Next: Suppressing wraparound artifacts of Up: Fourier approach Previous: Fourier approach

## Improving the accuracy of the t2 grid transform

The first problem is the loss of information in the transform to the t2 grid. As illustrated in Figure 2, the shallow part of the data gets severely compressed in the t2 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 t2 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 t2 grid to the original t grid. The right plots show the difference with the original model. Top: using the original grid size (Nt = 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 denote the data on the new grid and 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
 (8)
where LT denotes the adjoint interpolation operator. The operator (LT L)-1 provides a proper scaling of the result. If we use simple linear interpolation for the L operator, then LT L is a tridiagonal matrix, which can be easily inverted (in 8 N operations). If some parts in 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:
 (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 while preserving the efficient tridiagonal structure of . In this problem, the parameter can be chosen as small as possible, as long as it prevents the inversion from getting unstable.

Next: Suppressing wraparound artifacts of Up: Fourier approach Previous: Fourier approach
Stanford Exploration Project
7/5/1998