1-D Separation theory

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

$$\sv 0 \approx \st F_s \sv s',$$ (117)

where $\sv s$ is the scaled signal and $\st F_s$ is the matrix form of the filter. A single filter is used for the signal in all traces. The noise filter appears as

$$\sv 0 \approx \st F_n(x) \sv n.$$ (118)

$\st F_n(x)$ is a function of x because a separate filter is computed for each trace. For this section and the next, the noise filter is calculated on the data above the start times. In this section both these filters are one dimensional.

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 t², I multiply the signal by t² and force the signal to zero above the start time by using the function T'(x,t,v), which varies as t² below the start time and is a large fixed constant above the start time. The relative amplitudes of the noise and signal are also available from section . To minimize the signal and the noise together, I use  

$$\sv 0 \approx \frac{1}{\sigma_s} \st F_s T'(x,t,v) \sv s,$$ (119)

and  

$$\sv 0 \approx \frac{1}{\sigma_n(x)} \st F_n(x) \sv n.$$ (120)

Modifying equation () using $\sv n =\sv d -\sv s$ gives  

$$\frac{1}{\sigma_n(x)} \st F_n(x) \sv d \approx \frac{1}{\sigma_n(x)} \st F_n(x) \sv s.$$ (121)

Expressing both systems as a single system gives  

$$\left( \begin{array} {c} \sv 0 \\ \frac{1}{\sigma_n(x)} \s... ...) \\ \frac{1}{\sigma_n(x)} \st F_n(x) \end{array}\right) \sv s.$$ (122)

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 $\sv s=\sv d-\sv n$, equation () becomes  

$$\sv 0 \approx \frac{1}{\sigma_s} \st F_s T'(x,t,v) (\sv d-\sv n)$$ (123)

or  

$$\frac{1}{\sigma_s} \st F_s T'(x,t,v) \sv d \approx \frac{1}{\sigma_s} \st F_s T'(x,t,v) \sv n.$$ (124)

This expression, combined with equation (), gives  

$$\left( \begin{array} {c} \sv 0 \\ \frac{1}{\sigma_s} \st F... ...\ \frac{1}{\sigma_s} \st F_s T'(x,t,v)\end{array}\right) \sv n.$$ (125)

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 $\sv d=\sv n+\sv s$,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 $\st F_n(x)$ and $\st F_s$ 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 $\st F_n(x)$ and $\st F_s$ 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 $\st F_n(x)$ 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 $\st F_n(x)$.If any information in the signal falls in the null space created by $\st F_n(x)$,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 $\st F_s$ 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 $\st F_s$.If any information in the noise falls in the null space created by $\st F_s$,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 $\st F_n(x)$ and $\st F_s$ will not be removed from the signal. For system () initialized with the original data, any data in the common null space create by $\st F_n(x)$ and $\st F_s$ 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 $\st F_n(x)$ and $\st F_s$ 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  

$$\sv 0 \approx \frac{1}{\sigma_s} \sv s',$$ (126)

$$\sv 0 \approx \st F_s \sv s',$$ (127)

 

$$\sv 0 \approx \frac{1}{\sigma_n(x)} \sv n,$$ (128)

and

$$\sv 0 \approx \st F_n(x) \sv n.$$ (129)

Expanded with relative weights to predict the signal, these minimizations become the single system of regressions that follows:  

$$\left( \begin{array} {c} \sv 0 \\ \sv 0 \\ \lambda_3 \frac... ...}{\sigma_n(x)} \\ \lambda_4 \st F_n(x)\end{array}\right) \sv s.$$ (130)

This system has no effective null spaces. Unfortunately, () and () assume that the signal and noise are constant functions, not the sinusoidal functions normally found in seismic data. While system () worked well for the peaks of the noise and signal, it failed near the zero crossings. Because of these failures, I will no longer consider approaches such as that in system () in this thesis.