1-D Model Regularization: Discontinuity Penalty

In many applications, the interval velocity must be smooth across layer boundaries. To accomplish this, we incorporate a penalty on the change in velocity across the layer boundary, and effectively exchange quality in data fit for a continuous result. However, as mentioned by Lizarralde and Swift (1999), an accumulation of large residual errors would result if we forced continuity in the velocity function across layer boundaries with large velocity contrast. Therefore we ``turn off'' the discontinuity penalty at certain layers via a user-defined ``hard rock'' weight.

Let us write the weighted discontinuity penalty at the boundary between layers l and l+1:  

$$w^h_l \left[ v_{0,l} + k_l t_l - \left(v_{0,l+1} + k_{l+1}t_l \right) \right]$$ (6)

w^h_l is the hard rock weight. We suggest that the w^h_l be treated as a binary quantity: either 1 for soft rock boundaries or 0 for hard rock boundaries. As before, we write the misfits of equation (6) in fitting goal notation and combine with equation (5):  

$$\left[\begin{array} {c} \bold A \\ \epsilon \bold C \end{... ...n{array} {c} \boldsymbol \zeta \\ \bold 0 \end{array}\right]$$ (7)

$\bf C$ is simply the linear operator suggested by equation (6): a matrix with coefficients of $\pm 1$ and $\pm t_l$, with rows weighted by the w^h_l. Application of $\bf C$ is tantamount to applying a scaled, discrete first derivative operator to the model parameters in time. The scalar $\epsilon$ controls the trade off between model continuity and data fitting. Lizarralde and Swift (1999) give a detailed strategy for choosing $\epsilon$.