Background

Prediction-error filter (PEF) based interpolation can be cast as a two stage linear least-squares process Claerbout (1999), where a PEF is first estimated on the known data. Then, the output of convolution of the newly-found PEF with the desired model is minimized while fixing the known data. The first stage of the process can be described mathematically by  

$$\bold{W(DKf + d)} \approx \bold 0 ,$$ (1)

where the known data (d) is convolved (D) with a PEF with unknown coefficients (f), except for the first, which is constrained by K to be 1. If there are areas where the filter is being convolved with unknown data, those areas are weighted to by a diagonal weight W. The second stage can be described by

$$\begin{eqnarray} \bold{L_{data}m} &=& \bold{d} \nonumber \\ \bold{Fm} &\approx& \bold 0 .\end{eqnarray}$$ (2)

In the second fitting goal, L is a selector matrix that is 1 where there is a data point and where there isn't, m is the interpolated output, d is once again the known data, and F represents convolution with the newly-found PEF.

In the case of a non-stationary PEF, where the filter varies with position, a second fitting goal has to be added to the first stage of the interpolation process, so that the now much greater number of filter coefficients becomes adequately constrained. This fitting goal can be expressed as  

$$\bold{Af} \approx \bold 0 ,$$ (3)

where A is a regularization operator (typically a Laplacian) that operates spatially over each filter coefficient separately, and f is the non-stationary PEF. Fitting goal (1) is written identically for the non-stationary case, but each of the operators present (as well as the filter) are now non-stationary. A full description of what the matrices for non-stationary PEFs look like is given in SEP-113 Guitton (2003).

Typically, when interpolating data that are regularly-sampled, the filter is interlaced so that the filter skips over the missing traces, which allows a filter to be estimated Crawley (2000). Once the filter has been estimated, the interlacing of the filter is undone for the second stage of the interpolation process.

When the data are not regularly-sampled, the interlacing approach usually fails. In this case, a multi-scale approach can be used where a non-stationary PEF can be estimated on multiple regridded copies of the original data Curry and Brown (2001); Curry (2002, 2003). This can be expressed as  

$$\bf W \left( \left[ \begin{array} {c} \bf D_0 \\ \bf D_... ...... \\ \bf d_n \\ \end{array} \right] \right) \approx 0 .$$ (4)

Here, the different scales of data ${\bf D_{i}}$ are generated by the normalized adjoint of linear interpolation, which takes points from a fine grid and sprays them into the coarser grid, then normalizes by the fold. The weight W is now a diagonal weight for all scales of data, while the introduction of a sub-sampling operator P subsamples the non-stationary filter so that the spatial size of the filter will match the size of the rescaled data.

Another possible approach is to use a pair of non-stationary 2D PEFs which are estimated independently from the original unscaled data using fitting goals (1) and (3) in two different directions. Once these two PEFs have been estimated, they could be used in tandem to interpolate missing data by Claerbout (1999); Curry (2004):

$$\begin{eqnarray} \bold{L_{data}m} &\approx& \bold{d} \nonumber \\ \epsilon_{x} ... ...old 0 \nonumber \\ \epsilon_{y} \bold{F_{y}m} &\approx& \bold 0, \end{eqnarray}$$ (5)

where ${\bf F_{x}}$ and ${\bf F_{y}}$ are 2D non-stationary PEFs (compared to the typically-3D PEF shown before in fitting goal 2), ${\bf L_{data}}$ selects known data points, d is still the known data and m the unknown model. Unlike the multi-scale approach, this method requires that the data are evenly-sampled along tracks oriented in two different directions.