Test Case

A non-stationary, two-dimensional test case has been created as a proof of concept example. This test is based upon a simple plane wave model (, ).

 

testcase
Figure 3 The fully sampled version of the non-stationary test data.

In this case, there is one plane wave on each half of the example, and the plane wave on the right varies in amplitude from left to right. There is also Gaussian noise present in the data, which largely obscures the low amplitude portion of the plane wave on the right side of the example.

The data was randomly sub-sampled, keeping only 10-20 percent of the data. This sparse data, along with a mask describing the position of the known data, was used in the non-stationary interpolation scheme. The results of the interpolation are shown in Figure .

 

testfill
Figure 4 Results for three different sampling levels, from left to right: 10, 15 and 20 percent. From top to bottom: sparse data, sparse data filled with PEF trained on fully sampled original data, sparse data filled with PEF from sparse data with 64, 16 and 8 micro-patches, respectively.

The results are on the whole quite encouraging. The data was very heavily sub-sampled, and the interpolation scheme was able to identify the dips in the data and interpolate them properly. The dip from the left side of the example was smoothed over the area on the right side of the figure with the low signal-to-noise ratio, which was expected. As the number of micro-patches drops, the results deteriorate, as the regularization loses its effect and one patch becomes unstable. A higher number of micro-patches preserves the dips as well as the low amplitude area.

A more relevant case to interpolating seismic data is the qdome model (), which has been highly sub-sampled along two of three dimensions, with the vertical axis still fully sampled. The qdome model is a collection of folding layers, flat layers and a fault, which acts as an excellent overall test for this interpolation method. I have randomly removed 88 percent of the traces from the data set, and use the non-stationary multi-scale PEF-based interpolation to attempt to recreate the original model.

 

qdometest
Figure 5 Fully sampled and sub-sampled versions of the qdome model.

 

qdomefill
Figure 6 Above: Sparse data filled with a non-stationary PEF trained on all data. Below: Sparse data filled with a non-stationary PEF trained on sparse data.

The results for the qdome model are very promising. The smoothly varying dips were correctly estimated and interpolated almost everywhere, excluding very steep dips. This is due to two things, the size of the PEF might not have been large enough to capture the spatially aliased dips, and that the dips were changing rapidly within a small area, which was only covered by a small number of micro-patches.

The results for this 3D case are much more impressive than in the 2D case, even though more of the data was removed. This is due to several things: The extra dimension of data allows for more constraints to be applied by the regularization, the greater size of all of the dimensions allows for more fitting equations to be found, and most importantly that the well-sampled z-axis gives better coverage of the data space.