Abstract

Seismic waveform inversion aims at obtaining detailed estimates of subsurface medium parameters, such as the spatial distribution of soundspeed, from multiexperiment seismic data. A formulation of this inverse problem in the frequency domain leads to an optimization problem constrained by a Helmholtz equation with many right-hand sides. Application of this technique to industry-scale problems faces several challenges: First, we need to solve the Helmholtz equation for high wave numbers over large computational domains. Second, the data consist of many independent experiments, leading to a large number of PDE solves. This results in high computational complexity both in terms of memory and CPU time as well as input/output costs. Finally, the inverse problem is highly nonlinear and a lot of art goes into preprocessing and regularization. Ideally, an inversion needs to be run several times with different initial guesses and/or tuning parameters. In this paper, we discuss the requirements of the various components (PDE solver, optimization method, \dots) when applied to large-scale three-dimensional seismic waveform inversion and combine several existing approaches into a flexible inversion scheme for seismic waveform inversion. The scheme is based on the idea that in the early stages of the inversion we do not need all the data or very accurate PDE solves. We base our method on an existing preconditioned Krylov solver (CARP-CG) and use ideas from stochastic optimization to formulate a gradient-based (quasi-Newton) optimization algorithm that works with small subsets of the right-hand sides and uses inexact PDE solves for the gradient calculations. We propose novel heuristics to adaptively control both the accuracy and the number of right-hand sides. We illustrate the algorithms on synthetic benchmark models for which significant computational gains can be made without being sensitive to noise and without losing the accuracy of the inverted model.

Keywords

  1. seismic inversion
  2. Helmholtz equation
  3. preconditioning
  4. Kaczmarz method
  5. inexact gradient
  6. block-cg

MSC codes

  1. 65N21
  2. 65K10

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References

1.
A.Y. Aravkin, M.P. Friedlander, F.J. Herrmann, and T. van Leeuwen, Robust inversion, dimensionality reduction, and randomized sampling, Math. Program., 134 (2012), pp. 101--125.
2.
A.Y. Aravkin and T. van Leeuwen, Estimating nuisance parameters in inverse problems, Inverse Problems, 28 (2012), 115016.
3.
M. Arioli, I. S. Duff, D. Ruiz, and M. Sadkane, Block Lanczos techniques for accelerating the block Cimmino method, SIAM J. Sci. Comput., 16 (1995), pp. 1478--1511.
4.
G. Biros and O. Ghattas, Inexactness issues in the Lagrange-Newton-Krylov-Schur method for PDE-constrained optimization, in Large-Scale PDE-Constrained Optimization, Springer, Berlin, 2003, pp. 93--114.
5.
\AA. Björck and T. Elfving, Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations, BIT, 19 (1979), pp. 145--163.
6.
C. Bunks, Multiscale seismic waveform inversion, Geophysics, 60 (1995), pp. 1457--1473.
7.
R.H. Byrd, G.M. Chin, W. Neveitt, and J. Nocedal, On the use of stochastic Hessian information in optimization methods for machine learning, SIAM J. Optim., 21 (2011), pp. 977--995.
8.
H. Calandra, S. Gratton, J. Langou, X. Pinel, and X. Vasseur, Flexible variants of block restarted GMRES methods with application to geophysics, SIAM J. Sci. Comput., 34 (2012), pp. A714--A736.
9.
B. Engquist, Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Comm. Pure Appl. Math., 64 (2011), pp. 697--735.
10.
I. Epanomeritakis, V. Akçelik, O. Ghattas, and J. Bielak, A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion, Inverse Problems, 24 (2008), 034015.
11.
Y.A. Erlangga, C. Vuik, and C.W. Oosterlee, On a robust iterative method for heterogeneous Helmholtz problems for geophysics applications, Int. J. Numer. Anal. Model., 2 (2005), pp. 197--208.
12.
O.G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, and R. Scheichl, eds., Lect. Notes Comput. Sci. and Eng. 83, Springer, Berlin, 2012, pp. 325--363.
13.
M.P. Friedlander and M. Schmidt, Hybrid deterministic-stochastic methods for data fitting, SIAM J. Sci. Comput., 34 (2012), pp. A1380--A1405.
14.
D. Gordon and R. Gordon, Component-averaged row projections: A robust, block-parallel scheme for sparse linear systems, SIAM J. Sci. Comput., 27 (2005), pp. 1092--1117.
15.
D. Gordon and R. Gordon, CARP-CG: A robust and efficient parallel solver for linear systems, applied to strongly convection dominated PDEs, Parallel Comput., 36 (2010), pp. 495--515.
16.
D. Gordon and R. Gordon, Parallel solution of high-frequency Helmholtz equations using high-order finite difference schemes, Appl. Math. Comput., 218 (2012), pp. 10737--10754.
17.
D. Gordon and R. Gordon, Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers, J. Comput. Appl. Math., 232 (2012), pp. 182--196.
18.
A. Greenbaum, Iterative Methods for Solving Linear Systems, Front. Appl. Math. 17, SIAM, Philadelphia, 1997.
19.
E. Haber, U.M. Ascher, and D. Oldenburg, On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 16 (2000), pp. 1263--1280.
20.
E. Haber, M. Chung, and F. Herrmann, An effective method for parameter estimation with PDE constraints with multiple right-hand sides, SIAM J. Optim., 22 (2012), pp. 739--757.
21.
E. Haber and S. MacLachlan, A fast method for the solution of the Helmholtz equation, J. Comput. Phys., 230 (2011), pp. 4403--4418.
22.
M. Heinkenschloss and D. Ridzal, An inexact trust-region SQP method with applications to PDE-constrained optimization, in Numerical Mathematics and Advanced Applications, Springer, Berlin, 2008, pp. 613--620.
23.
M. Heinkenschloss and L.N. Vicente, Analysis of inexact trust-region SQP algorithms, SIAM J. Optim., 12 (2002), pp. 283--302.
24.
S. Kaczmarz, Angenäherte auflösung von systemen linearer gleichungen, Bull. Int. Acad. Polonaise Sci. Lett., 35 (1937), pp. 355--357.
25.
H. Knibbe, W.A. Mulder, C.W. Oosterlee, and C. Vuik, Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3D migration on parallel architectures, Geophysics, 79 (2014), pp. 547--561.
26.
X. Li, A.Y. Aravkin, T. van Leeuwen, and F.J. Herrmann, Fast randomized full-waveform inversion with compressive sensing, Geophysics, 77 (2012), pp. A13--A17.
27.
Y. Luo and G. Schuster, Wave-equation traveltime inversion, Geophysics, 56 (1991), pp. 645--653.
28.
P.P. Moghaddam, H. Keers, F.J. Herrmann, and W.A. Mulder, A new optimization approach for source-encoding full-waveform inversion, Geophysics, 78 (2013), pp. R125--R132.
29.
A.A. Nikishin and A.Yu. Yeremin, Variable block CG algorithms for solving large sparse symmetric positive definite linear systems on parallel computers, I: General iterative scheme, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1135--1153.
30.
J. Nocedal and S.J. Wright, Numerical Optimization, Springer, New York, 2000.
31.
S. Operto, J. Virieux, P. Amestoy, J.Y. L'Excellent, L. Giraud, and H.B.H. Ali, 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study, Geophysics, 72 (2007), pp. SM195--SM211.
32.
D. Osei-Kuffuor and Y. Saad, Preconditioning Helmholtz linear systems, Appl. Numer. Math., 60 (2010), pp. 420--431.
33.
R.-E. Plessix, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., 167 (2006), pp. 495--503.
34.
R.-E. Plessix, Y.-H. De Roeck, and G. Chavent, Waveform inversion of reflection seismic data for kinematic parameters by local optimization, SIAM J. Sci. Comput., 20 (1999), pp. 1033--1052.
35.
J. Poulson, B. Engquist, S. Fomel, S. Li, and L. Ying, A Parallel Sweeping Preconditioner for High Frequency Heterogeneous $3$D Helmholtz Equations, preprint, arXiv:1204.0111, 2012.
36.
G.R. Pratt, C. Shin, and G.J. Hicks, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion, Geophys. J. Int., 133 (1998), pp. 341--362.
37.
R.G. Pratt, Z.M. Song, P. Williamson, and M. Warner, Two-dimensional velocity models from wide-angle seismic data by wavefield inversion, Geophys. J. Int., 124 (1996), pp. 232--340.
38.
C. Riyanti, A. Kononov, Y.A. Erlangga, C. Vuik, C. Oosterlee, R.-E. Plessix, and W.A. Mulder, A parallel multigrid-based preconditioner for the $3$D heterogeneous high-frequency Helmholtz equation, J. Comput. Phys., 224 (2007), pp. 431--448.
39.
N.N. Schraudolph, J. Yu, and S Günter, A stochastic Quasi-Newton method for online convex optimization, in Proceedings of the 11th International Conference on Artificial Intelligence and Statistics, San Juan, Puerto Rico, 2007, pp. 436--443.
40.
F. Sourbier, A. Haidar, L. Giraud, H. Ben-Hadj-Ali, J. Virieux, and S. Operto, Three-dimensional parallel frequency-domain visco-acoustic wave modelling based on a hybrid direct/iterative solver, Geophys. Prospect., 59 (2011), pp. 834--856.
41.
W.W. Symes, Layered velocity inversion: A model problem from reflection seismology, SIAM J. Math. Anal., 22 (1991), pp. 680--716.
42.
W.W. Symes, Reverse time migration with optimal checkpointing, Geophysics, 72 (2007), pp. SM213--SM221.
43.
W.W. Symes, The seismic reflection inverse problem, Inverse Problems, 25 (2009), 123008.
44.
A. Tarantola and A. Valette, Generalized nonlinear inverse problems solved using the least squares criterion, Rev. Geophys. Space Phys., 20 (1982), pp. 129--232.
45.
I.S. Terentyev, T Vdovina, W.W. Symes, X. Wang, and D. Sun, IWAVE: A Framework for Wave Simulation, http://www.trip.caam.rice.edu/software/iwave/doc/html/ http://www.trip.caam.rice.edu/software/iwave/doc/html/ (2010).
46.
K. van den Doel and U.M. Ascher, Adaptive and stochastic algorithms for electrical impedance tomography and DC resistivity problems with piecewise constant solutions and many measurements, SIAM J. Sci. Comput., 34 (2012), pp. A185--A205.
47.
T. van Leeuwen, Fourier Analysis of the CGMN Method for Solving the Helmholtz Equation, preprint, arXiv:1210.2644, 2012.
48.
T. van Leeuwen, A.Y. Aravkin, and F.J. Herrmann, Seismic waveform inversion by stochastic optimization, Int. J. Geophys., 2011 (2011), 689041.
49.
T. van Leeuwen and F.J. Herrmann, Fast waveform inversion without source-encoding, Geophys. Prospect., 61, Suppl. (2012), pp. 10--19.
50.
T. van Leeuwen and F.J. Herrmann, Mitigating local minima in full-waveform inversion by expanding the search space, Geophys. J. Int., 195 (2013), pp. 661--667.
51.
T. van Leeuwen and W.A. Mulder, A comparison of seismic velocity inversion methods for layered acoustics, Inverse Problems, 26 (2010), 015008.
52.
T. van Leeuwen and W.A. Mulder, A correlation-based misfit criterion for wave-equation traveltime tomography, Geophys. J. Int., 182 (2010), pp. 1383--1394.
53.
J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74 (2009), pp. WCC1--WCC26.
54.
S. Wang, M.V. de Hoop, and J. Xia, On $3$D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver, Geophys. Prospect., 59 (2011), pp. 857--873.
55.
J. Young and D. Ridzal, An application of random projection to parameter estimation in partial differential equations, SIAM J. Sci. Comput., 34 (2012), pp. A2344--A2365.

Information & Authors

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S192 - S217
ISSN (online): 1095-7197

History

Submitted: 26 April 2013
Accepted: 14 March 2014
Published online: 30 October 2014

Keywords

  1. seismic inversion
  2. Helmholtz equation
  3. preconditioning
  4. Kaczmarz method
  5. inexact gradient
  6. block-cg

MSC codes

  1. 65N21
  2. 65K10

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