Velocity continuation by spectral methods |
The first problem is the loss of information in the transform to the grid. As illustrated in Figure 2, the shallow part of the data gets severely compressed in the 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 domain. The top plots in Figure 4 show the result of back transformation to the 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 grid to the original grid. The right plots show the difference with the original model. Top: using the original grid size (). Bottom: increasing the grid size by a factor of three. |
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To perform an accurate transform of the grid, I adopted the following
method, inspired by (Claerbout, 1986a). Let
denote the data on the new grid and
be the data on the old grid. If is the interpolation operator,
defined on the new grid, then the optimal least-square transformation
is
Velocity continuation by spectral methods |