Implementation of adaptive PEFs

Interpolating with adaptive PEFs means calculating a large volume of filter coefficients. It is possible to estimate all these filter coefficients by the same formulation as in the previous chapter, supplemented with some damping equations, like

$$\begin{eqnarray} \bold 0 &\approx& \bold Y \bold K \bold a + \bold r_0 \\ \bold 0 &\approx& \epsilon\ \bold R \bold a\end{eqnarray}$$ (23) (24)

where $\bold R$ is a roughening operator, $\bold Y$ is convolution with the data, and $\bold K$ is an adjustable filter coefficient selector. $\bold R$ does not roughen between coefficients within a single filter, but between coefficients at the same lag in different filters.

When the roughening operator $\bold R$ is a differential operator, the number of iterations can be large. To speed the calculation immensely, we can precondition the problem. Define a new variable $\bold p$ by $\bold a=\bold S\bold p$and insert it into () and () to get

$$\begin{eqnarray} \bold 0 &\approx & \bold Y \bold K \bold S\bold p + \bold r_0 \\ \bold 0 &\approx & \epsilon\ \bold R \bold S\bold p\end{eqnarray}$$ (25) (26)

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

$$\begin{eqnarray} \bold 0 &\approx & \bold Y \bold K \bold S\bold p + \bold r_0 \\ \bold 0 &\approx & \epsilon\ \bold I \bold p\end{eqnarray}$$ (27) (28)

We solve for $\bold p$ using conjugate gradients. To get $\bold a$, just use $\bold a=\bold S\bold p$.Once $\bold a$ is calculated, the missing traces are filled in as before. To simplify the formulation, one could drop the damping () and keep only (); then to control the null space, start from a zero solution and just limit the number of iterations. This is the way most of the examples later in this chapter are calculated.

Previously we solved for the PEFs $\bold a$, which have a fixed coefficient that is defined to have the value 1. We instead estimate $\bold p$, which is related by $\bold a=\bold S\bold p$.It appears troublesome that we do not necessarily know the fixed coefficient of $\bold p$.We can begin by applying $\bold p_0 = \bold S^{-1} \bold a_0$,putting some other value in the fixed coefficient of $\bold p$,that will be integrated by $\bold S$ to give 1's. But it is a hassle to then apply $\bold p$ to the data because our software has the value 1 built in. Luckily, the problem disappears by itself. Wherever the forward operator is applied, it looks like $\bold Y \bold K \bold S \bold p$, which is the same as $\bold Y \bold K \bold a$.We only need to apply $\bold S$ to the adjustable coefficients of $\bold p$,because we know the fixed coefficient of $\bold S\bold p$ equals one, even if we do not know the fixed coefficient of $\bold p$.We do not need to know or store the fixed coefficients of $\bold p$.