Model- and Operator-Based Representations for Directional and Non-Directional Wavefields

Abstract

Scattering and inverse scattering theories are based on the Lippmann-Schwinger equation and its linearised version, the so-called Born equation. In this chapter, we seek alternative representations that can be the basis for different methods of model- ing and inversion. One representation transforms the Lippmann-Schwinger equation into one that represents directional wave-fields, while another invokes operators in- stead model parameters. We investigate and derive different forms of the latter. At the end, we come to a classification of the different representations described in this chapter. We classify them as those that invoke solely model parameters and those that invoke operators. Each of these families can be further grouped into those that use directional wavefields and those that use non-directional wave-fields. This chapter derives formulas that can be the basis for methods such as a directional inverse scattering theory, full wavefield modeling, model-independent joint migration inversion and operator-based non-directional method that invokes contrasts of the so-called Helmholtz operator.