Properties of the spectral response cost functional employed in the inverse wave-scattering problem of the retrieval of the shear body wavespeed of the homogeneous solid occupying a half space bounded by a plane stress-free surface

Parseval's theorem leads to the finding that the minima of a least-squares spectral response cost functional K are at the same positions as the minima of a least squares signal response functional C. We describe the useful functional properties of K, in the context of a simple geophysical inverse problem pertaining to the retrieval of the shear wavespeed in a homogeneous underground, notably for enabling the location of its global minimum and dealing with the secondary minima. We show how the width of the search interval, the number and positions of the sensors, as well as the central frequency and bandwidth of the spectrum of the probe radiation, condition the aspect of the cost functional, particularly as regards the number of secondary minima and the depth of the trough associated with the global minimum. Finally, we evaluate the influence of prior uncertainties on the accuracy of the retrieval (via K) of the shear body wavespeed of the underground.