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Now we face the question, what are the best values of $\alpha$ and $\beta$to use in equation (2)? There is a well known least-squares answer, but I have been frustrated frequently enough, by lack of consideration to weighting functions, non-stationarity, and such that I chose a more careful investigation. To gain insight in the matter I prepared a movie sweeping through various values of $\alpha/\beta$.I'll be excruciatingly explicit about what I computed. Say $\beta$ is unity although it was really determined by the plotting program. For the missing traces define $d{\bf x}= {\bf x}_m$ where in conventional processing, $d{\bf x}=0$.
\v_k &=& {\bf B'}{\bf x}_k \\ d{\bf x}&=& \bold M {\bf B}\v_k \\ d\v &=& {\bf B'}d{\bf x}\end{eqnarray} (4)
where $\bold M$ is a diagonal matrix containing ones in locations of missing data and zeros in locations of known data. I studied side-by-side panels of

({\bf x}_k+\alpha \,d{\bf x}) \quad {\rm and} \quad (\v_k+\alpha \,d\v)\end{displaymath}

as a movie for various values of $\alpha$.

For data I began with a CMP gather used by Dave Hale. Denoting missing by m and known by k I interleaved and extended the data offsets by

m,m,m,m,m,m,k,m,k,m,k,\cdots, k,m,m,m,m,m,m\end{displaymath}

The movie illustrated that a concept familiar in Fourier analysis also applies to the velocity transform. This is the idea that roughness in the interior of the data corresponds to energy ``off ends'' in velocity space whereas energy ``off ends'' in offset space, controls resolution in the interior in velocity space.

The movie also illustrated that a subjective choice of the best value of $\alpha$ was a function of location in the (x,t)-plane. The best choices of $\alpha$ for internal regions and for external regions were quite different. Internally the best choice was one that would make interleaved traces look most like their neighbors which simply meant that they would be the same polarity and amplitude. Within the data the best fitting $\alpha$ was about $\alpha=.4$near the top and $\alpha=.2$ near the bottom. Off the sides of the data, a subjective analysis was not so simple, but it seemed clear to me that the best value of $\alpha$needed to be much larger, say about $\alpha=2.0$.

I was prepared to see a spectral difference between ${\bf x}$ and $d{\bf x}$, but none was apparent, indicating the rho filters in the definition of ${\bf B}$ had generally done their job.

The internal results can be explained by less than perfect quality of the operator ${\bf B}$.Clearly a more stationary value of $\alpha$ in the internal region could be achieved by a definition of the velocity transformation that put more effort to yielding a more unitary operator. The external behavior is more troublesome, and potentially far more interesting because it controls the resolution in velocity space.

From the results of this movie study I decided to separate the issues of trace interpolation from those of trace extrapolation though obviously there is some interaction near the ends of the cable. Of the two problems I chose the data extrapolation problem to have the most interest because it affects resolution of the geologically interesting part of our data. Since the ${\bf B'}$ operator chosen for this study works best at wide offsets, I chose to extend the far end of the cable.

next up previous print clean
Next: TRACE EXTRAPOLATION Up: Claerbout: Extending a CMP Previous: FORMULATON
Stanford Exploration Project