Limitations

The method described above isn't quite what we would like to have. Ideally we would rather phrase the problem as

$$\begin{eqnarray} \bf n&\approx&\bf G\bf A^{-1}\bf p\nonumber \\ \bf 0&\approx&\epsilon \bf p ,\end{eqnarray}$$ (13)

where $\bf G$ transforms our residual from the arbitrary distribution indicated by our semblance scan to a standard normal distribution. In this way we would be smoothing our semblance values rather than residual space shaping as indicated by fitting goals (12). The problem is that $\bf G$ can not be described in terms of a linear operator. If $\bf G$ was a stationary function we also wouldn't have a problem.