A fundamental limitation for the reconstruction of impedance profiles from seismic data

Waveform inversion aims at a quantitative estimation of the subsurface model. Such a quantitative estimation is a nonlinear problem: the relation between elastic parameters and waveform data is basically nonlinear. Perturbations in the velocity give rise to severe nonlinear perturbations in the wavefield. Other severe nonlinear effects appear when the impedance profile is very irregular (i.e., rapidly oscillating). In this paper, we avoid these drastic nonlinear effects by assuming the velocity distribution is known, and we restrict our analysis to the reconstruction of “regular” impedance profiles. Even in this simple framework, deviations between a linear and a nonlinear model are cumulative as time increases, and the deviations cannot be neglected for propagation times of the order of standard seismic recordings. A nonlinear approach appears to be essential for quantitative imaging of deep targets. However a nonlinear approach requires the wavelet to be known with an accuracy that cannot be reached today. Our sensitivity analysis shows that the wavelet/model ambiguity is much more severe than the one met in linear inversion. As a consequence of the error accumulation inherent in a nonlinear wave propagation model, a small disturbance in the wavelet leads to a strong disturbance in the deep parts of the solution model. In this context, the classical approach for estimating the wavelet by minimizing the energy of the primary reflection waveform is not likely to provide the required accuracy except for very special cases. Accurate wavelet measurements thus appear to be a major challenge for a sound reconstruction of impedance profiles.