Distribution of events not in the operator null space

In the discussion above, events were assumed to be either in the null space of an operator or not. In general, an operator will attenuate an event to some extent. The null space of an operator is completely attenuated, while other events are partially attenuated, depending on the situation in which the filter is designed. Operators derived from real seismic data are unlikely to perfectly predict either signal or noise. Events that are simply attenuated by the signal filter $\st S$ and the noise filter $\st N$ will be distributed between the calculated signal $\sv s$ and the calculated noise $\sv n$ depending on the relative attenuation of $\st S$ and $\st N$.