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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 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 *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 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.

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Stanford Exploration Project

7/5/1998