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An inverse problem of recovery the earth interior by multi-shot/multi- offset multicomponent seismic data is dealt with. The data are treated as an image of some nonlinear operator (forward map) which is implicitly introduced by initial-boundary value problem for elastic wave equations, so in order to recover elastic parameters nonlinear operator equation should be resolved.
The usual way to do this numerically is in implementation of some local gradient iterative minimization technique like conjugate gradient or Gauss-Newton methods and various their modifications. In seismic processing this method is known as full-waveform inversion. This approach was proposed in early 1980’s (Tarantola, 1984; 1986) and intensively studied and developed by many researches. The method seemed to be very promising at first, because of its apparent ability to recover elastic parameters under a few preprocessing of data. However its application to realistic synthetic and field data exposed its major drawbacks. The most significant of them were troubles in reconstruction of the smooth velocity constituent, the so- called macrovelocity or migration model. Certain difficulties were found in connection with coupling of elastic parameters and different sensitivity of the method to their perturbations (Assous and Collino, 1990).
Our belief is that all peculiarities of the inversion procedure are controlled mainly by the Frechet derivative of the forward map, because this is the derivative determines the gradient and Hessian of the L_2 data misfit functional. Singular Value Decomposition of this derivative is a powerful tool for analyzing its key features. In current work this analysis is applied to 2D elastic full waveform inversion. We show that all of the mentioned above drawbacks of the method are explained and predicted by the structure of right singular vectors. The notion of stable subspace in model space is introduced and discussed and its geometry is analyzed.
The results of synthetic data processing for surface, cross-well and offset vertical seismic profile (OVSP) acquisition systems are presented and discussed.
Acknowledgement. This research was partially supported by Russian Foundation of Basic Research, projects 10-05-00233 and 11-05-00947.
References
[1] F.Assous, F.Collino, 1990, A numerical method for the exploration of sensitivity: the case of identification of the 2D stratified elastic medium: Inverse Problems, 6(4), 487 - 513.
[2] A.Tarantola, 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259 - 1266.
[3] A.Tarantola, 1986, A strategy for nonlinear inversion of seismic reflection data: Geophysics, 51, 1893 - 1903.
Note. Abstracts are published in author's edition
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