Noise estimation by inversion

As described in chapter , a structure for creating an inverse may be  

$$\sv 0 \approx \st W(\st L\sv m-\sv d)$$ (76)

 

$$\sv 0 \approx \epsilon \st A \sv m,$$ (77)

where $\st W$, $\st L$, and $\st A$ are linear operators, and $\sv m$ and $\sv d$ correspond to a model and to the data. The value of $\epsilon$ is used to weight the relative importance of () and (). Replacing $\st W$ with the signal annihilation filter $\st S$,$\st L$ with $\st I$, the identity matrix, and ignoring $\sv 0 \approx \epsilon \st A \sv m$ for the moment gives an expression $\st S \sv d \approx \st S \sv n$ for calculating the noise from the data given the signal annihilation filter $\st S$.The expression $\st S \sv d \approx \st S \sv n$ is not useful in itself for calculating the noise $\sv n$,since the filter $\st S$ is not perfect and is unlikely to completely annihilate the signal to the point where the inversion for $\sv n$ could not restore it. Without additional constraints, the obvious solution to $\st S \sv d \approx \st S \sv n$ is $\sv d=\sv n$.In practice, I have found that, although the filter $\st S$ could attenuate the signal significantly, a simple inversion of $\st S\sv d = \st S\sv n$ for $\sv n$ restores much of the signal into the calculated noise $\sv n$.What is needed is a constraint to replace $\sv 0 \approx \epsilon \st A \sv m$in system ().

The constraint used here to keep signal out of the calculated noise is that the noise is approximately the noise estimated from prediction filtering $\st S\sv d$.This is a reasonable approximation, since $\st S\sv d$ should be somewhat close to the actual noise. The difference between the actual noise $\sv n$ and the approximated noise $S\sv d$should be fairly small and involves only the response of the noise to the filter $\st S$.The approximation is weighted as $\epsilon \sv n \approx \epsilon \st S\sv d$.The value for $\epsilon$ may be changed to account for the signal-to-noise ratio of the data.

The system of regressions to be solved is now  

$$\left( \begin{array} {c} \st S \sv d \\ \epsilon \st S \sv ... ... \begin{array} {c} \st S \\ \epsilon\end{array}\right) \sv n.$$ (78)

The results of solving this system are referred to as inversion prediction in the following discussion to distinguish it from prediction filtering.

Since this system estimates $\sv n$ from the approximation $\st S\sv d$,it is reasonable to initialize $\sv n$ to $\st S\sv d$ before entering the iterative solver. Another reason for initializing $\sv n$ to $\st S\sv d$ is that the filter $\st S$ is generally small and will pass only a limited range of spatial and temporal frequencies. In the case of a spike in the data, inversion for the noise with a small filter does not allow the complete restoration of the spike. Because the noise is expected to be almost white and in some cases dominated by spikes, initializing $\sv n$ to $\st S\sv d$ improves the calculation of $\sv n$ and reduces the number of iterations needed.

Equation () expressed as a minimization of the residual $\sv r$ is  

$$\sv r = \left( \begin{array} {c} \st S \\ \epsilon\end{ar... ...ay} {c} \st S \sv d \\ \epsilon \st S \sv d\end{array}\right).$$ (79)

Initializing $\sv n$ to $\st S\sv d$ involves adding

$$\left( \begin{array} {c} \st S \\ \epsilon\end{array}\right) \st S\sv d$$ (80)

to the right-hand side of equation () to produce, with some simplification,  

$$\sv r = \left( \begin{array} {c} \st S \\ \epsilon\end{ar... ...ray} {c} \st S \st S\sv d - \st S\sv d \\ 0\end{array}\right).$$ (81)

Since the iterative solver just updates $\sv n$ without regard to the initial value Claerbout (1995), the value of $\sv n$ in this equation may be considered as the change of the calculated noise from the first estimate of the noise $\st S\sv d$.This may be expressed as  

$$\sv r = \left( \begin{array} {c} \st S \\ \epsilon\end{ar... ... {c} \st S \st S\sv d - \st S\sv d \\ \sv 0\end{array}\right).$$ (82)

This is the effective system of regressions that is implimented in this chapter.

The results of inversion prediction are sensitive to the value of $\epsilon$.At the moment, the optimum value of $\epsilon$ is uncertain. It would seem that $\epsilon$ should decrease as the signal-to-noise ratio decreases, since the difference between the actual noise $\sv n$and the estimated noise $\st S\sv d$ is larger. However, in the presence of strong noise, the larger $\epsilon$ is, the more stable the inversion should be. If $\epsilon$ is relatively large, around 1.0, the amplitudes of the reflections are preserved and spurious events are somewhat suppressed. As $\epsilon$ gets very large, the result approaches the prediction filter result. When $\epsilon$ gets small, the amplitudes of the reflectors are attenuated, since the signal filter $\st S$ does not perfectly annihilate the signal before the inversion. For small $\epsilon$, the spurious events tend to return also. The best value of $\epsilon$appears to be different for samples with Gaussian noise than for samples with uniformly distributed noise. For most work, it appears that good values of $\epsilon$ vary from 0.1 to 3.0. Small values of $\epsilon$ remove background noise, but seem to introduce organized noise into the calculated signal. For the real data examined, the background noise increases as $\epsilon$ increases, and the continuity of the data increases as $\epsilon$ decreases. Further work is needed to determine how the strength and type of noise affects the value of $\epsilon$.

An example of the difference between prediction filtering and inversion prediction is seen in Figure . The filter $\st S$ is calculated from the data to predict the flat event. When $\st S$ is applied to the spike, the filter response can be seen in the prediction-filter result. The inversion prediction result has effectively eliminated the filter response.

 

onespikea
Figure 2 A comparison of the action of a t-x prediction filter and an inversion prediction on a spike.