Abstract
In this chapter, we introduce the basic integral representations for scalar and vector wavefields. We present the derivations of the integral equations for the acoustic (scalar) and vector wavefields using the properties of the corresponding Green’s functions. Based on these integral equations, we derive a family of integral approximations of the acoustic and vector wavefields, including Born, quasi-linear, quasi-analytical, and Kirchhoff approximations. These representations play an important role in forward modeling and in calculating the Fréchet derivative of the forward modeling operator, thus providing the foundation for solution of the inverse problem. For example, the Kirchhoff approximation has proved to be especially useful in forward and inverse wave propagation problems where the distribution of the reflecting boundaries is the main target.
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