Abstract
This paper tackles the issue of the computational load encountered in seismic imaging by Bayesian traveltime inversion. In Bayesian inference, the exploration of the posterior distribution of the velocity model parameters requires extensive sampling. The computational cost of this sampling step can be prohibitive when the first arrival traveltime prediction involves the resolution of an expensive number of forward models based on the eikonal equation. We propose to rely on polynomial chaos surrogates of the traveltimes between sources and receivers to alleviate the computational burden of solving the eikonal equation during the sampling stage. In an offline stage, the approach builds a functional approximation of the traveltimes from a set of solutions of the eikonal equation corresponding to a few values of the velocity model parameters selected in their prior range. These surrogates then substitute the eikonal-based predictions in the posterior evaluation, enabling very efficient extensive sampling of Bayesian posterior, for instance, by a Markov Chain Monte Carlo algorithm. We demonstrate the potential of the approach using numerical experiments on the inference of two-dimensional domains with layered velocity models and different acquisition geometries (microseismic and seismic refraction contexts). The results show that, in our experiments, the number of eikonal model evaluations required to construct accurate surrogates of the traveltimes is low. Further, an accurate and complete characterization of the posterior distribution of the velocity model is possible, thanks to the generation of large sample sets at a low cost. Finally, we discuss the extension of the current approach to more realistic velocity models and operational situations.
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References
Abramowitz, M., Stegun, I.A.: Orthogonal polynomials. In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edn, 55, U.S. Government Printing Office, Washington, D.C., chap 22, pp. 771–802 (1972)
Bayes, T.: An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53, 370–418 (1763)
Belhadj, J., Romary, T., Gesret, A., Noble, M., Figliuzzi, B.: New parameterizations for bayesian seismic tomography. Inverse Probl. 34(6), 065007 (2018). https://doi.org/10.1088/1361-6420/aabce7
Besag, J.E.: Comments on representations of knowledge in complex systems by u. grenander and m.i. miller. J. R. Stat. Soc. Ser. B 56, 591–592 (1994)
Billings, S.D., Sambridge, M.S., Kennett, B.L.: Errors in hypocenter location: picking, model, and magnitude dependence. Bull. Seism. Soc. Am. 84(6), 1978–1990 (1994)
Bodin, T., Sambridge, M.: Seismic tomography with the reversible jump algorithm. Geophys. J. Int. 178(3), 1411–1436 (2009). https://doi.org/10.1111/j.1365-246X.2009.04226.x
Bottero, A., Gesret, A., Romary, T., Noble, M., Maisons, C.: Stochastic seismic tomography by interacting Markov chains. Geophys. J. Int. 207(1), 374–392 (2016). https://doi.org/10.1093/gji/ggw272
Brynjarsdóttir, J., O‘Hagan, A.: Learning about physical parameters: the importance of model discrepancy. Inverse Probl. 30(11), 114007 (2014). https://doi.org/10.1088/0266-5611/30/11/114007
Cameron, R., Martin, W.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)
Cohen, A., Migliorati, G.: Optimal weighted least-squares methods. SMAI J. Comp. Math. 3(3716755), 181–203 (2017). https://doi.org/10.5802/smai-jcm.24
Conrad, P.R., Marzouk, Y.M.: Adaptive Smolyak pseudospectral approximations. SIAM J. Sci. Comp. 35(6), A2643–A2670 (2013). https://doi.org/10.1137/120890715
Constantine, P.G., Eldred, M.S., Phipps, E.T.: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229, 1–12 (2012)
Crestaux, T., Le Maître, O., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)
Cruz-Jiménez, H., Li, G., Mai, P., Hoteit, I., Knio, O.: Bayesian inference of earthquake rupture models using polynomial chaos expansion. Geosci. Model Dev. 11(7), 3071–3088 (2018). https://doi.org/10.5194/gmd-11-3071-2018
Cui, T., Marzouk, Y.M., Willcox, K.E.: Data-driven model reduction for the bayesian solution of inverse problems. Int. J. Numer. Methods Eng. 102(5), 966–990 (2015). https://doi.org/10.1002/nme.4748
Duane, S., Kennedy, A., Pendleton, B.J., Roweth, D.: Hybrid monte carlo. Phys. Lett. B 195(2), 216–222 (1987). https://doi.org/10.1016/0370-2693(87)91197-X
Earp, S., Curtis, A.: Probabilistic neural network-based 2d travel-time tomography. Neural Comput. Appl. 32, 216–222 (2020). https://doi.org/10.1007/s00521-020-04921-8
Efendiev, Y., Datta-Gupta, A., Ginting, V., Ma, X., Mallick, B.: An efficient two-stage markov chain monte carlo method for dynamic data integration. Water Resour. Res. (2005). https://doi.org/10.1029/2004WR003764
Eisner, L., Duncan, P.M., Heigl, W.M., Keller, W.R.: Uncertainties in passive seismic monitoring. Lead. Edge 28(6), 648–655 (2009)
Elsheikh, A.H., Hoteit, I., Wheeler, M.F.: Efficient bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates. Comput. Methods Appl. Mech. Eng. 269, 515–537 (2014). https://doi.org/10.1016/j.cma.2013.11.001
Fichtner, A., Zunino, A., Gebraad, L.: Hamiltonian monte carlo solution of tomographic inverse problems. Geophys. J. Int. 216(2), 1344–1363 (2018). https://doi.org/10.1093/gji/ggy496
Formaggia, L., Guadagnini, A., Imperiali, I., Lever, V., Porta, G., Riva, M., Scotti, A., Tamellini, L.: Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model. Comput. Geosci. 17(1), 25–42 (2013). https://doi.org/10.1007/s10596-012-9311-5
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 209–232 (1998)
Ghanem, R.G., Spanos, S.D.: Stochastic Finite Elements: A Spectral Approach. Springer (1991)
Giraldi, L., Le Maître, O., Mandli, K., Dawson, C., Hoteit, I., Knio, O.: Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate. Comput. Geosci. 21(4), 683–699 (2017). https://doi.org/10.1007/s10596-017-9646-z
Haario, H., Saksman, E., Tamminen, J.: An adaptive metropolis algorithm. Bernoulli 7(2), 223–242 (2001)
Hampton, J., Doostan, A.: Coherence motivated sampling and convergence analysis of least squares polynomial chaos regression. Comput. Methods Appl. Mech. Eng. 290, 73–97 (2015). https://doi.org/10.1016/j.cma.2015.02.006
Hastings, W.K.: Monte carlo sampling methods using markov chains and their applications. Biometrika 57(1), 97–109 (1970)
Hawkins, R., Sambridge, M.: Geophysical imaging using trans-dimensional trees. Geophys. J. Int. 203(2), 972–1000 (2015). https://doi.org/10.1093/gji/ggv326
Hawkins, R., Bodin, T., Sambridge, M., Choblet, G., Husson, L.: Trans-dimensional surface reconstruction with different classes of parameterization. Geochem. Geophys. Geosyst. 20(1), 505–529 (2019). https://doi.org/10.1029/2018GC008022
Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19, 293–325 (1948)
Iskandarani, M., Wang, S., Srinivasan, A., Thacker, W.C., Winokur, J., Knio, O.: An overview of uncertainty quantification techniques with application to oceanic and oil-spill simulations. J. Geophys. Res.: Oceans 121(4), 2789–2808 (2016). https://doi.org/10.1002/2015JC011366
Jeffreys, H.: An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 186(1007), 453–461 (1946). https://doi.org/10.1098/rspa.1946.0056
Jeffreys, H.: Theory of Probability, 3rd edn., Oxford (1961)
Karhunen, K.: Über lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae Ser. A I Math-Phys. 37, 1–79 (1947)
Kennedy, M.C., O‘Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 63(3), 425–464 (2001). https://doi.org/10.1111/1467-9868.00294
Kruschke, J.K.: Doing Bayesian Data Analysis: A Tutorial with R and BUGS, 1st edn. Academic Press Inc, Orlando (2010)
Le Bouteiller, P., Benjemaa, M., Métivier, L., Virieux, J.: An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media. Geophys. J. Int. 212(3), 1498–1522 (2017). https://doi.org/10.1093/gji/ggx463
Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer, Scientific Computation (2010)
Ley, C., Reinert, G., Swan, Y.: Distances between nested densities and a measure of the impact of the prior in bayesian statistics. Ann. Appl. Probab. 27(1), 216–241 (2017)
Li, G., Iskandarani, M., Le Hénaff, M., Winokur, J., Le Maître, O.P., Knio, O.M.: Quantifying initial and wind forcing uncertainties in the Gulf of Mexico. Comput. Geosci. 20(5), 1133–1153 (2016). https://doi.org/10.1007/s10596-016-9581-4
Loève M (1968) Probability Theory. The university series in higher mathematics, affiliated East-West-Press Pvt. Limited
Lucor, D., Le Maître, O.P.: Cardiovascular modeling with adapted parametric inference. ESAIM ProcS 62, 91–107 (2018). https://doi.org/10.1051/proc/201862091
Malinverno, A.: Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem. Geophys. J. Int. 151(3), 675–688 (2002). https://doi.org/10.1046/j.1365-246X.2002.01847.x
Marzouk, Y.M., Xiu, D.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. Commun. Comput. Phys. 6(4), 826–847 (2009)
Marzouk, Y.M., Najm, N.H., Rahn, L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224, 560–586 (2007)
Maxwell, S.: Microseismic: growth born from success. Lead. Edge 29(3), 338–343 (2010)
McKay, M., Conover, W., Beckman, R.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239–245 (1979)
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953). https://doi.org/10.1063/1.1699114
Morokoff, W.J., Caflisch, R.E.: Quasi-monte carlo integration. J. Comput. Phys. 122(2), 218–230 (1995). https://doi.org/10.1006/jcph.1995.1209
Mosegaard, K., Tarantola, A.: Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res.: Solid Earth 100(B7), 12431–12447 (1995). https://doi.org/10.1029/94JB03097
Navarro, M., Le Maître, O., Hoteit, I., George, D., Mandli, K., Knio, O.: Surrogate-based parameter inference in debris flow model. Comput. Geosci. 22(6), 1447–1463 (2018). https://doi.org/10.1007/s10596-018-9765-1
Noble, M., Gesret, A., Belayouni, N.: Accurate 3-d finite difference computation of traveltimes in strongly heterogeneous media. Geophys. J. Int. 199(3), 1572–1585 (2014). https://doi.org/10.1093/gji/ggu358
Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962). https://doi.org/10.1214/aoms/1177704472
Rawlinson, N., Sambridge, M.: Seismic traveltime tomography of the crust and lithosphere. Adv. Geophys. 46, 81–198 (2003)
Roberts, A., Hobbs, R., Goldstein, M., Moorkamp, M., Jegen, M., Heincke, B.: Crustal constraint through complete model space screening for diverse geophysical datasets facilitated by emulation. Tectonophysics 572–573, 47–63 (2012). https://doi.org/10.1016/j.tecto.2012.03.006
Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009). https://doi.org/10.1198/jcgs.2009.06134
Sethian, J.A.: A Fast Marching Level Set Method for Monotonically Advancing Fronts. In: Proc. Nat. Acad. Sci., pp. 1591–1595 (1995)
Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk SSSR 4(240–243), 123 (1963)
Sobol, I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model Comput. Exp. 1, 407–414 (1993)
Sochala, P., De Martin, F.: Surrogate combining harmonic decomposition and polynomial chaos for seismic shear waves in uncertain media. Comput. Geosci. (2017). https://doi.org/10.1007/s10596-017-9677-5
Sraj, I., Le Maître, O.P., Knio, O.M., Hoteit, I.: Coordinate transformation and polynomial chaos for the bayesian inference of a gaussian process with parametrized prior covariance function. Comput. Methods Appl. Mech. Eng. 298, 205–228 (2016). https://doi.org/10.1016/j.cma.2015.10.002
Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008). https://doi.org/10.1016/j.ress.2007.04.002
Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, Philadelphia (2004)
Vidale, J.: Finite-difference calculation of travel times. Bull. Seism. Soc. Am. 78(6), 2062–2076 (1988)
Wilkinson, D.J.: Parallel Bayesian computation. Statistics Textbooks and Monographs (2006)
Zhang, F., Dai, R., Liu, H.: Seismic inversion based on L1-norm misfit function and total variation regularization. J. Appl. Geophys. 109, 111–118 (2014). https://doi.org/10.1016/j.jappgeo.2014.07.024
Zhao, H.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005)
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Sochala, P., Gesret, A. & Le Maître, O. Polynomial surrogates for Bayesian traveltime tomography. Int J Geomath 12, 20 (2021). https://doi.org/10.1007/s13137-021-00184-0
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DOI: https://doi.org/10.1007/s13137-021-00184-0