A missing data example

Jest is designed for complex, large-scale problems. To hint at the problems we had in mind when designing Jest, Figure 2 shows the result of a medium-sized missing data estimation Claerbout (1994). The estimation almost exclusively involves tools used in the deconvolution example above.

 

seabeam
Figure 2 Estimation of missing bathymetry data. The image in the left panel is the bathymetry data collected by a side-scan sonar along a vessel's path across the Pacific mid-ocean ridge. The image in the center panel shows an estimation of the missing data based on the convolution with a Laplacian. The image in the right panel shows an estimation by a two-stage linear prediction-error filter technique.

The panel on the left shows bathymetry (water depth) measurements ${\bf d}_k$ above the mid-ocean ridge in the Pacific. The measurements were taken by a side-scan sonar hanging off the bottom of a ship. The vessel's path did not cover the entire image area and consequently we attempt to estimate the missing data. The masking operator ${\bf M}_k$ extracts the known data, ${\bf M}_k {\bf d} = {\bf d}_k$ from the total data set ${\bf d}$.

The solution in the center panel minimizes the output of the Laplace operator applied to the data ${\bf d}$, $\min_{\bf d}(\nabla^2 {\bf d})$, under the constraint that the known data does not change, ${\bf M}_k {\bf d} = {\bf d}_k$.

The solution in the right panel uses a two stage prediction-error technique. First, we minimize the convolution $\min_{\bf p}({\bf p} \ast {\bf d}_k)$ to estimate the prediction-error filter ${\bf p}$. Next, we replace the Laplacian in the earlier formulation by the prediction-error filter and solve again for the missing data. The missing data estimation is easily built from a combination of the objects shown in the deconvolution example, a specialized prediction-error filter class, and the mask operator ${\bf M}_k$.