Time-Domain Sensitivities for Elastic Wavefield Data

Exact mathematical expressions for the sensitivity of elastic wavefield data, with respect to small perturbations in mass density and Lame parameters, are derived by applying the reciprocity principle to solutions of the velocity-stress differential equations of isotropic elastodynamics. The formulae are valid for various types of sources (forces, moments, surface tractions) and receivers (displacement, velocity, acceleration, strain, rotation, pressure) used in seismic exploration, and apply to arbitrary data acquisition configurations in three spatial dimensions. Sensitivities with respect to variations in other elastic model parameters of seismological interest (wavespeeds, impedances, moduli) are readily obtained via linear combinations of the sensitivities due to perturbations in ρ , λ , and μ . Bandlimited sensitivity coefficients are numerically calculated with a time-domain finite-difference algorithm that solves the velocity-stress system on staggered spatial and temporal grids. For a given source-receiver pair located within an elastic earth model, two wavefields are generated; one is activated at the source position and the other is activated at the receiver position. At a specific gridpoint in the model, convolving the velocity vector components of the two wavefields gives a time-varying sensitivity function for perturbations in mass density. Convolution of the stress tensor components yields analogous sensitivities with respect to Lame parameter variations. Sensitivity coefficients are depicted and interpreted for a simple situation consisting of a single source-receiver pair situated within a homogeneous and isotropic elastic wholespace. Timeslice plots of λ sensitivity indicate that significant amplitudes are concentrated around the ellipsoidal locus of fixed-time P-to-P scattering from source to receiver. Plots of ρ and μ sensitivity are much more complex, containing multiple branches corresponding to P-P, P-S, S-P, and S-S scattering. Parameter sensitivities are used for solving the full waveform elastic inverse problem, and may provide useful information for design of seismic data acquisition experiments. Sandia National Laboratories is a multiprogram science and engineering facility operated by Sandia Corporation, a Lockheed-Martin company, for the US Department of Energy under contract DE-AC04-94AL85000