2-D signal and noise estimation theory

As before, the signal $\sv s$ is described by a two-dimensional signal annihilation filter $\st F_s$,so that $\st F_s \sv s \approx \sv 0$.The noise $\sv n$ is described by a noise annihilation filter $\st F_n$,so that $\st F_n \sv n \approx \sv 0$, where $\st F_n$ is now a single two-dimensional filter. As before, the recorded data $\sv d$ is the sum of the signal $\sv s$ and noise $\sv n$,making $\sv d=\sv s+\sv n$.

The use of filters to characterize the signal and noise is similar to the methods used to separate the signal and noise in the simple three-dip cases shown in chapter . Here, both the signal and noise are characterized by two-dimensional filters, one filter for the noise and one filter for the signal. Although the signal and noise in the three-dip case considered in chapter  did not overlap spatially, in the general case the noise and signal will always overlap. Because of this overlap, calculating the signal filter $\st F_s$ and the noise filter $\st F_n$ may be a problem, since the signal $\sv s$ and noise $\sv n$ are unavailable before the separation takes place. A combination of two techniques is used to produce reasonable estimates of $\st F_s$ and $\st F_n$here.

The first technique is to separate spatially the noise and the signal into regions where one or the other dominates. For example, in the case of the ground-roll noise considered later in this section, the noise may be isolated to a narrow wedge within a shot gather. The noise filter calculated over the data in this wedge will be influenced primarily by the ground roll. A similar technique is used to calculate the signal filter. After the ground roll and the first breaks are muted, the data is dominated by the signal, so that a filter calculated over this data will be influenced primarily by signal.

The second technique used to produce reasonable estimates of $\st F_s$ and $\st F_n$is to control the shape of the filters to produce the desired prediction. The signal filter $\st F_s$ is a purely lateral two-dimensional filter as described in chapter . If this filter is kept short in time, the steeply dipping ground roll becomes fairly difficult to predict. The noise filter $\st F_n$, on the other hand, is shaped to follow the dip of the ground roll and has the form

$$\begin{array} {cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0... ...\\ a & a & a & 0 \\ a & a & a & 0 \\ a & a & a & 0 \end{array},$$ (136)

where each of the as is a different filter coefficient. Notice that the output point is the upper right-hand side of the filter and there is a gap between the output point and the free filter coefficients. This gap prevents the prediction of reflection events that fall inside the window where the ground roll dominates. Also notice that the column that contains the one has no free filter coefficients in it. This prevents using the samples from within a given trace to predict anything within that trace. This is needed since ground roll is often an almost mono-frequency noise. If the predictions within a trace are allowed, the filter $\st F_n$ might become a simple frequency filter, and that frequency would also be removed from the signal. Not allowing predictions from within a trace also keeps the predictions from within the trace from overwhelming the predictions from the other traces. Since the noise is recognized by its coherence between traces, the trace-to-trace predictions must be maintained. The combination of the filter shapes and the spatial separation of the signal and noise provides good estimates of the signal and noise filters.

Since this method will generally be used on prestack data, either filter may be corrupted by high-amplitude noise. Following the method presented in chapter , high-amplitude noise may be removed by a trace-to-trace prediction method where samples that are not well predicted are thrown out. These missing data samples, as well as data not recorded, are then restored during the inversion for the signal and noise.

To review chapter , the data $\sv d$ is separated into the known data $\sv k$ and the missing data $\sv m$,so that $\sv d=\sv k+\sv m$.Two masks, $\st K$ and $\st M$, are used to describe the missing data, making $\sv k = \st K\sv d$ and $\sv m=\st M\sv d$, where $\st K+\st M=\st I$, where $\st I$ is the identity matrix.

Summarizing the previous definitions:
$\sv d$ = data
$\sv s$ = signal
$\sv n$ = noise
$\sv k$ = known data
$\sv m$ = missing data
$\st K$ = known data mask
$\st M$ = missing data mask
$\st F_s$ = signal annihilation filter
$\st F_n$ = noise annihilation filter.
The relationships between these factors are as follows:
$\st F_s \sv s \approx \sv 0$
$\st F_n \sv n \approx \sv 0$
$\sv d=\sv s+\sv n$
$\sv d=\sv k+\sv m$
$\st I=\st K+\st M$ or $\sv d = \st K\sv d + \st M\sv d$.

Taking $\st F_s \sv s \approx \sv 0$ and $\st F_n \sv n \approx \sv 0$ and replacing $\sv s$ with $\sv d-\sv n$produces the system  

$$\left( \begin{array} {c} \st F_s \sv d \\ \sv 0\end{array}\... ...\begin{array} {c} \st F_s \\ \st F_n\end{array}\right) \sv n.$$ (137)

Replacing the data $\sv d$ with $\sv k+\sv m$ and moving the terms that depend on the missing data $\sv m$ to the right-hand side gives  

$$\left( \begin{array} {c} \st F_s \st K \sv d \\ \sv 0\end{a... ...ght) \left( \begin{array} {c} \sv n \\ \sv m\end{array}\right).$$ (138)

Finally, system () is modified to weight the noise prediction so that the noise tends to fall only in the zone where the noise is expected. In this case, the weighting is done by a simple set of weights, $\st W_s$ and $\st W_n$, $\st W_s$ being small where signal is expected and large where only noise is expected, and $\st W_n$ being small where noise is expected and large where only signal is expected. The modified system then becomes  

$$\left( \begin{array} {c} \st F_s \st K \st W_s \sv d \\ \sv... ...ght) \left( \begin{array} {c} \sv n \\ \sv m\end{array}\right).$$ (139)

In the example shown here, the weight $\st W_n$has been set to follow the ground roll by a single velocity and a single window length. An alternative approach that might work well when the ground roll is very strong would be to make $\st W_n$ the inverse of the envelope of the data. This weighting function can also be used to describe the spatial extent of the noise for calculating the noise and signal filters. For signal and noise that vary as functions in time, the two sets of weights can become functions such as the t and t² in section .

To reduce the numbers of iterations required to solve system (), the initial estimate of $\sv n$ is set to be the wedge of data dominated by noise filtered by the signal annihilation filter $\st F_s$ to remove the signal that underlies the noise. This estimate reduces the number of iterations and improves the final estimate of $\sv n$ significantly.