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

摘要

Linear inversion is defined as the linear approximation of adirect-inverse solution. This definition leads to data requirementsand specific direct-inverse algorithms, which differwith all current linear and nonlinear approaches, and is immediatelyrelevant for target identification and inversion inan elastic earth. Common practice typically starts with a directforward or modeling expression and seeks to solve a forwardequation in an inverse sense. Attempting to solve a directforward problem in an inverse sense is not the same assolving an inverse problem directly. Distinctions include differencesin algorithms, in the need for a priori information,and in data requirements. The simplest and most accessibleexamples are the direct-inversion tasks, derived from the inversescattering series ISS, for the removal of free-surfaceand internal multiples. The ISS multiple-removal algorithmsrequire no subsurface information, and they are independentof earth model type.Adirect forward method solved in an inversesense, for modeling and subtracting multiples, wouldrequire accurate knowledge of every detail of the subsurfacethe multiple has experienced. In addition, it requires a differentmodeling and subtraction algorithm for each differentearth-model type. The ISS methods for direct removal ofmultiples are not a forward problem solved in an inversesense. Similarly, the direct elastic inversion provided by theISS is not amodeling formula for PPdata solved in an inversesense. Direct elastic inversion calls for PP, PS, SS, … data,for direct linear and nonlinear estimates of changes in mechanicalproperties. In practice, a judicious combination ofdirect and indirect methods are called upon for effective fielddata application.