Forward interpolation

  As I will illustrate in later chapters, the crucial part of data regularization problems is in the choice and implementation of the regularization operator $\bold{D}$ or the corresponding preconditioning operator $\bold{P}$. The choice of the forward modeling operator $\bold{L}$ is less critical. In this chapter, I discuss the nature of forward interpolation, which has been one of the traditional subjects in computational mathematics. Wolberg (1990) presents a detailed review of different conventional approaches. I discuss a simple mathematical theory of interpolation from a regular grid and derive the main formulas from a very general idea of function bases.

Forward interpolation plays only a supplementary role in this dissertation, but it has many primary applications, such as trace resampling, NMO, Kirchhoff and Stolt migrations, log-stretch, and radial transform, in seismic data processing and imaging. Two simple examples appear at the end of this chapter.