NOISY DATA

It is appealing to think of interpolating land data, because land data can be so expensive to acquire. Land data is more difficult to interpolate than marine data, however, because it tends to be much more noisy. We can guess that the PEFs will find the predictable part of the data, and that it will be the signal. However, even assuming we do estimate the correct PEF (one that captures the dips of the signal rather than the noise), energy from the noise will be carried along those dips to nearby interpolated traces. As an alternative, we can attempt to separate the signal from the noise while interpolating the missing traces. Taking the theory from Claerbout 1997, data $\bold d$ can be decomposed into known plus missing parts, $\bold d = \bold k + \bold m$.With known and unknown data selectors $\bold K$ and $\bold M$, write the data as

$$\bold d = \bold K \bold d + \bold M \bold m$$ (6)

where $\bold K\bold d$ is zero-padded known data and all the components of $\bold m$ are freely adjustable. The goal is to fill in the appropriate parts of $\bold m$.

The data can also be decomposed into signal plus noise, $\bold d = \bold s + \bold n$.Thus  

$$\bold s + \bold n = \bold K \bold d + \bold M \bold m$$ (7)

If we write fitting goals for signal and noise and then get rid of the noise with equation (7) we have

$$\begin{eqnarray} \bold 0&\approx&\bold N\bold n = \bold N(\bold K\bold d+\bold M\bold m-\bold s) \\ \bold 0&\approx&\bold S\bold s = \bold S \bold s\end{eqnarray}$$ (8) (9)

Solving (8) and (9) gives estimates for the signal component $\bold s$ of both the known and missing data, as well as an estimate of the noisy missing traces, $\bold M \bold m$.

This requires a signal predictor and a noise predictor. For the signal predictor I use the PEFs estimated from the data. In the example to follow I will throw out alternating midpoint gathers and attempt to interpolate them back, so I define noise as whatever is incoherent across midpoints and choose the noise predictor to be an average in a small window along the midpoint axis.