CONSTRAINED FILTERS

A common geophysical situation is a plane wave (signal) incident on a group of receivers. One expects to see the same waveform at each receiver. However, there is corrupting noise present at each receiver, and the noise may or may not be coherent from one receiver to the next. In fact, we may suppose there is so much noise on each receiver that the signal might not be detectable at all if there were only one receiver. This was the situation facing M.J. Levin when he was trying to detect weak underground nuclear explosions with an array of seismometers. He suggested a multichannel filter with constraints. First suppose that either all the signals arrive at the same time or that, if the times differ, at least they are known so that the data channels may be shifted into alignment. Now the problem is to filter each channel and then add up the channels; the noise should be rejected but the signal shape should be maintained. Let f_i(j) represent the filter weight on the ith channel at the jth lag. For illustration, consider two channels and three lag times. Then Levin's constraints which prevent signal distortion are

$$\begin{eqnarray} 1 &= & f_1(0) + f_2(0) \nonumber \\ 0 &= & f_1(1) + f_2(1) \\ 0 &= & f_1(2) + f_2(2) \nonumber\end{eqnarray}$$ (40)

That this does not cause signal distortion follows, since if the same signal s(Z) comes into each channel, the output is merely s(Z)[f₁(Z) + f₂(Z)]. But f₁ + f₂ is just (1, 0, 0) in this case or a delta function in general. We call the equation set (40) constraint equations because there are fewer equations than unknowns. The constraint equations may be written in usual form as  

$$\left[ \begin{array} {rllllll} -1 & 1 & 1 & 0 & 0 & 0 & 0 \... ...\eq \left[ \begin{array} {c} 0 \\ 0 \\ 0 \end{array} \right]$$ (41)

which may abbreviate as Gf = 0. If we use the method of least squares to minimize the total energy in the filter output, we will be attempting to suppress both signal and noise. But the constraint equations prevent the suppression of signal; hence only the noise is attenuated. If we let R denote the spectral matrix of the input data, then the filter f is determined by solving equations like  

$$\left[ \begin{array} {ll} {\bf R}& {\bf G}^T \\ \ & \ \\ ... ...left[ \begin{array} {c} V \\ 0 \\ \vdots \end{array} \right]$$ (42)

We have solved equations of this type in preceding sections.

EXERCISES:

1. In one application, where the channel amplifications were not well controlled, the lead terms of the filter were f₁ (0) = 100 and f₂ (0) = -99. Although this filter satisfied all that it was designed for, it was deemed inappropriate because the assumption of identical signals on each channel was a reasonable approximation but not exactly true. Can you suggest a more suitable constraint matrix?
2. Consider three seismometers in a row on the surface of the earth. The constraints considered so far have implied all signals arrive at the same time, i.e., vertically incident waves. Define a constraint matrix to pass both the vertically incident wave and the wave which causes x₁ (t) = x₂ (t+1) = x₃ (t+2). What is the sortest filter which can both satisfy the constraints and still have some possibility of rejecting noise?
3. Consider the Levin filter on m channels with filters containing k lags. What is the size of the matrix in (42)?