Covariance Modifications to Subspace Bases
Adaptive signal processing algorithms that rely upon representations of signal and noise subspaces often require updates to those representations when new data become available. Subspace representations frequently are estimated from available data with singular value (SVD) decompositions. Subspace updates require modifications to these decompositions. Updates can be performed inexpensively provided they are low-rank. A substantial literature on SVD updates exists, frequently focusing on rank-1 updates (see e.g. [Karasalo, 1986; Comon and Golub, 1990, Badeau, 2004]). In these methods, data matrices are modified by addition or deletion of a row or column, or data covariance matrices are modified by addition of the outer product of a new vector. A recent paper by Brand [2006] provides a general and efficient method for arbitrary rank updates to an SVD. The purpose of this note is to describe a closely-related method for applications where right singular vectors are not required. This note also describes the SVD updates to a particular scenario of interest in seismic array signal processing. The particular application involve updating the wideband subspace representation used in seismic subspace detectors [Harris, 2006]. These subspace detectors generalize waveform correlation algorithms to detect signals that lie in a subspace of waveforms of dimension d {ge} 1. They potentially are of interest because they extend the range of waveform variation over which these sensitive detectors apply. Subspace detectors operate by projecting waveform data from a detection window into a subspace specified by a collection of orthonormal waveform basis vectors (referred to as the template). Subspace templates are constructed from a suite of normalized, aligned master event waveforms that may be acquired by a single sensor, a three-component sensor, an array of such sensors or a sensor network. The template design process entails constructing a data matrix whose columns contain the master event waveform data, then performing a singular value decomposition on the data matrix to extract an orthonormal basis for the waveform suite. The template typically is comprised of a subset of the left singular vectors corresponding to the larger singular values. The application involves updating a subspace template when new data become available, i.e. when new defining events are detected for a particular source. It often is the case that the waveforms corresponding to a particular source drift over time [Harris, 2001]. The Green's functions describing propagation can be altered because of changes in the source region. For example, if the source is a mine, signals from explosions may change gradually as a pit is extended (the source moves) or the scattering topography is altered by excavation. This motivates a tracking adjustment to a subspace representation. This note also comments on SVD updates for a related problem. In realistic pipeline operations it often is the case that data from one or more channels of an array are unusable (dead channels, channels with prolific dropouts, etc.). In such cases it is desirable to modify an array subspace template to operate on data only from the remaining usable channels. Furthermore, it is desirable to modify the templates directly without recourse to the original data matrix. Usually the template design process is separate from the application of the template in a detector to a continuous data stream. Consequently, the original data matrix may not be available for template modification at detector run time.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 945871
- Report Number(s):
- LLNL-TR-409155; TRN: US200904%%181
- Country of Publication:
- United States
- Language:
- English
Similar Records
Singular value decomposition utilizing parallel algorithms on graphical processors
NORSAR Final Scientific Report Adaptive Waveform Correlation Detectors for Arrays: Algorithms for Autonomous Calibration