Covariance

With no other information, the Laplacian might be the best regularization operator that we could use. But if we know something else about out model, can we do better? According to Tarantola (1987), we should be using the inverse model covariance for our regularization operator. The statistics of the model vary spatially, but by breaking up into four patches, one above the anticline, one below the anticline, and two within the anticline we can at least approximate stationary statistics. If we calculate the covariance within patches where the statistics are relatively stable, we get Figure 3.

 

covar-change
Figure 3 The covariance at four different regions of our model (left panel of Figure 1.) The top left is above the upper unconformity; top right, the upper portion of the anticline; bottom left, lower portion of the anticline; and bottom right, bellow the lower anticline.