(6) |

Figure 6

I applied the inversion to the synthetic data displayed in figure after applying NMO. The NMO-corrected synthetic data is displayed in figure . The application of NMO has the effect of limiting the dip range in the data. This reduces the size of the model space necessary to represent the input data.

The inversion results for the NMO-corrected data appear in figures , + . Each figure displays all the slowness slices from the model space of the beam stack inversion. It is apparent from figures + that the primary energy is confined to the zero slowness slice of the model space, while figure shows that the primary energy resides in zero and negative slowness slices.

Figure 7

Figure 8

Figure 9

I forward modeled each inversion result to produce a reconstruction of the NMO-corrected synthetic data. The difference in total energy between the reconstructed synthetic data and original synthetic data is less than in each case. This difference was achieved in five iterations and required three minutes of computing time on one processor of an SGI power challenge.

An alternative to the iterative least-squares solution is a
direct inversion. To implement the direct inversion, the
data covariance matrix, , must be inverted. This matrix, and
the beam stack operator itself, can be
partitioned into a separate matrix for each frequency of interest, thus
reducing the size of the data covariance matrix significantly.
The partitioned beam stack operator in matrix form is of size
(*N*_{x})(*N*_{x}*N*_{p}) while the partitioned data covariance matrix is of size
(*N*_{x})(*N*_{x}). In order to apply a direct least
squares inversion, the data covariance matrix would have to be
inverted for each frequency of interest and applied as a
pre-conditioner to the application of the adjoint to the data.

Hampson (1986) used this approach to formulate the
direct least squares inverse of the parabolic radon transform. In the
case of the radon transform, the model covariance matrix must be inverted.
The model covariance matrix of the PRT is of size
(*N*_{p})(*N*_{p}) where *p* is the slowness parameter of the PRT. A
typical range of the PRT parameter *p* is 50.
Kostov (1990) has shown that the family of radon transforms
results in a model covariance matrix that is Toeplitz in
structure and as such
can be inverted with the Levinson method (and other methods) at a cost
proportional to *n ^{2}* instead of

11/11/1997