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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:

| |
(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):
| |
(7) |

is simply the linear operator suggested by equation (6):
a matrix with coefficients of and , with rows weighted by
the *w*^{h}_{l}. Application of is tantamount to applying a scaled,
discrete first derivative operator to the model parameters in time.
The scalar controls the trade off between model continuity and
data fitting. Lizarralde and Swift (1999) give a detailed strategy for
choosing .

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** Up:** Least Squares Formulation
** Previous:** Least Squares Formulation
Stanford Exploration Project

4/29/2001