Theory

Posing processing as the inverse of modeling irregular data from a regularly sampled model Ronen (1987), the relation between data and the model can be formulated with the simple system of linear equations:

$$\bf d = \bf L \bf m \EQNLABEL{equ1}$$ (44)

where the vector ${\bf d}$ represents the irregular input data, ${\bf m}$ represents a regularly sampled model and ${\bf L}$, in general, is any full or partial modeling operator.

Given the nature of multi-channel recording and the design of 3D surveys, it is expected that the number of data traces is different from the number of model traces. Most commonly, the number of observations is larger than the model parameters. One way to solve such a system of inconsistent equations is to look for a solution that minimizes the average error in the set of equations. This minimization can be done in a least-squares sense where the norm $\Vert \bf L \bf m- \bf d \Vert _2$ is minimized. The choice of ${\bf m}$ that makes this error a minimum gives the least-squares solution which can be expressed for the over-determined case as

$$\bf m = (\bf L^T \bf L)^{-1} \bf L^T \bf d \EQNLABEL{equ2}$$ (45)

for which ${\bf m}$ represents a minimum length solution.

When solving the under-determined problem, this solution takes a different expression:

$$\bf m = \bf L^T( \bf L \bf L^T)^{-1} \bf d \EQNLABEL{equ3}$$ (46)

where ${\bf m}$ is the minimum energy model that satisfies the linear equations.

These solutions define a least square inverse or pseudo-inverse for the operator ${\bf L}$. From equation equ2, we write this inverse in terms of ${\bf L}$ and its adjoint $\bf L^T$ as:

$$\bf L_m^{\dagger} = ( \bf L^T \bf L)^{-1} \bf L^T \EQNLABEL{equ4}$$ (47)

whereas in equ3 the inverse for the under-determined problem is:

$$\bf L_d^{\dagger} = \bf L^T( \bf L \bf L^T)^{-1} \EQNLABEL{equ5}$$ (48)

Applying the pseudo-inverse of equ4 is equivalent to applying the adjoint operator $\bf L^T$ followed by a spatial filtering of the model space by the inverse of $\bf L^T \bf L$. Therefore, I refer to this inverse as model-space inverse.

In equation equ5 the adjoint operator is applied after the data have been filtered with the inverse of $\bf L \bf L^T$ and, consequently, I refer to this inverse as data-space inverse.