Theoretical background for the inversion of seismic waveforms including elasticity and attenuation

To account for elastic and attenuating effects in the elastic wave equation, the stress-strain relationship can be defined through a general, anisotropic, causal relaxation function ^ijkl (x, ). Then, the wave equation operator is not necessarily symmetric (self-adjoint), but the reciprocity property is still satisfied. The representation theorem contains a term proportional to the history of strain. The dual problem consists of solving the wave equation withfinal time conditions and an anti-causal relaxation function. The problem of interpretation of seismic waveforms can be set as the nonlinear inverse problem of estimating the matter density (x) and all the functions ^ijkl (x, ). This inverse problem can be solved using iterative gradient methods, each iteration consisting of the propagation of the actual source in the current medium, with causal attenuation, the propagation of the residuals—acting as if they were sources—backwards in time, with anti-causal attenuation, and the correlation of the two wavefields thus obtained.