Block Models

The methodology also nicely lends itself to producing block velocity models. For example we can produce a velocity model with linear velocity functions within the layers by rewriting our fitting goals as:

$$\begin{eqnarray} \bold{0} &\approx& \bold{W'} (\bold C \bold C \bold G \bold p - \bold d) \ \bold{0} &\approx& \epsilon \bold p \nonumber\end{eqnarray}$$ (9)

where $\bold{G}$ is a diagonal matrix of 1s and 0s, 1s at layer boundaries and 0s everywhere else. As Figure  shows, our velocity function is approximately linear within the layers and has a reduced wiggliness compared to the smooth interval velocity curve.

 

bvint.stack Figure 4 The result of the inversion for a blocky velocity model using the fitting goals (9).

 

composite.stack
Figure 5 From left to right: CMP gather, velocity scan with picked RMS velocity within a fairway, smoothed and unsmoothed stacking power, the B function, continuous (dashed) and blocky (solid) interval velocity.