Blocky velocity inversion by hybrid norm

Dix inversion as an $L1$ -optimization problem

The Dix equation can be made linear by relating the square of interval velocity $v$ to the square of RMS velocity $V$ ,

$$v^2_\tau =  \tau V_\tau^2 - (\tau-1)V_{\tau-1}^2,$$ (1)

where $\tau$ is the two-way traveltime. By defining $u_\tau=v^2_\tau$ and $d_\tau=\tau V_\tau^2$ , we can set up the Dix inversion problem in an $L1$ sense as follows:

$$\vert\vert\mathbf W_d (\mathbf C \mathbf u-\mathbf d)\vert\vert _{\mathrm{hybrid}} \approx 0,$$ (2)

where $\mathbf W_d$ is a weight function proportional to the pick strength in the velocity scan divided by $\tau$ , $\mathbf C$ is the causal integration operator, and $\mathbf u$ and $\mathbf d$ are vectors containing all the values of $u_\tau$ and $d_\tau$ , respectively. The division by $\tau$ reduces the strength of the later events to balance the data fitting strength along the time axis.

The hybrid norm above defines the cost function as follows:

$$\mathbf C(\mathbf r) = \sqrt{\mathbf r^2+\mathbf R^2}-\mathbf R,$$ (3)

where $\mathbf r$ is the residual and $\mathbf R$ is a threshold which defines a smooth transition between the $L1$ and $L2$ norms (Claerbout, 2009).

Fitting goal (2) is not enough to fully constrain the inversion, because it has a large null space (Li and Maysami, 2009). Moreover, picking errors can lead to incorrect RMS velocities and unreasonable interval velocities. Therefore, a second fitting goal (i.e. a regularization term) is required to constrain this inversion. The regularization term can be written as follows:

$$\vert\vert\epsilon \mathbf A \mathbf u \vert\vert _{\mathrm{hybrid}} \approx 0,$$ (4)

where $\mathbf A$ is typically a roughening operator, and $\epsilon$ is a scalar to balance the two fitting goals.

Notice that the norm in fitting goal (2) has a different effect than the norm in fitting goal (4). Using the hybrid norm in data fitting makes the inversion less sensitive to outliers. On the other hand, using the hybrid norm in model styling affects the general shape of the estimated model, which is the goal of this paper.

Li and Maysami (2009) successfully produced blockiness in 1D when using the first derivative as a regularization operator. In the following sections, we will try different regularization operators to achieve the same goals in 2D.

Blocky velocity inversion by hybrid norm