Green’s Functions

Abstract

There are various approaches for developing a complete theory of F3DWI. The approach adopted in this book is based on the theory of Green’s functions. Compared with other equivalent formulations, the approach based on Green’s functions offers generality and additional flexibility. After a very brief summary of mathematical preliminaries in Sect. 3.1, I’ll give a tutorial about how to construct the adjoint of differential operators, with the emphasis on the scalar wave equation operator and the viscoelastic wave equation operator. The theoretical underpinning is the generalized Green’s identity. In Sect. 3.3, I’ll introduce the adjoint Green’s function and discuss its relations with the Green’s function. The reciprocity relation, the representation theorem and the adjoint representation theorem will be examined in the context of the generalized Green’s identity. In Sect. 3.4, I’ll discuss the receiver-side Green’s tensor (RGT) for the viscoelastic wave equation operator, its relation with the time-reversed adjoint Green’s tensor and give a tutorial about how to compute the RGT using the AWP code explained in Chap. 2. In Sect. 3.5, I’ll show how the Born series provides the first- and higher-order Frechet derivatives of the waveform with respect to the structural model and discuss the validity of the Born approximation.