Velocity continuation by spectral methods

Improving the accuracy of the $t^2$ grid transform

The first problem is the loss of information in the transform to the $t^2$ grid. As illustrated in Figure 2, the shallow part of the data gets severely compressed in the $t^2$ grid. The amount of compression can lead to inadequate sampling, and as a result, aliasing artifacts in the frequency domain. Moreover, it can be difficult to recover from the loss of information in the transformed domain when transforming back into the original grid. A partial remedy for this problem is to increase the grid size in the $t^2$ domain. The top plots in Figure 4 show the result of back transformation to the $t$ grid and the difference between this result and the original model (plotted on the same scale). We can see a noticeable loss of information in the upper (shallow) part of the data, caused by undersampling. The bottom plots in Figure 4 correspond to increasing the grid size by a factor of three. Some of the artifacts have been suppressed, at the expense of dealing with a larger grid.

fft-inv
Figure 4. The left plots show the reconstruction of the original data after transforming back from the $t^2$ grid to the original $t$ grid. The right plots show the difference with the original model. Top: using the original grid size ($N_t = 200$). Bottom: increasing the grid size by a factor of three.

To perform an accurate transform of the grid, I adopted the following method, inspired by (Claerbout, 1986a). Let $d_{\mbox{\tiny new}}$ denote the data on the new grid and $d_{\mbox{\tiny old}}$ be the data on the old grid. If $L$ is the interpolation operator, defined on the new grid, then the optimal least-square transformation is

$$d_{\mbox{\tiny new}} = (L^T L)^{-1} L d_{\mbox{\tiny old}}\;,$$ (8)

where $L^T$ denotes the adjoint interpolation operator. The operator $(L^T L)^{-1}$ provides a proper scaling of the result. If we use simple linear interpolation for the $L$ operator, then $L^T L$ is a tridiagonal matrix, which can be easily inverted (in $8 N$ operations). If some parts in $d_{\mbox{\tiny new}}$ are not fully constrained, then the tridiagonal matrix is not invertible. To obtain a solution in this case, we can include a regularization operator $D$ in (8), as follows:

$$d_{\mbox{\tiny new}} = (L^T L + \epsilon^2 D)^{-1} L d_{\mbox{\tiny old}}\;,$$ (9)

A convenient choice for $D$ is a second derivative operator, represented with the second-order finite-difference approximation. This operator allows the selection of the smoothest possible function $d_{\mbox{new}}$ while preserving the efficient tridiagonal structure of $L^T L + \epsilon^2 D$. In this problem, the parameter $\epsilon$ can be chosen as small as possible, as long as it prevents the inversion from getting unstable.

Velocity continuation by spectral methods