THEORETICAL DEFINITION OF A STACKING OPERATOR

In practice, integration of discrete data is performed by stacking, which requires special caution in the case of spatial aliasing Claerbout (1992). In theory, it is convenient to represent a stacking operator in the form of a continuous integral:  

$$S(t,y)= {\bf A}\left[M(z,x)\right]= \int w(x;t,y)\,M(\theta(x;t,y),x)\,dx\;.$$ (1)

Function M(z,x) is the input of the operator, S(t,y) is the output, $\theta$ represents the summation path, and w stands for the weighting function. The range of integration (the operator aperture) may also depend on t and y. Allowing x to be a two-dimensional variable, we can use definition (1) to represent an operator applied to three-dimensional data. Throughout this paper, I assume that t and z belong to a one-dimensional space, and that x and y have the same number of dimensions.

The goal of inversion is to reconstruct some function $\widehat{M}(z,x)$ for a given S(t,y), so that $\widehat{M}$ is in a particular sense close to M in equation (1).