Noise removal with missing data

A common problem in solving the previous systems is that high-amplitude noise overwhelms least-squares inversion techniques. Removing the worst of the noise before calculating a signal filter and before attempting to separate signal and noise can significantly improve the results. When doing the inversion after removing data, the samples that have been removed must be accounted for, otherwise the zeroed samples will be just another noise contaminating the process.

To allow for the zeroed samples, the previous inversions are recast to predict missing data while separating signal from the noise. This prediction of missing data is implemented by defining the data $\sv d$as the sum of the known data $\sv k$ and the missing data $\sv m$, or $\sv d=\sv k+\sv m$.The missing data $\sv m$ is then calculated along with the signal $\sv s$ and noise $\sv n$.These substitutions change the previous systems to  

$$\left( \begin{array} {c} \sv 0 \\ \st N \st K \sv d\end{arra... ...ight) \left( \begin{array} {c} \sv s \\ \sv m\end{array}\right)$$ (3)

and  

$$\left( \begin{array} {c} \st S \st K \sv d \\ \epsilon \st... ...ght) \left( \begin{array} {c} \sv n \\ \sv m\end{array}\right),$$ (4)

where $\st K\sv d=\sv k$ and $\st M\sv d=\sv m$, $\st K$ and $\st M$ being masks for the data describing the known and missing data positions such that $\st K+\st M=\st I$,where $\st I$ is the identity matrix.

An important additional advantage of this extension to the previous inversions is that data missing because of acquisition problems may also be estimated. Prediction filtering techniques have long been attempted in prestack data but have often failed because of missing data. Using the predictions from systems () and () allows these missing traces to be predicted while producing results that are not affected by missing traces that are otherwise treated as valid data.