Abstract
The goal of seismic velocity inversion is the estimation of seismic wave velocities inside the earth by attempting to predict, in a least-error sense, seismic waveforms measured at its surface. We present velocity inversion as a case study in the various ‘infinite-dimensional’ pathologies which may afflict practically important problems of distributed parameter identification, treated as optimization problems in function spaces. These features differentiate various problem formulations far beyond the degree one would expect for finite- (small-) dimensional problems. We illustrate this differentiation by comparing the characteristics of three different least-squares formulations of velocity inversion.
Similar content being viewed by others
References
R. Adams,Sobolev Spaces (Academic Press, New York, 1975).
K. Aki and P. Richards,Quantitative Seismology: Theory and Methods (Freeman, San Francisco, CA, 1980).
G. Beylkin, “Imaging of discontinuities in the inverse scattering problem,”Journal of Mathematical Physics 26 (1985) 99–108.
T. Chan, “Deflated Lanczos procedures for solving nearly singular linear systems,” Research Report YALEU/DCS/RR-403, Department of Computer Science, Yale University (New Haven, CT, 1985).
C. Chapman, “A new method for computing seismograms,”Geophysical Journal Royal Astrological Society 54 (1978) 481–518.
J.E. Dennis Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NY, 1983).
J. Duistermaat,Fourier Integral Operators (Conrant Institute Lecture Notes, 1975).
O. Gauthier, A. Tarantola and J. Virieux, “Two-dimensional nonlinear inversion of seismic waveforms,”Geophysics 51 (1986) 1387–1403.
S.H. Gray, “A second-order procedure for one-dimensional velocity inversion,”SIAM Journal on Applied Mathematics 39 (1980) 456–462.
S.H. Gray and F. Hagin, “Travel-time like variables and the solution of velocity inverse problems,” preprint.
Y. Hadjee and F. Collino, “A geometrical approach to the a-priori study of the 1-d inverse problem,” preprint, Institut Français due Pétrole (Rueil Malmaison, 1988).
D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, New York, 1973).
A. McAulay, “Prestack inversion with plane-layer point source modeling,”Geophysics 50 (1985) 77–89.
P. Mora, “Nonlinear 2-d elastic inversion of multi-offset seismic data,”Geophysics 52 (1987) 1211–1228.
P. Mora, “Nonlinear 2-d elastic inversion of real data,” Expanded abstract,57th Annual International Meeting (Society of Exploration Geophysicists, New Orleans, LA, 1987) pp. 430–432.
E.A. Robinson and S. Treitel,Geophysical Signal Analysis (Prentice-Hall, Englewood Cliffs, NY, 1980).
Rakesh, “A linearized inverse problem for the wave equation,”Communications on Partial Differential Equations 13 (1988) 573–601.
F. Santosa and W. Symes, “High frequency perturbational analysis for the point-source response of a layered acoustic medium,”Journal of Computational Physics 74 (1988) 318–381.
F. Santosa and W. Symes,An Analysis of Least-Squares Velocity Inversion, Society of Exploration Geophysicists Geophysical Monograph, Vol. 4 (Tulsa, 1989).
T. Steihaug, “Quasi-Newton methods for large-scale nonlinear problems,” Ph.D. Thesis, Yale University (New Haven, CT, 1981).
W. Symes, “Stability and instability results for inverse problems in several-dimensional wave propagation,” in:Proceedings of the Seventh International Conference on Computing Methods in Applied Science and Engineering (INRIA, 1985).
W. Symes, “Velocity Inversion by Coherency Optimization: Analysis,” preprint, Department of Mathematical Sciences, Rice University (Houston, TX, 1988a).
W. Symes, “Velocity Inversion by Coherency Optimization,” Technical Report 88–4, Department of Mathematical Sciences, Rice University (Houston, TX, 1988b).
W. Symes, “Bandlimited velocity inversion: a model problem in reflection seismology,” Technical Report 88–13, Department of Mathematical Sciences, Rice University (Houston, TX, 1988c).
R.A. Tapia, “Quasi-Newton methods for equality constrained optimization: equivalence of existing methods and a new implementation,” in:Nonlinear Programming, Vol. 3 (Academic Press, New York, 1978).
A. Tarantola,Inverse Problem Theory (Elsevier, New York, 1987).
A.N. Tihonov and V.Y. Arsenin,Solution of Ill-Posed Problems (Winston, New York, 1977).
Ph.L. Toint, “Towards an efficient sparsity exploiting Newton method for minimization,” in: I.S. Duff, ed.,Sparse Matrices and their Uses (Academic Press, London, 1981) pp. 57–88.
S. Treitel, P.R. Gutowski and D.E. Wagner, “Plane-wave decomposition of seismograms,”Geophysics 47 (1982) 1375–1401.
K.H. Waters,Reflection Seismology: A Tool for Resource Exploration (Wiley, New York, 1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Symes, W.W. Velocity inversion: A case study in infinite-dimensional optimization. Mathematical Programming 48, 71–102 (1990). https://doi.org/10.1007/BF01582252
Issue Date:
DOI: https://doi.org/10.1007/BF01582252