Full waveform inversion guided migration velocity analysis Thibaut Allemand* and Gilles Lambaré (CGG)

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1 Thibaut Allemand* and Gilles Lambaré (CGG) Summary While difficulties of full waveform inversion (FWI) using reflected waves only have been well identified, the combination of FWI with migration velocity analysis (MVA) appears as a promising solution. This is particularly appealing considering that resolution brought by both approaches now overlap when processing data from broadband acquisitions. We propose a new strategy for combining FWI and MVA with the purpose of velocity model building. The two cost functions are not minimized jointly as in formerly proposed methods but only the MVA cost function is considered as it is fully decoupled from amplitude aspects. The FWI update is, however, used as a guide during the optimization with the benefit of a stabilization of the MVA within the mid-frequency band. 2D synthetic results demonstrate the effectiveness of the approach. Introduction Full waveform inversion is now well established for velocity model building (Virieux and Operto, 2009). It is particularly appreciated for providing high resolution and structurally conformable velocity models in areas investigated by diving waves. In areas investigated by reflected waves only, the success of FWI requires a very accurate starting velocity model, and performing an accurate multi-parameter FWI requires taking into account density or even elastic behavior (Lu et al., 2013). In this context velocity model building by migration velocity analysis (MVA) (Stork 1992; Liu 1997; Woodward 1998; Guillaume et al. 2001, Woodward 2008; Guillaume et al. 2008) still dominates in industry. Moreover, far from being a declining technology these approaches have been considerably improved over the last six years (Lambaré et al., 2014), with a vertical resolution up to 6 Hz (Guillaume et al., 2012). With broadband acquisitions of down to 2 Hz, we have moved from the traditional mid-frequency gap (Claerbout, 1985) to an overlap of the resolution we can expect from ray based tomography and imaging. We propose here an innovative approach for velocity model building taking advantage of MVA and FWI in a combined approach. FWI is used for its structural conformity and high resolution, while MVA is used for its sensitivity to kinematic of reflected arrivals. FWI from reflected waves FWI has fully demonstrated its ability to invert shallow velocities. A theoretical analysis of respective contributions of reflected and diving waves shows that while short wavelength components of the velocity model are obtained from reflections, the long lateral wavelength components are obtained from diving waves. With reflected waves the resolution of FWI is more or less the resolution of a migration operator (Lailly, 1983; Tarantola, 1984a, 1984b). As a consequence the corresponding velocity update exhibits structural conformity and resolution but may be affected by mispositioning and erroneous amplitudes. In fact, reflected waves reveal the detailed velocity structures but have difficulties with their localization and quantification while diving waves, which mainly control the decrease of the cost function, provide the quantification of velocities but can t provide detailed structures. In areas investigated only by reflected waves a solution for a successful FWI is to replace the contribution of diving waves by another kinematic constraint. While Margrave et al. (2012) proposed to use well control, a more general strategy is the combination of FWI with MVA. This approach is developed with differential semblance optimization (DSO) (Symes and Carazone, 1992; Almomin and Biondi, 2012; Fleury and Perrone, 2012) using wave equation MVA rather than picking based MVA. The wave equation MVA cost function is then added to the traditional FWI cost function with some appropriate weightings, and, obs calc 2 C 2 TFWI v d d F I 2 traces 2 CIGs (v is the velocity model to optimize, d obs and d calc are the observed and calculated traces, I is the prestack migrated volume and F is an operator assessing the local focusing of migration). We propose an original strategy for combining MVA and FWI aiming at keeping the relevant information from both approaches: FWI for imaging the shape of structures and MVA for quantifying velocity variations from kinematic information. The idea is to keep structures obtained from FWI but determine their amplitude and localization from kinematic analysis. FWI guided MVA Compared to DSO mentioned above, our approach does not rely on a cost function combining FWI and MVA cost function terms. New cost function is purely MVA because it is expected to far better decouple kinematic effects from amplitude effects. The velocity model is updated iteratively by a local optimization scheme providing a set of velocity models [v n (x)], n = 0, N. The originality is that we constrain at each iteration the velocity update, v, with the velocity perturbation obtained by FWI, g(x) (we call it the

2 guide). The constraint preserves its structural conformity but modifies its amplitude by applying a smooth scaling, (x): v x x g x. The optimization scheme updates the smooth scaling factor, and then applies it to the guide to build the velocity update. The quality of the guide is very important for the success of the process. Considering that the problem is non-linear it has to be updated at each linear iteration (Figure 1). Figure 1: FWI guided MVA. A MVA criterion is used for assessing the quality of the velocity model but at each iteration the velocity update is guided by the result of some migration/inversion process. In the first iterations a constant function guide means absolutely no constraint is applied in a pure MVA manner. It provides long wavelength components of the velocity model and even starts to address mid-wavelength components of the velocity model. At this point the benefit of the guide appears. If it is properly done it will stabilize MVA in the mid-frequency gap and even add shorter wavelength components hardly accessible by pure MVA but covered by imaging (Figure 2). To do this the guide should cover wavelength components in the mid-frequency range relevant of the exact velocity model up to a smooth scaling factor. We believe gradient of FWI, migration/inversion results can provide such a guide. As a proof of concept in 2D we propose here to use ray+born migration/inversion (Thierry et al., 1999; Operto et al., 2000) for the computation of the guide and non-linear slope tomography (Chauris et al., 2002; Guillaume et al., 2008; Lambaré et al., 2008) for the determination of the velocity update by MVA. If both approaches are based on rays and consequently have some limitation in complex media, they offer in the meantime all the benefits of accurate and well established approaches. Figure 2: Resolution of FWI guided MVA. In black the famous sketch by Claerbout (1985) summarizing resolution we can expect from velocity analysis (Velocity) and imaging (reflectivity) and emphasizing the mid frequency gap (2-10 Hz). In solid line grey the improved resolution brought by modern MVA and in dash grey line the improved resolution brought by waveform of reflected broadband data. Our cost function requires picking locally coherent events and involves the square misfit between the local slope of the reflected event in the common mid-point direction observed on data, T obs / h, and computed from tomographic rays after common offset kinematic migration, T sr / h (Guillaume et al., 2008), (Figure 3) C v 1 T obs T sr 2 h h picks Figure 3: Non-linear slope tomography. The cost function consists of the misfit between the local slope of the event observed in the unmigrated time domain in the common mid point gather, T obs / h, and the corresponding slope obtained from tomographic rays, T sr / h,where T sr is the two-way travel time of the migration operator. Starting from a given velocity model, v n (x), each iteration involves four steps: 1) Computation of waveform residuals for velocity model v n (x); 2) Computation of a velocity perturbation by ray+born migration/inversion of waveform residuals to be used as a guide for the tomographic update, g n (x); 2.

3 3) Computation of the scaling of the velocity guide, n (x), to derive a velocity update improving focusing of ray based tomography, v n = n g n. 4) Addition of the scaled velocity perturbation to the velocity model, v n+1 = v n + v n Example We apply our FWI guided MVA to the synthetic Marmousi II case study (Martin et al., 2006). The model is extended laterally and a 500 m thick water layer is added at the top. The model is simplified to a constant density isotropic acoustic model and data are generated by finite differences for a marine towed streamer acquisition with offset ranging from 0 to 3 km. The source function is a Dirac function band path filtered within [3, 60] Hz. The workflow starts by a non-linear slope tomography with stereo-tomographic data picked on locally coherent events in the un-migrated domain (Billette et al., 2003) (Figure 4). The resulting velocity model is shown on Figure 5A (We limit our investigation to the non-complex part of the model). It is the starting velocity model for our FWI guided MVA process. model and to considering multi-parameter inversion (Lambaré et al., 1992; Jin et al., 1992; Ribodetti et al., 2000). In standard FWI this velocity perturbation would be scaled and added to velocity model in order to fit seismic traces. Here it is scaled and used to fit residual move-out of migrated gathers. We hypothesize that the constraint of the velocity perturbation by a kinematic criterion can be more accurate than a constraint based on amplitudes of waveform. The migration stack shown on Figure 5B is our guide. Within our FWI guided MVA process it is scaled by a smooth scaling function here defined from cubic cardinal Bsplines with nodes spaced by 500 m vertically and 1000 m horizontally. Non-linear slope tomography involves an iterative non-linear optimization of the stereo-tomographic data. During these iterations the velocity guide should be changed in order to take into account the modifications of the guide resulting from the change of the velocity model. Working in a pseudotime domain rather than in depth could be a solution (Plessix et al., 2012) but a more rigorous one would be recomputing the guide. Here considering that we had a good starting model we did not change the guide along the non-linear iterations of slope tomography. Final velocity model is shown on Figure 5C, while Figure 5D shows for comparison the exact Marmousi II velocity model slightly smoothed. We see a clear improvement in the delineation of the velocity structures. Conclusion Figure 4: stereo-tomographic picking. The location and slope of locally coherent events picked in the un-migrated time domain are superimposed on a zoom of a common shot gather. From this starting velocity model we compute residuals between observed and calculated data through numerical modelling by finite differences. These residuals are low pass filtered to 12 Hz and used as input for ray+born migration/inversion. The corresponding migration stack is shown on Figure 5B. Note that it does not represent a reflectivity distribution like in depth migration but directly a velocity perturbation determined from the amplitude and signature of reflected waveforms. Numerous published results have shown the potential accuracy of this approach but also its great sensitivity to the accuracy of the velocity We have proposed an original strategy for combining MVA and FWI in such a way that the relevant information coming from both approaches is preserved. FWI is used for imaging the shape of structures from waveforms while MVA is used for quantifying velocity variations from kinematic information. The overlap in resolution between MVA and FWI is an important prerequisite which can be obtained with both modern broadband acquisitions and modern MVA techniques. A 2D synthetic case study has demonstrated the potential of the approach using ray based approaches for FWI and MVA. Implementation of non-ray based approaches will be a natural extension especially considering the potential complexity of the velocity models resolved by the method. Further investigations are necessary, especially for the computation and the modification of the guide during the iterations, and for assessing the robustness of the method with multiparameter acoustic or elastic waves. Considering the fact that in this method the assessment of the velocity model is done using a purely kinematic criterion we have some hope about this last aspect. Acknowledgements We thank CGG for the authorization to present this work.

4 Figure 5: FWI guided MVA, application to Marmousi II. A) the velocity obtained by nonlinear slope tomography. It is used as starting model for our application of FWI guided MVA. B) the guide obtained by ray-born migration/inversion of residual data low-pass filtered to 12Hz. C) the velocity model result of FWI guided MVA. D) the exact Marmousi II model slightly smoothed for comparison to C.

5 References Almomin, A., and B. Biondi, Tomographic Full Waveform Inversion: Practical and Computationally Feasible Approach, SEG, Expanded Abstracts, 1-5. Chauris, H.,M. Noble, G. Lambaré, and P. Podvin, 2002, Migration velocity analysis from locally coherent events in 2D laterally heterogeneous media, Part I: Theoretical aspects: Geophysics, 67, Billette, F., S. Le Bégat, P. Podvin, and G. Lambaré, 2003, Practical aspects and applications of 2D stereotomography: Geophysics, 68, Claerbout, J.F., Imaging the Earth s interior. Vol Oxford: Blackwell scientific publications. Fleury, C. and F. Perrone, Integrated migration velocity analysis and full-waveform inversion, expanded abstract 2012 SEG/EAGE Summer research workshop Charleston. Guillaume, P., F. Audebert, P. Berthet, B. David, A. Herrenschmidt, X. Zhang, D finite-offset tomographic inversion of CRP-scan data, with or without anisotropy, SEG, Expanded Abstracts, 20, no. 1, Guillaume, P., G. Lambaré, O. Leblanc, P. Mitouard, J. Le Moigne, J. P. Montel, T. Prescott, R. Siliqi, N. Vidal, X. Zhang and S. Zimine, Kinematic invariants: an efficient and flexible approach for velocity model building, SEG, Expanded Abstracts, 27, no. 1, Guillaume, P., G. Lambaré, S. Sioni, X. Zhang, A. Prescott, D. Carotti, P. Dépré, S. Frehers, and H. Vosberg, Building Detailed Structurally Conformable Velocity Models with High Definition Tomography, EAGE extended abstract, W002. Jin, S., Madariaga, R., Virieux, J., and Lambaré, G., 1992, Two dimensional asymptotic iterative elastic inversion: Geophys. J. Internat., 108, Lailly, P., 1984, The seismic inverse problem as a sequence of before stack migrations: Conference on Inverse Scattering, Tulsa, Oklahoma: Philadelphia, SIAM. Lambaré, G., P. Guillaume and J.-P. Montel, Recent advances in ray-based tomography, EAGE extended abstract,????. Lambaré, G., Virieux, J., Madariaga, R., and Jin, S., 1992, Iterative asymptotic inversion of seismic profiles in the acoustic approximation: Geophysics, 57, Lu, R., S. Lazaratos, K. Wang, Y.H. Cha, I. Chikichev and R. Prosser, High-resolution elastic FWI for reservoir characterization, EAGE extended abstract, Th Margrave, G., R. Ferguson, and C. Hogan, Full Waveform Inversion by iterative depth migration and impedance estimation using well control, extended abstract WS5 From Kinematic to Waveform Inversion - Where Are we and where Do we Want to Go? A Tribute to Patrick Lailly. Martin, G. S., R. Wiley, and K. J. Marfurt, Marmousi 2: An elastic upgrade for Marmousi, The Leading Edge 25, Nichols, D., Resolution in seismic inversion. Spectral gap or spectral overlap, which is harder to handle? 74th EAGE Conference & Exhibition - Workshops From Kinematic to Waveform Inversion - Where Are we and where Do we Want to Go? A Tribute to Patrick Lailly. Operto, S., S. Xu, and G. Lambaré, Can we quantitatively image complex structures with rays? Geophysics, 65, 4, p Plessix, R.E., P. Milcik, C. Corcoran, H. Kuehl, K. Matson, Full waveform inversion with a pseudotime approach, EAGE extended abstract, W012. Ribodetti, A., P. Thierry, G. Lambaré, and S. Operto, Improved multiparameter ray+born migration/inversion, SEG, Expanded Abstracts, Stork, C., Reflection tomography in the postmigrated domain, Geophysics, 57, 5, Symes, W.W. and J.J. Carazzone, Velocity inversion by differential semblance optimization, Geophysics, 56, Tarantola, A., 1984a. Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 8, Tarantola, A., 1984b. Linearized inversion of seismic reflection data, Geophysical Prospecting, 32, 6,

6 Virieux, J. and S. Operto, 2009, An overview of full waveform inversion in exploration geophysics, Geophysics, 64, WCC1- WCC26. Thierry, P., S. Operto and G. Lambaré, Fast 2-D ray+born migration/inversion in complex media, Geophysics, 64, 1, Woodward, M., P. Farmer, D. Nichols and S. Charles, Automated 3D tomographic velocity analysis of residual moveout in prestack depth migrated common image point gathers, SEG, Expanded Abstracts, 17, 1, Woodward, M., D. Nichols, O. Zdraveva, P. Whitfield, and T. Johns, A decade of tomography, Geophysics, 73, 5, VE5 VE11.