Full Waveform Seismic Inversion With a Time-Variant Sensitivity Equation

A general approach toward solving the 3D full waveform seismic inverse problem entails iteratively refining a candidate earth model until an acceptable match is obtained between predicted and observed data. On each iteration, we calculate spatially distributed updates to the current model parameters by solving a large system of linear algebraic equations. This system is obtained by determining the sensitivity of each predicted seismic datum (particle displacement) to small perturbations in every model parameter (mass density and elastic moduli) as a function of recording time. These time-variant sensitivity equations are derived by applying the reciprocity principle and the first Born approximation to the velocity-stress equations governing isotropic elastic wave propagation. Construction of the time-variant sensitivity equations is computationally expensive, and thus requires parallel computation capability. For a given source-receiver pair, we obtain matrix elements by convolving velocity vector and strain tensor components of two wavefields, one activated at the source position and the other activated at the receiver position. Wave propagation through the current estimate of the 3D model is performed with a finite-difference algorithm that solves the elastodynamic velocity-stress system on staggered spatial and temporal grids. For all source-receiver pairs of the acquisition geometry, convolution products are stored at every subsurface gridpoint where parameter updates are desired. The right-hand vector in the linear system consists of displacement residual traces appended end-to-end. We can apply additional linear equations arising from a priori constraints imposed on the model (e.g., if the recording geometry leaves portions of the model space under-constrained). We then use an iterative algebraic solver (also implemented in a parallel computational environment) to seek the minimum-norm, least-squares solution of the entire. We demonstrate this inversion methodology with synthetic seismic data computed for several illustrative earth models, including isolated point diffractors embedded within a smooth background. Point diffractor models allow us to assess the spatial resolving power of the approach, as well as the algorithm's ability to distinguish perturbations in different material properties. For a reflection geometry with sufficient recording aperture, we can recover point perturbations to ρ , λ , or μ with as few as 10 sources recorded into 11 receivers. Sandia National Laboratories is a multiprogram science and engineering facility operated by Sandia Corporation, a Lockheed Martin company, for the United States Department of Energy under contract DE-AC04-94AL85000.