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