Reconstructed Full Waveform Inversion with the Extended SourceChao Wang, David Yingst, Paul Farmer, Ian Jones, Gary Martin, and Jacques Leveille, ION

SUMMARY

Conventional full waveform inversion (FWI) has been exten-sively applied to real seismic data and has successfully gener-ated high-fidelity earth models for better seismic imaging andstructural interpretation. Considering the nonlinearity and ill-posedness of the problem for conventional FWI, the successin providing reliable updated models heavily relies on the ac-curacy of the initial models and low frequency contents in theacquired data.

To relax the requirements of good initial models and adequatelow frequencies, we propose a novel approach to time do-main reconstructed full waveform inversion with the extendedsource (RFWI). RFWI relaxes the constraint that the forwardmodeled data exactly solve the wave equation as in conven-tional FWI, and instead uses an `2 approximate solution. RFWIestimates earth models and jointly reconstructs an extendedsource by minimizing an objective function that penalizes thewave equation error while fitting the data. RFWI extends thesolution space and therefore overcomes some of the problemswith local minima that prevent conventional FWI from obtain-ing a reliable solution with an inaccurate starting model and/orinsufficient low-frequency data .

INTRODUCTION

Conventional full waveform inversion (FWI) has been an es-sential tool to build high-fidelity earth models by minimiz-ing the misfit between the acquired and modeled data (Lailly,1983; Tarantola, 1984; Virieux and Operto, 2009). This least-squares problem has been implemented in both the time andfrequency domain (Sirgue and Pratt, 2004; Wang et al., 2013).However, it is a highly nonlinear, ill-posed inversion problemand mitigating convergence to local minima is a severe chal-lenge. The greatest limitation of FWI that affects the successin generating reliable solutions for large-scale production jobsis the critical requirement of low-frequency data coupled witha good starting model.

Over the last decade, various effort has been made to miti-gate the problems of local minima and many alternative meth-ods have been proposed (Shen and Symes, 2008; van Leeuwenand Hermann, 2013; Biondi and Almomin, 2014; Warner andGuasch, 2016; Wang et al., 2016; Huang et al., 2016). All theseprevious works and our proposed method in this paper aim toavoid convergence to local minima by adding additional pa-rameters to the models and expanding the solution space.

We now focus on the time-domain method and implementationusing finite difference scheme. To compute the misfit functionfor time-domain FWI, the conventional forward modeled dataare extracted from the source wavefield generated by solvingthe forward wave equation exactly with the given source sig-nature. Our reconstructed forward modeled data are obtained

from the reconstructed source wavefield by solving the waveequation with the extended or reconstructed source. We re-fer to this method as reconstructed full waveform inversionwith the extended source (RFWI). Having simulated the for-ward modeled data and source wavefield, conventional FWIsearches for earth models such that the synthetic data have thebest match to the field data in the least-squares sense. RFWIoptimizes over earth models and the source wavefield jointly tominimize the data misfit subject to the wavefield being consis-tent with the wave equation in an `2 sense. By reconstructinga better source wavefield from the extended source instead ofthe original source signature with the current model, RFWIrelaxes the severe requirement for conventional FWI and pro-vides a more reliable model update.

By including the source wavefield as an additional parame-ter to the search space, RFWI adds the wave equation erroras a penalty term to the original data misfit in conventionalFWI and formulates a joint minimization problem. We re-construct the source wavefield and estimate the earth modelsin an alternating fashion. We first reconstruct the extendedsource by minimizing the wave equation error together withthe data misfit. It is estimated from solving the normal equa-tion which is equivalent to the least-squares solution. Thesource wavefield is then reconstructed from forward propaga-tion of the extended source. With the reconstructed wavefieldand the extended source, models are updated with a gradientbased optimization method and inverted models are used forreconstructing another source and wavefield at next iteration.Time-domain implementation differentiates RFWI from otherprevious works that are related to wavefield reconstruction in-version in the frequency domain (van Leeuwen and Hermann,2013; Huang et al., 2016) and provides a more suitable solverfor processing 3D large-scale production data sets.

By adding additional parameter and expanding the search space,RFWI reconstructs the forward modeled data to better fit thefield data and avoid cycle skips. Therefore it mitigates some ofthe problems with local minima with inaccurate initial modelsand/or inadequate low-frequency contents that limit the suc-cess of conventional FWI. While FWI usually relies on divingwaves, RFWI takes advantages of reflected waves from wave-field reconstruction and produces deeper model updates. Italso compensates the wavefield errors that relates to the acous-tic assumptions and approximations during the wave propaga-tion. From the observations of both synthetic and field exam-ples, RFWI demonstrates more advantages in areas with sharpvelocity contrasts, especially with the presence of the salt.

This paper first presents the theory and methodology for 3Dtime domain RFWI. It also discusses the differences and sim-ilarities between conventional FWI and RFWI. The benefitsof RFWI over conventional FWI are demonstrated using a 2Dsynthetic example. Finally, the applicability of RFWI on fielddata is illustrated on both 2D and 3D streamer data from off-shore Mexico.

Reconstructed Full Waveform Inversion with the Extended Source

THEORY

Consider the following general wave equation,

2[m]u = f . (1)

where m represents the subsurface model parameters, 2[m] isthe wave operator or D’Alembert operator, u is the forwardpropagated wavefield, and f is the source signature. Let S[m]denote the solution operator of the forward propagated waveequation (1). At each iteration, conventional FWI solves thewave equation exactly with the given source and the currentmodel to obtain the source wavefield u = S[m] f . The objectivefunction for conventional FWI which uses the `2 norm of thedifference between the acquired field data and simulated for-ward modeled data that depends on the model m only is definedas

J[m] =12‖ PS[m] f −d0 ‖2

2 . (2)

Here P is the restriction operator (a projection) that records thesource wavefield u at the receiver locations and d0 is the fielddata.

Unlike conventional FWI, the idea of RFWI is to relax the con-straint that u be an exact solution of the wave equation to an `2approximation, by adding the wave equation error as a penaltyterm. Thus a new penalized objective function depending onboth the source wavefield u and the model m is introduced as

Jλ [u,m] =12‖ Pu−d0 ‖2

2 +λ 2

2‖2[m]u− f ‖2

2, (3)

where λ is a penalty factor that requires to be updated duringthe inversion. Source wavefield u should be forward goingand therefore in the range of S, i.e. u = S[m]g for some g.We refer to g as the reconstructed or extended source sinceit varies with space as well as with time for each shot and itis used to reconstruct the source wavefield u. Note then that2[m]u = g and so the objective function with the reconstructedor extended source and the earth model can be redefined as

Jλ [g,m] =12‖ PS[m]g−d0 ‖2

2 +λ 2

2‖ g− f ‖2

2 . (4)

To make this joint minimization problem computationally fea-sible, we solve g and m in an alternating fashion. We first min-imize the above objective function w.r.t. g using the currentmodel. Since the mismatch will be built into the reconstructedor extended source, the reconstructed wavefield better matchesthe true wavefield for both reflections and refractions, whichmitigates the issues of cycle skipping associated with conven-tional FWI. The least-squares problem for reconstructing theextended source g is equivalent to solving the following nor-mal equation

S∗P∗PSg+λ2g = S∗P∗d0 +λ

2 f .

Now assume that the extended source g is the solution of thenormal equation and write g as a perturbation of f

g = f +δ f ,

where

δ f = S∗P∗d0/λ2−S∗P∗PS f/λ

2 +O(1/λ4), (5)

according to the asymptotic expansion. To make this compu-tationally feasible, we ignore 1/λ 4 terms on the right handside of equation (5) and use the symbol˜for any wavefield thathas been approximately reconstructed. The extended or recon-structed source can be approximately computed using

g = f + ˜δ f , (6)

where˜δ f = S∗P∗(d0−PS f )/λ

2. (7)

Our next step is to minimize the above objective function (4)w.r.t. m using the reconstructed g. Note that we can now re-construct the forward source wavefield u = Sg. Then we needan explicit form of the wave operator and let’s consider the thefollowing operator for a VTI system of two coupled second-order partial differential equations in terms of P-wave verticalvelocity v, Thomsen parameters, ε and δ , assuming a constantdensity and zero shear velocity

2[v],1v2 ∂ t2−

(1+2ε 11+2δ 1

)(∂ 2

x +∂ 2y 0

0 ∂ 2z

), (8)

where v is the velocity model that will be inverted for, whileanisotropy parameters are fixed. The velocity model v then canbe updated using a conjugate gradient method and the gradientfor the objective function (4) w.r.t. v can be calculated using

∇vJλ [g(v),v] ≈ ∇vJλ [g,v]

= −⟨

2v3 ∂

2t Sg, S∗P∗d0−S∗P∗PSg

⟩≈ −

⟨2v3 ∂

2t Sg, S∗P∗d0−S∗P∗PSg

⟩= −

⟨2v3 ∂

2t S( f + ˜δ f ), λ

2 ˜δ f −S∗P∗PS ˜δ f⟩

≈ −⟨

2λ 2

v3 ∂2t u, ˜δ f

⟩+

⟨2v3 ∂

2t S f , S∗P∗PS ˜δ f

⟩.

This gradient computation can be easily extended to more gen-eral wave equations and used to perform multi-parameter in-version for velocity and other earth models, such as anisotropyparameters, attenuation quality factor, and/or density, either si-multaneously or sequentially. When the penalty factor λ islarge enough, RFWI and conventional FWI converge to verysimilar results. Thus the penalty factor has to be chosen care-fully to make RFWI produce favorable results and it varieswith iterations. The second term in the gradient computationmay be ignored to reduce the computation cost.

SYNTHETIC EXAMPLE

We first demonstrate the advantage of RFWI by applying it to a2D synthetic data set and draw a comparison with conventionalFWI. The true model is a modified SMAART Pluto syntheticsalt model as shown in Figure 1(a), which was used to generatethe field data set that has 250 shot gathers with a shot spacingof 20 m. Each shot gather contains 600 receivers with an inter-val of 20 m. The lowest frequency used for inversion was 4 Hzand maximum offset was 6000 m. The initial velocity modelis displayed in Figure 1(b), which is a smoothed version of

Reconstructed Full Waveform Inversion with the Extended Source

(a) True model (b) Initial model

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

Figure 1: Synthetic models

the true model. If we compare the inverted models from 87iterations of conventional FWI in Figure 1(c) and 54 iterationsfrom RFWI in Figure 1(d), we notice that RFWI provides amore detailed results with faster convergence, especially forthe sub-salt area.

FIELD EXAMPLES

The second example involves an application of RFWI to 2Dstreamer data from offshore Mexico. The acquisition lengthwas 148 km. 745 shots were used with a shot spacing of 200 m.The lowest frequency used for inversion was 3 Hz. Figure 2(a)shows the simple initial velocity model. Figure 2(b) shows theinversion result from RFWI. After the inversion using RFWI,not only the shallow velocity has been updated, but also deeperchanges have started building up the top of the salt with animproved salt velocity and refining the deeper structure. Wethen compare the stack images from the offset gathers to betterQC the results. Stack using the inverted model demonstratesbetter focusing compared to the stack using the initial modelas pointed by the red arrows in Figure 3. Finally, we forwardmodeled the data using the initial and inverted models and gen-erated shot gathers that are displayed in Figure 4(a) and 4(b).Comparing with field data in Figure 4(d) for the same shotrecord, the forward modeled data using the inverted model af-ter RFWI fit the field data much better than using the initialmodel, with clear improvement in both the near and far off-sets. If we further investigate RFWI results, the reconstructedforward modeled data that are extracted from the reconstructedsource wavefield using the RFWI inverted model shown in Fig-ure 4(c) has the best fit to the acquired data.

We finally present an application of RFWI to a 3D streamerdata set. This narrow azimuth seismic survey was locatedin the Campeche area from offshore Mexico. We used 2280sources with an interval of 500 m. The maximum offset was6200 m and the lowest frequency used for inversion was 3Hz. Figure 5(a) shows the legacy velocity model that wasused as initial model for RFWI and Figure 5(b) shows theinverted model from RFWI which includes updates for both

(a) Initial

(b) RFWI inverted

Figure 2: Velocity models

(a) Using initial model

(b) Using RFWI inverted model

Figure 3: Stack images

shallow and deep sections. We then compare the offset gath-ers to QC the inversion results. Comparing the shallow andtop-salts events, the offset gathers using RFWI inverted modelshown in Figure 6(b) are more flattened than using the initialmodel shown in Figure 6(a). Another tool for QC RFWI re-sutls is using reverse time migration (RTM). Comparing withRTM image using the initial model in Figure 7(a), image usingthe RFWI inverted model in Figure 7(b) shows better imagefocusing and event continuity at the top salt and also at thedeeper area below the salt as indicated by the red rectangles.RFWI demonstrates advantages in providing higher-resolutionvelocity for areas with strong velocity contrasts.

Reconstructed Full Waveform Inversion with the Extended Source

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

(c) Using reconstruction (d) Field data

Figure 4: Shot gathers

(a) Initial

(b) RFWI inverted

Figure 5: Velocity models

CONCLUSION

We presented the methods and applications of our proposednovel inversion method - time domain RFWI. RFWI helpsavoid cycle skipping issues to overcome some of the prob-lems with local minima and relaxes the requirement for con-ventional FWI. It demonstrates more advantages in areas with

strong velocity contrasts, which makes it a beneficial methodfor velocity model building with the presence of salt.

(a) Using initial velocity model

(b) Using RFWI inverted velocity model

Figure 6: Offset gathers

(a) Using initial velocity model

(b) Using RFWI inverted velocity model

Figure 7: RTM images

ACKNOWLEDGEMENTS

We would like to thank SMAART for providing the pluto syn-thetic model. The Campeche reimaging program data was re-processed and reimaged by ION in partnership with Schlum-berger, who holds data licensing rights. We also thank ION forpermission to publish the results and our colleagues for pro-viding valuable discussion and support.

Reconstructed Full Waveform Inversion with the Extended Source

REFERENCES

Biondi, B. and Almomin, A. [2014] Simultaneous inversionof full data bandwidth by tomographic fullwaveform inver-sion. Geophysics, 79(3), WA129–WA140.

Huang, H., Symes, W. and Nammour, R. [2016] Matchedsource waveform inversion: Volume extension. ExpandedAbstracts, Society of Exploration Geophysicists, Dallas,TX, 1364–1368.

Lailly, P. [1983] The Seismic Inverse Problem as a Sequence ofBefore-Stack Migrations. In: Bednar, J. (Ed.) Conferenceon Inverse Scattering: Theory and Applications, Societyfor Industrial and Applied Mathematics, Philadelphia, 206–220.

van Leeuwen, T. and Hermann, F. [2013] Mitigating local min-ima in full-waveform inversion by expanding the searchspace. Geophysical Journal International, 195(1).

Shen, P. and Symes, W. [2008] Automatic velocity analysis viashot profile migration. Geophysics, 73(5), VE49–VE59.

Sirgue, L. and Pratt, R. [2004] Efficient waveform inversionand imaging: A strategy for selecting temporal frequencies.Geophysics, 69(1), 231–248.

Tarantola, A. [1984] Inversion of seismic reflection data in theacoustic approximation. Geophysics, 49(8), 1259–1266.

Virieux, J. and Operto, S. [2009] An overview of full-waveform inversion in exploration geophysics. Geophysics,74(6), 127–152.

Wang, C., Yingst, D., Bai, J., Leveille, J., Farmer, P. andBrittan, J. [2013] Waveform inversion including well con-straints, anisotropy, and attenuation. The Leading Edge,32(9), 1056–1062.

Wang, C., Yingst, D., Farmer, P. and Leveille, J. [2016]Full-waveform inversion with the reconstructed wavefieldmethod. Expanded Abstracts, Society of Exploration Geo-physicists, Dallas, TX, 1237–1241.

Warner, M. and Guasch, L. [2016] Adaptive waveform inver-sion: Theory. Geophysics, 81(6), R429–R445.