3D elastic FWI for land seismic data: a graph space OT approachWeiguang He1⇤, Romain Brossier1, Ludovic Métivier1,2

1 Univ. Grenoble Alpes, ISTerre, Grenoble, France, 2 Univ. Grenoble Alpes, CNRS, LJK, Grenoble, France ,

SUMMARYIntegrating surface wave information is challenging for landseismic full waveform inversion (FWI). Cycle-skipping of sur-face waves can easily occur due to their highly dispersive andoscillating properties. While this issue can be mitigated usingwider basin objective functions, a more severe difficulty is re-lated to the unbalanced amplitude distribution between surfacewaves and body waves. The energetic surface waves dominatethe objective function and drive the inversion to update onlythe shallow structure. The contribution from body waves ismasked and the deep structures are not recovered. In a recentstudy, we have shown how an optimal-transport based misfitfunction can help mitigating this issue, providing naturally abetter balance between events (KR-OT approach). Here, weapply a newly introduced OT based misfit function, relying ona graph space approach (GS-OT), in this framework of elas-tic FWI for land data. GS-OT better handles cycle skippingthan KR-OT. We show here that it also helps to balance theamplitude of different seismic events. We design a practicalworkflow based on the GS-OT misfit function, coupled withan on-the-fly source estimation wavelet and a Gaussian-timewindow strategy. The method is applied to a synthetic casestudy from the SEAM II Foothill model. From crude velocitymodels, high resolution of VP and especially VS are obtained,using only sixteen seismic sources.

INTRODUCTIONFull waveform inversion (FWI) reconstructs the subsurface struc-ture by fitting the synthetics to the observed seismic data (Lailly,1983; Tarantola, 1984). Due to years of accumulation of tech-nological improvement and theoretical development, applica-tion of FWI to marine data has achieved success. However,application to land data is still under development mainly dueto three difficulties: first, surface waves; second, varying to-pography; third, uncertainty on source wavelet reconstruction.

Surface waves are energetic but shallow penetrating. The mag-nitude of the amplitude of surface waves is several orders higherthan that of body waves due to the geometrical spreading. Whilebeing energetic, the surface waves only propagate along theshallow depth. The information carried by surface waves aboutthe near-surface complexities is accumulated and mixed alongthe source-receiver path in the shallow depth. Surface wavescan easily cycle-skip being highly dispersive and oscillating.Yet, the challenge is not only about the cycle-skipping issue,but also the unbalanced amplitude distribution between sur-face and body waves. When the entire seismograms are incor-porated to the objective function, the surface waves dominatethe inversion and the contribution from body waves becomesnegligible. At the end, only the very shallow depth is recon-structed. The varying topography further complicate the seis-mic wave propagation (Bleibinhaus and Rondenay, 2009) andthe land seismic inversion. The surface waves are scattered andmay even convert to body waves. Both of the surface waves

and body waves become more complex due to the varying to-pography. The non-linearity of the inversion is thus increased.An additional difficulty for inverting land data is also relatedto the source wavelet estimation due to non-repeatability andcomplex source coupling effects. Since FWI is a data drivenstrategy, the inaccuracy of source wavelet is compensated bymodifying the velocity structures. Bias in the source waveletsare thus mapped as model anomalies.

Recent attempts to mitigate the surface wave issue are based ona hierarchical data-based inversion strategy. Trinh et al. (2018)interpret the early body waves first with a conventional timewindow, then incorporate the entire seismograms. Brossieret al. (2009) propose an exponentially decaying time window(Shin and Cha, 2008) centered around the first break as weight-ing function to gradually incorporate the entire seismograms.To tackle cycle-skipping issue, modifying the objective func-tion can help interpreting the surface waves, such as envelopefunction (Borisov et al., 2016) and FK domain function (Ma-soni et al., 2013; Pérez Solano et al., 2014). However, besidesthe loss of resolution induced by these misfit function, the am-plitude balancing problem is not solved.

Optimal transport (OT) misfit functions have been recently pro-posed to deal with cycle-skipping issue in the frame of FWI(Engquist and Froese, 2014; Engquist et al., 2016; Métivieret al., 2016a,b, 2018; Yang et al., 2018). In a recent study(He et al., 2018a), we have illustrated how a specific OT misfitfunction (Kantorovich-Rubinstein OT, KR-OT in the follow-ing), coupled with a Gaussian-time windowing strategy, helpsbalancing the amplitudes of different types of events, suggest-ing that the high amplitudes of surface waves will have less im-pact into the gradient build-up. However, KR-OT still suffersfrom cycle skipping if too large time shifts are considered. Agraph space OT method has been recently introduced in FWI toovercome the limitations of KR-OT with respect to cycle skip-ping for large time shifts (Métivier et al., 2019). This promptsus to investigate GS-OT in the frame of elastic FWI for landdata. Our aim is to design a practical workflow applicable toreal data. We consider a spectral element method to accuratelysimulate seismic wave propagation in presence of varying to-pography, using the code SEM46 (Trinh et al., 2019). To betterhandle the bias due to potential errors in source wavelet esti-mation, we update them on the fly along with velocity modelsat each iteration (Plessix and Cao, 2011; He et al., 2018b).

In the following, we first summarize the GS-OT approach, andanalyze a simple synthetic trace example to exhibit its proper-ties in terms of balancing events with different amplitudes. Wethen illustrate the interest for our on-the-fly wavelet estimationon a simple 2D elastic example. Finally, we consider the in-version of a realistic synthetic example based on the SEAMII Foothill model (Oristaglio, 2016). Starting from crude ini-tial velocity models, our GS-OT based approach converges to-wards a satisfactory estimation of both VP and VS models, us-

10.1190/segam2019-3216485.1Page 1320

© 2019 SEGSEG International Exposition and 89th Annual Meeting

Interpreting the entire seismograms with GS-OT

Figure 1: OT (KR, GS) balances events.

ing only 16 seismic sources.

METHODOLOGYOptimal transport function for event balancing

The GS-OT strategy is based on a trace-by-trace comparison.Each trace is considered as a point cloud in a 2D space (dis-crete graph). For the discretized synthetic (ti ,pi ) and observedtrace (t j ,d j ), the GS-OT distance is defined as finding the per-mutation � which minimizes

h(p,d) = min�

NX

i=1(ti � t� (i) )2 +

⇣ ⌧A

⌘2(pi � d� (i) )2, (1)

where� ranges over all the possible permutations of (1,2, ...,N ),⌧ is user-defined parameter, which should be set as the max-imum estimated time-shift between synthetics and observeddata, while A is the maximum amplitude range for each pairof seismic trace. Equation 1 is a linear assignment problem,which can be solved efficiently by the auction algorithm (Bert-sekas and Castanon, 1989). The corresponding adjoint sourcesi at time ti is given by

si = 2⇣ ⌧

A

⌘2(pi � d� (i) ). (2)

Equation 1 and 2 apply to each seismic trace. The completemisfit function over the acquisition built by Nr receivers andNs sources writes

f =NsX

s=1

NrX

r=1ws,r hs,r (ps,r [m],ds,r ), (3)

where [m] indicates the dependency of synthetic data ps,r onmodel parameters m. the weight ws,r has been introduced forflexibility. In the following, ws,r is used either based on theenergy norm ws,r =

PNi d2

i of the observations in order topreserve the natural amplitude decay over distances, either asws,r = 1 in order to give the same weight to each seismictrace.

More details on the GS-OT approach can be found in Métivieret al. (2018, 2019). In particular, the GS-OT approach over-comes the limitation of KR-OT in terms of convexity with re-spect to time shifts. In (He et al., 2018a) we have shown that

Figure 2: Impact of source wavelet estimation strategy. Wepresent only the model increment for VS : we expect a constantincrement of 100 m.s�1 (homogeneous red). L2 inversion: (a)with true source wavelet, (b) source estimation once and fixed,(c) source estimation on the fly starting with �(t), (d) sourceestimation on the fly starting with good initial guess. (e) KR-OT inversion. (f) GS-OT inversion.

the adjoint source associated with KR-OT naturally balancesthe contribution from surface waves and body waves. In thisstudy, we further illustrate that GS-OT presents this propertyas well. In Figure 1, we generate an observed trace and a syn-thetic trace with 3 Hz Ricker wavelets by mimicking the real-istic situation of misaligned and missing events (�t=7 m, 512sampling points). The adjoint sources of L2, KR-OT and GS-OT in Figure 1 reveal that both KR-OT and GS-OT balancethe amplitude distribution. This is a good property for landseismic FWI.

Inversion with unknown source wavelet

We further investigate the impact of source wavelet estimationstrategies in the frame of elastic FWI. We design a simple ex-periment by considering a homogeneous true model (VP=4.774 km/s,VS=2.8 km/s, ⇢=2.4 g/cm3) and a homogeneous initial modelwith a 100 m.s�1 slower vs . The observed data are generatedwith a modified Ricker wavelet with energy between 2 Hz and6 Hz. During the inversion, VP and VS are updated simulta-neously. We compare 4 source wavelet estimation strategiescombined with a L2 misfit function. The results are presentedin Figure 2(a-d).

1. Figure 2 (a): True source wavelet is used. This inversion isa reference to evaluate other tests.

2. Figure 2 (b): Estimate the source wavelets in the initialmodel and keep them fixed during the inversion. The shal-low area is not correct. The inaccuracy of source wavelet ismapped as model anomalies.

3. Figure 2 (c): Source estimation on the fly starting from�(t). The updating is not uniform. Within the range X=10 kmand 12 km, the updating is even in opposite direction.

4. Figure 2 (d): Source estimation on the fly starting from pre-viously estimated source wavelets. The updating is moreuniform albeit the recovered depth is shallow.

The source estimation in test 2-4 is done per shot. From thesetests the strategy 4 appears the best. We repeat the same inver-sion but with KR-OT (Fig. 2 (d)) and GS-OT (Fig. 2 (f)). The

10.1190/segam2019-3216485.1Page 1321

© 2019 SEGSEG International Exposition and 89th Annual Meeting

Interpreting the entire seismograms with GS-OT

(a) Initial source wavelet (b) Final source wavelet

Figure 6: Source wavelet comparison in the 1st band.

best model in case of unavailable source wavelet is retrievedby GS-OT.

SEAM MODEL 3D SYNTHETIC INVERSIONExperimental setupThe true model shown in Figure 3 is extracted from 3D SEAMFoothill model. To simulate the geological process, layeredstructures are twisted and displaced; also low velocity alluvialzones are padded at the surface. We smooth the true model toremove the layered structures and the near-surface complexity,and take it as initial model (Figure 3). During the inversion,the density is fixed to the smooth initial value to mimic a re-alistic situation, VP and VS are updated simultaneously. Theobserved data corresponds to a flat-spectrum response: we dothe forward simulation in the true model with a 3 Hz Rickerwavelet, then we deconvolve the data with the same Rickerwavelet. Time interval is �t = 0.6 ms, and total recording timeis around 4.8 s. We use 16 sources (vertical force, inter-sourcedistance is 900 m), and 720 mono-component geophones (ver-tical velocity, inter-receiver distance is 20 m). In Figure 4 (a),we show the observed data in the full frequency band [1.5,6.5] Hz. The body waves are hardly visibly despite the use ofa strong clip to display the seismogram.

For the inversion, we use the same mesh and same time incre-ment. We estimate the first source wavelets in the initial model(Figure 6 (a)). For the shots in the low velocity alluvial zones,the source wavelets have bigger amplitudes and later startingtimes than other shots. For data comparison, we show the syn-thetics on given offset ranges, and mirror the correspondingobserved traces in the same range. In these plots, each trace isnormalized by the maximum amplitude of the observed tracein a given offset range (Fig. 4 (a-1)). For offsets larger thanX=5 km, the phase shift of surface waves is already significant.For body waves (after X=8 km), the fitting is even worse.

In the following inversions, a Bessel filter (Trinh et al., 2017) isused to smooth the gradient, the filter length being adapted de-pending on the local wavelength. Model parameters VP and VSare simultaneously updated using the l-BFGS algorithm (No-cedal, 1980; Métivier and Brossier, 2016).

FWI workflow

The first frequency band is [1.5, 2.5] Hz. In the first step, weapply our GS-OT based strategy on the entire seismograms.In this first step, we apply ws,r =

PNi d2

i in GS-OT (Equa-tion 3) and choose ⌧=0.2 s. A second step is designed to en-hance the recovery of body waves. We re-estimate the initialsource wavelets based on the first step inversion. Then we ap-

ply a Gaussian time window roughly around the early bodywaves (for each seismic trace, the arrival time t is approxi-mated as t=0.6 s+0.3 �sr where �sr is offset in km). Thewidth of Gaussian window is determined by the standard vari-ation 0.96 s (0.2 times of the whole recording time). To furtherenhance the role of the body waves, we apply ws,r = 1 to nor-malize the energy over traces. After the first frequency stage,we enlarge the frequency band to [1.5, 6.5] Hz. We repeat thetwo-step inversion strategy with ws,r = 1 in these two stepsand decrease ⌧ to 0.1 s.

FWI resultsAfter the first step of the first frequency band, the surface wavesare well fitted, but not the body waves (figures not shown); Fig-ure 5 (a-1) and (a-2) show the inverted VP and VS respectively.VS is recovered down to 3 km, while VP is almost not updated.After the second step in the first frequency band, both of thebody waves and surface waves are correctly predicted (Figure4 (b-2)). The body waves contribute to the model reconstruc-tion at depth, (see inverted models in Figure 5 (b-1) and (b-2)).Both VP and VS are updated significantly. The final sourcewavelets are shown in Figure 6 (b). As we apply the same de-convolution to each gather, we expect to retrieve similar sourcesignatures, which we observe

Moving to the second frequency band, more events includingscattered surface waves come out in Figure 4 (a). Followingthe same two-step workflow, the mismatched body and surfacewaves (Figure 4 (c-1)) are correctly predicted (Figure 4 (c-2)).The final models shown in Figure 5 (c-1) and (c-2) reveals highresolution details, especially for VS . VP is also recovered downto 3 km. Note that using a larger number of sources could im-prove the smoothness of the recovered VS .

CONCLUSIONOur FWI workflow based on the use of GS-OT and on-the-flywavelet estimation, combined with a two-steps process usinga Gaussian time-window, seems to provide satisfactory resultfor a challenging 3D elastic synthetic case study based on theSEAM II Foothill model. Comparisons with the use of KR-OTand L2 misfit function (not shown here) reveal that the inver-sion would stop, in this case, after the first step in the firstfrequency band, indicating probable cycle skipping issues.

These relatively good results obtained with a rather sparse ac-quisition (16 sources) prompt us to investigate the interest ofthis workflow on real data. We have started to work on a realdataset acquired in a mountainous area of south China. Thepreliminary inversion seems consistent with an independentEM survey.

ACKNOWLEDGMENTS

This study was partially funded by the SEISCOPE consortium(https://seiscope2.osug.fr), sponsored by AKERBP, CGG, CHEVRON,EQUINOR, EXXON-MOBIL, JGI, PETROBRAS, SCHLUM-BERGER, SHELL, SINOPEC and TOTAL. This study wasgranted access to the HPC resources of CIMENT infrastruc-ture (https://ciment.ujf-grenoble.fr) and CINES/IDRIS/ TGCCunder the allocation 046091 made by GENCI.

10.1190/segam2019-3216485.1Page 1322

© 2019 SEGSEG International Exposition and 89th Annual Meeting

Interpreting the entire seismograms with GS-OT

(a-1) True vp (a-2) True vs (a-3) True ⇢

(b-1) Initial vp (b-2) Initial vs (b-3) Initial ⇢

Figure 3: True model (top row) and initial model (bottom row) from the SEAM II Foothill model. During the inversion the densityis fixed to the initial value, VP and VS are simultaneously updated.

(b-1) 1st initial comparison (c-1) 2nd initial comparison

(a) Broadband Obs [1.5, 6.5] Hz (b-2) After 1st band (c-2) After 2nd band

Figure 4: Observed and synthetic data comparison.

(a-1) First vp (b-1) Second vp (c-1) Final vp

(a-2) First vs (b-2) Second vs (c-2) Final vs

Figure 5: In the first step (top row), surface waves contribute to the inversion. In the second step (second row), body wavescontribute to deep reconstruction. In the second frequency band (third row), the final model reveals fine details.

10.1190/segam2019-3216485.1Page 1323

© 2019 SEGSEG International Exposition and 89th Annual Meeting

REFERENCES

Bertsekas, D. P., and D. Castanon, 1989, The auction algorithm for the transportation problem: Annals of Operations Research, 20, 67–96, doi: https://doi.org/10.1007/bf02216923.

Bleibinhaus, F., and S. Rondenay, 2009, Effects of surface scattering in full-waveform inversion: Geophysics, 74, no. 6, WCC69–WCC77, doi: https://doi.org/10.1190/1.3223315.

Borisov, D., R. Modrak, H. Rusmanugroho, Y. Yuan, F. Simons, J. Tromp, and F. Gao, 2016, Spectral-element based 3d elastic full-waveforminversion of surface waves in the presence of complex topography using an envelope-based misfit function: 86th Annual International Meeting,SEG, Expanded Abstracts, doi: https://doi.org/10.1190/segam2016-13843759.1.

Brossier, R., S. Operto, and J. Virieux, 2009, Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform in-version: Geophysics, 74, no. 6, WCC105–WCC118, doi: https://doi.org/10.1190/1.3215771.

Engquist, B., and B. D. Froese, 2014, Application of the Wasserstein metric to seismic signals: Communications in Mathematical Science, 12, 979–988, doi: https://doi.org/10.4310/cms.2014.v12.n5.a7.

Engquist, B., B. D. Froese, and Y. Yang., 2016, Optimal transport for seismic full waveform inversion: Communications in Mathematical Sciences, 14,2309–2330, doi: https://doi.org/10.4310/cms.2016.v14.n8.a9.

He, W., R. Brossier, L. Métivier, and R.-É. Plessix, 2018a, Land seismic multi-parameter full waveform inversion in elastic VTI media by simulta-neously interpreting body waves and surface waves with optimal transport based objective function: Geophysical Journal International, submitted.

He, W., R. Brossier, L. Métivier, and J. Virieux, 2018b, A practical workflow of full waveform inversion to process land seismic data: Synthetic study:80th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: https://doi.org/10.3997/2214-4609.201801376.

Lailly, P., 1983, The seismic problem as a sequence of before-stack migrations: Conference on Inverse Scattering: Theory and Applications, SIAM.Masoni, I., R. Brossier, J. Virieux, and J. Boelle, 2013, Alternative misfit functions for FWI applied to surface waves: 75th Annual International

Conference and Exhibition, EAGE, Extended Abstracts, Th P10 13, doi: https://doi.org/10.3997/2214-4609.20130297.Métivier, L., A. Allain, R. Brossier, Q. Mérigot, E. Oudet, and J. Virieux, 2018, Optimal transport for mitigating cycle skipping in full waveform

inversion: A graph space transform approach: Geophysics, 83, no. 5, R515–R540, doi: https://doi.org/10.1190/geo2017-0807.1.Métivier, L., and R. Brossier, 2016, The SEISCOPE optimization toolbox: A large-scale nonlinear optimization library based on reverse commu-

nication: Geophysics, 81, no. 2, F11–F25, doi: https://doi.org/10.1190/geo2015-0031.1.Métivier, L., R. Brossier, Q. Mérigot, and E. Oudet, 2019, A graph space optimal transport distance as a generalization of Lp distances: Application to

a seismic imaging inverse problem: Inverse Problems, submitted.Métivier, L., R. Brossier, Q. Mérigot, E. Oudet, and J. Virieux, 2016a, Measuring the misfit between seismograms using an optimal transport distance:

Application to full waveform inversion: Geophysical Journal International, 205, 345–377, doi: https://doi.org/10.1093/gji/ggw014.Métivier, L., R. Brossier, Q. Mérigot, E. Oudet, and J. Virieux, 2016b, An optimal transport approach for seismic tomography: Application to 3D full

waveform inversion: Inverse Problems, 32, 115008, doi: https://doi.org/10.1088/0266-5611/32/11/115008.Nocedal, J., 1980, Updating quasi-Newton matrices with limited storage: Mathematics of Computation, 35, 773–782, doi: https://doi.org/10.2307/

2006193.Oristaglio, M., 2016, Land seismic challenges wraps up with Foothills model: The Leading Edge, 35, 292–293, doi: https://doi.org/10.1190/

tle35030292.1.Pérez Solano, C., D. Donno, and H. Chauris, 2014, Alternative waveform inversion for surface wave analysis in 2D media: Geophysical Journal

International, 198, 1359–1372, doi: https://doi.org/10.1093/gji/ggu211.Plessix, R. E., and Q. Cao, 2011, A parametrization study for surface seismic full waveform inversion in an acoustic vertical transversely isotropic

medium: Geophysical Journal International, 185, 539–556, doi: https://doi.org/10.1111/j.1365-246x.2011.04957.x.Shin, C., and Y. H. Cha, 2008, Waveform inversion in the Laplace domain: Geophysical Journal International, 173, 922–931.Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266, doi: https://doi.org/10.1190/1

.1441754.Trinh, P. T., R. Brossier, L. Métivier, L. Tavard, and J. Virieux, 2018, Data-windowing hierarchy in multi-parameter elastic FWI: 3D synthetic foothills

case study: 80th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: https://doi.org/10.3997/2214-4609.201801570.Trinh, P. T., R. Brossier, L. Métivier, L. Tavard, and J. Virieux, 2019, Efficient 3D time-domain elastic and viscoelastic full waveform inversion using a

spectral-element method on flexible Cartesian-based mesh: Geophysics, 84, no. 1, R75–R97, doi: https://doi.org/10.1190/geo2018-0059.1.Trinh, P. T., R. Brossier, L. Métivier, J. Virieux, and P. Wellington, 2017, Bessel smoothing filter for spectral element mesh: Geophysical Journal

International, 209, 1489–1512, doi: https://doi.org/10.1093/gji/ggx103.Yang, Y., B. Engquist, J. Sun, and B. F. Hamfeldt, 2018, Application of optimal transport and the quadratic Wasserstein metric to full-waveform

inversion: Geophysics, 83, no. 1, R43–R62, doi: https://doi.org/10.1190/geo2016-0663.1.

10.1190/segam2019-3216485.1Page 1324

© 2019 SEGSEG International Exposition and 89th Annual Meeting