Real data Results

We used 125 CMP's from a 2-D prestack dataset acquired in the Gulf of Mexico. This data is suitable for using Dix equation, since the main reflectors are flat. The area is heavily faulted which may imply strong lateral velocity variations with sharp edges to preserve.

First, we performed velocity analysis on each CMP, and then use an auto-picker to pick the maximum stacking power that corresponds to the best RMS velocity at each CMP location. The value of the stacking power at the auto-picked RMS velocity was used as a quality measure of the data, and used as the data residual weight (W) in equations (2), (5), and (8). Clapp (2003) shows an alternative way to calculate the data residual weights based on a multiple realization of the RMS velocity. His approach looks promising since the the RMS velocity average is used as the data and the RMS velocity variance is used as the data residual weight. Figure shows the auto-picked RMS velocity and a stack of the CMP's.

 

vrms2d
Figure 1 A) Auto-picked RMS velocity of one CMP from a 2-D prestack dataset, B) raw RMS velocity, C) the CMP in question, D) and the stacked data using the raw RMS velocity.

Figures , , and show the interval velocities resulting from solving the inverse problems stated in equations (2), (5), and (8) respectively. We also show in figure  a graph comparing the three methods and the RMS velocity used as input data at two CMP locations.

 

vint2d_2fit
Figure 2 Interval velocity computed by 2-D inversion of the RMS velocity (equation (2)).

 

vint2d_2fit_L1
Figure 3 Interval velocity computed by 2-D inversion of the RMS velocity using Cauchy norm (equation (5)).

 

vint2d_GradMag
Figure 4 Interval velocity computed by 2-D inversion of the RMS velocity using gradient magnitude in the weighting function (equation (8)) .

 

comp1
Figure 5 Comparison of the results of solving the inverse problems stated in equations (2) Smooth, (5) Cauchy norm, (8) Gradient Magnitude, and the RMS velocity at the midpoint positions 8.04 and 12.194 km.

The resulting interval velocity models show what the regularization was designed to do. In figure  the resulting interval velocity is smooth in time and space. Figure  shows sharp edged rectangular shapes all over the image, looking reasonable in the faults but in general geological unappealing. Figure  shows sharp objects with more geological appeal.

The preferential shapes can also be seen in the diagonal weight operator. Figures  and show ${\bf Q_{\tau}}^{N}$ and ${\bf Q_{x}}^{N}$, the last nonlinear iteration diagonal weight operator in equation (7). Notice the two preferential directions in what the edges are preserved. Figure  shows ${\bf Q_{\vert\vert\nabla\vert\vert}}^{N}$, the last nonlinear iteration diagonal weight operator in equation (9). Notice the isotropic behavior of the diagonal weight calculated using the gradient magnitude operator.

 

qx_2fit
Figure 6 ${\bf Q_{\tau}}^{N}$ is the last nonlinear iteration diagonal weight operator in equation (6).

 

qz_2fit
Figure 7 ${\bf Q_{x}}^{N}$ is the last nonlinear iteration diagonal weight operator in equation (7).

 

q_GradMag
Figure 8 ${\bf Q_{\vert\vert\nabla\vert\vert}}^{N}$ is the last nonlinear iteration diagonal weight operator in equation (9).