2. A Numerical Study of Linear Viscoacoustic Inversion

Abstract

Attenuation and dispersion of the seismic pulse can strongly affect the outcome of amplitude versus offset (AVO) analysis. These anelastic aspects of wave propagation also influence the outcome of linear inversion for short scale fluctuations in elastic parameters, the model based analogue of AVO analysis. The viscoacoustic model provides a framework for illustration of these effects. An efficient time domain viscoacoustic finite difference scheme for synthesis of primaries only plane-wave (1-D model) and line source (2-D model) reflection records is the basis for linear inversion algorithms via iterative minimization of mean square error. This approach to inversion requires computation of the mean square error gradient. The gradient computation becomes feasible through the use of the adjoint state technique, together with a checkpointing scheme due to Griewank. Inversion of synthetic data in the plane-wave domain illustrates the sensitivity to attenuation of even qualitative aspects of inversion. Viscoacoustic 2-D inversion of a small part of the Mobil AVO data illustrates applicability to field data in the offset domain.

A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

“…Newton's method is particularly efficient for solving an equation equivalent to (6) and this approach, due to Moré and Sorensen [28], is the method of choice whenever it is affordable to compute the Cholesky factorization of matrices of the form H − λI. However, in some applications this computation may be prohibitive either because of storage considerations or because the matrix H is not explicitly available.…”

“…As ex- plained in [6], this model describes the behavior of an anelastic fluid, in which the strain response to a change of stress is linear but not completely instantaneous. A relaxation function G(t, x) is used to express the stress-strain relation.…”

Reverse time migration with optimal checkpointing

“…I believe that the object-oriented approach to expression of high-level algorithms such as those described here has much to recommend it. However, it is relatively straightforward to use the codes from Griewank and Walther (2000) to add optimal checkpointing to a procedural implementation of the adjoint state method for a particular system, such as acoustic reverse time migration, see for example Blanch et al (1998).…”

“…The author and his collaborators used optimal checkpointing in a viscoacoustic least-squares inversion scheme (Blanch et al, 1998). Ghattas and his coworkers have also used the algorithm in the context of least-squares inversion of basin structure from earthquake data (Akcelik et al, 2003).…”

“…The purpose of this short paper is to (re)introduce to the seismic imaging community an algorithm devised by Griewank which largely ameliorates the complexity problem just described (Griewank, 1992;Griewank and Walther, 2000;Griewank, 2000;Blanch et al, 1998;Akcelik et al, 2003). This optimal checkpointing algorithm trades floating point arithmetic for memory.…”