2-D velocity estimation

An attractive feature of the velocity-estimation procedure defined in this paper is that it is linear. This means there should be no unpleasant surprises associated with multiple minima or with convergence speed. If bad results are found, they should arise directly from bad RMS velocity picks or from the possibly inappropriate association of stack power with the statistical weighing function.

Another attractive feature of the velocity-estimation procedure defined here is that it leads directly to a rational approach to finding interval velocity as a function of midpoint. Traditionally, velocity at each midpoint is estimated separately. Thus one midpoint cannot help neighboring estimates, nor are unrealistically rapid variations with midpoint suppressed in a natural way. For example, suppose we redefine our fitting goals as:

$$\begin{eqnarray} 0 &\approx& \bold W' \left[ \bold C^2 \bold L^{-1} \bold p - \bold d \right] \ \bold 0 &\approx& \epsilon \bold p .\end{eqnarray}$$ (10) (11)

where $\bold L$ is a roughening operator such as the Laplacian (in the space of midpoint and traveltime depth) Starting with picked RMS velocities (Figure data2d) from a North Sea salt structure, attempted to invert for interval velocity. As Figure vint2d shows, for small, long wavelength variations in lateral velocity (Dix being approximately correct) we obtain a smooth, reasonable interval velocity model with minimal effort.

 

data2d
Figure 10 2-D RMS velocity function picked by finding the maximum semblance within a fairway. Note that for some CMP locations, the picked velocity is completely inaccurate. The inversion algorithm takes these inaccuracies into account and leads to a smooth model (vint2d).

 

vint2d
Figure 11 2-D interval velocity function obtained from applying fitting goals (10) and (11) after 30 iterations.

For a further test we replaced the Laplacian of equation (11) with steering filters Clapp et al. (1997) built from dips off an interval interval velocity model (Figure ) built with a more sophisticated scheme (nearest offset section could just have easily been used). We then applied the 2-D inversion and obtained Figure . Generally, we did a good job recovering the salt structure, but are still susceptible to multiple infected velocity picks.

 

reflectors-cor
Figure 12 Supplied interval velocity model with picked reflectors superimposed.

 

reflectors-est
Figure 13 Estimated interval velocity model with picked reflectors superimposed.