Singular values

To compare the various ways of sampling in velocity domain, we should choose some criterion. A condition number of a matrix appears to be useless here, as we can see from Figure . All the matrices are singular. Figure c gives us a clue to the means of comparison. Two curves are plotted in this figure. One represents singular values of the matrix ${\bf H^TH}$ for ten velocities (the smoother one), and the other, for two velocities. The matrix for two velocities is evidently singular and the data cannot be restored when this matrix is used. The curve of singular values of this matrix drops faster to the zero level. The distribution of singular values for the matrix of ten velocities is more uniform.

Let us analyze the curves in Figure a from this point of view. There are four curves in the figure, which from the left to the right (upper parts of the curves) represent singular values of matrices computed for even sampling in sloth, slowness, velocity and alacrity domains. This is the succession according to the above criterion. The velocity and alacrity curves have evidently worse behavior. The difference between the sloth and slowness curves is much less. The difference between the sloth and slowness curves in Figure d is almost nil. In this figure high velocities were not included. In the region of high velocities slowness has worse behavior than sloth.

Singular values for matrices $\bf H$ are in Figure b. They should be the square roots of those in Figure a. This relationship holds for about the first 150 singular values. It does not hold for the rest of the values because of the limited computational precision (double precision used).