Theory

Similarly to Ronen 1987, we pose processing as the inverse of modeling irregular data from regularly sampled model. In mathematical notation we formulate the problem with the following system of equations:

 

d = Lm (1)

where the vector d represents the irregular input data, L represents the modeling operator, and the vector m is the model.

Given the nature of multi-channel recording, the design of 3D surveys and the acquisition problems mentioned earlier, it must be expected that the number of data traces is different from the number of model traces, most likely the number of observations is larger than the number of 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 $\Vert Lm-d \Vert _2$ is minimized. The choice of m that makes this error a minimum gives the least-squares solution which can be expressed for the overdetermined case as

 

m = (L^TL)^-1L^Td (2)

When solving the underdetermined problem, this solution takes a different expression:

 

m = L^T(LL^T)^-1d (3)

where m is the minimum energy model that satisfies the linear equations.

These solutions define a least square inverse or pseudo-inverse to the operator L. From equation (2), we write this inverse in terms of L and its adjoint L^T as:

 

$$L_m^{\dagger} = (L^TL)^{-1}L^T$$ (4)

whereas in (3) the inverse for the underdetermined problem is:

 

$$L_d^{\dagger} = L^T(LL^T)^{-1}$$ (5)

Applying the pseudo-inverse of (4) is equivalent to applying the adjoint operator L^T followed by a spatial filtering of the model space by the inverse of L^TL. Therefore, we will refer to this inverse as model-space inverse.

In equation (5) the adjoint operator is applied after the data have been filtered with the inverse of LL^T and, consequently, we will refer to this inverse as data-space inverse.

In the next section we discuss the connection between the data-space inverse solution and the inversion problem in seismology using irregularly sampled data.