DEFINITIONS

Ultimately I hope to do this problem exactly the way theory says to do inverse problems. Meanwhile, I need short cuts to convince me that the iterations I hope to do will head in the right direction.

The traditional formulation of geophysical inverse problems requires a noise covariance matrix and a model covariance matrix. In my experience these are hard to define. I have recently made considerable progress defining the model covariance matrix from a model or a model estimate. (See my companion paper: Information from smiles, monoplane-annihilator weighted regression.)

For starters, I reorganize inverse theory to define the image as the migration operator applied to the complete data set and I take the unknowns to be the missing data. This formulation enables me to work with data and model covariances instead of model and noise covariances. If it shows promise we can always come back and look for the noise covariance.

Start with the definitions:

k known data

u unknown values in data space

d=k+u the complete data space.

d₀=k+g raw data with gaps filled by gapfill program

K Kirchhoff migration operator (for example)

M matrix diagonal with 1 where data missing, 0 otherwise

m=Kd model or image, migration of the complete data space.

A_d array of 2-D prediction-error filters in data space

A_m array of 2-D prediction-error filters in model space

$$\Lambda$$ gain control function in model space

r_d=A_d d data residual

$$r_m=\Lambda A_m m$$ model residual

$$\Delta d = MK'{A'}_m A_mK d_0$$ perturbation of the unknown data

A regression I might try for the unknown u is

$$\begin{eqnarray} 0 &\approx \quad r_d \ =& A_d ( k+u) \\ 0 &\approx \quad r_m \ =& \Lambda A_m m \quad = \quad \Lambda A_m K (k+u)\end{eqnarray}$$ (1) (2)

Of these two criteria, I will need to choose a balance. I can solve for u with either alone, or with some weighted combination. Naturally, I'll try all combinations. If I get it to work, I'll reorder it to make m the unknowns.