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Seismic iterative migration velocity analysis: two strategies to update the velocity model. (English) Zbl 1369.86006

Summary: The objective of seismic imaging is to recover properties of the Earth from surface measurements recorded during active seismic surveys. Migration Velocity Analysis techniques aim at determining a background velocity model (smooth part of the pressure wave velocity model) using the redundancy of seismic data and consist of solving a nested optimisation problem. In the inner loop, an extended reflectivity model (detailed part of the model) is determined from recorded primary reflections through a data-fitting procedure depending on a given background model. In the outer loop, a coherency criterion defined on the extended reflectivity assesses the quality of the background model. The inner problem is usually solved with a single iteration of gradient optimisation, leading to artefacts in the velocity updates. We study the benefits of further iterating on the reflectivity in the inner loop, which also allows the introduction of multiple reflections in the procedure. We propose two strategies for the computation of the gradient of the outer objective function. In the first case, we compute the exact numerical gradient by taking care of the background dependency of all inner iterations. In the second case, we derive an approximate gradient by assuming the optimal reflectivity has been obtained. Both methods are compared on their computational merits and through simple numerical examples on 2D synthetic data sets. The examples illustrate that regularisation of the inner problem is essential to obtain coherent velocity updates. The second approach displays a smaller sensitivity to regularisation and is simpler to implement.

MSC:

86A15 Seismology (including tsunami modeling), earthquakes

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[1] Bell, B.M., Burke, J.V.: Algorithmic differentiation of implicit functions and optimal values. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., Bischof, C.H., Bücker, H. M., Hovland, P., Naumann, U., Utke, J. (eds.) Advances in Automatic Differentiation, vol. 64, pp 67-77. Springer, Berlin (2008) · Zbl 1152.65434
[2] Blazek, K.D., Stolk, C.C., Symes, W.W.: A mathematical framework for inverse wave problems in heterogeneous media. Inverse Probl. 29(6), 065001 (2013) · Zbl 1273.35309 · doi:10.1088/0266-5611/29/6/065001
[3] Brown, M.P., Guitton, A.: Least-squares joint imaging of multiples and primaries. Geophysics 70(5), S79-S89 (2005) · doi:10.1190/1.2052471
[4] Bunks, C., Saleck, F., Zaleski, S., Chavent, G.: Multiscale seismic waveform inversion. Geophysics 60(5), 1457-1473 (1995) · doi:10.1190/1.1443880
[5] Chauris, H., Cocher, E.: From migration to inversion velocity analysis. Geophysics 82(3), S207-S223 (2017) · doi:10.1190/geo2016-0359.1
[6] Chauris, H.; Lameloise, CA; Cocher, E., Inversion velocity analysis - the importance of regularisation, ws05-a02 (2015)
[7] Chauris, H., Noble, M.: Two-dimensional velocity macro model estimation from seismic reflection data by local differential semblance optimization: applications to synthetic and real data sets. Geophys. J. Int. 144(1), 14-26 (2001) · doi:10.1046/j.1365-246x.2001.00279.x
[8] Chavent, G., Jacewitz, C.: Determination of background velocities by multiple migration fitting. Geophysics 60(2), 476-490 (1995) · doi:10.1190/1.1443785
[9] Fichtner, A.: Full Seismic Waveform Modelling and Inversion. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin (2011)
[10] Friedlander, M.P., Schmidt, M.: Hybrid deterministic-stochastic methods for data fitting. SIAM J. Sci. Comput. 34(3), A1380-A1405 (2012) · Zbl 1262.90090 · doi:10.1137/110830629
[11] Gauthier, O., Virieux, J., Tarantola, A.: Two-dimensional nonlinear inversion of seismic waveforms: numerical results. Geophysics 51(7), 1387-1403 (1986) · doi:10.1190/1.1442188
[12] Hou, J., Symes, W.W.: An approximate inverse to the extended Born modeling operator. Geophysics 80 (6), R331-R349 (2015) · doi:10.1190/geo2014-0592.1
[13] Hou, J., Symes, W.W.: Accelerating extended least-squares migration with weighted conjugate gradient iteration. Geophysics 81(4), S165-S179 (2016) · doi:10.1190/geo2015-0499.1
[14] Huang, Y.: Born waveform inversion in shot coordinate domain. Ph.D. thesis, Rice University (2016)
[15] Lailly, P.: The seismic inverse problem as a sequence of before stack migrations. In: Conference on Inverse Scattering: Theory and Application, pp. 206-220. Society for Industrial and Applied Mathematics, Philadelphia (1983)
[16] Lambaré, G., Virieux, J., Madariaga, R., Jin, S.: Iterative asymptotic inversion in the acoustic approximation. Geophysics 57(9), 1138-1154 (1992) · doi:10.1190/1.1443328
[17] Lameloise, C.A., Chauris, H., Noble, M.: Improving the gradient of the image-domain objective function using quantitative migration for a more robust migration velocity analysis. Geophys. Prospect. 63(2), 391-404 (2014) · doi:10.1111/1365-2478.12195
[18] MacKay, S., Abma, R.: Imaging and velocity estimation with depth-focusing analysis. Geophysics 57(12), 1608-1622 (1992) · doi:10.1190/1.1443228
[19] Métivier, L., Brossier, R., Virieux, J., Operto, S.: Full waveform inversion and the truncated Newton method. SIAM J. Sci. Comput. 35(2), B401-B437 (2013) · Zbl 1266.86002 · doi:10.1137/120877854
[20] Moré, J. J., Thuente, D.J.: Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20(3), 286-307 (1994) · Zbl 0888.65072 · doi:10.1145/192115.192132
[21] Mulder, W.A.: Automatic velocity analysis with the two-way wave equation. In: 70Th EAGE Conference & Exhibition, p. P165. Rome (2008)
[22] Mulder, W.A.: Subsurface offset behaviour in velocity analysis with extended reflectivity images. Geophys. Prospect. 62(1), 17-33 (2014) · doi:10.1111/1365-2478.12073
[23] Nocedal, J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999) · Zbl 0930.65067
[24] Plessix, R. E. ́: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int. 167(2), 495-503 (2006) · doi:10.1111/j.1365-246X.2006.02978.x
[25] Pratt, R.G., Song, Z.M., Williamson, P., Warner, M.: Two-dimensional velocity models from wide-angle seismic data by wavefield inversion. Geophys. J. Int. 124(2), 323-340 (1996) · doi:10.1111/j.1365-246X.1996.tb07023.x
[26] Rickett, J.E., Sava, P.C.: Offset and angle-domain common image-point gathers for shot-profile migration. Geophysics 67(3), 883-889 (2002) · doi:10.1190/1.1484531
[27] Robein, E.: Seismic imaging: A Review of the Techniques, their Principles, Merits and Limitations. EAGE Publications bv (2010)
[28] Shen, P., Symes, W.W.: Automatic velocity analysis via shot profile migration. Geophysics 73(5), VE49-VE59 (2008) · doi:10.1190/1.2972021
[29] Shen, P., Symes, W.W.: Horizontal contraction in image domain for velocity inversion. Geophysics 80(3), R95-R110 (2015) · doi:10.1190/geo2014-0261.1
[30] Shen, P., Symes, W.W., Stolk, C.C.: Differential semblance velocity analysis by wave-equation migration. In: SEG Technical Program Expanded Abstracts 2003, pp. 2132-2135 (2003) · Zbl 1136.86305
[31] Stolk, C.C., Symes, W.W.: Smooth objective functionals for seismic velocity inversion. Inverse Prob. 19 (1), 73-89 (2003) · Zbl 1136.86305 · doi:10.1088/0266-5611/19/1/305
[32] Symes, W.W.: Migration velocity analysis and waveform inversion. Geophys. Prospect. 56(6), 765-790 (2008) · doi:10.1111/j.1365-2478.2008.00698.x
[33] Symes, W.W.: Seismic inverse problems: recent developments in theory and practice. In: Inverse Problems—From Theory to Applications, pp. 2-6. IOP Publishing Ltd (2014)
[34] Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation. Geophysics 49(8), 1259-1266 (1984) · doi:10.1190/1.1441754
[35] ten Kroode, F.: A wave-equation-based Kirchhoff operator. Inverse Prob. 28(11), 115013 (2012) · Zbl 1282.86014 · doi:10.1088/0266-5611/28/11/115013
[36] van Leeuwen, T., Mulder, W.A.: A comparison of seismic velocity inversion methods for layered acoustics. Inverse Prob. 26(1), 015008 (2010) · Zbl 1180.35576
[37] Verschuur, D.J., Berkhout, A.J.: From removing to using multiples in closed-loop imaging. Lead. Edge 34 (7), 744-759 (2015) · doi:10.1190/tle34070744.1
[38] Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 74 (6), WCC1-WCC26 (2009) · doi:10.1190/1.3238367
[39] Weibull, W., Arntsen, B.: Automatic migration velocity analysis using reverse time migration. In: 73Rd EAGE Conference & Exhibition, p. B012 (2011)
[40] Wong, M., Biondi, B., Ronen, S.: Imaging with multiples using least-squares reverse time migration. Lead. Edge 33(9), 970-976 (2014) · doi:10.1190/tle33090970.1
[41] Yilmaz, Ö.Z.: Seismic data analysis. Society of Exploration Geophysicists (2001)
[42] Zhang, D., Schuster, G.T.: Least-squares reverse time migration of multiples. Geophysics 79(1), S11-S21 (2014) · Zbl 1333.49036 · doi:10.1190/geo2013-0156.1
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