Robust estimation of the cross‐spectral density matrix from ambient noise using random matrix theory.

2010 ◽  
Vol 128 (4) ◽  
pp. 2302-2302
Author(s):  
R. Menon ◽  
P. Gerstoft ◽  
W. S. Hodgkiss
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Robust Estimation ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Ambient Noise ◽  
Matrix Theory ◽  
Spectral Density Matrix ◽  
Cross Spectral Density ◽  
The Cross

scholarly journals Matrix Methods for Estimating the Coherence Functions from Estimates of the Cross-Spectral Density Matrix

1996 ◽  
Vol 3 (4) ◽  
pp. 237-246 ◽  
Author(s):  
D.O. Smallwood
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Multiple Input Multiple Output ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Cross Spectral Density ◽  
Input Multiple Output ◽  
The Cross ◽  
Value Decomposition ◽  
Physical Order

It is shown that the usual method for estimating the coherence functions (ordinary, partial, and multiple) for a general multiple-input! multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross-spectral density matrix of the input and output records. The results can be equivalently obtained using singular value decomposition (SVD) of the cross-spectral density matrix. Using SVD suggests a new form of fractional coherence. The formulation as a SVD problem also suggests a way to order the inputs when a natural physical order of the inputs is absent.

scholarly journals Generation of Stationary Non-Gaussian Time Histories with a Specified Cross-spectral Density

1997 ◽  
Vol 4 (5-6) ◽  
pp. 361-377 ◽  
Author(s):  
David O. Smallwood
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Probability Density ◽  
Complex Frequency ◽  
Spectral Density Matrix ◽  
Cross Spectral Density ◽  
Time Histories ◽  
The Cross ◽  
Non Gaussian ◽  
Skewness And Kurtosis

The paper reviews several methods for the generation of stationary realizations of sampled time histories with non-Gaussian distributions and introduces a new method which can be used to control the cross-spectral density matrix and the probability density functions (pdfs) of the multiple input problem. Discussed first are two methods for the specialized case of matching the auto (power) spectrum, the skewness, and kurtosis using generalized shot noise and using polynomial functions. It is then shown that the skewness and kurtosis can also be controlled by the phase of a complex frequency domain description of the random process. The general case of matching a target probability density function using a zero memory nonlinear (ZMNL) function is then covered. Next methods for generating vectors of random variables with a specified covariance matrix for a class of spherically invariant random vectors (SIRV) are discussed. Finally the general case of matching the cross-spectral density matrix of a vector of inputs with non-Gaussian marginal distributions is presented.

On surface-generated ambient noise in an upward refracting ocean

1994 ◽  
Vol 346 (1680) ◽  
pp. 321-352 ◽  
Keyword(s):  
Spectral Density ◽  
Green’S Function ◽  
Normal Mode ◽  
Green's Function ◽  
Sound Speed ◽  
Ambient Noise ◽  
Direct Path ◽  
Noise Field ◽  
Cross Spectral Density ◽  
The Cross

Naturally generated ambient noise in the ocean is created by breaking waves, spray and precipitation. Each of these mechanisms produces a pulse of sound that propagates down into the depths of the ocean, and the superposition of all such pulses from across the whole sea surface constitutes the ambient noise field. Since the noise is a stochastic phenomenon, its properties are described in terms of statistical quantities, the most useful being the power spectral density at a point and the cross-spectral density between two points in the field. If these second-order statistical measures are independent of absolute position, the noise field is said to be spatially homogeneous. In the rare case of an isovelocity, deep ocean, the noise field at depths greater than a wavelength or so beneath the surface is spatially homogeneous, consisting of a random superposition of plane waves. A non-uniform sound speed profile, however, introduces wave-front curvature which modifies the situation significantly. the noise exhibits strong spatial homogeneity over length scales that are comparable with the apertures of typical acoustic arrays. Apart from the implications with regard to array performance, this is important in connection with certain aspects of acoustical oceanography, whereby information on the oceanographic environment is extracted from the noise field (Buckingham et al. 1992). Such information is accessible only if the structure of the noise field is well understood. The problem lies in determining the spatial and spectral properties of the noise in a profile. Fundamental to the noise analysis is the Green’s function for the channel, which characterizes the propagation conditions; and yet for most non-uniform sound speed profiles the analysis of the Green's function is intractable. However, there is one profile, designated the inverse-square profile, for which a complete, exact solution for the field has been developed (Buckingham 1991). The inverse-square profile is monotonic increasing with depth, giving rise to upward refractive propagation. Such a profile is found in several ocean environments: the polar oceans, where the temperature and hence the sound speed show a minimum at the surface; the mixed surface layer, extending to a depth of order 100 m in the open ocean; and the ocean-surface bubble layer, occupying the first ten metres or so beneath the surface. An analysis of the noise field in the presence of an inverse square profile, based on the solution for the Green’s function, shows that the cross-spectral density of the noise in the vertical consists of three components: a normal mode sum, representing noise originating largely in distant sources; a direct path contribution, from sources that are more or less overhead; and a near-surface term that is negligible at depths greater than a wavelength. In the theoretical noise spectrum , the normal mode and direct path components are prominent, dominating, respectively, at low and high frequencies. The cross-over frequency depends on the parameters of the profile and attenuation in the medium, but for polar oceans is in the region of several hundred hertz. At a much lower frequency, around 10 Hz, where the polar profile ceases to support normal mode propagation, a minimum appears in the theoretical spectrum . This is the result of a very rapid fall off in the normal mode component of the noise and a slow rise of the direct path component with decreasing frequency. Each of the three components of the vertical cross-spectral density exhibits strong spatial inhomogeneity. This is exemplified by the dramatic dependence of the cross-spectrum on both the mean depth of the sensors and frequency. Although such behaviour adds complexity to the structure of the noise field, this could be advantageous since it allows the possibility of performing inversions on noise cross-spectral data to determine properties of the medium. Recent measurements of low-frequency (50-2000 Hz) and very low-frequency (5-200 Hz) ambient noise spectra in the marginal ice zone of the Greenland Sea, where the sound speed profile is of the inverse-square form, have been compared with the predictions of the new noise theory. There is evidence in the measured spectra that both the normal mode and direct path components of the noise are present with the predicted relative levels. A minimum around 10 Hz is a ubiquitous feature of the VLF spectra, and the LF spectra show a change of slope close to 400 Hz, both of which are in accord with the theory. Along the ice edge a highly non-uniform (spatial) distribution of energetic sources is known to be present, whose effects in the observed spectra are consistent with arguments developed from the inverse-square noise analysis.

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