Inverse Problems Related To Fields From Localized Sources Over A Stratified Half-Space

Abstract

Abstract Inverse problems involving a localized source over a stratified half-space are relevant in many different contexts, such as non-invasive investigations of subsurface physical properties by means of acoustic, electromagnetic or elastodynamic waves. In the present chapter the previous treatment is generalized inasmuch as instead of normally or obliquely incident plane waves, we now consider more general incident fields generated within a bounded region. The main assumption about the half-space to be investigated is the same as before, i.e., it is assumed to be plane-stratified. For these more general incident fields it is shown how the use of suitably defined moments of the fields and the reflection kernel can be used to solve inverse problems. In particular it is shown how the use of the zeroth, second and fourth moments of the three-dimensional fields reduce the inverse problem to a set of one-dimensional problems. When one has additional symmetry properties of the incident field, such as axial symmetry, it is shown how the inverse problem is reduced to a one-dimensional problem through the use of a Hankel transform. In this case an explicit form of the wave-splitting operator is found in the Hankel transformed space. The Green’s function formalism for this case is developed. The explicit form of the wave-splitting is also used to treat wave propagation in homogeneous plasmas, dispersive media and waveguides.