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Title: Subspace Detectors: Efficient Implementation

The optimum detector for a known signal in white Gaussian background noise is the matched filter, also known as a correlation detector [Van Trees, 1968]. Correlation detectors offer exquisite sensitivity (high probability of detection at a fixed false alarm rate), but require perfect knowledge of the signal. The sensitivity of correlation detectors is increased by the availability of multichannel data, something common in seismic applications due to the prevalence of three-component stations and arrays. When the signal is imperfectly known, an extension of the correlation detector, the subspace detector, may be able to capture much of the performance of a matched filter [Harris, 2006]. In order to apply a subspace detector, the signal to be detected must be known to lie in a signal subspace of dimension d {ge} 1, which is defined by a set of d linearly-independent basis waveforms. The basis is constructed to span the range of signals anticipated to be emitted by a source of interest. Correlation detectors operate by computing a running correlation coefficient between a template waveform (the signal to be detected) and the data from a window sliding continuously along a data stream. The template waveform and the continuous data stream may bemore » multichannel, as would be true for a three-component seismic station or an array. In such cases, the appropriate correlation operation computes the individual correlations channel-for-channel and sums the result (Figure 1). Both the waveform matching that occurs when a target signal is present and the cross-channel stacking provide processing gain. For a three-component station processing gain occurs from matching the time-history of the signals and their polarization structure. The projection operation that is at the heart of the subspace detector can be expensive to compute if implemented in a straightforward manner, i.e. with direct-form convolutions. The purpose of this report is to indicate how the projection can be computed efficiently for continuous multichannel seismic data. The speed of the calculation is significant as it may become desirable to deploy subspace detectors numbering in the thousands. One application contemplated for these detectors is as screens against signals from repeating sources such as mines or aftershocks of large earthquakes. With many tens of stations and potentially hundreds of sources to screen, efficient implementations are desirable. Speed, of course, can be achieved by procuring faster computers or special-purpose hardware. The approach we examine here is the development of two efficient algorithms that can make the calculations run faster on any machine. In the first section, we describe the subspace detector as we use it for the detection of repeating seismic events, defining terms and the parameterization used in succeeding sections. This section also reviews how the correlation computations central to the matched filter and subspace detectors can be implemented as a collection of convolution operations. Convolution algorithms using fast Fourier transforms, such as the overlap-add and overlap-save methods, have long been known as efficient implementations of discrete-time finite-impulse-response filters [e.g. Oppenheim and Schafer, 1975]. These may be extended in a straightforward manner to implement multichannel correlation detectors. In the second section, we describe how multichannel data can be multiplexed to compute the required convolutions with a single pair of FFT operations instead of a pair for each channel. This approach increases speed approximately twofold. Seismic data, almost invariably, are oversampled. This characteristic provides an opportunity for increased efficiency by decimating the data prior to performing the correlation calculations. In the third section, we describe a bandpass transformation of the data that allows a more aggressive decimation of the data without significant loss of fidelity in the correlation calculation. The transformation computes a complex-analytic representation for the template waveforms and the multichannel data, followed by a demodulation for both to base-band (i.e. a single band around zero frequency). This approach provides a factor of two to four increase in speed depending on the details of data sampling rate and the desired pass band of the correlation calculation. The approaches described in the last two sections can be used simultaneously to compound efficiencies.« less
Publication Date:
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Report Number(s):
TRN: US200708%%101
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Resource Type:
Technical Report
Research Org:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA
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Country of Publication:
United States