Methodology

The simplest inverse interpolation approach outlined in Claerbout (1999) can be written in least squares fitting goals as follows.

$$\begin{eqnarray} \bf Bm - d &\approx& \bf 0 \nonumber \\ \bf \epsilon Am &\approx& \bf 0\end{eqnarray}$$ (1)

boldB is nearest neighbor interpolation and maps a gridded model (m) to the irregular data space (d). The model grid is 860x500 points, while the data space consists of over 132,000 (x,y,z) triples. A is a model regularization operator, which penalizes model roughness. For all examples contained herein, $\bf A = \nabla$. $\epsilon$ balances the tradeoff between data fitting and spatial model smoothness.

To handle non-gaussian noise, Fomel and Claerbout (1995) implement an Iteratively Reweighted Least Squares (IRLS) scheme to nonlinearly estimate a residual weight which automatically reduces the importance of ``bad'' data in least squares estimation. Adding a diagonal residual weight to equation (1) gives

$$\begin{eqnarray} \bf W(Bm - d) &\approx& \bf 0 \nonumber \\ \bf \epsilon Am &\approx& \bf 0.\end{eqnarray}$$ (2)

To handle systematic errors between data tracks, Fomel and Claerbout (1995) supplement the residual weight in system (1) with a first derivative filter to decorrelate the residual. Lomask (1998) used a single prediction error filter (PEF). Karpushin and Brown (2001) use a bank of PEF's, one for each acquisition track. Whatever the case, we can refer to the differential operator as D and modify equation (2) to obtain a new system of equations:

$$\begin{eqnarray} \bf WD(Bm - d) &\approx& \bf 0 \nonumber \\ \bf \epsilon Am &\approx& \bf 0.\end{eqnarray}$$ (3)

W is the same as in equation (2), except for the addition of zero weights at track boundaries.