Example

One example of vector quantities with spatial correlation is in migrating from topography using Riemannian coordinate systems Shragge and Sava (2004); Shragge (2005). The approach attempts to migrate from a coordinate system that is consistent with the recording surface. The resulting coordinate system has therefore variable spatial density. Figure  shows both the scaling of the orthogonal wave number (squeezing and stretching of the coordinate system) and a velocity component for the 2-D problem.

 

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Figure 4 The left panel shows velocity field, the right panel shows the stretching of the coordinate system from Cartesian.

At first glance these two fields do not show an obvious correlation. If we form a 2-D histogram (shown in Figure ) another story emerges. The parameters are correlated, and concentrated in top portion of the histogram. Figure  shows the histogram from several different depths steps. The correlation and clustering of the parameters becomes even more obvious. Figure  shows the same depths overlaid by the N-D Lloyd codebook. Note how the histograms features are easily identified. Figure  shows the error percentage in the parameters by using the N-D Lloyd approach and by evenly partitioning the parameter range at each depth step. Not only is the evenly spaced partitioning scheme eight times more expensive, the difference between the true parameters at each model location and the selected parameters are significantly higher. Tang and Clapp (2006) use a 3-D version of Lloyd's algorithm for selected $v,\delta$ and $\epsilon$ parameters for anisotropic migration.

 

jeff.histo Figure 5 A 2-D histogram of the parameters shown in Figure . The vertical axis is velocity (A), the horizontal axis is the stretching parameter (B). The amplitude shows the number of times a given (A,B) combination exists in the data shown in Figure . Note the correlation in the parameters.

 

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Figure 6 2-D histograms (black and grey indicates the number of parameters with a given value) at three different depths. Note the correlation of the parameters.

 

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Figure 7 2-D histograms (black and grey indicates the number of parameters with a given value) at three different depths overlaid by the Lloyd codebook (*). Note how well the features are detected.

 

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Figure 8 The percent error between the selected reference parameter and the true parameter. The top two panels show the result of using evenly spaced A,B parameters. The bottom two panels show the result of using the Lloyd approach to select the parameters. Note the significantly smaller error with the Lloyd's approach with 1/8 as many parameters.