Examples of matrices

Some matrices were generated to compare the properties of inversion dependent on sampling in velocity space. When we want to obtain a velocity panel by solving equation (7), then the matrix ${\bf HH^T}$ should be inverted. When we want to get data back from the velocity stack panel, then the matrix ${\bf H^TH}$should be inverted.

It would be best to take the parameters of some real CMP gather and generate the corresponding matrices. The amount of computer time required for such computations would be, however, too large. With the following parameters I was able to obtain the results in a reasonable time: n1=20, o1=0 sec, d1=0.1 sec, n2=10, o2=0 km, d2=0.2 km. Number of velocities in most computations was chosen to be 10. The value that differs most from the real values is the sampling interval in time. This is probably the reason why so many singular values are practically equal to zero (Figure ).

Let us begin with the operator ${\bf H^TH}$, i.e. the operator needed for inversion of velocity stack. The matrix $\bf H$ is in Figure a. We can see clearly how it is composed of NMO operators. Using only one velocity $v=\infty$ km/sec in the velocity stack leads to the matrix in Figure b. This is a singular matrix, which corresponds to the fact that the original data cannot be recovered from this stack. The velocity stack for ten velocities in the range 1.5-8.5 km/sec with even sampling in velocity and alacrity domains leads to the matrices in Figure c and Figure d. The high limit of velocity v=8.5 km/sec was chosen, because it is the maximum resolvable velocity in the profile from Southern California in 4 msec sampling. Figure e and Figure f represent matrices with even sampling in the slowness and sloth domains. We can see that for sampling in velocity and alacrity spaces, the elements of matrices are pushed closer to the diagonals, which makes them more similar to the singular matrix in Figure b. From this point of view sampling in slowness or sloth domains seems to have better properties.

The same argument holds for matrices ${\bf HH^T}$ (needed if inversion is used to compute the velocity panel) (Figure a,b,c,d). By comparing Figure c and Figure d we see that sloth blurs elements of the matrix around diagonals slightly more than slowness and thus should be preferred.

${\bf HH^T}$ for one velocity $v=\infty$ km/sec is a unit matrix (Figure e) and hence regular, which agrees with the fact that horizontal stack of a gather is uniquely determined. The matrix for two velocities is in Figure f. In this matrix, the number of diagonals linearly increases with the number of velocities, in contrast to the matrix ${\bf H^TH}$, in which the number of diagonals depends on the number of offsets.