Full Waveform Inversion with the Reconstructed Wavefield MethodChao Wang, David Yingst, Paul Farmer, and Jacques Leveille, ION

SUMMARY

Over the last decade, conventional full waveform inversion(FWI) has been widely applied to real seismic data for bothproduction and research purposes. The underlying theory hasbeen well established and produces high resolution subsurfacemodels by minimizing the misfit between the seismic dataand simulated seismograms obtained by solving the waveequation exactly. However, in practice, it is still a challenginginversion method for updating the model parameters. Alocal optimization scheme is used to solve the minimizationproblem and it does not prevent convergence towards localminima because of the nonlinearity and ill-posedness of theproblem. For example, FWI may converge to a local minimumbecause of the lack of low frequencies in the recorded data oran inaccurate starting model.

We propose a novel approach to time domain full waveforminversion with the reconstructed wavefield method (RFWI).RFWI relaxes the constraint that the forward modeled dataexactly solve the wave equation as in conventional FWI, andinstead uses an `2 approximate solution. RFWI estimatesearth models and jointly reconstructs the forward wavefieldby minimizing an objective function that includes penaltiesfor both the data misfit and wave equation error. By extendingthe search space, RFWI offers potential benefits of avoidingcycle skipping and overcoming some of the problems withlocal minima.

This paper first presents the theory and implementation of timedomain RFWI. It also discusses the differences and similaritiesbetween conventional FWI and RFWI. The benefits of RFWIover conventional FWI are demonstrated using a 2D syntheticexample. Finally, the applicability of RFWI on field data isillustrated on a 2D streamer data set from offshore Congo anda 3D ocean bottom seismic data set from the Gulf of Mexico.

INTRODUCTION

The goal of conventional full waveform inversion (FWI) is toestimate earth properties from the information acquired on thesurface. FWI has been an important method to build high-fidelity earth models for seismic imaging (Lailly, 1983; Taran-tola, 1984; Virieux and Operto, 2009). It minimizes the misfitof the difference between the acquired and modeled data andhas been implemented in both the time and frequency domain(Sirgue and Pratt, 2004; Wang et al., 2013). However, it isa highly nonlinear, ill-posed problem and mitigating conver-gence to local minima is a severe challenge. For example, itmay suffer from cycle skipping problems if there is a lack oflow frequency data. It may also converge to a local minimumwithout a good starting model.

For time domain FWI, the synthetic data are extracted from

the wavefield generated by solving the wave equation, with anexact numerical solver using a finite difference scheme. In thispaper, we propose a novel approach to time domain FWI. Thismethod, referred to as full waveform inversion with a recon-structed wavefield (RFWI), replaces the exact solution of thewave equation with an `2 approximation. While conventionalFWI searches for earth models such that the simulated wave-field solves the wave equation exactly and the simulated datahave the best match to the field data, RFWI optimizes overearth models and the wavefield jointly to minimize the datamisfit subject to the wavefield being consistent with the waveequation in an `2 sense.

Unlike the misfit (objective) function of conventional FWI, theidea of RFWI is to add the wave equation error as a penaltyterm to the original data misfit. Instead of solving for one un-known, which is an earth model, now we are solving for twounknowns a model and a forward propagated wavefield. Wereconstruct the wavefield and estimate the model parameter inan alternating fashion. We first reconstruct the wavefield byminimizing the wave equation error and the data misfit. Thisleast sqaures solution is computed by solving the normal equa-tion. The reconstructed wavefield is then used for updating themodel parameter with a gradient based optimization method.Recently, wavefield reconstruction inversion has been intro-duced in the frequency domain (van Leeuwen and Hermann,2013). Here we introduce our novel method and implementa-tion of time domain RFWI, which is based on finite differencescheme and can be applied to 3D large-scale data sets.

By expanding the search space, RFWI forces the forward mod-eled data to better fit the field data and avoid cycle skips. Modelparameters can then be updated by enforcing the wave equa-tion in an `2 sense. RFWI may mitigate some of the problemswith local minima that occur in conventional FWI when thereis a lack of low frequency data or the initial model is inade-quate. It also takes advantages of reflected seismic waves andreconstructs deeper portions of the model than conventionalFWI that usually relies on diving waves. In general, RFWIdemonstrates more advantages in areas with strong velocitycontrasts.

THEORY

In this paper, we only consider inversion for the velocity modelin the isotropic acoustic wave equation. The idea can be easilyextended to more general wave equations. Consider the fol-lowing isotropic acoustic wave equation,

2[v]u = (1v2

∂ t2−∆)u = f . (1)

Here v is the velocity model, 2[v] is the wave operator orD’Alembert operator, u is the forward propagated wavefield,and f is the source wavelet. Let S[v] denote the solution op-erator of the forward propagated wave equation (1). Then we

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Full Waveform Inversion with the Reconstructed Wavefield Method

have forward simulated wavefield u = S[v] f . Inversion can beformulated as the following constrained minimization problem

minu,v

12 ‖ Pu−d0 ‖22,

s.t. 2[v]u = f .(2)

Here P is the restriction operator (a projection) that records thewavefield u at the receiver locations and d0 is the field data.

The idea of conventional FWI is to solve the wave equationexactly with the given source. By replacing the forward prop-agated wavefield u with the exact solution S[v] f , we eliminatethe constraint in (2) and obtain the objective function for con-ventional FWI which uses the norm of the difference betweenthe acquired field data and computer simulated forward mod-eled data that depends on velocity only

J[v] =12‖ PS[v] f −d0 ‖22 . (3)

The idea of RFWI is to relax the constraint in (2) that u be anexact solution of the wave equation to an `2 approximation,by adding the wave equation error as a penalty term. Thus anew penalized objective function depending on both u and v isintroduced

J̄λ [u,v] =12‖ Pu−d0 ‖22 +

λ 2

2‖2[v]u− f ‖22, (4)

for a penalty scalar λ . Wavefield u should be forward goingand therefore in the range of S, i.e. u = S[v]g for some g. Notethen that 2[v]u = g and so this objective function can be castas

Jλ [g,v] =12‖ PS[v]g−d0 ‖22 +

λ 2

2‖ g− f ‖22 . (5)

This problem is a joint minimization with respect to both gand v. To make it computationally feasible, we first minimizethe above objective function w.r.t. g for a fixed v, which isthe current velocity model. This is equivalent to solving thefollowing least squares problem

ming

∥∥∥∥(

PS[v]λ I

)g−

(d0λ f

)∥∥∥∥2

2,

which reduces to the following normal equation

S∗P∗PSg+λ 2g = S∗P∗d0 +λ 2 f .

Now assume that the reconstructed source g is the solution ofthe normal equation. Write g as a perturbation of f ,

g = f + ḡ/λ 2,

where ḡ satisfies

ḡ = S∗P∗d0−S∗P∗PS f −S∗P∗PSḡ/λ 2. (6)

To make this computationally feasible, we ignore the thirdterm on the right hand side of equation (6) and redefine

g = f + g̃/λ 2, (7)

where g̃ satisfies

g̃ = S∗P∗d0−S∗P∗PS f . (8)

To proceed we simply assert without proof that this is suffi-ciently close to the minimum that ∇gJλ [g(v),v]≈ 0. Introduc-ing a new objective function J̃λ [v] = Jλ [g(v),v], it follows that

∇vJ̃λ =∂g∂v

∇gJλ +∇vJλ ≈ ∇vJλ .

Our next step is to minimize the above objective function J̃λ [v]w.r.t. v. Having reconstructed g, we can now reconstruct theforward wavefield ũ = Sg. Letting u1 = S f and u2 = Sg̃, thefinal forward wavefield ũ can be reconstructed by adding u1 tou2/λ 2 according to equation (7), i.e.

ũ = u1 +u2/λ 2.

Note that the fixed forward reconstructed wavefield ũ = Sg andg = 2ũ, so the velocity model v then can be updated using aconjugate gradient method and the gradient for this objectivefunction w.r.t. v can be calculated using

∇vJ̃λ [v] ≈ ∇vJλ [g,v]

= −〈

2v3

∂ 2t ũ, S∗P∗d0−S∗P∗PSg

〉

= −〈

2v3

∂ 2t ũ, λ2(g− f )

〉

= −〈

2v3

∂ 2t ũ, g̃〉.

When the penalty scalar λ is large enough, RFWI and conven-tional FWI would converge to very similar results. In orderto make RFWI produce favorable model updates, the scalar λneeds to be chosen carefully.

SYNTHETIC EXAMPLE

(a) True model (b) Initial model

(c) Inverted model from conventional FWI (d) Inverted model from RFWI

Figure 1: Synthetic models

We first verify our proposed time domain RFWI by applyingit to a 2D synthetic data set and draw a comparison with con-ventional time domain FWI. The true model for this syntheticexample is a modified BP 2004 model as shown in Figure 1(a).The field data set was generated using 531 shots with an inter-val of 50 m. Each shot gather contains 531 receivers with aninterval of 50 m. The initial model is a simple velocity modelwith linear velocity gradient shown in Figure 1(b). Figure 1(c)is the updated model from conventional FWI and Figure 1(d)

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Full Waveform Inversion with the Reconstructed Wavefield Method

is the updated model from RFWI. From these results we no-tice that the salt is much better resolved using RFWI than con-ventional FWI. This is because RFWI takes advantages of thereflected seismic waves. Cycle-skipping is avoided by RFWIsince it is forcing the forward data to match the field data afterwavefield reconstruction and it is less likely for RFWI to betrapped into local minima than conventional FWI with a badstarting model. This makes RFWI a technique that will workbetter in areas with strong velocity contrasts.

FIELD EXAMPLE 1

The second example involves an application of time domainRFWI to a 2D streamer data set from offshore Congo. Theacquisition length was 300 km and no preprocessing has beenapplied to the input data. The field data set has been resam-pled to 16 ms. The inversion used 500 shots with a shot spac-ing of 600 m. The maximum offset is 10200 m and the lowestfrequency used is 5 Hz. We simulated synthetic data using aRicker wavelet with a peak frequency of 10 Hz. The modelin Figure 2(a) shows a simple initial velocity for this exam-ple and the model in Figure 2(b) is the inverted velocity fromRFWI. The maximum depth is 6012 m. The updated velocityincludes more structural details that follow the geology. Afterthe inversion using RFWI, not only the shallow velocity hasbeen updated, but also deeper changes have started building upthe top of the salt with an improved salt velocity. Finally, we

(a) Initial

(b) RFWI inverted

Figure 2: Velocity models

forward modeled the data using the initial and inverted mod-els and generated shot gathers that are displayed in Figure 3(a)and 3(b). Comparing with field data in Figure 3(d) for thesame shot record, the forward modeled data using the invertedmodel after RFWI fit the field data much better than using theinitial model. The synthetic data also demonstrate improve-ment in the near and far offsets. Figure 3(c) shows the re-constructed seismogram using the reconstructed wavefield andinverted model 2(b), which matches the shot gather of the fielddata in Figure 3(d) the best.

(a) using initial model (b) using RFWI model

(c) using reconstruction (d) field data

Figure 3: Shot gathers

FIELD EXAMPLE 2

Finally we present an application of time domain RFWI to 3Dmarine data. This deep water ocean bottom seismic survey islocated in the Green Canyon area of the Gulf of Mexico. Theacquisition area was 160 km2. This survey has 19901 sourceswith an interval of 50 m. Maximum offset used is 7000 m.The lowest frequency in the observed data is about 5 Hz. Thesource signature was derived from the down-going wavefieldon a zero offset section. The inversion proceeded with a singlefrequency band up to 8 Hz. The initial velocity model was builtfrom tomography and its maximum depth is 12000 m. We firstcompare the gradients of conventional FWI and RFWI withthis initial model. Figure 4(a) shows the gradient for conven-tional FWI using the initial model. Cycle skipping preventsit from getting any updates except for the shallow sedimentarea. The deeper portion of the velocity model will not havea reasonable update because of limitations imposed by usingthe refraction data. However, the RFWI gradient using the ini-tial model in Figure 4(b) has a clear base salt boundary at theright location. The subsalt gradient looks reasonable and fol-lows the geology. The shallow updates for the sediment areashould be comparable between the two methods as is observed.RFWI does improve the structure of the salt and the sedimentsbelow it. Since the initial gradient demonstrates the potentialbenefits of RFWI, we now want to iterate and update the veloc-ity model using RFWI. The inverted velocity model is shownin Figure 5(b) after 13 iterations of RFWI. The maximum up-dated depth is 12000 m. It demonstrates a reasonable shallowupdate above the salt. It also shows a deep update below 7000

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Full Waveform Inversion with the Reconstructed Wavefield Method

(a) Conventional FWI

(b) RFWI

Figure 4: Gradients using the initial velocity model shown inFigure 5(a)

m that follows the salt boundary and sub salt geology. Thelocation of the salt body interface is well preserved by impos-ing a salt mask as in Figure 5. By improving the velocity, the

(a) Initial

(b) RFWI inverted

Figure 5: Velocity models

offset gathers 6(b) after RFWI show improvement in flatnesscompared to the gathers 6(a) using the initial velocity for sedi-ment area above 2707 m. To further evaluate the RFWI result,we generated the images using RTM and compare the subsaltimaging using the initial and inverted velocity models. Figure7 shows zoomed images below the salt from 6035 m to 11883m migrated with the two models. Comparing with the imageusing the initial model in Figure 7(a), the structures below thesalt are improved with better continuity for the updated imagein Figure 7(b).

CONCLUSION

We presented the methodology and results of our proposednovel inversion method - time domain RFWI. RFWI helpsavoid cycle skipping issues and overcomes some of the prob-lems with local minima that occur in conventional FWI. Itdemonstrates more advantages in areas with strong velocitycontrasts.

(a) Using initial velocity model

(b) Using RFWI inverted velocity model

Figure 6: Offset gathers for depth from 702 m to 2707 m

ACKNOWLEDGMENTS

We would like to thank ION for permission to publish the re-sults and thank our colleagues for providing valuable support,especially Carlos Calderon, Ian Jones, Nick Bernitsas, IvanBerranger, Junyong Chang, Wei Huang, and Guoquan Chen.

(a) Using initial velocity model (b) Using RFWI inverted velocity model

Figure 7: Migrated images for depth from 6035 m to 11883 m

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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2016

SEG Technical Program Expanded Abstracts have been copyedited 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 Lailly, P., 1983, The seismic inverse problem as a sequence of before stack migrations: Proceedings of

the Conference on Inverse Scattering, Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, 206–220.

Sirgue, L., and R. Pratt, 2004, Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies: Geophysics, 69, 231–248, http://dx.doi.org/10.1190/1.1649391.

Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266, http://dx.doi.org/10.1190/1.1441754.

van Leeuwen, T., and F. Hermann, 2013, Mitigating local minima in full-waveform inversion by expanding the search space: Geophysical Journal International, 195, 661–667, http://dx.doi.org/10.1093/gji/ggt258.

Virieux, J., and S. Operto, 2009, An overview of full-waveform inversion in exploration geophysics: Geophysics, 74, no. 6, WCC1–WCC26, http://dx.doi.org/10.1190/1.3238367.

Wang, C., D. Yingst, J. Bai, J. Leveille, P. Farmer, and J. Brittan, 2013, Waveform inversion including well constraints, anisotropy, and attenuation: The Leading Edge, 32, 1056–1062, http://dx.doi.org/10.1190/tle32091056.1.

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http://dx.doi.org/10.1190/1.1649391http://dx.doi.org/10.1190/1.1441754http://dx.doi.org/10.1093/gji/ggt258http://dx.doi.org/10.1190/1.3238367http://dx.doi.org/10.1190/tle32091056.1