DISCRETE KIRCHHOFF IMPLEMENTATIONS

Kirchhoff operators represent a class of linear operators based on integral solutions to the wave equation. The linearity of the transformation follows from the linear properties of integrals. The implementation of integrals as discrete summation reduces to matrix-vector multiplication where we hardly ever write down these matrices. The linear operation transforms a space to another space (e.g., a data space ${\bf d}$ to a model space ${\bf m}$). These spaces are simply represented by vectors whose components are packed with numbers. The relation between data and model is then represented by the linear system of equations:

$$\bold d = \bold L \bold m. \EQNLABEL{equ-forward}$$ (48)

This is often referred to as the forward modeling relation, where the goal of imaging is to perform the inverse of these calculations, i.e., to find models from the data. Mathematically, this is equivalent to estimating the inverse of the operator ${\bf L}$.