Correction of an effect of using a faulty model for the retrieval of a bulk wave slowness of a layered half space

Abstract

This study concerns the retrieval of a single, real constitutive parameter (the bulk shear wave inverse velocity) of a simple, although representative, geophysical configuration involving two homogeneous, non-dissipative media, from its simulated response to pulsed plane wave probe radiation. This nonlinear inverse problem is solved exactly, at all frequencies, by equating the simulated frequency domain response incorporating the true real inverse velocity to the assumed response incorporating a trial complex inverse velocity. Due to discordance, caused by prior (a parameter of a model that is not retrieved, but rather assumed to be known, although its value may be wrong) uncertainty, between the models associated with the assumed and trial responses (the model giving rise to the latter is therefore qualified as being ‘faulty’), the imaginary part of the retrieved inverse velocity turns out to be non-nil, and, in fact, to be all the greater, the larger is the prior uncertainty. Moreover, the retrieved inverse velocity is found to be dispersive and its real part nearly equal to the true inverse velocity at all frequencies. The reconstructed signals (obtained by inserting the retrieved complex parameter into the faulty response model) are found to coincide exactly with the true signals for three types of probe pulses, even when the prior uncertainty is large.