Blocky velocity inversion by hybrid norm

Regularization by the Laplacian Operator

First, we will use the Gulf of Mexico data provided by WesternGeco. This is a four-second dataset, at a sampling interval of 4 ms. The offset axis has 24 traces starting at 264 m with an increment of 134 m. There are 125 CMP gathers with a spacing of 67 m. Figure 1(b) shows the results of autopicking the dataset, and Figure 1(a) shows the results of direct Dix conversion. The direct conversion is done trace by trace after vertical smoothing which prevents the minimum interval velocity to go below water velocity. Figure 2 shows the strength of the picks in the velocity scans, which will be used as a weight as described above.

lab0
Figure 1. The WG dataset. (a) The input RMS velocity, which is automatically picked from the CMP gathers. (b) The interval velocity by direct dix conversion.

lab-weight2d Figure 2. The strength of the picks in the velocity scan of the WG dataset, which is used as the weight before dividing by time.

We start by choosing the symmetric Laplacian operator. Figure 3 shows the results of using the $L2$ norm and Figure 4 shows the results of using the hybrid norm. Although the two Figures look similar, the hybrid norm shows less smoothing and more detail than the $L2$ norm. The hybrid norm results are still not blocky, because a linear trend in velocity will also result in a zero second derivative. In the next section we attempt to more closely approach the first derivative by using the helix derivative.

l2-lab18
Figure 3. The WG dataset. (a) The interval velocity estimated by using the Laplacian operator as a regularization in the $L2$ norm. (b) The reconstructed RMS velocity.

hbe-lab19
Figure 4. The WG dataset. (a) The interval velocity estimated by using the Laplacian operator as a regularization in the hybrid norm. (b) The reconstructed RMS velocity.

Blocky velocity inversion by hybrid norm