SEGREGATING P AND S CROSSTALK

Signals can be contaminated by other signals, and images can be contaminated by other images. This contamination is called ``crosstalk." An everyday example in seismology is the mixing of pressure waves and shear waves. When waves come straight up, vertical detectors record their pressure-wave component, and horizontal detectors record their shear-wave component. Often, however, waves do not come exactly straight up. In these cases, the simple idealization is contaminated and there is crosstalk. Here we study a simplified form of this signal-corruption problem, as given by the equations

$$\begin{eqnarray} \bold v &=& \bold p + \alpha \bold s + \bold n \ \bold h &=& \bold s + \alpha' \bold p + \bold n'\end{eqnarray}$$ (1) (2)

where $\bold v$ and $\bold h$ represent vertical and horizontal observations of earth motion, $\bold p$ and $\bold s$ represent theoretical pressure and shear waves, $\bold n$ and $\bold n'$ represent noises, and $\alpha$ and $\alpha'$ are the cross-coupling parameters. You can think of $\bold v$, $\bold h$, $\bold p$, $\bold s$,$\bold n$ and $\bold n'$ as collections of numbers that can be arranged into a signal or into an image. Mathematically, they are abstract vectors. In our notation, boldface $\bold v$ represents the vector as a whole, and italic v represents any single component in it. (Traditionally, a component is denoted by v_i.)