WHY SMOOTH RADIALLY

Dips change quickly along every axis in seismic data. As a result a single PEF has trouble characterizing it, even in small patches Crawley (1999). By estimating a space-varying PEF, we can overcome this deficiency. Unfortunately, this changes our estimation problem from something overdetermined to something, at times, grossly underdetermined. To stabilize our filter estimation we must apply some type of regularization to the standard PEF estimation optimization goals:

$$\begin{eqnarray} \bf 0&\approx&\bold Y \bold a \\ \nonumber \bf 0&\approx&\epsilon \bold F \bold a \end{eqnarray}$$ (1)

where $\bold a$ is our space-varying filter, $\bold Y$ is convolution with our data, and $\bold F$ is a roughener. To speed up convergence, we can take advantage of helix theory Claerbout (1998c) and reformulate our regularized problem into a preconditioned one

$$\begin{eqnarray} \bf 0&\approx&\bold Y {\bold F^{-1} \bf A^{-1}} \bf p\\ \nonumber \bf 0&\approx&\epsilon \bf p\end{eqnarray}$$ (2)

where

$$\bold p = \bold F \bold a .$$ (3)

Our choice for $\bold F$ can have significant influence on both the speed and quality of our filter estimation.

The character of seismic data itself gives us a clue on what type of regularization we should use. A PEF filter is most successful when the statistics of the data it is being estimated from are stationary. Logically, our rougher $\bold F$,or ${\bold F^{-1}}$, should tend to smooth along a region with consistent dips, or along Snell traces Claerbout (1978). Figure 1 shows several constant velocity hyperbolas, with the same dips highlighted. These dips all fall along a radial line through zero time and zero offset. If we look at hyperbolas in v(z), Figure 2, we see that there is deviation from a simple line, but generally this trend is preserved.

 

dips.constant Figure 1 Constant velocity curves. The thick lines are the same dip on all the reflectors. Note how they form a line.

 

dips.vz Figure 2 V(z) medium curves. The thick lines represent the same dip. Note how they are not perfectly linear but generally lay along a line.

 

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Figure 3 The effect of dip smoothing. The top-left panel is the input, the top-right is the result of applying the forward operator, bottom-left is the adjoint response; and bottom-right is the cascade of forward and the adjoint.