The first test dataset contains rainfall measurements from Switzerland on the 8th of May 1986. The dataset was used in the Spatial Interpolation Comparison Dubois (1999) for comparing different spatial interpolation methods. Figure shows the data area: the Digital Elevation Model of Switzerland and the country's borders. A total of 467 rainfall measurements were taken. A subset of randomly selected 100 measurements was used in the 1997 Spatial Interpolation Comparison in order to compare the results with the known data. Figure shows the spatial location of the selected data samples.
Figure 15 Digital Elevation Model of Switzerland and the country's borders. The country borders are extracted from the Digital Chart of the World (DCW) provided by ESRI.
Rainfall level is generally a smoothly varying quantity. We cannot expect it to be represented a priori by a simple function. Therefore, it is reasonable to take the regularization operator to be a convolution with the Laplacian filter. The corresponding preconditioning operator is then a deconvolution with the minimum-phase Laplacian constructed in the previous section. The interpolation result using the model-space regularization scheme (-) is shown in Figure . The input irregular data were regularized on a 376 by 253 grid, which corresponds to the digital elevation model in Figure . Similarly to what happens in the one-dimensional synthetic examples, the solution converges steadily but with a slow spread of information away from the known data points. It takes about 10,000 iterations to achieve full convergence. Figure is a correlation plot of the observed and interpolated data points for the 367 points that were not used in the interpolation experiment. If we take into account the fairly unpredictable distribution of rainfall, the correlation is relatively good in comparison with analogous results of the Spatial Interpolation Contest Dubois (1999).
Figure 18 Correlation between observed and predicted rainfall data values.
The result of applying recursive filter preconditioning with the minimum-phase Laplacian operator is shown in Figure . Full convergence is achieved after only 100 iterations. The result after 10 iterations (the left plot in Figure ) is already close to the final solution. Recursive preconditioning speeded up the iteration count by a factor of 1000. The actual gain in execution time is several times smaller because of the correspondingly longer filter, but it is still impressively large.