is possibleto introduce jumps in velocity across the boundaries ofthe blocks, this model can be conveniently used to gen-erate smooth velocity fields required by many migrationalgorithms (in particular, those based on ray tracing).

Since our goal is to estimate the relevant VTIparameters and carry out depth imaging for modelswith significant lateral and vertical velocity variationand considerable structural complexity, velocity modelbuilding is conveniently implemented in the prestackdepth-migrated domain (e.g., Stork, 1991; Liu, 1997).Parameter estimation in the post-migrated domain, usu-ally referred to as migration velocity analysis (MVA),consists of two main steps: (1) parameter update de-signed to minimize the residual moveout of events in theimage gathers, and (2) prestack depth migration thatcreates an image of the subsurface using the updatedparameters estimated in step (1). These two steps areiterated until the events in the image gathers are suffi-ciently flat. Note that MVA is quite robust in the pres-ence of random noise because migration improves thesignal-to-noise ratio (Gardner et. al., 1974; Liu, 1997).

The parameter-estimation methodology employedhere is based on the results of Sarkar and Tsvankin(2002), hereafter referred to as Paper I. Although depthimaging of P-wave data requires knowledge of the fiveparameters listed above (VP0, kz, kx, ε, and δ) in eachblock, Paper I shows that the moveout of events in imagegathers is governed by the following four combinationsof these parameters:(1) the NMO velocity at a certain point on the surfaceof the factorized layer or block: Vnmo ≡ VP0

√1 + 2δ;

(2) the vertical velocity gradient kz;(3) the lateral velocity gradient combined with the pa-rameter δ: kx = kx

√1 + 2δ;

(4) the anellipticity parameter η ≡ (ε − δ)/(1 + 2δ).If prestack migration is performed with the correct val-ues of Vnmo, kz, kx, and η, the image gathers for reflec-tions from both horizontal and dipping interfaces areflat. To decouple the horizontal gradient kx from thecoefficient δ and determine the other anisotropic coeffi-cient ε, the velocity VP0 has to be known at a certainpoint within the factorized block (Paper I).

The paper starts with a discussion of the minimuminformation required to estimate the model parametersfrom P-wave moveout data. Then we give a descrip-tion of the MVA methodology including nonhyperbolicmoveout analysis on image gathers needed to constrainthe anisotropic velocity field. The accuracy of the algo-rithm and its robustness in the presence of random noiseare assessed by synthetic tests for a single layer and amultilayered factorized VTI medium. We also discussdifferent ways to specify the vertical velocity VP0 andthe influence of errors in VP0 on the inverted values ofthe other parameters and on the quality of the migratedimage.

2 PARAMETER ESTIMATION IN A

FACTORIZED VTI LAYER

Here, we use the results of Paper I to evaluate the feasi-bility of estimating the parameters of a factorized VTIlayer from P-wave reflection data. By replacing the ac-tual factorized v(x, z) model with narrow vertical stripsof factorized v(z) media, Paper I demonstrates that themoveout of a single horizontal event in an image gatheris governed by the effective values of the NMO velocityand the parameter η:

v2

nmo(x, t0) = V 2(x)(1 + 2δ)e2kzt0 − 1

2kz t0, (1)

η(x, t0) =1

8

{

(1 + 8η) (e2kz t0 − 1) kz t02(ekzt0 − 1)2

+ 1

}

; (2)

V (x) ≡ VP0 + kx x is the vertical P-wave velocity atthe surface, and t0 ≡ t0(x, z) is the zero-offset time atlocation x from a horizontal reflector at depth z.

If long-offset data needed to constrain η (Grechka

Migration velocity analysis and anisotropy 121

and Tsvankin, 1998) have been acquired, moveout anal-ysis of a single event can yield estimates of bothvnmo(x, t0) and η(x, t0). Next, suppose that P-wave trav-eltimes from two horizontal reflectors sufficiently sepa-rated in depth are available. Then the ratio of the NMOvelocities for these two events (vnmo,1 and vnmo,2) canbe used to find [equation (1)]

v2

nmo,1(x, t0,1)

v2nmo,2(x, t0,2)

=t0,2 (e2kz t0,1 − 1)

t0,1 (e2kz t0,2 − 1), (3)

where t0,1 and t0,2 are the zero-offset times for the twoevents. According to equation (3), conventional hyper-bolic moveout analysis of two horizontal events locatedin the same factorized block can provide an estimateof the vertical gradient kz. Knowledge of kz and thezero-offset time t0 is sufficient for obtaining the anellip-ticity parameter η from equation (2) applied to one orboth reflection events. The remaining two key quanti-ties, Vnmo = VP0

√1 + 2δ and kx = kx

√1 + 2δ, can then

be computed from equation (1), if the effective NMOvelocities are determined at two or more locations x.

We conclude that the moveout of horizontal eventsat two different depths and two image locations pro-vides enough information to estimate the parametersVnmo, kz, kx, and η. For the special case of a factorizedv(z) medium with a constant vertical gradient kz, themoveouts of two horizontal events at a single image lo-cation can be inverted for the parameters Vnmo, kz, andη.

As shown in Paper I, reflection moveout of dippingevents in factorized v(x, z) VTI media is controlled bythe same parameters (Vnmo, kz, kx, and η) as that ofhorizontal events. Most importantly, NMO velocity ofevents dipping at 25-30◦ or more is highly sensitiveto the parameter η (Alkhalifah and Tsvankin, 1995;Tsvankin, 2001), whereas the inversion of nonhyper-bolic moveout from horizontal reflectors for η may sufferfrom instability (Grechka and Tsvankin, 1998). There-fore, the inclusion of dipping events in velocity analysisis helpful in obtaining accurate estimates of η; also, dip-dependent reflection moveout provides additional infor-mation about the parameters Vnmo, kz, and kx.

Still, even if both horizontal and dipping events areavailable, the parameters VP0, kx, ε, and δ remain gener-ally unconstrained by P-wave reflection traveltimes. Inparticular, the vertical velocity VP0 is needed to definethe depth scale of the VTI model in the migration ofP-wave data. Hence, to build an anisotropic model fordepth imaging, at least one medium parameter must beknown a priori. Unless specified otherwise in the syn-thetic data examples discussed below, the velocity VP0 isassumed known at some location on the surface of eachfactorized layer. Given this information about VP0, wecan use velocity analysis of P-wave data to estimate theparameters kz, kx, ε, and δ.

3 ALGORITHM FOR MIGRATION

VELOCITY ANALYSIS

Inversion of seismic data is a nonlinear problem thatcan be solved through an iterative application of migra-tion and velocity updating. Migration creates an imageof the subsurface for trial values of the medium param-eters, and then velocity analysis is used to update themodel for the next run of the migration code. This itera-tive procedure, conventionally called migration velocity

analysis (MVA), is continued until a certain criterion(e.g., small residual moveout of events in image gath-ers) is satisfied.

Here, we apply anisotropic prestack depth migra-tion (the migration algorithm is described in detail inPaper I) and tomographic velocity update to P-wavedata acquired over the subsurface composed of factor-ized v(x, z) VTI blocks. The iterations are stopped whenthe residual moveout for at least two reflectors in eachfactorized block is close to zero (i.e., the migrated depthstays the same to within a specified fraction of the wave-length for different offsets). The overall organization ofour MVA algorithm is similar to that developed by Liu(1997) for isotropic media, but the VTI model is char-acterized, for P-waves, by two additional parameters –ε and δ.

The tomographic update of the medium parametersis based entirely on the residual moveout of events inimage gathers. For horizontal reflectors embedded in aweakly anisotropic homogeneous VTI medium, the mi-grated depth z

Min image gathers can be written as

(Paper I):

z2

M(h) ≈ z2

M(0) + h2V 2

P0,M

(

1

V 2

nmo,T

− 1

V 2

nmo,M

)

+

2h4

h2 + z2

T

(

ηM

V 2

nmo,T

V 2

nmo,M

− ηT

V 2

nmo,M

V 2

nmo,T

)

, (4)

where the subscripts T and M denotes the true and mi-gration medium parameters, respectively, h is the half-offset, and z

Tis the true zero-offset depth of the re-

flector. Equation (4) is nonhyperbolic and governed bytwo independent parameters – Vnmo and η. The NMOvelocity Vnmo controls the hyperbolic (described by theh2–term) part of the moveout curve and also contributesto the nonhyperbolic (h4) term, while η influences non-hyperbolic moveout only. A similar closed-form expres-sion is not available for dipping reflectors, but both thehyperbolic and nonhyperbolic portions of the residualmoveout curve for dipping events also depend on Vnmo

and η (Paper I).As discussed above, the residual moveout of P-

waves in factorized v(x, z) VTI media is a function of theparameters Vnmo, kz, kx, and η. Although it is difficultto express the migrated depth z

Min laterally heteroge-

neous media analytically in terms of these parameters,the residual moveout equation can be cast in a form


similar to that in equation (4):

z2

M(h) ≈ z2

M(0) + Ah2 + B

2h4

h2 + z2

M(0)

. (5)

A and B are dimensionless constants that describe thehyperbolic and nonhyperbolic portions of the moveoutcurve, respectively. Numerical tests (see below) confirmthat the functional form in equation (5) with fitted co-efficients A and B provides a good approximation forP-wave moveout in long-spread image gathers.

To apply equation (5) in velocity analysis, we firstpick an approximate value of the zero-offset reflectordepth z

M(0) on the migrated stacked section. The pa-

rameters A and B are obtained by a 2-D semblancescan on image gathers at each migrated zero-offset depthpoint. The best-fit combination of A and B that max-imizes the semblance value is substituted into equa-tion (5) to describe the residual moveout. It should beemphasized that the coefficients A and B in our algo-rithm are not directly inverted for the parameters Vnmo,kz, kx, and η. Rather, the only role of A and B is inproviding an adequate functional approximation for theresidual moveout.

After estimating the residual moveout in imagegathers, we update the N -element parameter vector λ

using the algorithm described in Appendix A. The up-date ∆λ of the parameter vector is obtained by solvingthe system of linear equations,

AT

A∆λ = AT

b . (6)

Here A is a matrix with M ·P rows (P is the total num-ber of image gathers used in the velocity analysis and Mis the number of offsets) and N columns that includesthe derivatives of the migrated depth with respect tothe medium parameters. The superscript T denotes thetranspose, and b contains the migrated depths that de-fine the residual moveout. The full definitions of thematrix A and vector b are given in Appendix A.

For all examples described below, each iteration ofthe MVA consists of the following four steps:(1) prestack depth migration with a given estimate ofthe medium parameters;(2) picking along two reflectors in each VTI block todelineate the reflector shapes;(3) semblance scanning using equation (5) to estimateA and B for image points along each reflector;(4) application of equation (6) to update the mediumparameters in such a way that the variance of the mi-grated depths as a function of offset is minimized (seeAppendix A for more details about the minimizationprocedure).

Steps 1–4 are repeated until the magnitude of resid-ual moveout of events in image gathers becomes suffi-ciently small.

0

1000

2000

dept

h (m

)

2000 4000cmp coordinate (m)surface coordinate (m)

dep

th (

m)

Figure 1. True image of two reflectors embedded in a fac-torized v(x, z) VTI medium with the parameters VP0(x =3 km, z = 0) = 2600 m/s, kz = 0.6 s−1, kx = 0.2 s−1,ε = 0.1, and δ = −0.1. The corresponding effective parame-ters are Vnmo(x = 3 km, z = 0) = 2326 m/s, kz = 0.6 s−1,kx = 0.18 s−1, and η = 0.25.

0

1000

2000

dept

h (m

)

2000 4000cmp coordinate (m) a

bB

A

(a) (b)

surface coordinate (m)

dep

th (

m)

Figure 2. (a) Image of the model from Figure 1 obtainedusing a homogeneous isotropic velocity field with VP0 = 2600m/s. (b) Semblance contour plot computed from equation (5)for the shallow reflector at the surface location 3 km.

4 EXAMPLE WITH A SINGLE

FACTORIZED LAYER

First, we consider two irregular reflectors embedded ina factorized v(x, z) VTI medium with kz > kx > 0and a positive value of η typical for shale formations(Figure 1). For the first application of prestack depthmigration, we choose a homogeneous, isotropic medium(VP0 = 2600 m/s, kz = kx = ε = δ = 0) as the initial ve-locity model. The migrated stacked image in Figure 2ais clearly inferior to the true image in Figure 1. We startthe velocity-updating process by manually picking alongboth imaged reflectors to outline their shapes. Then


0

1000

2000

dept

h (m

)

2000 4000cmp coordinate (m)

0

1000

2000

dept

h (m

)


(a) (b)

surface coordinate (m) surface coordinate (m)

dep

th (

m)

dep

th (

m)


Figure 3. Stacked image after (a) four iterations; (b) eightiterations.

1000

1200

1400

1600

dept

h (m

)

2000 4000

a b c d e

dep

th (

m)

Figure 4. Residual moveout in image gathers for both re-flectors at the surface location 3 km: (a) for the initial model;(b) after two, (c) four, (d) six, and (e) eight iterations. Theresidual moveout is minimized during the velocity-updatingprocess.

equation (5) is used to compute two-parameter sem-blance scans for each reflector and evaluate the residualmoveout in the image gathers.

One such semblance scan computed for the shallowreflector at the surface coordinate 3 km is displayed inFigure 2b. The values of A and B that correspond tothe maximum semblance coefficient in Figure 2b pro-vide an accurate description of residual moveout at thislocation. Although a certain degree of trade-off existsbetween A and B, any pair of values inside the inner-most semblance contour gives almost the same varianceof the migrated depths. Note that the interplay betweenA and B is similar to that between the NMO velocityand parameter η in the inversion of P-wave nonhyper-bolic reflection moveout (Grechka and Tsvankin, 1998;Tsvankin, 2001).

0

1000

2000

dept

h (m

)


0

1000

2000

dept

h (m

)


(a) (b)


dep

th (

m)

dep

th (

m)


Figure 5. (a) Stacked image obtained after velocity anal-ysis with the wrong value of the vertical velocity VP0(x =3 km, z = 0) = 2000 m/s. The estimated medium parametersare kz = 0.58± 0.02 −1, kx = 0.15 ± 0.0 s−1, ε = 0.51 ± 0.0,δ = 0.17±0.01. (b) Stacked image for the correct medium pa-rameters (Figure 1). Since VP0 in section (a) is smaller thanthe true value, both reflectors are shifted up with respect totheir correct positions in section (b).

For purposes of velocity analysis, we use the imagegathers at 12 equally spaced surface locations between3 km and 4.2 km. The maximum offset-to-depth ratiofor the selected image gathers at the shallow reflector isclose to two, which is marginally suitable for estimatingthe parameter η. Tighter constraints on η are providedby the NMO velocities of reflections from the dippingsegments of the shallow reflector (the dips exceed 30◦

in the middle of the section).After the residual moveout has been evaluated, we

fix the vertical velocity VP0(x = 3000 m, z = 0) = 2600m/s at the correct value and update the parameters kz,kx, ε, and δ using equation (6). The stacked images afterfour (Figure 3a) and eight (Figure 3b) iterations illus-trate the improvements in the focusing and positioningof both reflectors during the velocity update. The mag-nitude of the residual moveout for both reflectors de-creases as the model parameters converge toward theiractual values (Figure 4). The velocity-updating proce-dure is stopped after eight iterations because events inall analyzed image gathers are practically flat.

The inverted model parameters are close to the cor-rect values: kz = 0.58 ± 0.02 s−1, kx = 0.2 ± 0.0 s−1,ε = 0.12 ± 0.01, and δ = −0.09 ± 0.01. The errorbars were computed by assuming a standard deviationof ±5 m in picking migrated depths on the selectedimage gathers and substituting this picking error intoequation (6) to find the corresponding deviations of themodel parameters near the actual solution.

The accurate results of the above test were ob-tained with the correct value of the vertical velocity at agiven point on the surface of the factorized layer. Next,we apply the MVA method with an erroneous value ofVP0(x = 3 km, z = 0) = 2000 m/s, which is 23% smallerthan the true velocity (2600 m/s). The stacked images ofboth reflectors obtained after the velocity analysis (Fig-ure 5a) are well focused, which indicates that the im-


0

1

time

(s)

0 1000offset (m)

0

1000

2000

dept

h (m

)


dep

th (

m)

tim

e (s

)

offset (m)

(a) (b)

Figure 6. Influence of noise on the velocity analysis andmigration. (a) A shot gather from the dataset in Figure 1after the addition of Gaussian noise; the signal-to-noise ratiois 1.5. (b) The image obtained for the noisy dataset.

0

1

time

(s)

0 1000offset (m)

0

1000

2000

dept

h (m

)


dep

th (

m)

tim

e (s

)

(a) (b)

offset (m)

Figure 7. Influence of noise on the velocity analysis and

migration. (a) The same shot gather as in Figure 6, but witha more severe noise contamination (signal-to-noise ratio is1). (b) The image obtained for the noisy dataset.

age gathers have been flattened. Indeed, although theestimated medium parameters listed in the caption ofFigure 5 are distorted, the effective parameters respon-sible for the residual moveout are close to their actualvalues: Vnmo(x = 3 km, z = 0) = 2315 m/s, kz = 0.58s−1, kx = 0.17 s−1, and η = 0.25.

This result corroborates the analysis of residualmoveout in Paper I and confirms that our algorithmconverges toward the correct parameters Vnmo, kz, kx,and η, even if the vertical velocity VP0 on the surface ofthe layer is poorly known. Since VP0 assumed in the ve-locity analysis is too low, however, both reflectors inFigure 5a are imaged at depths that are about 23%smaller than the actual ones in Figure 5b. The depthdistortion also leads to the rotation of the dipping seg-ments of the reflecting interfaces, which is discussed inmore detail below.

5 MVA IN THE PRESENCE OF NOISE

To evaluate the influence of noise on the estimation ofthe medium parameters and the quality of imaging, weadded Gaussian noise to the data set from Figure 1. The

signal-to-noise ratio, measured as the ratio of the peakamplitude of the signal to the root-mean-square (rms)amplitude of the background noise, is about 1.5, andthe frequency bands of the noise and signal are identi-cal (Figure 6a). The estimates of the medium parame-ters obtained after the migration velocity analysis withthe correct value of VP0 at the surface location 3 kmare as follows: kz = 0.56 ± 0.04 s−1, kx = 0.2 ± 0.0s−1, ε = 0.12 ± 0.02, and δ = −0.09 ± 0.02. The errorbars were computed in the same way as those for thenoise-free synthetic example above (Figure 3), but thedepth-picking error for all offsets and image locationswas assumed to be 15 m instead of 5 m. Clearly, thenoise contamination did not cause measurable errors inthe medium parameters or noticeable distortions in thestacked image (Figure 6b).

Even for the much more severely contaminateddata set in Figure 7, the inverted medium parametersare close to the actual values: kz = 0.52 ± 0.07 s−1,kx = 0.2±0.01 s−1, ε = 0.13±0.03, and δ = −0.07±0.03,Here the error bars were computed under the assump-tion that the noise increased the depth-picking error to20 m. (Since the dominant wavelength in this examplewas about 80 m, picking errors are unlikely to exceed 20m, even for a substantial level of noise.) Also, despite thelow signal-to-noise ratio, the migrated stacked section inFigure 7b has a sufficiently high quality, comparable tothat of the true image in Figure 1.

We conclude that, the migration velocity analysisemployed here gives reliable estimates of the anisotropicparameters and velocity gradients in the presence of ran-dom noise. One aiding factor is that the MVA operateson migrated data, which have a higher signal-to-noiseratio than do those of the original records because ofpartial stacking applied to the data during the migrationstep. The semblance (coherency) operator used to eval-uate the residual moveout on image gathers also con-tributes to the robustness of the parameter estimationby suppressing remaining random noise in the migrateddata.

6 SENSITIVITY STUDY

The above results demonstrate that, in principle, theresidual moveout from two reflectors in a factorized layeris sufficient to estimate the four key parameters Vnmo,kz, kx, and η. This section is devoted to two importantpractical issues related to the implementation of our al-gorithm. By performing a series of numerical tests, weestablish the minimum depth separation between thetwo reflectors and the minimum lateral spread of theimage gathers (i.e., the difference between the largestand smallest surface coordinates of the image locations)needed for stable parameter estimation.

Consider two horizontal reflectors embedded in thefactorized v(x, z) medium with the parameters listed in


0 500 10000

0.05

0.1

0.15

thickness (m)

erro

r in

kz (

1/s)

offset/depth=2offset/depth=1.5

d (m)

Figure 8. Influence of the vertical distance d between thetwo horizontal reflectors used in the velocity analysis on theabsolute error in the vertical gradient kz . The depth of theshallow reflector is 1 km; the maximum offset is 2 km forthe upper curve and 1.5 km for the lower curve. The modelparameters are VP0(x = 3 km, z = 0) = 2600 m/s, kz = 0.6s−1, kx = 0.2 s−1, ε = 0.2, and δ = 0.1.

0 500 1000−0.02

0

0.02

0.04

d (m)

erro

r in

kx (

s−1 ) offset/depth=2 & 1.5

Figure 9. Influence of the distance between the two horizon-tal reflectors on the absolute error in the horizontal gradientkx. The parameters are the same as in Figure 8.

0 500 10000

0.02

0.04

d (m)

erro

r in

ε


Figure 10. Influence of the distance between the two hor-izontal reflectors on the absolute error in the parameter ε.The parameters are the same as in Figure 8.

0 500 10000

0.02

0.04

0.06

d (m)

erro

r in

δ


Figure 11. Influence of the distance between the two hor-izontal reflectors on the absolute error in the parameter δ.The parameters are the same as in Figure 8.

0 200 400 6000

0.05

0.1

0.15

lateral extent (m)

erro

r in

kx (

s−1 ) offset/depth=2

offset/depth=1.5

Figure 12. Influence of the lateral spread of the image gath-ers on the absolute error in kx. The reflector depths are 1 kmand 1.2 km; the other parameters are the same as in Fig-ure 10. The velocity analysis is performed on 12 image gath-ers, each with 20 offsets.

the caption of Figure 8. The depth of the shallow re-flector is fixed at 1000 m, while the depth of the secondreflector varies from 1050 m to 2000 m. Figures 8–9 il-lustrate the dependence of the error in the estimatedparameters kz, kx, ε, and δ on the distance between thereflectors. The errors in the parameters were computedfrom equation (6) assuming that the error in pickingthe migrated depths is ±5 m. The velocity analysis op-erated with the residual moveout on 12 image gathers(with 20 offsets each) whose horizontal coordinates spana distance of 1200 m. For all tests, the vertical velocityat one location on the surface was held at the correctvalue.

For the parameters kz, ε and δ, the dependence ofthe estimated error on the distance d between the reflec-tors has a similar character (Figures 8, 10, and 11). Theerror initially decreases rapidly with increasing d andthen becomes almost constant as d approaches 500 m.For a maximum offset-to-depth ratio (at the shallow re-flector) of two, the error curves flatten out for d ≈ 250m, which is equal to 1/5 of the depth of the bottom re-


flector. If the maximum offset-to-depth ratio is 1.5, thecurve flattens out for a larger depth d ≈ 350 m (≈1/4of the depth of the bottom reflector).

This behavior of the error curves is in good agree-ment with the analysis of the effective NMO velocityand parameter η in Paper I. Accurate estimation fothe vertical gradient kz, and then the NMO velocity atthe surface of the factorized layer, requires a sufficientlylarge difference between the NMO velocities of the twoevents used in the velocity analysis [see equation (3)].In other words, the reflectors should be sufficiently sep-arated in depth to resolve the interval NMO velocity,which carries information about the gradient kz. An ac-curate estimate of kz makes it possible to obtain Vnmo atthe surface and then, using the nonhyperbolic portionof the moveout curve, the parameter η. The minimumsuitable vertical distance d found here is close to theminimum layer thickness conventionally assumed in in-terval velocity estimation based on the Dix equation.

In contrast, the error in the horizontal gradient kx

is practically insensitive to variations in the distance be-tween the two reflectors (Figure 9) because the lateralspread of the coordinates of the image gathers is keptconstant at 1.2 km. The influence of the maximum hor-izontal distance between the image gathers on the errorin kx is shown in Figure 12. As expected, the gradientkx becomes better constrained with increasing lateralspread of the image gathers, with the error curve flat-tening out for spreads exceeding 300-400 m.

Note that the errors in all parameters reduce withincreasing number of offsets in the image gathers, whichcan influence the sensitivity estimates. Although the re-sults of the error analysis also depend on the anisotropiccoefficients ε and δ and the velocity gradients, this de-pendence is not significant if the velocity update is per-formed with reasonable constraints on the model pa-rameters.

7 TEST FOR A MULTILAYERED MODEL

After performing a series of tests for a single factorizedlayer, we apply the algorithm to a three-layer modelshown in Figure 13. Each layer contains two reflectinginterfaces, as requried in our method, with every sec-ond reflector serving as the boundary between layers.The first and third layers are vertically heterogeneous[v(z)] and isotropic, while the second layer is a factor-ized, laterally heterogeneous [v(x, z)] medium. All inter-faces are quasi-horizontal, with the largest dips (at theflanks of the syncline) of 10◦ or less. The model is de-signed to represent a typical depositional environmentin the Gulf of Mexico, where anisotropic shale layers(the middle layer in Figure 13) are often embedded be-tween isotropic sands.

For the velocity analysis we use image gathers lo-cated along the left flank of the syncline with the sur-face coordinates ranging from 4400 m to 5600 m; the

0

2000

dept

h (m

)

5 10surface coordinate (km)

Figure 13. True image of a three-layer factorized medium.Every second reflector (indicated here with arrows) repre-sents the bottom of a layer. The parameters of the first sub-surface layer are VP0(x = 4000 m, z = 0 m) = 1500 m/s,kz = 1.0 s−1, and kx = ε = δ = 0; for the second layer,VP0(x = 4000 m, z = 800 m) = 2300 m/s, kz = 0.6 s−1,kx = 0.1 s−1, ε = 0.1, and δ = −0.1; for the third layer,VP0(x = 4000 m, z = 1162 m) = 2718 m/s, kz = 0.3 s−1,and kx = ε = δ = 0.

0

0.5

1

kx

kz ε δ

Figure 14. Estimated (◦) and true (?) parameters of thefirst layer obtained using the correct VP0(x = 4000 m, z =0 m) = 1500 m/s on the surface.

maximum offset-to-depth ratio for the image gathers isclose to two. The medium parameters are estimated inthe layer-stripping mode starting at the surface. For thefirst (top) layer, the vertical velocity is assumed to beknown at a single surface location [VP0(x = 4000 m, z =0 m) = 1500 m/s]. The chosen value of VP0 correspondsto that for water-bottom sediments; on land, VP0 at the

−0.2

0

0.2

0.4

0.6

kx

kz ε δ

Figure 15. Estimated (◦) and true (?) parameters of thesecond layer obtained using the correct VP0(x = 4000 m, z =800 m) = 2300 m/s at the layer’s top.


−0.2

0

0.2

0.4

kx

kz ε δ

Figure 16. Estimated (◦) and true (?) parameters of thethird layer obtained using the correct VP0(x = 4000 m, z =1162 m) = 2718 m/s at the layer’s top.

0

2000

dept

h (m

)


Figure 17. Stacked image obtained after prestack depth mi-gration using the estimated parameters from Figures 14–16.The vertical velocity VP0 at the top of each layer was known.

top of the model may be estimated from near-surfacevelocity measurements. Starting with a homogeneousisotropic model (VP0 = 1500 m/s) the parameters kz,kx, ε, and δ in the first layer, obtained from the migra-tion velocity analysis with the correct vertical velocityVP0(x = 4000 m, z = 0 m), are close to the true values(Figure 14).

To estimate the medium parameters in the secondand third layers, we need to fix the vertical velocity ata certain spatial location in each layer. Three differentscenarios for choosing VP0 in the second and third layersare examined below.

−0.2

0

0.2

0.4

0.6

kx

kz ε δ

Figure 18. Estimated (◦) and true (?) parameters of thesecond layer obtained with an inaccurate value of the verticalvelocity at the top of the second layer [VP0(x = 4000 m, z =800 m) = 2600 m/s].

−0.2

0

0.2

0.4

kx

kz ε δ

Figure 19. Estimated (◦) and the true (?) parameters of thethird layer obtained with an inaccurate value of the verticalvelocity at the top of the second layer [VP0(x = 4000 m, z =800 m) = 2600 m/s] but the correct VP0(x = 4000 m, z =1208 m) = 2732 m/s at the top of the third layer.

0

2000

dept

h (m

)


Figure 20. Stacked image obtained after prestack depth mi-gration using the estimated parameters from Figures 14, 18,and 19. The vertical velocity VP0 at the top of the secondlayer was inaccurate.

7.1 VP0 at the top of each layer is known

Suppose a vertical borehole was drilled at the surfacelocation 4000 m, and the vertical velocity at the top ofthe second and third layers was measured from soniclogs or check shots. Prestack depth migration with theestimated parameters of the first layer yields the depthof the top of the second layer at the surface location4000 m. Using the correct value of the vertical velocityat this point VP0(x = 4000 m, z = 800 m) = 2300 m/s,

−0.2

0

0.2

0.4

0.6

kx

kz ε δ

Figure 21. Estimated (◦) and true (?) parameters of the sec-ond layer obtained assuming that VP0 is continuous betweenthe first and second layers at the point (x = 3900 m, z =800 m).


−0.2

0

0.2

0.4

0.6

kx

kz ε δ

Figure 22. Estimated (◦) and true (?) parameters ofthe third layer obtained assuming that VP0 is continuousbetween the first and second layers at the point (x =3900 m, z = 800 m) and between the second and third lay-ers at the point (x = 5937 m, z = 1483 m).

0

2000

dept

h (m

)


Figure 23. Stacked image obtained after prestack depthmigration using the estimated parameters from Figures 14,21, and 22. The vertical velocity was assumed to be con-tinuous between the first and second layers at the point(x = 3900 m, z = 800 m) and between the second and thirdlayers at the point (x = 5937 m, z = 1483 m). True (•) andestimated (◦) points of continuity are also indicated.

we carry out the velocity analysis for the second layer,which results in good estimates of all four parameters(Figure 15). Repeating the same procedure for the thirdlayer with the velocity VP0(x = 4000 m, z = 1162 m) =2718 m/s, we obtain the interval parameters close to thetrue values (Figure 16).

The shapes and depths of the reflectors imaged forthe reconstructed velocity model (Figure 17) closely re-semble those on the true image (Figure 13). This testconfirms that migration velocity analysis in layered fac-torized VTI v(x, z) media can be used to invert for thethe velocity gradients kz and kx and the anisotropic co-efficients ε and δ if the vertical velocity is known at asingle point in each layer.

7.2 VP0 in the second layer is incorrect

Now suppose that the vertical velocity VP0(x =4000 m, z = 800 m) used for the top of the second layer

has error (2600 m/s instead of 2300 m/s). Althoughthis error in VP0 causes distortions in the inverted val-ues of the other parameters (Figure 18), the effectivequantities Vnmo(x = 4000 m, z = 800 m) = 2080 m/s,kz = 0.56 s−1, kx = 0.09 s−1, and η = 0.23 do not signif-icantly differ from the true values, which corroboratesour results for a single layer (Figure 5). Since the as-sumed value of VP0(x = 4000 m, z = 800 m) is higherthan the correct velocity, the second layer is stretchedin depth by about 13%, and the bottom of the synclineis imaged at a depth that is 80m too large (Figure 20).This depth stretch in the second layer also causes a tiltof the syncline’s flanks whose dips in Figure 20 exceedthe true values.

To continue the velocity analysis, we fix the verticalvelocity at the imaged top of the third layer at the cor-rect value. Despite the depth shift of the top of the thirdlayer, the algorithm yields accurate values of all four in-terval parameters (Figure 19).Because of the depth anddip distortions in the second layer, however, the two bot-tom reflectors are imaged at somewhat greater depthsand are slightly deformed (Figure 20). In particular, onthe left side of the section the fifth and sixth reflectorsare no longer horizontal; they have acquired mild dipsto conform to the stretched synclinal structure above.

7.3 VP0 is continuous across the boundaries

If no borehole information is available, one assump-tion that might be made is that the velocity VP0 isa continuous function of depth at a certain horizon-tal coordinate. To identify this point of continuity atthe boundary between the first and second layers, weexamine the moveout along the third and fourth reflec-tors (only for offsets smaller than 1000 m) after migra-tion with an isotropic homogeneous velocity field in thesecond layer. The migration velocity was chosen to beequal to the estimated velocity at the bottom of thefirst layer (i.e., at the second reflector). To select thepoint of continuity, we pick the surface coordinate withthe smallest residual moveout on the image gathers atthe third and fourth reflectors. This criterion yieldedx = 3900 m, which is sufficiently close to the true pointof continuity for the second reflector (x = 4000 m).Using the estimated vertical velocity at x = 3900 m[VP0(x = 3900 m, z = 800 m) = 2316 m/s], we estimatethe parameters of the second layer with high accuracy(Figure 21).

To find the point of continuity between the secondand third layers, we again perform prestack depth mi-grations assuming that the third layer is homogeneousand isotropic. Since the second layer is laterally hetero-geneous, the migration velocities range from 2400 m/sto 3400 m/s. Applying the criterion of minimum residualmoveout for the fifth and the sixth reflectors, the pointof continuity was found at (x = 5937 m, z = 1483 m),where the vertical velocity is VP0 = 2900 m/s. Although


the location (x = 5937 m, z = 1483 m) is shifted by al-most 1000 m from the true continuity point betweenthe second and third layers, the results of the velocityanalysis (Figure 22) and imaging (Figure 23) are quitesatisfactory.

In the absence of borehole data, the assumption ofcontinuous vertical velocity provides a practical way tobuild an anisotropic heterogeneous model for prestackmigration. Depending on the complexity of the model,however, the point of continuity may be estimated witha substantial lateral shift or may not exist at all. Still,our tests show that for models without steep dips orstrong lateral heterogeneity, an error in identifying thepoint of continuity does not distort the effective param-eters Vnmo, kz, kx, and η. Therefore, the migrated sec-tion would still be well focused, although the imagedreflectors would be subject to a depth stretch.

8 DISCUSSION AND CONCLUSIONS

Approximating heterogeneous VTI models by factorizedblocks or layers with linear velocity variation, provides aconvenient way to reconstruct anisotropic velocity fieldsfor P-wave prestack imaging. The migration velocityanalysis (MVA) algorithm introduced here estimates theanisotropic parameters and velocity gradients in eachblock by minimizing the residual moveout of P-wave re-flection events in image gathers.

The residual moveout of both horizontal and dip-ping events in factorized VTI media is governed by foureffective parameters – the NMO velocity Vnmo at thesurface of the factorized block, the vertical velocity gra-dient kz, the quantity kx = kx

√1 + 2δ that contains the

lateral velocity gradient kx and the anisotropic param-eter δ, and the anellipticity parameter η. Applicationof our MVA method confirms the conclusion of Sarkarand Tsvankin (2002; Paper I) that stable recovery ofthe parameters Vnmo, kz, kx, and η requires reflectionmoveout from at least two interfaces within each blocksufficiently separated in depth.

Numerical tests indicate that the velocity-analysisalgorithm yields robust estimates of the four parame-ters if the vertical distance between the two interfacesexceeds 1/4 of the depth of the bottom reflector. Fora specific model, which may be typical of the subsur-face, we also determined the minimum lateral spread inthe image gathers for a stable recovery of the lateralgradient kx. Another essential condition for stable esti-mation of the parameter η is either the presence of dip-ping interfaces (dips should exceed 25◦) or acquisitionof long-spread data from subhorizontal reflectors pro-viding maximum offset-to-depth ratios of at least two.

The residual moveout on image gathers for largeoffset-to-depth ratios was described by a nonhyperbolicfunction that depends on two independent moveout pa-rameters. Although these parameters are not directlyused in the velocity analysis, their best-fit values found

from semblance search give an accurate approximationfor the residual moveout. The MVA is implemented inan iterative fashion, with the residual moveout mini-mized at each iteration step by solving a system of lin-ear equations for the parameter updates. Since the pa-rameter estimation is performed in the post-migrateddomain, the algorithm is robust in the presence of ran-dom noise and does not lose accuracy for models withsignificant lateral heterogeneity and dipping structures.

The main problem in the application of P-wave ve-locity analysis for VTI media is that the vertical ve-locity VP0, needed to build velocity models for depthmigration, is generally unconstrained by P-wave reflec-tion moveout (Alkhalifah and Tsvankin, 1995; Grechkaet. al., 2002; Tsvankin, 2001). Also, the lateral gradi-ent kx is always coupled to the anisotropy coefficientδ through the parameter kx = kx

√1 + 2δ. A priori

knowledge of VP0 at any single point in the factor-ized block, however, is sufficient for estimating the truelateral gradient kx and, therefore, reconstructing thespatially varying vertical-velocity field, as well as theThomsen anisotropic parameters ε and δ.

The vertical velocity can often be estimated fromborehole data using either check shots or sonic logs. Ifno borehole information is available, a suitable modelfor depth imaging can be constructed by assuming thatVP0 is continuous across layer boundaries. Then, giventhe value of the vertical velocity at a single point on thesurface, the entire velocity model in depth can be esti-mated from the residual moveout of P-wave reflectionevents. The examples presented in the paper demon-strate that the assumption of continuity of VP0 offers apractical way to build reasonably accurate anisotropicvelocity models that are particularly suitable for migra-tion codes that require a smooth velocity field. As thelevel of structural complexity increases, however, themigration result becomes more dependent on the lat-eral location of the assumed continuity point, and theadopted continuous velocity field may cause errors inthe final image.

For relatively simple models with subhorizontal in-terfaces, the distortions related to an error in the ver-tical velocity are limited to a depth stretch that canvary from one layer to another. In the presence of dip-ping interfaces, an overstated value of VP0 causes theimaged ones to be larger than the true dips; if VP0 isunderstated, the imaged dips are too small. In multilay-ered media, a depth stretch for dipping interfaces in theoverburden can distort the shape of the underlying re-flectors, even if the parameters immediately above thesereflectors are estimated correctly.

Still, the examples given above show that the move-out of events in image gathers is not influenced by anincorrect choice of VP0, and the migrated image remainswell focused as long as the algorithm yields accuratevalues of Vnmo, kz, kx, and η. This conclusion, however,may break down if the subsurface contains interfaces


with significant dip or curvature. Then P-wave reflec-tion moveout and, therefore, residual moveout on imagegathers become dependent on the vertical velocity andthe parameters ε and δ (Le Stunff et al., 2001; Grechkaet al., 2002). For models of this type, the layer-strippingapproach adopted in our MVA algorithm is not alwaysadequate because the parameters of a given layer mayremain unconstrained in the absence of reflection datafrom deeper interfaces (Le Stunff et al., 2001).

ACKNOWLEDGMENTS

We are grateful to members of the A(nisotropy)-Teamof the Center for Wave Phenomena (CWP), ColoradoSchool of Mines, for helpful discussions and to KenLarner for his review of the manuscript. Partial supportfor this work was provided by the Chemical Sciences,Geosciences and Biosciences Division, Office of BasicEnergy Sciences, U.S. Department of Energy.

References

Alkhalifah, T. and Tsvankin, I., 1995, Velocityanalysis for transversely isotropic media: Geophysics,60,1550–1566.

Alkhalifah, T., 1997, Seismic data processing in ver-tically inhomogeneous TI media: Geophysics, 62, 662–675.

Cerveny, V., 1972, Seismic rays and ray intensities ininhomogeneous anisotropic media: Geophys. J. R. Astr.Soc., 29, 1–13.

Gardner, G.H.F., French, W.S., and Matzuk,T., 1974,Elements of migration and velocity analysis: Geo-physics, 39, 811–825.

Grechka, V., and Tsvankin, I., 1998, Feasibilityof nonhyperbolic moveout inversion in transverselyisotropic media: Geophysics, 63, 957–969.

Grechka, V., and Tsvankin, I., 1999, 3-D moveout in-version in azimuthally anisotropic media with lateral ve-locity variation: Theory and a case study: Geophysics,64, 1202–1218.

Grechka, V., Pech, A., and Tsvankin, I., 2002, P-wavestacking-velocity tomography for VTI media: Geophys.Prosp., 50, 151–168.

Han, B., Galikeev, T., Grechka, V., Le Rousseau,J., and Tsvankin, I., 2000, A synthetic example ofanisotropic P-wave processing for a model from the Gulfof Mexico, in Ikelle, L., and Gangi, A., Eds., Anisotropy2000: Fractures, converted waves and case studies: Pro-ceedings of the Ninth International Workshop on Seis-mic Anisotropy (9IWSA), Soc. Expl. Geophys.

Liu, Z., 1997, An analytical approach to migrationvelocity analysis: Geophysics, 62, 1238–1249.

Le Stunff, Y., and Jeannot, J.P., 1998, Pre-stackanisotropic depth imaging: 60th EAGE Conference, Ex-tended Abstracts.

Le Stunff, Y., Grechka, V., and Tsvankin, I., 2001,Depth-domain velocity analysis in VTI media using sur-face P-wave data: Is it feasible?: Geophysics, 66, 897–903.

Sarkar, D., and Tsvankin, I., 2002, Analysis of imagegathers in factorized VTI media: CWP Project Review(CWP-425).

Stork, C., 1991, Reflection tomography in the postmigrated domain: Geophysics, 57, 680–692.

Thomsen, L., 1986, Weak elastic anisotropy: Geo-physics, 51, 1954–1966.

Tsvankin, I., 2001, Seismic signatures and analysisof reflection data in anisotropic media: Elsevier SciencePubl. Co., Inc.

APPENDIX A: ALGORITHM FOR

VELOCITY UPDATE

Following the approach suggested by Liu (1997), we de-sign the velocity-updating algorithm to minimize thevariance in the migrated depths of events in image gath-ers. To simplify a generally nonlinear inverse (minimiza-tion) problem, we perform the velocity analysis itera-tively, with a set of linear equations being solved ateach iteration. Below we discuss the velocity update per-formed at a single (lth) step of the iterative process.

Suppose that prestack migration after the (l − 1)th

iteration of the velocity analysis resulted in the migrateddepths z0(xj, hk) (xj is the surface coordinate of thejth image gather, and hk is the half-offset). Then themigrated depths z(xj , hk) after the lth iteration can berepresented as a linear perturbation of z0(xj , hk):

z(xj, hk) = z0(xj , hk) + ΣNi=1

∂z0(xj , hk)

∂λi

∆λi , (A1)

where ∂z0(xj , hk)/∂λi are the derivatives of the mi-grated depths with respect to the medium parametersλi (i = 1, 2, 3, ... N), and ∆λi = λ′

i − λi are the desiredparameter updates. The goal of the updating procedureis to estimate ∆λi and, therefore, find the parametersλ′

i to be used for the migration after the lth iteration.The variance V of the migrated depths for a single

reflection event at all offsets and image gathers is

V = ΣPj=1 ΣM

k=1 [z(xj , hk) − z(xj)]2 , (A2)

where z(xj) = (1/M) ΣMk=1 z(xj , hk) is the average mi-

grated depth of the event at surface coordinate xj , P isthe number of image gathers used in the velocity update,and M is the number of offsets in each image gather.The minimization at each iteration step is accomplishedby searching for the parameter updates that satisfy thecondition ∂V/∂(∆λr) = 0 (r = 1, 2, 3, ... N). Subsitut-ing equation (A1) in equation (A2), differentiating thevariance with respect to the parameter updates, and


setting ∂V/∂(∆λr) = 0 yields

− ΣPj=1Σ

Mk=1Σ

Ni=1(gjk,i

− gj,i)(gjk,r− gj,r )∆λi

= ΣPj=1Σ

Mk=1[z0(xj

, hk) − z0(xj

)](gjk,r

− gj,r

) , (A3)

where gjk,r ≡ ∂z0(xj , hk)/∂λr, gjk,i ≡ ∂z0(xj , hk)/∂λi,and gj,i ≡ (1/M)ΣM

k=1gjk,i; all derivatives are evaluatedfor the medium parameters λi.

Equation (A3) can be rewritten in matrix form as

AT

A∆λ = AT

b , (A4)

where A is a matrix with M · P rows and N columnswhose elements are gjk,r − gj,r, and b is a vector withM ·P elements defined as z0(xj , hk)− z0(xj). A

TA is a

square N × N matrix, and the vector AT

b has N ele-ments, so the problem has been reduced to a system ofN linear equations with N unknowns ∆λ. We solve thesystem (A4) using a linear conjugate gradient scheme toobtain ∆λ and the updated parameters λ

′ = ∆λ + λ.The derivatives of the depths z(xj , hk) with respect

to the medium parameters λi (and, therefore, the matrixA) can be determined from the imaging equations (e.g.,Liu, 1997; Paper I):

τs(y, h, x, z, ~λ) + τr (y, h, x, z, ~λ) = t(y, h) , (A5)

∂τs(y, h, x, z, ~λ)

∂y+

∂τr(y, h, x, z, ~λ)

∂y=

∂t(y, h)

∂y. (A6)

Here y is the common-midpoint location at the surface,h is the half-offset, τs is the traveltime from the sourcelocation xs (xs = y + h) to the diffractor location (x, z)that was obtained after prestack depth migration withthe medium parameters λi, τr is the traveltime fromthe receiver location xr (xr = y−h) to the point (x, z),and t(y, h) is the observed reflection traveltime. Notethat y, x, and z depend on the medium parameters λi,while h is an independent variable. Because x is fixedat the surface location where a particular image gatheris analyzed, however, the derivative of x with respect toλi is set to zero.

Differentiating equation (A5) with respect to λi

gives

[

∂τs

∂y+

∂τr

∂y

]

dy

dλi

+

[

∂τs

∂z+

∂τr

∂z

]

dz

dλi

+

[

∂τs

∂λi

+∂τr

∂λi

]

=∂t

∂y

dy

dλi

. (A7)

Taking equation (A6) into account simplifies equa-tion (A7) to

[

∂τs

∂z+

∂τr

∂z

]

dz

dλi

= −∂τs

∂λi

− ∂τr

∂λi

, (A8)

or

dz

dλi

= −[

∂τs

∂λi

+∂τr

∂λi

]

1

qs + qr

, (A9)

where qs = ∂τs/∂z and qr = ∂τr/∂z are the vertical

slownesses evaluated at the diffractor for the specularrays connecting the diffractor with the source and thereceiver, respectively.

To find the derivatives dz/dλi, we perform ray trac-ing using the prestack-migrated image after the (l−1)th

iteration. First, the dip of the reflector needed to definethe specular reflected rays is estimated by manual pick-ing on the image. Then, for a given diffraction pointon the reflector and a fixed source-receiver offset, thespecular ray is traced through two models, one of whichis defined by the parameters λi and the other by pa-rameters slightly deviating from λi (i.e., λi are slightlyperturbed). The corresponding perturbation of the trav-eltime between the source and the diffractor is dividedby the perturbation in λi to obtain ∂τs/∂λi, while thesame quantity for the traveltime leg between the diffrac-tor and the receiver gives ∂τr/∂λi. The slownesses qr

and qs at the diffraction point are part of the output ofthe ray-tracing algorithm (Cerveny, 1972).


Documents

Migration velocity analysis in factorized VTI media · and considerable structural complexity, velocity model building is conveniently implemented in the prestack depth-migrated domain