Failure of straightforward methods

The conventional answer to the crosstalk question is to choose $\alpha$ so that $\bold p = \bold v - \alpha \bold h$ has minimum power. The idea is that since adding one signal $\bold p$to an independent signal $\bold s$is likely to increase the power of $\bold p$,removing as much power as possible may be a way to separate the independent components. The theory proceeds as follows. Minimize the dot product  

$${\rm Energy} \eq \bold p \cdot \bold p \eq (\bold v - \alpha \bold h) \cdot (\bold v - \alpha \bold h)$$ (5)

by differentiating the energy with respect to $\alpha$, and set the derivative to zero. This gives  

$$\alpha \eq { \bold v \cdot \bold h \over \bold h \cdot \bold h }$$ (6)

Likewise, minimizing $(\bold s \cdot \bold s)$ yields $\alpha ' = ( \bold h \cdot \bold v )/( \bold v \cdot \bold v )$.

In equation (5) the ``fitting function'' is $\bold h$,because various amounts of $\bold h$ can be subtracted to minimize the power in the residual $(\bold v-\alpha \bold h)$.Let us verify the well-known fact that after the energy is minimized, the residual is orthogonal to the fitting function. Take the dot product of the fitting function $\bold h$and the residual $(\bold v-\alpha \bold h)$,and insert the optimum value of $\alpha$ from equation (6): Results for both $\bold p$ and $\bold s$are shown in Figure 1.

 

uniform
Figure 1 Left shows two panels, a ``Pressure Wave'' contaminated by crosstalk from ``Shear'' and vice versa. Right shows a least-squares attempt to remove the crosstalk. It is disappointing to see that the crosstalk has become worse.

At first it is hard to believe the result: the crosstalk is worse on the output than on the input. Our eyes are drawn to the weak signals in the open spaces, which are obviously unwanted new crosstalk. We do not immediately notice that the new crosstalk has a negative polarity. Negative polarity results when we try to extinguish the strong positive polarity of the main signal. Since the residual misfit is squared, our method tends to ignore small residuals and focus attention on big ones: hence the wide-scale growth of small residuals.

The least-squares method is easy to oversimplify, and it is not unusual to see it give disappointing results. Real-life data are generally more complicated than artificial data like the data used in these examples. It is always a good idea to test programs on such synthetic data since the success or failure of a least-squares method may not be apparent if the method is applied to real data without prior testing.