1-D Super Dix

A relatively simple, but more realistic, example is estimating interval velocities $\bf v_{int}$ from RMS velocities $\bf v_{rms}$.Clapp et al. (1998) did this by taking advantage of the linear relation between $\bf v_{rms}^2$ and $\bf v_{int}^2$.We can keep our interval velocities relatively smooth by adding a roughening operator $\bf D$.The fitting goals then become

$$\begin{eqnarray} \bf 0&\approx&\bf r_{n} = \bf T \bf v_{rms}^2 - \bf C \bf v_{in... ...mber \\ \bf 0&\approx&\bf r_{m} = \epsilon \bf D \bf v_{int}^2 ,\end{eqnarray}$$ (13)

where $\bf C$ is causal integration and $\bf T$ is the result of causal integration with a vector of ones.

Figure shows the result of applying this procedure on a simple CMP gather. The left panel shows the initial CMP gather and the center panel shows the stack power of various $\bf v_{rms}$ values. We use the maximum within a reasonable fairway (the solid lines overlaying the stack power scan) as our data (dashed lines). The right panel of Figure  shows: our auto-picked $\bf v_{rms}$ (solid line), our inverted $\bf v_{int}$(dashed lines), and our interval velocity converted back to RMS velocity (dotted line).

 

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Figure 12 The left panel shows the initial CMP gather. The center panel shows the stack power of various $\bf v_{rms}$ values. Overlaid is a fairway (the solid lines overlaying the stack power scan) that we used for automatically picking RMS values (dashed lines). The right panel of Figure  shows: our auto-picked $\bf v_{rms}$ (solid line), our inverted for $\bf v_{int}$(dashed lines), and our interval velocity converted back to RMS velocity (dotted line).

Fitting goals (13) again assume a constant variance in our data. This is an incorrect assumption in this case for two very obvious regions. First, the $\bf T$ operator applied to our data means that late times are going to be given a much larger weight in our inversion. A solution to this problem is introduce a weighting operator $\bf D_1$,which is simply $\frac{1}{\bf T}$.A second error in the assumption of constant variance is that we know that not all our data ($\bf v_{rms}$ measurements) are of the same quality. The center panel of Figure  shows that there are areas where there are no significant reflectors. In addition, there are areas where our stack power results show an obvious maximum at a given $\bf v_{rms}$ value and other areas where the maximum is much less clear. To try to take into account both of these phenomena I calculated a weighted variance within the fairway shown in the center panel of Figure ,

$$v(i) = \frac{\sum_{j=b(i)}^{e(i)} (v(j)-v_{max}(i))^2 s(i,j)^4} { \sum_{j=b(i)}^{e(i)} s(i,j)^4} ,$$ (14)

where

b(i)

is the beginning sample of the fairway at a given sample i,

e(i)

is the ending sample of the fairway at a given sample i,

v(j)

is the v_rms at a given stack power location,

v_max(i)

is the velocity associated with the maximum stack power value (our data), and

s(i,j)

is the semblance value at time sample i and some v_vrms value j.

The left panel of Figure  shows our stack power scan overlaid by $\bf v_{rms}$ (dashed line) and $\bf v_{rms}+- \sqrt{\bf v}$ (dashed lines). Note how at areas with a sharp stack power blob the variance is small, while when the stack power blob is wide, where we have little coherent energy, the variance is large. We can now estimate new interval velocity model using,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{n} = \bf N( \bf T \bf v_{rms}^2 - \bf C \b... ...mber \\ \bf 0&\approx&\bf r_{m} = \epsilon \bf D \bf v_{int}^2 ,\end{eqnarray}$$ (15)

where $\bf N$ is $\bf V \bf D_1$. The right panel of Figure  shows our data $\bf v_{rms}$, $\bf v_{int}$, and $\bf C \bf v_{int}$.

 

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Figure 13 The left panel shows our stack power scan overlaid by $\bf v_{rms}$ (dashed line) and $\bf v_{rms}+- \sqrt{\bf v}$ (dashed lines). The right panel of Figure  shows our data $\bf v_{rms}$, $\bf v_{int}$, and $\bf C \bf v_{int}$.