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 it is assumed 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.

I decrease the patch size, to as small as a single data sample, changing the problem from overdetermined to very underdetermined. It is possible to 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. To speed the calculation immensely and make the equations somewhat neater, we can ``precondition'' the problem. Define a new variable $\bold p$ by $\bold a=\bold S\bold p$and insert it into (1) to 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 $\bold a$, just use $\bold a=\bold S\bold p$.To simplify things, one could just drop the damping (5) and keep only (4); then to control the null space, start from a zero solution and limit the number of iterations.

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 later.