Discrete Kirchhoff implementations

Kirchhoff operators represent a class of linear transformations based on integral solutions to the wave equation. The implementation of integrals as discrete summation reduces to a matrix-vector multiplication where we hardly ever write down the 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 given by the linear system of equations:

$$\bold d = \bold L \bold m. \EQNLABEL{equ-forward}$$ (39)

Equation equ-forward represents a 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 ${\bf L}$.