Blocky velocity inversion by hybrid norm

Regularization by the First Derivative Operator in Two Directions

The previous regularizations show that only a first derivative can create blockiness. However, using the first derivative means that we must pick a direction each time we apply the derivative. As a first test, we pick two directions: the vertical and horizontal as follows:

$$\vert\vert\epsilon \mathbf D_z \mathbf u \vert\vert _{\mathrm{hybrid}} \approx 0,$$ (5)

$$\vert\vert\epsilon \mathbf D_x \mathbf u \vert\vert _{\mathrm{hybrid}} \approx 0,$$ (6)

where $\mathbf D_z$ and $\mathbf D_x$ are the first derivative operators along the z- and x-axis, respectively . The derivative of each direction is applied in a separate regularization equation (i.e. we have two regularization equations in this case) in order to maintain symmetry. Combining these two filters in one regulariztion will cause an asymmetry in blockiness, similar to the previous result from the helix derivative regularization.

Figure 7 shows the results of using the $L2$ norm with two first derivative applications, and Figure 8 shows the results of using the hybrid norm. Blockiness is clearly present in the hybrid norm results. However, there seems to be a preference for the sharp boundaries to be either horizontal or vertical, which is due to the directions of the derivatives we chose.

l2-lab39
Figure 7. The WG dataset. (a) The interval velocity estimated by using the first derivative operator in two directions as a regularization in the $L2$ norm. (b) The reconstructed RMS velocity.

hbe-lab37
Figure 8. The WG dataset. (a) The interval velocity estimated by using the first derivative operator in two directions as a regularization in the hybrid norm. (b) The reconstructed RMS velocity.

Blocky velocity inversion by hybrid norm