Fast multi-parameter anisotropic full waveform inversion with irregular shot samplingChao Wang∗, David Yingst, John Brittan, Paul Farmer, and Jacques Leveille, ION Geophysical

SUMMARY

The goal of full waveform inversion (FWI) is to derive high-fidelity earth models for seismic imaging by fitting the ac-quired data. One of the major drawback of FWI is that it ishighly compute-intensive. In this paper, we propose a fastmulti-parameter FWI to dramatically reduce the computationcost. Considering the often significant numbers of sources(and receivers) in 3D seismic data acquisition, we propose amethod to largely reduce the simulation time per iteration byusing a reasonably small subset of sources instead of the fullvolume. We choose the subset by randomly picking a num-ber of sequential sources, designed to follow an irregular sam-pling pattern. This subset will be re-selected and different ateach iteration and needs to be sufficiently sampled to avoid ar-tifacts. The results achieved from this subsampling approachare comparable to the conventional method, with a highly re-duced computation cost at each iteration.

Our forward modeling and its adjoint computation are basedon the acoustic wave equation in vertical transversely isotropic(VTI) media and our target model includes three parametersthat are P-wave velocity and Thomsen’s anisotropy parameters(epsilon and delta). During the iterative process of the multi-parameter FWI, we update three parameters simultaneously ateach iteration. Multi-parameter VTI FWI does not require anyextra wavefield computation than mono-parameter VTI FWI.Hense it is more efficient and speeds up the process if our targetmodel includes more than one parameter.

This paper presents the time domain implementation for a fastmulti-parameter VTI FWI. This approach will be illustratedon 3D marine data from the Green Canyon area of the Gulf ofMexico.

INTRODUCTION

For the conventional FWI, forward and back propagation areperformed for each source individually, which means the costof conventional FWI for simulating the wavefields is propor-tional to the number of sources. Taking into account of theexcessive growth in the number of sources (and receivers) inlarge-scale 3D data acquisition, one of the main challenges ofconventional FWI is the high computation cost. Recently, var-ious researchers have investigated different methods to speedup FWI. The source-encoding technique has been proposedand studied for the purpose of cost saving in FWI (Krebs et al.(2009); van Leeuwen et al. (2011)). While replacing the se-quential sources by a number of simultaneous sources, a source-encoding technique can significantly reduce the computationcost per iteration. However, simultaneous sources will intro-duce noisy cross-talk in the gradient and can thus damage themodel update. Therefore, the cross-talk must be suppressedduring the iterations. However averaging out the cross-talk

leads to slow convergence.

In 2013, van Leeuwen and Hermann (2013) proposed to userandomly chosen sequential sources instead of encoded sourcesto eliminate the source cross-talk for frequency domain FWI.In their paper, they showed that we do not need to use simul-taneous sources to reap the benefits of stochastic optimization.In order to eliminate the source cross-talk and make it easilyfit into a general acquisition framework, we select a subset ofsequential sources to reduce the simulation cost at each iter-ation. Our picked sources follow an irregular sampling pat-tern. This subset of sources will be re-selected and different ateach iteration and needs to be appropriately sampled to avoidartifacts. We apply our source selection scheme to our timedomain multi-parameter FWI with our optimization and regu-larization techniques.

As considering that including the effects of anisotropy oftenhelps to improve FWI results, our forward modeling and its ad-joint computation are based on the time domain VTI acousticwave equations. The definition of a suitable parameterizationis a crucial issue for multi-parameter FWI. We choose to pa-rameterize our VTI FWI by the P-wave velocity and anisotropyparameters ε and δ . Our sensitivity analysis of acoustic VTIFWI has shown that velocity can be updated successfully witha rough guess of ε and δ . However ε and δ updates rely ona relatively good starting velocity model. Based on the sensi-tivity analysis, we designed a two-step practical workflow. Itconsists of one step of mono-parameter inversion for velocityonly, followed by another step of multi-parameter inversion forvelocity, ε and δ simultaneously. This joint inversion increasesthe convergence rate for updating three parameter simultane-ously at no extra wavefield cost.

This paper presents the objective function, its gradient, andmodel update for fast multi-parameter VTI FWI. It also dis-cusses the randomized techniques using irregular shot sam-pling and compares the results with uniform shot sampling.This approach will be illustrated on 3D marine data from theGreen Canyon area of the Gulf of Mexico.

METHOD: OBJECTIVE FUNCTION

The full misfit is a slightly modified misfit function from Taran-tola (1987). Our objective function is an approximation tothe full misfit that depends on the chosen subset of sequentialsources with additional well constraints.

minm

J[m] =1|S|∑

xi∈S

Φi[m] (1)

s.t. Pm = m0,

where Φi[m] = 12

∥∥∥T(

dobsi −ζi dpred

i [m])∥∥∥

2

2is the misfit for

source xi that belongs to the chosen subset S. Sources thatare not in the subset S will not be included in the objective

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Fast multi-parameter anisotropic FWI

(a) (b)

(c) (d)

Figure 1: (a) initial velocity, (b) inverted velocity with 25% uniform picking, (c) inverted velocity with 25% irregular picking, and(d) inverted velocity with all the sources

function. dobsi is the observed seismic data and dpred

i [m] is thepredicted data for the model m at source locations xi. The pre-dicted data are obtained by sampling the extrapolated wave-field p generated by a high-order finite difference scheme tothe receiver locations, based on the acoustic VTI wave equa-tions (3). T is a data preconditioner and ζi is a normalizationscalar. m0 is the extended model generated from well logsbased on the neighboring structure. P is the projection opera-tor that maps the model m to m0’s grid.

This approximation relies on the selection of sequential sourcesbased on a desired irregular pattern. In order to avoid insuffi-cient sampling, we need to constrain the picking criteria witha maximum lag between sources. Then we choose the sequen-tial source randomly within the area that satisfy the constraints.For the next iteration, we re-select a different subset of sources.

We can solve the constrained problem (1) by minimizing thefollowing unconstrained objective function with respect to musing an Augmented Lagrangian Method (Hestenes, 1969; Pow-ell, 1969; Li et al., 2013):

L [m] =1|S|∑

xi∈S

Φi[m]−〈λ ,Pm−m0〉+µ2‖Pm−m0‖2

2 ,

(2)where λ is a Lagrange multiplier and µ is a penalty scalar.The major advantage of the method is that unlike the penaltymethod, it is not necessary to take µ → ∞ in order to solve theoriginal constrained problem and avoids numerical issue forlarge µ .

METHOD: GRADIENT COMPUTATION

In 2000, Alkhalifah (2000) derived pseudo-acoustic wave equa-tions for anisotropic media that kinematically model the com-pressional wave propagation. A number of variations of pseudo-acoustic wave equations have been developed since then (Zhouand Bloor, 2006). Here we use a VTI system of two coupledsecond-order partial differential equations in terms of P-wavevertical velocity v, Thomsen parameters, ε and δ , assuminga constant density and zero shear velocity, with initial andboundary conditions:

1v2 ∂ 2

t

(qp

)=

(1+2ε 11+2δ 1

)(∂ 2

x +∂ 2y 0

0 ∂ 2z

)(qp

)+

(0f

),

(3)where f is the input source wavelet, p is the forward-propagatedwavefield and q is the auxiliary wavefield. We then solve theadjoint equations of the forward equations (3) and obtain theback-propagated wavefields p+ and q+ by back-propagatingthe residual.

In this case, the approximated misfit gradient ∇J with respectto model m that includes three parameters (velocity v, Thom-sen parameters ε and δ ), are given by

∇J(x) =1|S|∑

xi∈S

∑

t

2v3(x) (∂

2t pp++∂ 2

t qq+)(x, t;xi)

2((∂ 2x q+∂ 2

y q)q+)(x, t;xi)

2((∂ 2x q+∂ 2

y q)p+)(x, t;xi)

.

(4)

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Fast multi-parameter anisotropic FWI

The total gradient is given by

∇L (x) = ∇J(x)− (P∗λ )(x)+µ(P∗(Pm−m0))(x).

Our forward and adjoint equations provide a very simple formof gradient calculation. Convergence can be accelerated usinggradient preconditioning. The current preconditioning normal-izes the gradient by the amplitude of the forward propagatedwave with a whitening factor.

METHOD: MODEL UPDATE

The basic scheme for updating the model using an iterativeoptimization method is

mk+1 = mk +αkBdk,

where αk is the step length computed using line search and dkis the search direction at the k-th iteration. It uses the gradientat an initial point for an initial direction estimate and updatesthat direction using nonlinear conjugate gradient method.

mk =

(vkεkδk

),B =

(bv 0 00 bε 00 0 bδ

). (5)

mk is the joint model including three parameters at the k-thiteration. B is a scaling factor that is chosen to correct theweights for each direction component.

GULF OF MEXICO EXAMPLE

We present an application of fast multi-parameter FWI to 3Dmarine data. This deep water survey is located in the GreenCanyon area of the Gulf of Mexico. The acquisition area was160 km2 and used four-component ocean bottom seismic re-ceivers in deep water over relatively shallow salt bodies with19901 shots. Maximum offset used is 7000 m. The lowestfrequency observed in the data is about 3 Hz. The source sig-nature was derived from the down-going wavefield on a zerooffset section.

We ran three multi-parameter FWI tests to illustrate the bene-fits of irregular shot sampling using exactly the same workflowwith same number of iterations. First we chose 25% of sequen-tial sources on a uniform sampling pattern. Second, we picked25% of sequential sources using our randomized irregular sam-pling pattern. For the third test, we used all the sources. Fromthe velocity update shown in Figure 1, we illustrate that theirregular shot sampling using 25% of sources shown in Figure1(c) reduced the simulation cost by a factor of four comparingto the conventional method shown in Figure 1(d) and produceda comparable result. However, the uniform shot sampling us-ing 25% of sources shown in 1(b) introduced strong artifactdue to insufficient sampling. For this data example, 25% ofsources are the minimum subset required for successful inver-sion due to frequency and source spacing.

The inverted epsilon and delta models in Figure 2(b) and 3(b)show reasonable shallow updates up to a depth of 2500 m in-cluding some detailed structures above the salt. Therefore 25%

(a)

(b)

Figure 2: (a) initial epsilon, (b) inverted epsilon with 25% ir-regular picking

of sequential sources using irregular shot sampling generatereliable anisotropy updates as well.

In the FWI workflow for this data, we first applied mono-parameter FWI for 33 iterations and then we parameterized thetarget model by three parameters for 7 iterations of simultane-ous inversion. The initial models, Figure 1(a), 2(a), and 3(a),were built from anisotropic VTI tomographic inversions. Var-ious strategies such as multi-scale, layer stripping, and offsetweighting have been applied to minimize the risk of converg-ing to local minima.

We validate our results by comparing the flatness of the mi-grated gathers. Gather 4(b) after randomized FWI shows over-all improvement in flatness compared to gather 4(a) beforeFWI. To further evaluate randomized FWI results, we gener-ated stack images with the initial models and the inverted mod-els obtained by FWI with irregular shot sampling. The stackimage after FWI in Figure 5(b) shows improvement comparedto the initial stack shown in Figure 5(a) with better focus andevent consistency.

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Fast multi-parameter anisotropic FWI

(a)

(b)

Figure 3: (a) initial delta, (b) inverted delta with 25% irregularpicking

CONCLUSION

In this paper, we have presented a methodology and strate-gies for fast multi-parameter FWI using a randomized irreg-ular shot sampling technique. Picking a small subset of se-quential sources helps to reduce the computation cost signifi-cantly. These approaches were illustrated on a 3D marine dataset from the Green Canyon area of the Gulf of Mexico. Fromthe results, we showed that simultaneous inversion for multipleparameters using a small subset produced good results compa-rable with using the entire dataset with only 25% simulationcost of conventional FWI.

ACKNOWLEDGMENTS

We would like to thank ION-GXT for permission to publishthe results and SINBAD consortium for stimulating the inno-vation. We also thank our colleagues at ION GeoScience Teamfor providing us valuable support in this work, especially IanJones, Helen Delome, Guoquan Chen, Jianyong Bai, and Mo-hamed Dolliazal.

(a)

(b)

Figure 4: Gather using (a) initial models, (b) inverted modelswith 25% irregular picking

(a)

(b)

Figure 5: Stack image using (a) initial models, (b) invertedmodels with 25% irregular picking

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http://dx.doi.org/10.1190/segam2014-0234.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Alkhalifah, T., 2000, An acoustic wave equation for anisotropic media : Geophysics, 65, 1239–1250, http://dx.doi.org/10.1190/1.1444815.

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