To improve the separation, more information is required. Here I attempt to use the time spectrum of the signal and the noise as the extra information. In a manner similar to the process used to determine the amplitudes, the spectrum of each trace is calculated from the data above the first break, where the data above the first breaks are assumed to be noise. Each trace has a separate filter calculated. For the noise, a single filter is derived over the data after the first breaks.

The filters used here annihilate the signal and noise. The signal filter is represented by

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I would like to incorporate what I know about the amplitudes
of the signal and noise from section
with the filters derived
here to improve the separation of signal and noise.
Since the signal loses amplitude as about ** t^{2}**,
I multiply the signal by

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An alternative approach is to solve for the noise. By repeating the same set of steps, but solving for the noise instead of the signal with , equation () becomes

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The results of either system () or () may be used. Appendix A of Abma 1994 shows that the results of either system of equations will produce equivalent results when ,but it can be seen that this does not apply when null spaces containing the signal and noise occur. Although it might be argued that the filters and are never perfect and so actual null spaces are not created, given a reasonable number of iterations of the least-squares solver and the finite capabilities of the floating point number representation used, effective null spaces will be created in these cases. Since the filters and are fairly effective in removing the noise and signal, they create effective null spaces in systems () and ().

Another way of looking at this null space problem is to examine the information that is available in the system to calculate a solution. In system (), any information about the noise is eliminated from the system since the filter has removed the noise from the data going into the left-hand side of the system. Therefore, the signal calculated from system () will not contain any information removed by .If any information in the signal falls in the null space created by ,the solution for the signal from system () will not contain that information. In system (), any information about the signal is eliminated from the system since the filter has removed the signal from the data going into the left-hand side of the system. Therefore, the noise calculated from system () will not contain any information removed by .If any information in the noise falls in the null space created by ,the solution for the noise from system () will not contain that information. In short, if the signal and noise have overlapping null spaces, the overlap is eliminated in systems () and ().

Another aspect of solving these inversions involves initialization of the inversionAbma (1995). This initialization reduces the number of iterations of the solver significantly, thus reducing the cost of the inversion. It also modifies the action of the inversion. If system () has the signal initialized with the original data, any data in the common null space create by and will not be removed from the signal. For system () initialized with the original data, any data in the common null space create by and will not be removed from the noise. It can then be seen that the noise calculated from system () without initialization subtracted from the data will be equal to the signal calculated from system () with initialization. Also, the signal calculated from system () without initialization subtracted from the data will be equal to the noise calculated from system () with initialization.

For one-dimensional filters, the overlap of the action of and is likely to be a problem, especially when both the signal and noise are broadband. One approach to solving the problem of this overlap is to modify the systems so that the effective null spaces do not occur. One method I have tried is to separate the amplitude and the filtering effects into separate equations. These equations were

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2/9/2001