Initial estimates of the calculated noise and signal

For the discussion that follows, the calculation of the signal from equation () is labeled $\sv s_1$. The calculation of the noise from equation () is then $\sv n_1 = \sv d - \sv s_1$.The calculation of the noise from equation () is labeled $\sv n_2$. The calculation of the signal from equation () is then $\sv s_2 = \sv d - \sv n_2$.Is it true that $\sv s_1 = \sv s_2$ and is $\sv n_1 = \sv n_2$? Ideally, yes, but actually, the situation is more complicated. The two signals and the two noises calculated need not be equal, as will be shown in the examples below.

To imagine how $\sv s_1 \neq \sv s_2$,consider an event that is eliminated by both filters $\st S$ and $\st N$.This event is in the null space of both $\st S$ and $\st N$Menke (1989); Nichols (1994a); Strang (1988). In both equation () and equation () that event will be eliminated from the system, and no information about this event will be available for the solver. Therefore, no part of that event will occur in the calculated solutions $\sv s_2$ and $\sv n_1$.The event will then be completely contained in $\sv n_2$ and $\sv s_1$.

The initial estimates of $\sv s_2$ and $\sv n_1$ might be set to values varying from zero to the data $\sv d$ when using iterative methods for solving equations () and (). If the initial estimates of $\sv s_1$ and $\sv n_1$ are zero, the problem will appear as equations () and (). If the initial value of the signal $\sv s$ in equation () is the data $\sv d$,the constant  

$$\left( \begin{array} {c} \st S \\ \st N\end{array}\right) \sv d$$ (90)

should be subtracted from the residual that is minimized in solving equation ()  

$$\sv r \approx \left( \begin{array} {c} \st S \\ \st N\end... ...\left( \begin{array} {c} \sv 0 \\ \st N\sv d\end{array}\right),$$ (91)

$\sv r$ being the residual. If the constant is added to the right-hand side instead of subtracted from the residual, the expression  

$$\sv r \approx \left( \begin{array} {c} \st S \\ \st N\end... ...eft( \begin{array} {c} \st S \\ \st N\end{array}\right) \sv d$$ (92)

results. Simplifying this gives  

$$\sv r \approx \left( \begin{array} {c} \st S \\ \st N\end... ...ft( \begin{array} {c} \st S \sv d \\ \sv 0\end{array}\right).$$ (93)

When this equation is compared to equation (), it might be supposed that the $\sv s$ in equation () is $-\sv n$ from equation (), which was previously labeled as $-\sv n_2$.When the initial value $\sv d$ of $\sv s$ is added, the result becomes $\sv d-\sv n_2$.Instead of the previously calculated value of $\sv s_1$ from equation (), using the initial estimate of d for $\sv s$ in equation () gives the value $\sv s_2 = \sv d - \sv n_2$, which is the same answer as equation (). A similar relationship is true for equation () and equation () if the estimated noise is set to $\sv d$.

The difference between solving with zero as the initial solution and solving with the data as the initial solution is simply where to put the null space. If the initial solution contains no null space data, the final solution will not contain any of the null space data. If the initial solution contains data that falls in the null space, the final solution will leave this null space unchangedNichols (1994a). The difference between equations () and () is the placement of events that fall in the null space.

To summarize the previous discussion, the solutions for the signal and noise derived from equations (), (), and () are the same whether the noise or signal is calculated, provided the initial estimates of the signal and noise are the same and the estimates for the signal and noise sum to the data $\sv d$.For example, equation () solved with an initial estimate of the signal of zero assumes the noise has an initial value of the data. Solving equation () with the same initial values of the signal being zero and the noise being the data gives the same results for the calculated noise and data. For a more symmetrical result, the noise and signal might both be initialized with half the data. This choice of initial values gives us a useful tool in specifying how data in the null space of both $\st S$ and $\st N$ are distributed.