Blocky velocity inversion by hybrid norm

Regularization by the First Derivative Operator in Four Directions

To reduce the bias in blockiness directions, we increased the directions of the first derivative to four: the previous two directions, plus two directions at 45 degrees to the vertical and horizontal axes. Figure 9 shows the results of using the $L2$ norm, and Figure 10 shows the results of using the hybrid norm. By comparing Figure 10 to Figure 8, we can clearly see a large improvement in the model. The details and blockiness are still preserved. However, the inversion now has more ``freedom'' to pick the direction of blockiness out of four directions instead of two directions.

l2-lab46
Figure 9. The WG dataset. (a) The interval velocity resulted by using the first derivative operator in four directions as a regularization in the $L2$ norm. (b) The reconstructed RMS velocity.

hbe-lab45
Figure 10. The WG dataset. (a) The interval velocity resulted by using the first derivative operator in four directions as a regularization in the hybrid norm. (b) The reconstructed RMS velocity.

Blocky velocity inversion by hybrid norm