Elastic full-waveform inversion of transmission data in 2D VTI media · 2017. 5. 3. · Elastic...

CWP-804

Elastic full-waveform inversion of transmission data in 2D VTImedia

Nishant Kamath & Ilya TsvankinCenter for Wave Phenomena, Colorado School of Mines

ABSTRACTFull-waveform inversion (FWI) has proved effective in significantly improving thespatial resolution of seismic velocity models. However, it is implemented mostly forisotropic media and the applications to anisotropic models are typically limited toacoustic approximations. Here, we develop a foundation for elastic FWI in laterallyheterogeneous VTI (transversely isotropic with a vertical symmetry axis) media. Themodel is parameterized in terms of the P- and S-wave vertical velocities and the P-wavenormal-moveout and horizontal velocities. To derive the gradients of the objectivefunction with respect to the VTI parameters, we employ the adjoint-state method. Theiterative inversion is performed in the time domain using the steepest-descent methodand a finite-difference modeling code. To test the algorithm, we introduce Gaussiananomalies in the Thomsen parameters of a homogeneous VTI medium and performFWI of transmission data for different configurations of the source and receiver arrays.The inversion results strongly depend on the acquisition geometry and the aperturebecause of the parameter trade-offs. In contrast to acoustic FWI, the elastic inversionhelps constrain the S-wave vertical velocity, which for our model is decoupled fromthe other parameters.

1 INTRODUCTION

Full-waveform inversion (FWI) is a technique for estimat-ing subsurface properties by using entire seismic waveformsrecorded at the surface or in a borehole. Depending on theproblem and availability of forward-modeling algorithms,FWI can be performed in the time domain (Kolb et al., 1986;Gauthier, 1986; Mora, 1987; Bunks et al., 1995) or frequencydomain (Song and Williamson, 1995; Song et al., 1995; Pratt,1999; Pratt and Shipp, 1999). Evaluation of the gradient of theobjective function is often based on the adjoint-state method,as described in Tarantola (1984a), Fichtner et al. (2006), andLiu and Tromp (2006).

Kohn et al. (2012) discuss the influence of parameteriza-tion on elastic isotropic FWI and conclude that it is preferableto describe the model in terms of the P- and S-wave velocitiesand density rather than the impedances. Liu and Tromp (2006)derive the gradients of the objective function with respect tothe stiffness coefficients and the perform FWI for earthquakedata from an elastic isotropic model. FWI has been extendedto anisotropic media, but typically in the acoustic approxima-tion (Plessix and Rynja, 2010; Gholami et al., 2011; Plessixand Cao, 2011; Shen, 2012). Such “anisotropic acoustic” algo-rithms, however, do not properly handle reflection amplitudesand cannot be applied to multicomponent data. Elastic FWIof synthetic multicomponent surface data (consisting of bothdiving waves and reflections) for VTI media is performed byLee et al. (2010), but suboptimal parameterization in terms ofthe stiffness coefficients causes ambiguity in their results.

In our previous work (Kamath and Tsvankin, 2012; here-after, referred to as Paper I), we invert multicomponent re-flection data (PP- and PSV-waves) from a horizontally layeredVTI model for the interval Thomsen parameters − the P- andS-wave vertical velocities (VP0 and VS0) and anisotropy co-efficients ε and δ. Inversion for density makes the objectivefunction highly nonlinear, thereby causing the algorithm to gettrapped in local minima. Therefore, the interval densities arefixed at the correct values. Although PP-waves alone may besufficient to resolve VP0, VS0, ε, and δ, stable parameter esti-mation for layers at depth requires employing long-offset data(with the spreadlength-to-depth ratio reaching at least two) orthe addition of PS-waves. Inversion of multicomponent databenefits from using a multiscale approach, which helps reducethe sensitivity to the choice of the initial model (Bunks et al.,1995).

Here, we introduce an expansion of elastic FWI to later-ally heterogeneous VTI media. The model is parameterized interms of VP0, VS0, and the P-wave normal-moveout (Vnmo,P )and horizontal (Vhor,P ) velocities. To compute the gradient ofthe objective function, we adapt the results of Liu and Tromp(2006) obtained with the adjoint-state method. FWI is per-formed in the time domain with the wavefield generated usinga 2D elastic finite-difference modeling code. Then the algo-rithm is applied to transmission data generated for homoge-neous VTI models with Gaussian anomalies in the parametersVP0, VS0, and ε.

160 Nishant Kamath & Ilya Tsvankin

2 METHODOLOGY

2.1 Full-waveform inversion for VTI media

FWI in the time domain is designed to minimize the followingobjective function:

F =1

2

N∑r=1

∫ T

0

‖u(xr, t)− d(xr, t)‖2dt , (1)

where N is the number of receivers, T is the trace length,u(xr, t) is the displacement computed for a trial model, andd(xr, t) is the displacement recorded at receiver location xr .Because the relationship between the data and the modelis nonlinear, the inversion is performed iteratively, with themodel update at each iteration found as:

∆m = [JTJ]−1 JT ∆d, (2)

where J is the Frechet derivative matrix obtained by perturb-ing each model parameter, JTJ is the approximate Hessianmatrix, T denotes transposition, and ∆d is the difference be-tween the observed data and those computed for a trial model.

In Paper I, we use PP reflections or a combination of PPand PS events generated for a horizontally layered VTI modelto estimate the interval parameters VP0, VS0, ε, and δ. Forthe purpose of inversion, it is convenient to use quantities thatare directly constrained by the data and have the same units.Hence, instead of ε and δ Paper I operates with the P-waveNMO velocity (Vnmo,P = VP0

√1 + 2δ) and the horizontal

velocity (Vhor,P = VP0

√1 + 2ε). If the number of layers

(which is fixed during the inversion) is not large, it is possibleto compute the Frechet matrix and the approximate Hessianexplicitly by perturbing each model parameter.

2.2 Inverse problem in 2D

In the case of laterally heterogeneous media, computationof the Frechet derivatives becomes prohibitively expensivebecause it involves calculating as many forward models ateach iteration as the number of parameters (typically de-fined on a grid). It is more practical to calculate the gra-dient (JT ∆d in equation 2) of the objective function withthe adoint-state method, which has been widely used in FWI(Tarantola, 1984b; Plessix, 2006; Liu and Tromp, 2006; Ficht-ner et al., 2006). The model update, which is a scaled ver-sion of the gradient, is then calculated using steepest-descentor conjugate-gradient algorithms. Alternatively, either the so-called BFGS (Broyden-Fletcher-Goldfarb-Shanno) method orits limited-memory equivalent, the L-BFGS method (both arequasi-Newton techniques), can be employed to scale the gradi-ent by the inverse of an approximate Hessian matrix (Virieuxand Operto, 2009).

The adjoint-state method is designed to compute the gra-dient of the objective function using the so-called “adjointwavefield.” Because the wave equation is self-adjoint, it can besolved for the adjoint wavefield with the data residuals treatedas sources. The residuals at each time step are injected “back-ward in time” (i.e., starting from the last time sample), which

is commonly described as back-propagation of data residuals.For 2D multicomponent data, the vertical and horizontal dis-placement components of the data residuals should be injectedinto the medium simultaneously. The gradient is obtained byapplying the imaging condition to the spatial derivatives of theforward and adjoint wavefields.

Here, we assume that the properties of the VTI mediumvary in 2D and consider only in-plane polarized waves (P andSV). Hence the model is described by four stiffness coeffi-cients (written in the Voigt notation): C11, C33, C13, and C55.However, it is certain combinations of the stiffnesses that con-trol traveltimes and amplitudes of seismic waves (Tsvankin,2012). In particular, description of wave propagation and in-version of seismic data can be facilitated by employing Thom-sen parameters and their simple combinations (e.g., the anel-lipticity coefficient η). Lee et al. (2010), who parameterize theVTI model in terms of the stiffnesses, are unable to resolvethe coefficient C13, likely because of the tradeoff betweenC13 and C55 in P-wave kinematic signatures. In Paper I wecould constrain the relevant Thomsen parameters (VP0, VS0,ε, and δ), although the algorithm operated with the vertical,NMO and horizontal velocities. Likewise, here we parameter-ize the model in terms of the velocities VP0, VS0, Vnmo,P , andVhor,P .

The gradients of the objective function (equation 1) withrespect to the elements of the stiffness tensor are derived inAppendix A using the results of Liu and Tromp (2006):

∂F∂cijkl

= −∫ T

0

∂ui

∂xj

∂ψk

∂xldt , (3)

where u andψ are the forward and adjoint displacement wave-fields, respectively. Using the chain rule, we can find the gra-dient for each velocity Vn (VP0, VS0, Vnmo,P , and Vhor,P ):

∂F∂Vn

=∑ijkl

∂F∂cijkl

∂cijkl∂Vn

. (4)

The stiffness coefficients are expressed in terms of the ve-locities in equations A18−A21. Combining equations 3, 4,and A18−A21 yields the gradients with respect to velocities(equations A22− A25).

Here, FWI is implemented in the time domain, primarilybecause the finite-difference modeling software available to usperforms time-domain computations. We employ the steepest-descent method to update the model at each iteration withequal step length for all parameters.

3 NUMERICAL TESTS FOR TRANSMISSION DATA

Next, we perform tests for simple synthetic models to verifythe accuracy of the gradient computation. Because the initialstage of FWI typically involves diving waves, the data aregenerated for transmission experiments. The model includesGaussian anomalies in the Thomsen parameters VP0, VS0, andε inserted into a homogeneous VTI background between linearrays of sources and receivers (Figure 1).

In the first test, the model includes an anomaly in ε, while

Elastic full-waveform inversion of transmission data in 2D VTI media 161

Figure 1. VTI model with a Gaussian anomaly (standard deviation σ = 300m) in the anisotropy parameter ε. The background and maximumvalues of ε are 0.1 and 0.142, respectively. The other Thomsen parameters are spatially invariant: VP0 = 3000m/s, VS0 = 1500m/s, andδ = −0.05. The dots on the left mark the source locations and the vertical line on the right represents an array of receivers placed at each grid point(6.6 m apart).

(a) (b)

Figure 2. (a) Vertical and (b) horizontal displacements for the model in Figure 1 generated by a shot at z = 1.5 km.

the other parameters are spatially invariant (Figure 1). Thewavefield is generated by a point displacement source polar-ized in the horizontal direction with a peak frequency of 10 Hz.The vertical and horizontal displacements (“recorded data”)from a shot in the center of the array are shown in Figure2. The “modeled” data are then generated in the backgroundmedium without the anomaly, and the adjoint source is ob-tained as the difference between the two wavefields.

Figure 3 displays the gradients with respect to the modelparameters (velocities) calculated from equations A22−A25.For the source-receiver geometry in Figure 1, waves travel rel-atively close to the isotropy plane, and are influenced primarilyby Vhor,P , which is a function of ε. Hence, the largest gradientis that for the velocity Vhor,P , which correctly identifies ε asthe parameter that needs updating. However, the initial gradi-ent with respect to VS0 is also significant because ε influencesthe SV-wave amplitude.

Starting from the homogeneous background model, weperform the inversion using the steepest-descent method. Werun 50 iterations or stop the inversion if the objective func-

tion flattens out sufficiently (Figure 4). The gradient for VS0

has opposite signs in subsequent iterations, and the differencebetween the inverted VS0 and the actual (background) valueis negligible (Figure 5(b)). Likewise, the inverted and initialvalues of VP0 and δ are close, which confirms that FWI con-verges toward the actual model. The updates in Vhor,P , com-bined with negligible changes in VP0, ensure the reconstruc-tion of the anomaly in ε.

The shape of the anomaly (i.e., it is stretched along thehorizontal axis), however, is somewhat distorted because ofthe source-receiver configuration. For this acquisition geome-try, spatial resolution should indeed be higher in the verticaldirection than horizontally, as discussed by Wu and Toksoz(1987). Even though the objective function decreases to just0.04% of the initial value, the maximum estimated value ofε is about 0.12, whereas the actual ε reaches 0.14. When theaperture is increased by reducing the distance between the ar-rays by about one-half and increasing the vertical extent of thearrays by 0.5 km (Figure 6), the shape of the anomaly is betterresolved (Figure 7). In addition, because the inverted veloci-


(a) (b)

(c) (d)

Figure 3. Gradients of the objective function (equation 1) with respect to (a) VP0, (b) VS0, (c) Vnmo,P , and (d) Vhor,P for the model from Figure1. All four gradients are plotted on the same scale.

Figure 4. Change in the normalized objective function with iterations for the model from Figure 1.

ties are closer to the actual values, the estimated ε (maximumvalue of 0.13) is slightly more accurate than in the previousexample. Because this configuration yields better inversion re-sults, it is used in all subsequent tests.

Next, we introduce an anomaly in the P-wave vertical ve-locity VP0, with the sources still polarized horizontally (Figure8). The anomaly in VP0 also causes perturbations in the NMOvelocity Vnmo,P and the horizontal velocity Vhor,P . As a re-sult, the largest update (about 74% of the actual anomaly) forthe given configuration is the one with respect to the horizontalvelocity (Figure 9(c)), whereas the updates for VP0 (45%) andVnmo,P (44%) are much smaller. An update in Vhor,P with-

out the corresponding change in VP0 results in an undesiredupdate in ε (Figure 9(c)) and steers the inversion away fromrecovering the true anomaly in VP0 (Figure 9(a)). Because theparameter δ depends on the velocity ratio Vnmo,P /VP0, andthe inversion updates both the velocities proportionately, thereis no significant change in δ (Figure 9(d)). These results indi-cate that the inversion for VP0 fails because of a lack of prop-agation directions near the vertical.

In contrast, when the source-receiver configuration is ro-tated by 90 (Figure 10) and the wave propagation is predom-inantly vertical, the largest update is the one for VP0 (Figure11). Since the actual ε and δ remain unperturbed, the inversion


(a) (b)

(c) (d)

Figure 5. Difference between the inverted parameters (a) VP0, (b) VS0, (c) ε, and (d) δ and their initial values for the model from Figure 1. Thevelocities have units of km/s.

Figure 6. VTI model with the same parameters as in Figure 1, but the source and receiver arrays are longer and closer to each other.

needs to update Vnmo,P and Vhor,P along with VP0, whichwould keep the anisotropy coefficients unchanged. However,whereas the change in VP0 is about 85% of the required up-date, those in Vnmo,P and Vhor,P are only about 12.5% and

11%, respectively. This results in relatively large incorrect up-dates in the parameters ε (-0.075) and δ (-0.07).

From these tests, it is clear that the inversion resultsstrongly depend on how the wavefields probe the model. Be-cause of inherent trade-offs between the parameters for some


(a) (b)

(c) (d)

Figure 7. Difference between the inverted parameters (a) VP0, (b) VS0, (c) ε, and (d) δ and their initial values for the model from Figure 6. Thevelocities have units of km/s.

source-receiver configurations, even elastic FWI suffers fromnonuniqueness. Incorporating reflection data might providemore constraints on the VTI parameters.

The last test is performed for a Gaussian anomaly in VS0

embedded between horizontal source and receiver arrays (Fig-ure 12). The maximum perturbation in VS0 with respect tothe background is the same as that for VP0 in the previoustests (Figures 8 and 10), but the percentage perturbation inVS0 is two times higher. Hence, to avoid the problem of cycle-skipping, the peak frequency of the source is reduced to 5 Hz.The inversion results, which include an update only in VS0

(Figure 13), indicate that there is no apparent trade-off be-tween the model parameters. As was the case in the inversionfor ε, despite the significant decrease in the objective func-tion (to 0.03% of the initial value), the estimated VS0 is offby about 3%. Interestingly, the inversion for VS0 yields simi-lar results when the source and receiver arrays are vertical andwave propagation is predominantly horizontal. This is likelydue to the fact that the SV-wave velocity in VTI media is thesame in the vertical and horizontal directions.

4 CONCLUSIONS

One of the most important steps in performing full-waveforminversion is calculation of the gradient of the objective func-tion with respect to the model parameters. Here, we employedthe adjoint-state method to develop gradient computation forelastic multicomponent wavefields from 2D VTI media. ThenFWI was implemented in the time domain for transmittedwaves from point displacement sources.

The in-plane polarized waves (P and SV) are con-trolled by combinations of four stiffness coefficients:C11, C13, C33, and C55. The adjoint-state method helpedderive analytic expressions for the gradients of the least-squares objective function with respect to the stiffnesses. Amore convenient parameterization used here includes the P-wave vertical (VP0), NMO (Vnmo,P ), and horizontal (Vhor,P )velocities and the S-wave vertical velocity (VS0).

Numerical tests were conducted for Gaussian anomaliesin one of the Thomsen parameters added to a homogeneousVTI background. The magnitude of the update with respect toeach model parameter was shown to be governed by the lo-


Figure 8. VTI model with a Gaussian anomaly in VP0. The background and maximum values of VP0 are 3000 m/s and 3283 m/s, respectively. Theother Thomsen parameters are spatially invariant: VS0 = 1500m/s, δ = −0.05, and ε = 0.1.

cation and configuration of the source and receiver arrays. Ananomaly just in ε (with unperturbed VP0, VS0, and δ) producesthe corresponding anomaly in a single velocity estimated byFWI (Vhor,P ). When the source and receiver arrays are ver-tical, the inversion algorithm correctly updates only Vhor,P ,thus recovering the spatial distribution of ε.

In another test, introducing an anomaly in VP0 alsochanges the velocities Vnmo,P and Vhor,P . In crosshole geom-etry, the inversion updates predominantly the horizontal ve-locity without the corresponding changes in VP0 and Vnmo,P ,which results in distorted estimates of VP0 and ε. The ratioVnmo,P /VP0, however, remains practically unchanged duringthe inversion, so the coefficient δ stays close to the backgroundvalue (as it should be). The same anomaly in the P-wave ver-tical velocity, but with horizontal arrays of sources and re-ceivers, does result in a significant update in VP0. The veloc-ities Vnmo,P and Vhor,P , however, remain almost unchanged.Hence, although the inversion recovers most of the anomaly inVP0, it incorrectly updates ε and δ.

The best-constrained parameter in this series of testsproved to be the S-wave velocity VS0, which has negligibleinfluence on P-wave data. The inversion for an anomaly inVS0 produces an accurate update in that parameter with smallchanges in VP0, ε, and δ. In addition, estimation of VS0 worksequally well for horizontal and vertical source and receiver ar-rays, likely because the SV-wave velocities in VTI media co-incide in the horizontal and vertical directions.

The tests confirm that increasing the aperture improvesthe spatial resolution and magnitude of the recovered anoma-lies. The observed trade-offs in the inversion results may beexplained in terms of the ‘radiation pattern’ exhibited by thedifferent model parameters, as will be discussed in a sequelpaper.

5 ACKNOWLEDGMENTS

We are grateful to the members of the A(nisotropy) andi(maging) teams at CWP and to Gerhard Pratt (University ofWestern Ontario), Tariq Alkhalifah (KAUST), Daniel Kohn(University of Kiel), Ken Matson (Shell), and Jon Sheiman(contractor, Shell), for fruitful discussions. We would also liketo thank John Stockwell (CWP) and Paul Martin (Dept. ofMathematics, CSM) for help with mathematical issues. Thiswork was supported by the Consortium Project on SeismicInverse Methods for Complex Structures at CWP and by theCIMMM Project of the Unconventional Natural Gas Insti-tute at CSM. The reproducible numerical examples in this pa-per are generated with the Madagascar open-source softwarepackage freely available from http://www.ahay.org.

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(a) (b)

(c) (d)

Figure 9. Difference between the inverted parameters(a) VP0, (b) VS0, (c) ε, and (d) δ and their initial values for the model with an anomaly inVP0 (Figure 8).

Figure 10. VTI model with a Gaussian anomaly in VP0. The model parameters are the same as in Figure 8, but the source and receiver arrays arehorizontal. The sources are polarized vertically.


(a) (b)

(c) (d)

Figure 11. Difference between the inverted parameters (a) VP0, (b) VS0, (c) ε, and (d) δ and their initial values for the model with an anomaly inVP0 (Figure 10).

Figure 12. VTI model with a Gaussian anomaly in VS0. The background and maximum values of VS0 are 1500 m/s and 1783 m/s, respectively.The other Thomsen parameters are spatially invariant: VP0 = 3000m/s, δ = −0.05, and ε = 0.1. The sources are polarized horizontally.

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Plessix, R.-E., and H. Rynja, 2010, VTI full waveform in-version: a parameterization study with a narrow azimuthstreamer data example: SEG Technical Program ExpandedAbstracts, 29, 962–966.


(a) (b)

(c) (d)

Figure 13. Difference between the inverted parameters (a) VP0, (b) VS0, (c) ε, and (d) δ and their initial values for the model with an anomaly inVS0 (Figure 12).

Plessix, R. ., 2006, A review of the adjointstate method forcomputing the gradient of a functional with geophysicalapplications: Geophysical Journal International, 167, 495–503.

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APPENDIX A: GRADIENT COMPUTATION FOR VTI MEDIA USING THE ADJOINT STATE METHOD

As discussed in Plessix (2006) and Liu and Tromp (2006), the objective function in equation 1 should be minimized under the con-straint that the modeled displacement u(xr, t) satisfies the wave equation. Here, we use the elastic wave equation for heterogeneous,arbitrarily anisotropic media:

ρ∂2ui

∂t2− ∂

∂xj

(cijkl

∂uk

∂xl

)= fi , (A1)

where ρ is the density, cijkl are the components of the stiffness tensor, and f is the body force per unit volume. All indices changefrom 1 to 3 and summation over repeated indices is implied. The displacement wavefield is subject to the initial conditions,

u(x, 0) = 0,∂u(x, 0)

∂t= 0, (A2)

and the radiation boundary condition,

u(x, t)|x→∞ → 0. (A3)

The method of Lagrange multipliers (Strang, 1991) is used to define the Lagrangian Λ:

Λ =1

2

∑r

T∫0

‖u(xr, t)− d(xr, t)‖2dt−

T∫0

∫Ω

λi

[ρ∂2ui

∂t2− ∂

∂xj

(cijkl

∂uk

∂xl

)− fi

]dV dt, (A4)

where r = 1, 2 . . . N denotes the receivers, Ω is the integration domain (which includes the entire 3D space), ∂Ω is the surfaceof Ω, and λ(x, t) is the vector Lagrange multiplier that needs to be determined. The objective is to find the stationary points ofthe Lagrangian, which is done by calculating the variation in Λ when u, λ, and cijkl are perturbed. After integration by parts andapplication of the Gauss divergence theorem, we obtain the change in the Lagrangian,

δΛ =

T∫0

∫Ω

N∑r=1

(ui(x, t)− di(x, t)

)δ(x− xr)δui dV dt

−T∫

0

∫Ω

δcijkl∂uk

∂xl

∂λi

∂xjdV dt−

T∫0

∫Ω

[ρ∂2λi

∂t2− ∂

∂xj

(cijkl

∂λk

∂xl

)]δui dV dt

−T∫

0

∫Ω

[ρ∂2λi

∂t2− ∂

∂xj

(cijkl

∂λk

∂xl

)− fi

]δλi dV dt

−∫Ω

[ρλi

∂(δui)

∂t− ρ δui

∂λi

∂t

]∣∣∣∣∣∣T

0

dV

+

T∫0

∫∂Ω

λi

[δcijkl

∂uk

∂xl+ cijkl

∂(δuk)

∂xl

]nj dS dt−

T∫0

∫∂Ω

δui cijkl∂λk

∂xlnj dS dt , (A5)

where n is the vector normal to the surface ∂Ω. Perturbing u(x, t) in equations A2 and A3 yields the initial and boundary conditionsfor δu(x, t):

δu(x, 0) = 0,∂δu(x, t)

∂t= 0, δu(x, t)|x→∞ → 0. (A6)

The Lagrange multiplier λ is constrained by the “final” conditions (i.e., those at time T ),

λ(x, T ) = 0,∂λ(x, T )

∂t= 0, (A7)


and the boundary condition,

λ(x, t) |x→∞ → 0. (A8)

Equation A5 then reduces to

δΛ =

T∫0

∫Ω

N∑

r=1

(ui(x, t)− di(x, t)

)δ(x− xr)−

[ρ∂2λi

∂t2− ∂

∂xj

(cijkl

∂λk

∂xl

)]δui dV dt

−T∫

0

∫Ω

∂uk

∂xl

∂λi

∂xjδcijkl dV dt

−T∫

0

∫Ω

[ρ∂2λi

∂t2− ∂

∂xj

(cijkl

∂λk

∂xl

)− fi

]δλi dV dt . (A9)

The Lagrangian is stationary with respect to the variables u, λ, and cijkl when the coefficients of δui, δλi, and δcijkl in theintegrands of equation A9 go to zero. For a given model (i.e., fixed cijkl), setting the coefficient of δλi to zero gives the stateequation, which coincides with the elastic wave equation A1. Setting the coefficient of δui to zero yields the adjoint-state equation:

ρ∂2λi

∂t2− ∂

∂xj

(cijkl

∂λk

∂xl

)=

N∑r=1

[ui(xr, t)− di(xr, t)

], (A10)

subject to the conditions at time T (equation A7) and boundary conditions (equation A8). Equations A1 and then A10 are solved toobtain the wavefields u and λ respectively.

Substituting u and λ into the coefficient of the term containing δcijkl in equation A9 gives the variation in the Lagrangianwith the stiffnesses. Since Λ = F when u satisfies the wave equation (from equation A4), the change in the objective function δFcaused by perturbations of the stiffness coefficients is given by:

δF = −T∫

0

∫Ω

∂ui

∂xj

∂λk

∂xlδcijkl dV dt . (A11)

This is a general result for an anisotropic medium described by the complete stiffness tensor cijkl. Expressions for models withspecific symmetries can be derived from equation A11 by substituting the appropriate stiffness tensors or matrices.

Note that the boundary conditions for u(x, t) and λ(x, t) can be modified to include a free surface where the tractions dueto u and λ go to zero. However, the addition of the free surface causes complications in finite-difference modeling and producessurface multiples. Instead, we impose the radiation condition to create absoring boundaries on all sides of the model.

To simulate the Lagrange-multiplier wavefield, it is convenient to define an “adjoint wavefield” ψ (Liu and Tromp, 2006):

ψ(x, t) ≡ λ(x, T − t). (A12)

The wavefield ψ satisfies the wave equation A10 but with the source function reversed in time:

ρ∂2ψi

∂t2− ∂

∂xj

(cijkl

∂ψk

∂xl

)=

N∑r=1

[ui(xr, T − t)− di(xr, T − t)

]. (A13)

The initial conditions for ψ (using equations A7 and A12) are as follows:

ψ(x, 0) = 0,∂ψ(x, 0)

∂t= 0. (A14)

The wavefield ψ also satisfies the radiation boundary condition:

ψ(x, t)|x→∞ → 0. (A15)

From equations A11 and A12, we can find the gradient of the objective function with respect to the stiffness coefficients:

∂F∂cijkl

= −T∫

0

∂ui

∂xj

∂ψk

∂xldt. (A16)


If, instead of cijkl, the model is described by parameters mn, the gradient of F can be found from the chain rule:

∂F∂mn

=∑ijkl

∂F∂cijkl

∂cijkl∂mn

. (A17)

Here, we parameterize the model in terms of the velocities VP0, VS0, Vnmo,P , and Vhor,P . The stiffness coefficients (written in thetwo-index notation) represent the following functions of the velocities (Tsvankin, 2012):

C11 = ρ V 2hor,P , (A18)

C33 = ρ V 2P0, (A19)

C13 = ρ√

(V 2P0 − V 2

S0)(V 2nmo,P − V 2

S0)− ρ V 2S0, (A20)

C55 = ρ V 2S0. (A21)

Using equations A16, A17, and A18 − A21, we obtain the following gradients of the objective function with respect to thevelocities:

∂F∂VP0

= −2VP0

T∫0

ρ

∂ψ3

∂x3

∂u3

∂x3+

1

2

√V 2

nmo,P − V 2S0

V 2P0 − V 2

S0

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

) dt, (A22)

∂F∂VS0

= −2VS0

T∫0

ρ

2V 2

S0 − V 2P0 − V 2

nmo,P

2√

(V 2nmo,P − V 2

S0)(V 2P0 − V 2

S0)− 1

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

)

+

(∂ψ1

∂x3+∂ψ3

∂x1

)(∂u1

∂x3+∂u3

∂x1

)dt, (A23)

∂F∂Vnmo,P

= −2Vnmo,P

T∫0

ρ

2

√V 2P0 − V 2

S0

V 2nmo,P − V 2

S0

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

)dt, (A24)

∂F∂Vhor,P

= −2Vhor,P

T∫0

ρ∂ψ1

∂x1

∂u1

∂x1dt, (A25)

Elastic full-waveform inversion of transmission data in 2D VTI media · 2017. 5. 3. · Elastic...

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