Hierarchical decomposition and inversion

Abstract

In inverse scattering one attempts to reconstruct the material composition of a domain whose interior is inaccessible to direct measurement by probing it from the outside. To this end the domain is considered as a contrasting domain in a known background configuration. The probing is carried out by exciting the object with a number of sources, while the resulting wavefield is detected at a number of receiver positions. In the corresponding mathematical description of the experiment the wavefield quantities are subject to a spatial-temporal differential operator, and to the boundary conditions, such as, for example, source conditions and the radiation conditions. In general terms inversion can be formulated as a non-linear expression where the measurements are related to the contrast-function in the medium. This representation is equivalent to a volume integral over the contrasting domain where the contrast function, together with the actual field, acts as weights of the kernel function. This kernel function depends on the position of two points in the contrasting domain and is known as the Green's function. The Green's function represents the inverse of the differential operator. In the usual formulation of the inverse problem the wave-theoretical character of the inverse operator is predetermined; only the constitutive parameters are allowed to vary. In this sense inversion is equal to inverse forward modelling. However, this approach leads to a restriction on the inversion process. The data to be inverted are harnessed due to this assumption. The parameters do not have enough flexibility to compensate for the discrepancies between observed and calculated data when the observed data cannot be attributed to such a wave problem. In the hierarchical approach it is proven that any wave problem can be decomposed into a set of subproblems. By arranging this set of subproblems in increasing order of complexity the associated inverse process is divided into two steps. The first step consists of determining the contribution of the sub-set members to the whole data set. In the second step a linear inversion is performed to each sub-set member. In this process the influence of less complex and previously determined members is taken into account. This procedure is not equal to inverse forward modelling.