INTERPOLATING MISSING TRACES

We estimate missing data in two steps of linear least squares Claerbout (1992). The first step is estimation of PEFs. After the PEFs have been estimated they are used to fill in the empty trace bins. This is the second step of least squares. We want the recorded and estimated data to have the same dips. Since the dip information is now carried in the PEFs, this is once again specifying that the convolution of the filter and data should give the minimum output, except that now the filters are known and the data is unknown. We constrain the data by specifying that the originally recorded data cannot change. To separate the known and unknown data we have a known data selector $\bold K$ and an unknown data selector $\bold U$, with $\bold U + \bold K = \bold I$. These multiply by 1 or depending on whether the data were originally recorded or not. With $\bold A$ signaling convolution with the PEF and $\bold y$ the vector of data, the regression is $0 \approx \bold A(\bold U+\bold K)\bold y$, or $\bold A\bold U\bold y \approx -\bold A\bold K\bold y$.

While a PEF at every sample works well for destroying the data, it is not the best choice for reconstructing it; interpolation with PEFs estimated at every data point gives poor results and requires extravagant memory allocation. One answer is just that zero is not the correct value of $\epsilon$ in (5); but we can greatly improve the results and decrease the memory usage without adding equations, by using very small patches, such as $2\times2\times2$; small enough that the assumption of stationarity within a patch is reasonable. This is similar to putting an extra roughener in the damping equation, in that it is essentially an infinite penalty on variations of $\bold p$ between small groups of samples, and it has the important economizing effect of reducing the memory allocation. In the method where the patches are independent Crawley (1998), the number of filter coefficients puts a lower bound on patch size; the problem has to stay well overdetermined to produce a useful PEF. Using smoothly varying filters effectively reduces the minimum patch size, so that the filter estimation problem can be underdetermined, and still produce useful PEFs.