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 *i*th channel at the *j*th lag.
For illustration,
consider two channels and three lag times.
Then Levin's constraints which prevent signal distortion are

(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*)[*f _{1}*(

(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

(42) |

- In one application,
where the channel amplifications were not well controlled,
the lead terms of the filter were
*f*(0) = 100 and_{1}*f*(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
*x*(_{1}*t*) =*x*(_{2}*t*+1) =*x*(_{3}*t*+2). What is the sortest filter which can both satisfy the constraints and still have some possibility of rejecting noise? - Consider the Levin filter on
*m*channels with filters containing*k*lags. What is the size of the matrix in (42)?

10/30/1997