CWP-806

Gradient calculation for waveform inversion of microseismicdata in VTI media

Oscar Jarillo Michel and Ilya TsvankinCenter for Wave Phenomena, Colorado School of Mines

ABSTRACT

In microseismic data processing, source locations and origin times are usually obtainedusing kinematic techniques, whereas moment-tensor estimates are typically based onlinear inversion of P- and S-wave amplitudes. Waveform inversion (WI) can potentiallyprovide more accurate source parameters along with an improved velocity model byincorporating information contained in the entire trace including the coda. Here, weaddress one of the key issues in implementing WI for microseismic surveys — effi-cient calculation of the gradient of the objective function with respect to the modelparameters. Application of the adjoint-state method helps obtain closed-form expres-sions for the gradient with respect to the source location, origin time, and momenttensor. Computation of the forward and adjoint wavefields is performed with a finite-difference algorithm that handles elastic VTI (transversely isotropic with a verticalsymmetry axis) models. Numerical examples illustrate the properties of the gradientfor multicomponent data recorded by a vertical receiver array placed in homogeneousand horizontally layered VTI media.

1 INTRODUCTION

Microseismicity has developed in recent years as an efficienttechnique for monitoring of hydraulic fractures and the sur-rounding reservoir volume (Maxwell, 2010; Kendall et al.,2011). Microseismic experiments typically involve recordingthe seismic response to hydraulic fracturing of tight reservoirs,most often shales. The location of the induced microfractures,as well as the origin time of the corresponding seismic eventscan be inferred from the data acquired in an observation bore-hole or at the surface. Usually only a single borehole is avail-able, but it is highly beneficial to record microseismic data inseveral boreholes.

Accurate location of microseismic events requires knowl-edge of the background velocity model. This initial model isusually obtained from sonic logs and traveltimes of the directP- and S-waves excited by perforation shots and recorded bygeophones deployed in a monitor borehole. Afterwards, themodel can be updated using the traveltimes or waveforms ofmicroseismic events. Velocity analysis without adequately ac-counting for seismic anisotropy may lead to significant errorsin event location Van Dok et al. (2011). In particular, shalesare known to be transversely isotropic and may become or-thorhombic or possess an even lower symmetry due to fractur-ing (Tsvankin and Grechka, 2011; Tsvankin, 2012).

Depending on the receiver geometry and spatial distribu-tion of sources, microseismic data can help estimate the per-tinent anisotropy parameters (Grechka and Duchkov, 2011)simultaneously with event location. For example, Grechkaet al. (2011) demonstrate that anisotropic velocity models con-

structed from traveltimes while locating microseismic eventsprovide more accurate source locations than those obtainedwithout accounting for anisotropy. Also, Li et al. (2013) showthat building VTI velocity models simultaneously with eventlocation significantly reduces the traveltime residuals com-pared to standard location techniques that employ isotropic ve-locity models obtained from sonic logs and perforation shots.Grechka and Yaskevich (2013, 2014) demonstrate that for mi-croseismic surveys with sufficient angle coverage it is possibleto construct layered triclinic models and substantially improvethe accuracy of event location.

Still, kinematic inversion essentially replaces a seismictrace with the δ-functions corresponding to the traveltimes ofthe direct arrivals, which restricts the resolution of event lo-cation according to the Rayleigh criterion (i.e., two sourcesare indistinguishable if the distance between them is smallerthan one-half of the predominant wavelength). Considering therapidly increasing usage of back-projection techniques and therich information content of microseismic data, improved re-sults can be expected from waveform inversion (WI.) Indeed,WI operates with the entire trace including scattered waves,so location results are not subject to the Rayleigh criterion.Hence, for wavelengths typical in downhole microseismic sur-veys, one can expect substantially reduced event-location er-rors. Another potential benefit of WI, not explored in this pa-per, is an improved accuracy of the velocity model.

The source mechanism of microseismic events can revealimportant information about the rupture process. Point earth-quake sources are described by the second-rank symmetric

184 Oscar Jarillo Michel and Ilya Tsvankin

seismic moment tensor M. As discussed by Vavrycuk (2007),all six independent elements of M for microseismic events canbe retrieved from the amplitudes of P-waves recorded in threeboreholes or from P- , SV-, and SH-wave amplitudes measuredin two boreholes. For sources and receivers located in the [x1,x3]-plane of a VTI model, the in-plane polarized waves dependon the elements M11, M13, and M33, which potentially can befound by inverting P- and SV-waveforms recorded in a singleborehole. In practice, M is obtained independently of othermicroseismic parameters by linear inversion of P- and S-waveamplitudes. Since WI incorporates waveform information, itmay provide improved estimates of M and, at the same time,perform event location and constrain the origin time.

Waveform inversion is used to build high-resolution ve-locity models from seismic data using phase and, sometimes,amplitude information (Tarantola, 1984; Gauthier et al., 1986;Mora, 1987; Pratt, 1999; Virieux and Operto, 2009). RecentlyWI has been extended to elastic and anisotropic media (Leeet al., 2010; Kamath and Tsvankin, 2013), which makes it suit-able for multicomponent reflection and microseismic data.

The objective function in WI quantifies the difference be-tween observed and predicted data in the time or frequencydomain. Efficient inversion requires application of iterative op-timization schemes such as gradient-based methods, which in-volve calculation of the gradient of the objective function withrespect to the model parameters. In principle, the gradient canbe found from the Frechet derivatives obtained by differentiat-ing the wavefield with respect to each model parameter. How-ever, if the number of unknowns is large, computation of theFrechet derivatives becomes prohibitively expensive.

A computationally efficient alternative for computing thegradient without the Frechet derivatives is the adjoint-statemethod (Plessix, 2006; Fichtner, 2006, 2009; Kohn, 2011).This method makes it possible to calculate the gradient us-ing just two forward-modeling simulations: first to generatethe forward wavefield (predicted data), and second to computethe adjoint wavefield.

There has been significant progress in applying ad-joint methods to tomographic velocity analysis and source-parameter inversion in global seismology. Tromp et al. (2005)and Liu and Tromp (2006) employ an adjoint formulationbased on the Lagrangian-multiplier method to derive the gra-dient for estimating the P- and S-wave velocities in isotropicmedia. They also analyze the sensitivity (Frechet) kernels for2D and 3D velocity models. Kim et al. (2011) obtain gradientexpressions for the source parameters using the adjoint-statemethod. They also implement conjugate-gradient inversion toestimate the source location and moment tensor for two earth-quakes in Southern California using a known isotropic velocitymodel. Morency and Mellors (2012) follow the same approachto evaluate the source parameters of a geothermal event.

Here, we discuss gradient calculation for waveform inver-sion of microseismic data in heterogeneous VTI media. Thecurrent algorithm is designed to estimate the source location,origin time, and moment tensor from 2D microseismic exper-iments.

We start by describing 2D elastic finite-difference model-

ing for dislocation-type sources embedded in a VTI medium.Then we discuss application of the adjoint-state method tocompute the gradient for waveform inversion. Efficient gra-dient calculation is implemented by adapting to our problemthe general expressions for the gradient obtained by Kim et al.(2011). Although the interval Thomsen parameters of layeredVTI media can be estimated by WI as well, the current formu-lation is limited to the gradient for the source parameters. Theperformance of the algorithm is illustrated by synthetic testsfor homogeneous and layered VTI media. The examples dis-play multicomponent wavefields obtained by forward and ad-joint finite-difference modeling and help understand the prop-erties of the objective-function derivatives for the source loca-tion, origin time, and moment tensor.

2 GRADIENT FROM THE ADJOINT-STATEMETHOD

2.1 Forward problem

The wave equation for a heterogeneous anisotropic mediumcan be written as:

ρ∂2ui

∂t2 −∂

∂x j

(ci jkl

∂uk

∂xl

)= fi , (1)

where u(x, t) is the displacement field, t is time, ci jkl is thestiffness tensor (i, j,k, l = 1,2,3), ρ(x) is density, and f(x, t) isthe body force.

Dislocation-type sources are described by the seismicmoment tensor,

M =

M11 M12 M13M21 M22 M23M31 M32 M23

, (2)

which can be incorporated into the source term of equation 1by using the notion of equivalent force (Aki and Richards,1980; Dahlen and Tromp, 1998):

ρ∂2ui

∂t2 −∂

∂x j

(ci jkl

∂uk

∂xl

)=−Mi j

∂[δ(x−xs)]∂x j

S(t) , (3)

where xs is the source location, S(t) is the source time func-tion, and δ(x− xs) is the spatial δ-function. We use a finite-difference (FD) algorithm to obtain exact solutions of equa-tion 3.

2.2 Modeling examples

Forward modeling for gradient calculation in VTI media iscarried out with the elastic finite-difference program sfewein MADAGASCAR . The model is described by the intervalThomsen parameters — the P- and S-wave vertical velocitiesVP0 and VS0 and the anisotropy coefficients ε and δ defined as(Thomsen, 1986; Tsvankin, 2012):

Gradient calculation for waveform inversion of microseismic data in VTI media 185

𝑥1

𝑥3

𝜃

Fault plane

𝝂

s

Figure 1. 2D fault geometry used in forward modeling. The dip angleθ (0◦ ≤ θ ≤ 90◦) is measured down from the horizontal axis. The in-cidence plane [x1,x3] is assumed to coincide with the dip plane of thefault and contain the slip s.

ε≡ c11− c33

2c33, (4)

δ≡ (c13 + c55)2− (c33− c55)2

2c33(c33− c55). (5)

Note that P- and SV-waves propagating in the [x1,x3]-plane are not influenced by the Thomsen parameter γ. An im-portant combination of Thomsen parameters is the coefficientσ ≡ (VP0/VS0)2(ε− δ), which is largely responsible for thekinematic signatures of SV-waves.

The relevant elements of M for the in-plane polarizedwaves (P and SV) from a dislocation source in a VTI mediumcan be represented as (Aki and Richards, 1980; Vavrycuk,2005):

M11 = Σ uν3 s3 (c13− c11) , (6)

M13 = Σ u(ν1 s3 +ν3 s1)c55 , (7)

M33 = Σ uν3 s3 (c33− c13) , (8)

where Σ is the fault area, u is the magnitude of slip (displace-ment discontinuity), ν is the unit fault normal, s is the unit vec-tor in the slip direction, and c11, c13, c33, and c55 are the stiff-ness coefficients in the two-index Voigt notation (Figure 1).Because the vectors ν and s are confined to the [x1,x3]-plane,equations 6 – 8 for dip-slip sources can be expressed as a func-tion of the fault dip angle θ. Taking into account that ν1=sinθ,ν3=cosθ, s1=cosθ, and s3=−sinθ, we rewrite quations 6 –8 as:

M11 =−Σ u2

sin2θ(c13− c11) , (9)

M13 = Σ u cos2θc55 , (10)

M33 =−Σ u2

sin2θ(c33− c13) . (11)

The wavefields generated by dislocation-type sources ina homogeneous VTI medium are shown in Figure 2. Theamplitude distribution and intensity of the P- and S-waveschange substantially with the fault orientation. Because themagnitude of σ is relatively large (close to unity), the SV-wavefront exhibits triplications at oblique propagation direc-tions (Tsvankin, 2012).

2.3 Inverse problem

As mentioned above, the P- and SV-waves recorded in the[x1,x3]-plane depend on the components M11, M13, and M33of the moment tensor. Here, our goal is to invert just for thethree moment-tensor elements, the source coordinates xs

1 andxs

3, and the origin time t0 assuming that the velocity model isknown.

The observed data dobs and predicted data dpre are pro-duced by two forward simulations, where dpre is obtained af-ter perturbing the source parameters used to generate dobs. Inboth cases, the elastic displacement field u(xs,xrn , t) excitedby a source located at xs is recorded by N receivers locatedat xrn (n = 1,2, ...,N). The data residuals are measured by theleast-squares objective function, which has to be minimized bythe inversion algorithm:

F (m) =12‖dpre(m)−dobs‖2 . (12)

2.4 Application of the adjoint-state method

The objective function depends on the model parametersthrough the state variables, which represent the solution of theforward-modeling equations. In our case, the state variable isthe elastic displacement field u(x, t) generated by a microseis-mic source and governed by equation 3.

Iterative optimization techniques involve calculation ofthe model update at each iteration. The update direction isdetermined by the gradient (the derivatives of the objectivefunction with respect to the model parameters), ∂F (m)/∂m.The adjoint-state method is designed to find this gradient forthe entire set of model parameters in just two modeling simu-lations. However, because this method does not calculate theFrechet derivatives, it is impossible to evaluate the sensitivityof the solution to perturbation errors.

The adjoint-state method involves the following mainsteps (Perrone and Sava, 2012):

(i) Computation of the state variable (forward wavefield).

186 Oscar Jarillo Michel and Ilya Tsvankin

(a) (b)

(c) (d)

Figure 2. Vertical displacement generated by dip-slip sources with different orientation (defined by the angle θ) in a homogeneous VTI medium.The medium parameters are ρ = 2 g/cm3, VP0 = 4047 m/s, VS0 = 2638 m/s, ε = 0.4, and δ = 0 (σ = 0.94). The moment tensor is computed fromequations 9 – 11 with Σ u = 1m3 and (a) θ = 0◦, (b) θ = 30◦, (c) θ = 60◦, and (d) θ = 90◦.

(ii) Computation of the adjoint source.(iii) Computation of the adjoint-state variable (adjoint

wavefield).(iv) Computation of the gradient.

In addition to equation 3, the method requires solving theadjoint wave equation:

ρ∂2u†

i∂t2 − ci jkl

∂2u†k

∂x j ∂xl=

N

∑n=1

(dobs−dpre)(T − t)δ(x−xrn) ,

(13)where u† is called the “adjoint wavefield.” The so-called ad-joint source on the right-hand side of equation 13 is obtained

by differentiating the objective function F (m) with respect tothe forward wavefield u, and consists of the time-reversed dataresiduals. Therefore, the adjoint simulation can be carried outwith the same modeling code by “injecting” the adjoint sourceat the receiver locations and then running the forward simula-tion.

The derivatives of the objective function with respect tothe moment-tensor elements, source coordinates, and origintime can be found as (Kim et al., 2011):

∂F∂Mi j

=Z T

0ε

†i j(x

ts, t)S(T − t)dt , (14)

Gradient calculation for waveform inversion of microseismic data in VTI media 187

∂F∂xs

i=

Z T

0

∂[M : ε†(xts, t)]∂xi

∣∣∣∣xts

S(T − t)dt , (15)

∂F∂t0

=Z T

0M : ε

†(xts, t)∂S(T − t)

∂tdt , (16)

where xts is the trial source location, T is the recording time,ε† = 1

2 [∇u† +(∇u†)T] is defined as the adjoint strain tensor,and M : ε† is the double inner product of the tensors M and ε†.Equations 14 – 16 applied to the 2D problem yield the gradientvector g for the six source parameters:

g ={

∂F∂M11

,∂F

∂M13,

∂F∂M33

,∂F∂xs

1,

∂F∂xs

3,

∂F∂t0

}. (17)

It is interesting that in contrast to the gradient forvelocity-related parameters (Liu and Tromp, 2006), whichdepends on the interaction between the forward and adjointwavefields, equations 14 – 16 include only the adjoint wave-field. This means that there is no need to store or recalculatethe forward wavefield during gradient calculation.

When the adjoint-state method is applied to velocity anal-ysis, u† is supposed to “illuminate” the erroneous parts of thevelocity model. Likewise, for the source-inversion problem,u† reveals the difference in M, xs, and t0 that produces themismatch between the observed and predicted data.

In principle, the number of sources in the adjoint prob-lem is unrestricted. If the forward wavefield is excited by mul-tiple sources, the adjoint wavefield may focus at the actual aswell as perturbed (trial) source locations. The areas of focus-ing make the main contribution to the gradient. However, forinversion purposes we only need the derivatives in equations14 – 16 at the trial source position.

3 SYNTHETIC TESTS

Next, we present the results of synthetic tests for a homoge-neous VTI medium and a stack of three constant-density VTIlayers, with the wavefield modeled using the FD code men-tioned above.

In the first experiment (Figure 3), the observed and initialpredicted data (Figures 4 and 5) are generated for a single mi-croseismic event recorded in a vertical “borehole” by receiverslocated at each grid point. Whereas the predicted field is gen-erated by a horizontal dip-slip source (θ = 0◦), the predictedfield is computed with a vertical source (θ = 90◦). Next, theadjoint source “injected” at the receiver locations is used tocompute the adjoint wavefield (Figure 6). This wavefield fo-cuses at the time t = 0.44 s near the actual source, where themodel perturbation is located. The focusing time correspondsto the raypath between the source and the receivers located atthe end of the array.

After applying equations 14 – 16, we obtain the follow-ing derivatives for the five source parameters: ∂F /∂M11 =0.00038, ∂F /∂M13 =1.28, ∂F /∂M33 =−0.0022, ∂F /∂xs

1 =

Figure 3. Source (white dot) and a vertical line of receivers at x1 = 1.2km embedded in a homogeneous VTI medium. The receivers areplaced vertically at each grid point (every 6 m). The medium parame-ters are ρ = 2 g/cm3, VP0 = 4047 m/s, VS0 = 2638 m/s, ε = 0.4, andδ = 0. The source is located at x1 = 0.3 km, x3 = 0.75 km.

2.21, ∂F /∂xs3 =−0.0039, and ∂F /∂t0 =−0.82. The sign of

the derivatives (i.e., of the components of the gradient) definesthe update direction in the model space, whereas their abso-lute values determine the magnitude of the update of the cor-responding model parameter. The units of the derivatives arenot included here because we can compare only the deriva-tives for the same type of parameters (e.g., for the coordinatesxs

1 and xs3). The main contribution to the gradient comes from

the focusing of the adjoint wavefield near the source location.It may look surprising that the derivatives for xs

1, xs3, and t0 are

nonzero because the values of xs and t0 for the actual and trialsources coincide. However, the derivatives for xs and t0 de-pend on the components of M, which deviate from the actualvalues (equations 15 and 16). Hence, to avoid cycle-skippingproblems during the joint inversion for xs, t0, and M, a goodinitial guess is needed for all three types of parameters.

The second test is performed for the three-layer VTImedium in Figure 7. The actual and trial models have thesame source parameters as in the previous experiment. Thedata are generated by a source placed in the middle layer andrecorded by receivers in a vertical “borehole.” The deriva-tives calculated using the adjoint-state method are ∂F /∂M11=0.004, ∂F /∂M13=2.44, ∂F /∂M33=−0.0044, ∂F /∂xs

1=0.82,∂F /∂xs

3 =−0.0071, and ∂F /∂t0 =−0.21. These derivativesare similar to the ones obtained for the previous test becausethe tensor M is perturbed in the same way for both cases. Thelayer boundaries, however, create a number of reflected andmode-converted waves that should make waveform inversionfor source parameters better constrained.

For the third experiment, we keep the actual source at thesame location as in the previous tests and use the homoge-neous model from Figure 3. This time, the initial trial sourceis moved horizontally, which generates pronounced gradient

188 Oscar Jarillo Michel and Ilya Tsvankin

(a) (b)

Figure 4. Snapshots of the vertical displacement for the model in Figure 3 at t = 0.14 s. Σ u = 1m3. (a) The observed wavefield produced by adip-slip source with θ = 0◦. (b) The predicted wavefield from a source with θ = 90◦.

(a) (b)

Figure 5. Vertical displacement of the (a) observed and (b) predicted data for the model in Figure 3 generated with the source parameters fromFigure 4.

contributions at both source locations (Figure 8). However,as mentioned above, we are interested only in the derivativescomputed for the trial source location.

The obtained derivatives are ∂F /∂M11 = −0.00017,∂F /∂M13 =−0.25, ∂F /∂M33 = 0.00033, ∂F /∂xs

1 =−4.28,∂F /∂xs

3 = 0.0061, and ∂F /∂t0 = 1.51. Although the trialsource has the correct moment tensor, the derivatives for M11,M13, and M13 do not vanish because the wavefield substan-tially changes with source location. The derivative for t0 is alsononzero because moving the source creates traveltime shiftssimilar to those due to a change in the origin time. Still, the in-version should be able to resolve the trade-off between xs andt0 because they influence the traveltimes in a different fashion.

The large value of ∂F /∂xs1 compared to ∂F /∂xs

3 cor-rectly indicates that the trial horizontal source coordinate xs

1should be updated more significantly than xs

3. The negativesign of ∂F /∂xs

1 will lead to a smaller value of xs1 for the next

iteration of inversion, so that the source will move toward itsactual position (x1 = 0.3 km.)

The last experiment demonstrates a potential applica-tion of the adjoint wavefield. The model includes two actualsources with the same tensor M and origin time t0 that generatethe wavefield u(x, t) (Figure 9). However, we assume that thedata dobs are produced by a single event and specify a singletrial source. Then the adjoint wavefield u† is expected to focusat two different locations, corresponding approximately to theactual and trial source position. Instead, the adjoint wavefieldactually focuses at three locations (Figure 10), which helpsidentify the second source missing in the trial model.

In addition to finding “hidden sources,” the field u† canbe used to improve the initial trial source position because theadjoint wavefield focuses (for the correct velocity model) nearthe actual source location. This starting source position can belater refined during WI.

Gradient calculation for waveform inversion of microseismic data in VTI media 189

(a) (b)

Figure 6. Snapshots of the vertical component of the adjoint wavefield for the model in Figure 3 at times (a) t = 0.35 s and (b) t = 0.44 s. Theadjoint wavefield focuses at the actual and trial source locations (which are the same) on plot (b).

Figure 7. Three-layer VTI model used in the second experiment. Thesource-receiver geometry is the same as in Figure 3. The distance be-tween receivers is 5 m. The parameters ρ = 2 kg/m3, ε = 0.4, andδ = 0 are the same in all three layers. The vertical velocities in the firstlayer are VP0 = 4047 m/s and VS0 = 2638 m/s; for the second layer,VP0 = 4169 m/s and VS0 = 2320 m/s; for the third layer, VP0 = 4693m/s and VS0 = 2682 m/s.

4 CONCLUSIONS

Waveform inversion is a potentially powerful tool to solve si-multaneously two of the most important problems in micro-seismic monitoring, event location and source-mechanism es-timation. Here, we employed the adjoint-state method to findanalytic expressions for the gradient of the WI objective func-tion following the results of Kim et al. (2011). Then adjoint

Figure 8. Actual source (white dot), trial source (red dot) and a verticalline of receivers (spacing is 6 m) embedded in a homogeneous VTImedium. The medium parameters are the same as in Figure 3. Theactual source is located at x1 = 0.3 km, x3 = 0.75 km and the trialsource is at x1 = 0.32 km, x3 = 0.75 km. The moment tensor for bothsources corresponds to a horizontal (θ = 0◦) dip-slip fault with Σ u =1m3.

simulations were implemented to calculate the WI gradient forthe source location, origin time, and moment tensor using aknown VTI velocity model.

Synthetic tests were carried out for multicomponentwavefields from one and two sources recorded by a dense ar-ray of receivers in a vertical “borehole.” The first experimentwas performed for a trial source with a perturbed moment ten-sor correctly positioned in a homogeneous VTI medium. The

190 Oscar Jarillo Michel and Ilya Tsvankin

Figure 9. Two actual sources (white dots), a trial source (red dot),and a vertical line of receivers (the spacing is 6 m) embedded in ahomogeneous VTI medium. The medium parameters are the same asin Figure 3. The actual sources are located at x1 = 0.3 km, x3 = 0.75km, and x1 = 0.7 km, x3 = 1 km; the trial source is at x1 = 0.42 km,x3 = 0.75 km. The moment tensor for all three sources corresponds toa horizontal (θ = 0◦) dip-slip fault with Σ u = 1m3.

adjoint wavefield focuses near the trial source location, whichidentifies the perturbed area. Although in this experiment thesource position was correct, the derivatives for the source co-ordinates and origin time do not vanish because they dependon errors in the moment-tensor elements.

In the second test, a trial source with a distorted momenttensor was placed in a three-layer VTI model. The additionalinterfaces produce a more complicated wavefield with a num-ber of reflected and converted waves. These additional eventsdo not significantly influence gradient calculation but shouldimprove the accuracy and stability of parameter estimation.

The algorithm was also tested for a source with the cor-rect moment tensor and origin time but erroneous location in ahomogeneous VTI medium. As in the previous examples, thederivatives of the objective function with respect to the correctparameters (in this case, the moment tensor and origin time)are nonzero. The dependence of the gradient on unperturbedmodel parameters implies that it is essential to have accurateinitial guesses for all unknowns.

Finally, the wavefield was generated by two sources setoff simultaneously, while the predicted wavefield was excitedby a single source. Focusing of the adjoint wavefield helpedus identify the approximate location of the “missing” source.In general, if the velocity model is not strongly distorted, theadjoint wavefield can provide an accurate starting trial sourceposition for waveform inversion.

To take full advantage of the potential of WI in micro-seismic studies, it can be applied to anisotropic velocity modelbuilding as well. This extension will be discussed in a futurepublication.

5 ACKNOWLEDGMENTS

We are grateful to Vladimir Grechka (Marathon Oil), who hasgenerously provided valuable insights and feedback regardingvarious aspects of this work, and to Jeroen Tromp (PrincetonUniversity) for his advice on gradient calculation. We thankEsteban Dıaz Pantin, Francesco Perrone, Nishant Kamath, andSteven Smith (all from CWP) for their technical assistanceand help with the implementation of the adjoint computations.The numerical examples in this paper are produced with theMADAGASCAR open-source software package freely availableat www.ahay.org. This work was supported by the ConsortiumProject on Seismic Inverse Methods for Complex Structures atthe Center for Wave Phenomena (CWP).

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