Multi-Scale Filters

Missing data is estimated in two steps of linear least squares (). The first step is estimation of PEFs. Prediction-error filters are good at estimating lines or planes of constant dip, but not curves. Individual filters are estimated on small patches so events appear as dipping planes of approximately constant dip.

The shape of a 3-D PEF with five points on the time axis, and three and two points respectively on two space axes is shown in Figure notExpandedAltBW. The dark shaded box is constrained to hold the value 1, the light boxes are adjustable coefficients. The filter forms a half volume, which can be oriented in different ways. To find the values of the adjustable coefficients, specify that the convolution of the filter with the known data gives the minimum power. This means solving the regression $0 \approx \bold Y\bold a$, where $\bold a$ is a vector containing the filter and $\bold Y$ is convolution with the recorded data. Constraining one filter coefficient to 1 (the shaded box in Figure notExpandedAltBW) prevents the trivial answer $\bold a = 0$.

 

dataPic Figure 1 Representation of input data cube. Recorded traces are dark, traces to be interpolated are light.

The data are sorted so the recorded traces and the traces to be interpolated are arranged in the input cube in a checkerboard pattern. In the case of missing shots (or dual-source marine geometry), this corresponds to a cube with time, receiver location, and offset coordinates. For comparison with the filter shapes, the input cube is represented by Figure dataPic, with the shaded squares corresponding to empty trace bins. The checkerboard arrangement has the intuitively pleasing quality that every missing trace is surrounded on four sides by a recorded trace. It also poses the problem that there is no densely sampled patch for estimating a filter such as the one in Figure notExpandedAltBW. Any regression equations that have zeroes in them should be dropped. For the filter shown in Figure notExpandedAltBW, this eliminates all the regression equations. To make the filter estimation work, we scale all the axes of the filter by two. This gives the expanded filter shown in Figure expandedAltBW. Convolved across a checkerboard of present and missing data, this filter alternately hits exclusively filled and exclusively empty bins. Because all the axes are scaled equally, the expanded and compressed filters are self-similar and have the same dip information ().

 

expandedAltBW Figure 2 Representation of expanded PEF. Dark square is constrained to be a 1. Light squares are coefficients adjusted to minimize output power when convolved with grid of Figure dataPic.

 

notExpandedAltBW Figure 3 Representation of PEF. Coefficients are same as Figure expandedAltBW. This filter is applied to grid of Figure dataPic to find missing traces.

After the filter has been estimated it is 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 PEF, this is once again specifying that the convolution of the filter and data should give the minimum output, except that now the filter is 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 was 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 $\bold A\bold U\bold y \approx -\bold A\bold K\bold y$.To fill the missing bins, the filter must touch both filled and empty bins together, so it gets compressed to look like Figure notExpandedAltBW.