20-07-2012, 03:03 PM
Introduction to matched filters
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ABSTRACT
Matched filters are a basic tool in electrical engineering for extracting known wavelets
from a signal that has been contaminated by noise. This is accomplished by crosscorrelating
the signal with the wavelet. The cross-correlation of the vibroseis sweep (a
wavelet) with a recorded seismic signal is one geophysical application. Another
geophysical application is found in Kirchhoff migrations, where the summing of energy
in diffraction patterns is equivalent to a two-dimensional cross-correlation.
The basic concepts of matched filters are presented with figures illustrating the
applications in one and two dimensions.
INTRODUCTION
1D model for matched filtering
Matched filtering is a process for detecting a known piece of signal or wavelet that is
embedded in noise. The filter will maximize the signal to noise ratio (SNR) of the signal
being detected with respect to the noise.
Consider the model in Figure 1 where the input signal is s(t) and the noise, n(t). The
objective is to design a filter, h(t), that maximizes the SNR of the output, y(t).
Example with a longer chirp wavelet
The above experiment is repeated with a chirp wavelet shown in Figure 5 and the
noisy signals in Figure 6. This wavelet is longer and contains higher frequencies than the
short wavelet. These higher frequencies also define the shape of the matched filter and
will allow higher frequencies of the noise to pass as observed in the matched-filter result
2D data
The same principles apply to detecting a 2D wavelet in a 2D signal. Such applications
are used to detect potential military equipment in video or optical images. In geophysics,
it can be used to detect seismic diffractions in a process we know as seismic migration.
Note that the matched filter concept tells us that we need to cross-correlate with the same
size and amplitude of the object being detected.
Consider the noisy section, the 2D wavelet, and the matched filter result in Figure 8.
The noisy section in (a) has 100 by 100 samples, and contains one barely detectable
wavelet. The wavelet in (b) has 21 by 21 samples and is much narrower when inserted
into the grid of the noisy section. The cross-correlation in © shows a significant
improvement in the matched filter result.
CONCLUSION
Matched filters are designed to extract the maximum SNR of a signal that is buried in
noise. Some applications in geophysics are in the cross-correlation of vibroseis data, and
in Kirchhoff migration.