Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data

Abstract

Linear inversion is defined as the linear approximation of a direct-inverse solution. This definition leads to data requirements and specific direct-inverse algorithms, which differ with all current linear and nonlinear approaches, and is immediately relevant for target identification and inversion in an elastic earth. Common practice typically starts with a direct forward or modeling expression and seeks to solve a forward equation in an inverse sense. Attempting to solve a direct forward problem in an inverse sense is not the same as solving an inverse problem directly. Distinctions include differences in algorithms, in the need for a priori information, and in data requirements. The simplest and most accessible examples are the direct-inversion tasks, derived from the inverse scattering series (ISS), for the removal of free-surface and internal multiples. The ISS multiple-removal algorithms require no subsurface information, and they are independent of earth model type. A direct forward method solved in an inverse sense, for modeling and subtracting multiples, would require accurate knowledge of every detail of the subsurface the multiple has experienced. In addition, it requires a different modeling and subtraction algorithm for each different earth-model type. The ISS methods for direct removal of multiples are not a forward problem solved in an inverse sense. Similarly, the direct elastic inversion provided by the ISS is not a modeling formula for PP data solved in an inverse sense. Direct elastic inversion calls for [Formula: see text], [Formula: see text], [Formula: see text], … data, for direct linear and nonlinear estimates of changes in mechanical properties. In practice, a judicious combination of direct and indirect methods are called upon for effective field data application.