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# Noise estimation by inversion

As described in chapter , a structure for creating an inverse may be
 (76)

 (77)
where , , and are linear operators, and and correspond to a model and to the data. The value of is used to weight the relative importance of () and (). Replacing with the signal annihilation filter , with , the identity matrix, and ignoring for the moment gives an expression for calculating the noise from the data given the signal annihilation filter .The expression is not useful in itself for calculating the noise ,since the filter is not perfect and is unlikely to completely annihilate the signal to the point where the inversion for could not restore it. Without additional constraints, the obvious solution to is .In practice, I have found that, although the filter could attenuate the signal significantly, a simple inversion of for restores much of the signal into the calculated noise .What is needed is a constraint to replace 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 .This is a reasonable approximation, since should be somewhat close to the actual noise. The difference between the actual noise and the approximated noise should be fairly small and involves only the response of the noise to the filter .The approximation is weighted as .The value for may be changed to account for the signal-to-noise ratio of the data.

The system of regressions to be solved is now
 (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 from the approximation ,it is reasonable to initialize to before entering the iterative solver. Another reason for initializing to is that the filter 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 to improves the calculation of and reduces the number of iterations needed.

Equation () expressed as a minimization of the residual is
 (79)
 (80)
to the right-hand side of equation () to produce, with some simplification,
 (81)
Since the iterative solver just updates without regard to the initial value Claerbout (1995), the value of in this equation may be considered as the change of the calculated noise from the first estimate of the noise .This may be expressed as
 (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 .At the moment, the optimum value of is uncertain. It would seem that should decrease as the signal-to-noise ratio decreases, since the difference between the actual noise and the estimated noise is larger. However, in the presence of strong noise, the larger is, the more stable the inversion should be. If is relatively large, around 1.0, the amplitudes of the reflections are preserved and spurious events are somewhat suppressed. As gets very large, the result approaches the prediction filter result. When gets small, the amplitudes of the reflectors are attenuated, since the signal filter does not perfectly annihilate the signal before the inversion. For small , the spurious events tend to return also. The best value of 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 vary from 0.1 to 3.0. Small values of remove background noise, but seem to introduce organized noise into the calculated signal. For the real data examined, the background noise increases as increases, and the continuity of the data increases as decreases. Further work is needed to determine how the strength and type of noise affects the value of .

An example of the difference between prediction filtering and inversion prediction is seen in Figure . The filter is calculated from the data to predict the flat event. When 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.

Next: Improving the signal-prediction filter Up: Random noise removal enhanced Previous: Shortcomings of prediction filtering
Stanford Exploration Project
2/9/2001