TIME- AND SPACE-VARYING PEFS

The time dip of seismic data changes rapidly along many axes, so a single PEF can only represent a small amount of data. Often we divide the data into patches, where we assume the data have constant dips. Because seismic data have curvature and may not be well represented by piecewise-constant dips, it is appealing to extend the idea of time-variable filtering to include spatial dimensions as well, and have smoothly varying PEFs to represent curved events.

Instead of one PEF per patch, we estimate a PEF for every output data point; changing the problem from overdetermined to very underdetermined. We can estimate all these filter coefficients by the usual formulation, supplemented with some damping equations, say  

$$\begin{array} {lll} \bold 0 &\approx& \bold Y \bold K \bold ... ...d r_0 \\ \bold 0 &\approx& \epsilon\ \bold R \bold a\end{array}$$ (1)

where $\bold R$ is a roughening operator, $\bold Y$ is convolution with the data, and $\bold K$ is a known filter coefficient mask.

When the roughening operator $\bold R$ is a differential operator, the number of iterations can be large. We can speed the calculation immensely and make the equations somewhat neater by ``preconditioning''. When we define a new variable $\bold p$ by $\bold a=\bold S\bold p$and insert it into (1) we get

$$\begin{eqnarray} \bold 0 &\approx & \bold Y \bold K \bold S\bold p + \bold r_0 \\ \bold 0 &\approx & \epsilon\ \bold R \bold S\bold p\end{eqnarray}$$ (2) (3)

Now, because the smoothing and roughening operators are somewhat arbitrary, we may as well replace $\bold R \bold S$ by $\bold I$ and get

$$\begin{eqnarray} \bold 0 &\approx & \bold Y \bold K \bold S\bold p + \bold r_0 \\ \bold 0 &\approx & \epsilon\ \bold I \bold p\end{eqnarray}$$ (4) (5)

We solve for $\bold p$ using conjugate gradients. To see $\bold a$, we just use $\bold a=\bold S\bold p$.

To reduce clutter, we could drop the damping (5) and keep only (2); then to control the null space, we need only to start from a zero solution and limit the number of iterations. As a practical matter, without (5) we must find a good number of iterations and with it we must find a good value for $\epsilon$.

For $\bold S$ we can use polynomial division by a Laplacian or by filters with a preferred direction. If the data are CMP gathers, it is attractive to use radial filters, which are explained further down.