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Tollefsen, D, Gerstoft P, Hodgkiss WS.  2017.  Multiple-array passive acoustic source localization in shallow water. Journal of the Acoustical Society of America. 141:1501-1513.   10.1121/1.4976214   AbstractWebsite

This paper considers concurrent matched-field processing of data from multiple, spatially-separated acoustic arrays with application to towed-source data received on two bottom-moored horizontal line arrays from the SWellEx-96 shallow water experiment. Matched-field processors are derived for multiple arrays and multiple-snapshot data using maximum-likelihood estimates for unknown complex-valued source strengths and unknown error variances. Starting from a coherent processor where phase and amplitude is known between all arrays, likelihood expressions are derived for various assumptions on relative source spectral information (amplitude and phase at different frequencies) between arrays and from snapshot to snapshot. Processing the two arrays with a coherent-array processor (with inter-array amplitude and phase known) or with an incoherent-array processor (no inter-array spectral information) both yield improvements in localization over processing the arrays individually. The best results with this data set were obtained with a processor that exploits relative amplitude information but not relative phase between arrays. The localization performance improvement is retained when the multiple-array processors are applied to short arrays that individually yield poor performance. (C) 2017 Acoustical Society of America.

Gemba, KL, Hodgkiss WS, Gerstoft P.  2017.  Adaptive and compressive matched field processing. Journal of the Acoustical Society of America. 141:92-103.   10.1121/1.4973528   AbstractWebsite

Matched field processing is a generalized beamforming method that matches received array data to a dictionary of replica vectors in order to locate one or more sources. Its solution set is sparse since there are considerably fewer sources than replicas. Using compressive sensing (CS) implemented using basis pursuit, the matched field problem is reformulated as an underdetermined, convex optimization problem. CS estimates the unknown source amplitudes using the replica dictionary to best explain the data, subject to a row-sparsity constraint. This constraint selects the best matching replicas within the dictionary when using multiple observations and/or frequencies. For a single source, theory and simulations show that the performance of CS and the Bartlett processor are equivalent for any number of snapshots. Contrary to most adaptive processors, CS also can accommodate coherent sources. For a single and multiple incoherent sources, simulations indicate that CS offers modest localization performance improvement over the adaptive white noise constraint processor. SWellEx-96 experiment data results show comparable performance for both processors when localizing a weaker source in the presence of a stronger source. Moreover, CS often displays less ambiguity, demonstrating it is robust to data-replica mismatch. (C) 2017 Acoustical Society of America.

Tippmann, JD, Sarkar J, Verlinden CMA, Hodgkiss WS, Kuperman WA.  2016.  Toward ocean attenuation tomography: Determining acoustic volume attenuation coefficients in seawater using eigenray amplitudes. Journal of the Acoustical Society of America. 140:EL247-EL250.   10.1121/1.4962348   AbstractWebsite

A deep-water experiment in the Pacific made in situ measurements of the volume attenuation coefficients of sea water in the mid-frequency range. The frequency, temperature, salinity, pH, and pressure dependent seawater attenuation coefficients were determined using a vertical line array that received and identified over 2000 unique paths from 1200 mid-frequency 3-9 kHz LFM source transmissions at a convergence zone range and depth up to 400 m. The results show no change in attenuation coefficients in this band compared to estimates from 30-year-old models previously determined from a combination of long-range ocean acoustic and laboratory experiments. The inversion also explores the feasibility of ocean acoustic attenuation tomography to further separate the depth-dependent chemical components responsible for the total attenuation loss through by isolating a group of deep-water refracting acoustic paths. (C) 2016 Acoustical Society of America

Cho, C, Song HC, Hodgkiss WS.  2016.  Robust source-range estimation using the array/waveguide invariant and a vertical array. The Journal of the Acoustical Society of America. 139:63-69.   10.1121/1.4939121   Abstract
Verlinden, CMA, Sarkar J, Hodgkiss WS, Kuperman WA, Sabra KG.  2015.  Passive acoustic source localization using sources of opportunity. Journal of the Acoustical Society of America. 138:EL54-EL59.   10.1121/1.4922763   AbstractWebsite

The feasibility of using data derived replicas from ships of opportunity for implementing matched field processing is demonstrated. The Automatic Identification System (AIS) is used to provide the library coordinates for the replica library and a correlation based processing procedure is used to overcome the impediment that the replica library is constructed from sources with different spectra and will further be used to locate another source with its own unique spectral structure. The method is illustrated with simulation and then verified using acoustic data from a 2009 experiment for which AIS information was retrieved from the United States Coast Guard Navigation Center Nationwide AIS database. (C) 2015 Acoustical Society of America

Song, HC, Hodgkiss WS.  2015.  Self-synchronization and spatial diversity of passive time reversal communication. Journal of the Acoustical Society of America. 137:2974-2977.   10.1121/1.4919324   AbstractWebsite

Time reversal (TR) achieves spatial and temporal focusing in complex environments. In the context of communications, passive (uplink) TR is equivalent to active (downlink) TR with the communications link being in opposite directions but with the same theoretical performance. The benefit of passive TR, however, is its natural self-synchronization, making TR communications robust and readily extended to cases involving temporal diversity and synthetic aperture communications. Self-synchronization is examined analytically and demonstrated with shallow water experimental data. In addition, the impact of spatial diversity on communications performance is investigated in terms of the aperture and element spacing of the 64-element vertical receive array. (C) 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

Tan, BA, Gerstoft P, Yardim C, Hodgkiss WS.  2015.  Change-point detection for recursive Bayesian geoacoustic inversions. Journal of the Acoustical Society of America. 137:1962-1970.   10.1121/1.4916887   AbstractWebsite

In order to carry out geoacoustic inversion in low signal-to-noise ratio (SNR) conditions, extended duration observations coupled with source and/or receiver motion may be necessary. As a result, change in the underlying model parameters due to time or space is anticipated. In this paper, an inversion method is proposed for cases when the model parameters change abruptly or slowly. A model parameter change-point detection method is developed to detect the change in the model parameters using the importance samples and corresponding weights that are already available from the recursive Bayesian inversion. If the model parameters change abruptly, a change-point will be detected and the inversion will restart with the pulse measurement after the change-point. If the model parameters change gradually, the inversion (based on constant model parameters) may proceed until the accumulated model parameter mismatch is significant and triggers the detection of a change-point. These change-point detections form the heuristics for controlling the coherent integration time in recursive Bayesian inversion. The method is demonstrated in simulation with parameters corresponding to the low SNR, 100-900 Hz linear frequency modulation pulses observed in the Shallow Water 2006 experiment [Tan, Gerstoft, Yardim, and Hodgkiss, J. Acoust. Soc. Am. 136, 1187-1198 (2014)]. (c) 2015 Acoustical Society of America.

Tan, BA, Gerstoft P, Yardim C, Hodgkiss WS.  2014.  Recursive Bayesian synthetic aperture geoacoustic inversion in the presence of motion dynamics. Journal of the Acoustical Society of America. 136:1187-1198.   10.1121/1.4892788   AbstractWebsite

A low signal to noise ratio (SNR), single source/receiver, broadband, frequency-coherent matched-field inversion procedure recently has been proposed. It exploits coherently repeated transmissions to improve estimation of the geoacoustic parameters. The long observation time improves the SNR and creates a synthetic aperture due to relative source-receiver motion. To model constant velocity source/receiver horizontal motion, waveguide Doppler theory for normal modes is necessary. However, the inversion performance degrades when source/receiver acceleration exists. Furthermore processing a train of pulses all at once does not take advantage of the natural incremental acquisition of data along with the ability to assess the temporal evolution of parameter uncertainty. Here a recursive Bayesian estimation approach is developed that coherently processes the data pulse by pulse and incrementally updates estimates of parameter uncertainty. It also approximates source/receiver acceleration by assuming piecewise constant but linearly changing source/receiver velocities. When the source/receiver acceleration exists, it is shown that modeling acceleration can reduce further the parameter estimation biases and uncertainties. The method is demonstrated in simulation and in the analysis of low SNR, 100-900 Hz linear frequency modulated (LFM) pulses from the Shallow Water 2006 experiment. (C) 2014 Acoustical Society of America.

Menon, R, Gerstoft P, Hodgkiss WS.  2014.  On the apparent attenuation in the spatial coherence estimated fromseismic arrays. Journal of Geophysical Research-Solid Earth. 119:3115-3132.   10.1002/2013jb010835   AbstractWebsite

Several recent studies have used the coherence of seismic noise between stations to retrieve the phase slowness and attenuation. However, there is considerable debate on the feasibility of attenuation retrieval, its interpretation, and its dependence on the noise directionality and has been the subject of several analytical and numerical studies. In this article, we perform a detailed analysis of the various factors that play a role in the estimation of spatial coherence and attenuation from seismic arrays using data from the Southern California Seismic Network. For instance, certain common preprocessing steps such as averaging neighboring frequencies to improve the estimate are sufficient to introduce attenuation-like effects. The presence of first-mode surface Rayleigh wave and P waves in addition to the fundamental mode in Southern California (at frequencies 0.05-0.2 Hz) suggests that the underlying spatial coherence is better modeled as a linear combination of the above wave types. Although this describes the observed coherence better than a simple zeroth-order Bessel function, the resulting phase cancelations due to the multiple seismic waves can be misconstrued as attenuation if not taken into consideration. Using simulations, we show that due to the slowness inhomogeneity, azimuthally averaging the coherence is not equivalent to homogenizing the medium and instead introduces apparent attenuation in the coherence due to interference. Trying to fit an exponential decay model to this apparent attenuation results in an attenuation coefficient which is similar to previously published results.

Yardim, C, Gerstoft P, Hodgkiss WS, Traer J.  2014.  Compressive geoacoustic inversion using ambient noise. Journal of the Acoustical Society of America. 135:1245-1255.   10.1121/1.4864792   AbstractWebsite

Surface generated ambient noise can be used to infer sediment properties. Here, a passive geoacoustic inversion method that uses noise recorded by a drifting vertical array is adopted. The array is steered using beamforming to compute the noise arriving at the array from various directions. This information is used in two different ways: Coherently (cross-correlation of upward/downward propagating noise using a minimum variance distortionless response fathometer), and incoherently (bottom loss vs frequency and angle using a conventional beamformer) to obtain the bottom properties. Compressive sensing is used to invert for the number of sediment layer interfaces and their depths using coherent passive fathometry. Then the incoherent bottom loss estimate is used to refine the sediment thickness, sound speed, density, and attenuation values. Compressive sensing fathometry enables automatic determination of the number of interfaces. It also tightens the sediment thickness priors for the incoherent bottom loss inversion which reduces the search space. The method is demonstrated on drifting array data collected during the Boundary 2003 experiment. (C) 2014 Acoustical Society of America

Carriere, O, Gerstoft P, Hodgkiss WS.  2014.  Spatial filtering in ambient noise interferometry. Journal of the Acoustical Society of America. 135:1186-1196.   10.1121/1.4863658   AbstractWebsite

Theoretically, the empirical Green's function between a pair of receivers can be extracted from the cross correlation of the received diffuse noise. The diffuse noise condition rarely is met in the ocean and directional sources may bias the Green's function. Here matrix-based spatial filters are used for removing unwanted contributions in the cross correlations. Two methods are used for solving the matrix filter design problem. First a matrix least-square problem is solved with a low-rank approximation of the pseudo-inverse, here, derived for linear and planar arrays. Second, a convex optimization approach is used to solve the design problem reformulated with ad hoc constraints. The spatial filter is applied to real-data cross correlations of elements from a linear array to attenuate the contribution of a discrete interferer. In the case of a planar array and simulated data, a spatial filter enables a passive upgoing/downgoing wavefield separation along with an efficient rejection of horizontally propagating noise. The impact of array size and frequency band on the filtered cross correlations is discussed. (C) 2014 Acoustical Society of America.

Menon, R, Gerstoft P, Hodgkiss WS.  2013.  Effect of Medium Attenuation on the Asymptotic Eigenvalues of Noise Covariance Matrices. Ieee Signal Processing Letters. 20:435-438.   10.1109/lsp.2013.2250500   AbstractWebsite

Covariance matrices of noise models are used in signal and array processing to study the effect of various noise fields and array configurations on signals and their detectability. Here, the asymptotic eigenvalues of noise covariance matrices in 2-D and 3-D attenuating media are derived. The asymptotic eigenvalues are given by a continuous function, which is the Fourier transform of the infinite sequence formed by sampling the spatial coherence function. The presence of attenuation decreases the value of the large eigenvalues and raises the value of the smaller eigenvalues (compared to the attenuation free case). The eigenvalue density of the sample covariance matrix also shows variation in shape depending on the attenuation, which potentially could be used to retrieve medium attenuation properties from observations of noise.

Lani, SW, Sabra KG, Hodgkiss WS, Kuperman WA, Roux P.  2013.  Coherent processing of shipping noise for ocean monitoring. Journal of the Acoustical Society of America. 133:EL108-EL113.   10.1121/1.4776775   AbstractWebsite

Ambient noise was recorded on two vertical line arrays (VLAs) separated by 450m and deployed in shallow water (depth similar to 150 m) off San Diego, CA continuously for 6 days. Recordings were dominated by non-stationary and non-uniform broadband shipping noise (250 Hz to 1.5 kHz). Stable coherent noise wavefronts were extracted from ambient noise correlations between the VLAs during all 6 days by mitigating the effect of discrete shipping events and using array beamforming with data-derived steering vectors. This procedure allows the tracking of arrival-time variations of these coherent wavefronts during 6 days and may help in developing future passive acoustic tomography systems. (C) 2013 Acoustical Society of America

Cho, SE, Song HC, Hodgkiss WS.  2013.  Multiuser acoustic communications with mobile users. Journal of the Acoustical Society of America. 133:880-890.   10.1121/1.4773267   AbstractWebsite

A multiuser receiver is developed that is capable of separating receptions from independent, mobile users whose transmissions overlap in both time and frequency. With respect to any one user's Doppler corrected signal, the other communication signals appear as multiple-access interference distributed across the Doppler dimension. A previously developed receiver composed of an adaptive time-reversal processor embedded within a successive interference cancellation framework is limited to stationary users. This paper extends the receiver to properly remove the interference from moving sources by modeling the effects of Doppler through the interference cancellation receiver. The combined receiver has the ability to remove interference in both the temporal and spatial domains, and this property is shown to be preserved even when users are in motion. When applied to data collected during a recent shallow water experiment (KAM11), the receiver is shown to be capable of separating packets in a two user system where one user is moving while the other is stationary. (C) 2013 Acoustical Society of America. []

Song, HC, Hodgkiss WS.  2013.  Efficient use of bandwidth for underwater acoustic communication. Journal of the Acoustical Society of America. 134:905-908.   10.1121/1.4812762   AbstractWebsite

In a recent shallow water experiment, acoustic communication transmissions were carried out over the 10 to 32 kHz band in similar to 100m deep water over a 3 km range. A natural question is how best to utilize that bandwidth. In one multiband approach discussed previously, the band was divided into four smaller subbands that were processed independently using time reversal decision-feedback equalizers (TR-DFEs). This letter presents a complementary wideband approach using data from the same experiment achieving a data rate of up to 60 kbits/s with 32 quadrature amplitude modulation. These results suggest that a wideband approach can be beneficial in terms of spectral efficiency with modest computational complexity using a TR-DFE. (C) 2013 Acoustical Society of America.

Tan, BA, Gerstoft P, Yardim C, Hodgkiss WS.  2013.  Broadband synthetic aperture geoacoustic inversion. Journal of the Acoustical Society of America. 134:312-322.   10.1121/1.4807567   AbstractWebsite

A typical geoacoustic inversion procedure involves powerful source transmissions received on a large-aperture receiver array. A more practical approach is to use a single moving source and/or receiver in a low signal to noise ratio (SNR) setting. This paper uses single-receiver, broadband, frequency coherent matched-field inversion and exploits coherently repeated transmissions to improve estimation of the geoacoustic parameters. The long observation time creates a synthetic aperture due to relative source-receiver motion. This approach is illustrated by studying the transmission of multiple linear frequency modulated (LFM) pulses which results in a multi-tonal comb spectrum that is Doppler sensitive. To correlate well with the measured field across a receiver trajectory and to incorporate transmission from a source trajectory, waveguide Doppler and normal mode theory is applied. The method is demonstrated with low SNR, 100-900 Hz LFM pulse data from the Shallow Water 2006 experiment. (C) 2013 Acoustical Society of America.

Karimian, A, Yardim C, Haack T, Gerstoft P, Hodgkiss WS, Rogers T.  2013.  Towards assimilation of atmospheric surface layer using numerical weather prediction and radar clutter observations. Journal of Applied Meteorology and Climatology. : American Meteorological Society   10.1175/JAMC-D-12-0320.1   AbstractWebsite

Radio wave propagation on low-altitude paths over the ocean above 2 GHz is significantly affected by negative refractivity gradients in the atmospheric surface layer (ASL) which form what is often referred to as an evaporation duct (ED). Refractivity from clutter (RFC) is an inversion approach for estimation of the refractivity profile from radar clutter, and RFC- ED refers to its implementation for the case of evaporation ducts. An approach for fusing RFC-ED output with evaporation duct characterization based on ensemble forecasts from a numerical weather prediction (NWP) model is examined here. Three conditions of air sea temperature difference (ASTD) are examined. Synthetic radar clutter observations are generated using the advanced propagation model (APM). The impact of ASTD on the evaporation duct refractivity profile, atmospheric parameter inversion, and propagation factor distributions are studied. Relative humidity at a reference height and ASTD are identified as state variables. Probability densities from NWP ensembles, RFC-ED, and joint inversions are compared. It is demonstrated that characterization of the near surface atmosphere by combining RFC-ED and NWP reduces the estimation uncertainty of ASTD and relative humidity in an evaporation duct, with respect to using either method alone.

Yardim, C, Gerstoft P, Hodgkiss WS.  2013.  Particle smoothers in sequential geoacoustic inversion. The Journal of the Acoustical Society of America. 134:971-981. AbstractWebsite

Sequential Bayesian methods such as particle filters have been used to track a moving source in an unknown and space/time-evolving ocean environment. These methods treat both the source and the ocean parameters as non-stationary unknown random variables and track them via the multivariate posterior probability density function. Particle filters are numerical methods that can operate on nonlinear systems with non-Gaussian probability density functions. Particle smoothers are a natural extension to these filters. A smoother is appropriate in applications where data before and after the time of interest are readily available. Both past and “future” measurements are exploited in smoothers, whereas filters just use past measurements. Geoacoustic and source tracking is performed here using two smoother algorithms, the forward-backward smoother and the two-filter smoother. Smoothing is demonstrated on experimental data from both the SWellEx-96 and SW06 experiments where the parameter uncertainty is reduced relative to just filtering alone.

Roux, P, Kuperman WA, Cornuelle BD, Aulanier F, Hodgkiss WS, Song HC.  2013.  Analyzing sound speed fluctuations in shallow water from group-velocity versus phase-velocity data representation. Journal of the Acoustical Society of America. 133:1945-1952.   10.1121/1.4792354   AbstractWebsite

Data collected over more than eight consecutive hours between two source-receiver arrays in a shallow water environment are analyzed through the physics of the waveguide invariant. In particular, the use of vertical arrays on both the source and receiver sides provides source and receiver angles in addition to travel-times associated with a set of eigenray paths in the waveguide. From the travel-times and the source-receiver angles, the eigenrays are projected into a group-velocity versus phase-velocity (Vg-Vp) plot for each acquisition. The time evolution of the Vg-Vp representation over the 8.5-h long experiment is discussed. Group speed fluctuations observed for a set of eigenrays with turning points at different depths in the water column are compared to the Brunt-Vaisala frequency. (C) 2013 Acoustical Society of America.

Fried, SE, Walker SC, Hodgkiss WS, Kuperman WA.  2013.  Measuring the effect of ambient noise directionality and split-beam processing on the convergence of the cross-correlation function. Journal of the Acoustical Society of America. 134:1824-1832.   10.1121/1.4816490   AbstractWebsite

Measurements of ambient noise have been used to infer information about the ocean acoustic environment. In recent years the correlation of ambient noise has been shown to give estimates of the travel time of acoustic paths between the sensors recording the noise. A number of issues affect the results of the noise correlation. This paper presents the results of noise correlation of the two horizontally separated arrays of sensors in the 2010 ambient noise experiment. Using the experimental data, the effects on the convergence of the noise correlation are examined with respect to the size and shape of the arrays, the length of time used, and the directionality of the noise field. (C) 2013 Acoustical Society of America.

Karimian, A, Yardim C, Gerstoft P, Hodgkiss WS, Barrios AE.  2012.  Multiple grazing angle sea clutter modeling. IEEE Transactions on Antennas and Propagation. 60:4408-4417.   10.1109/tap.2012.2207033   AbstractWebsite

Radar clutter in a non-standard atmosphere usually is modeled based on a single grazing angle at each range. Instead, the angular distribution of incident power can be used to obtain a more accurate model of the clutter. Angular spectral estimation provides the grazing angle distribution of propagating power. However, a large gradient in the refractivity profile, e. g., an evaporation duct, distorts plane wave propagation which in turn violates assumptions of plane wave spectral estimation. Ray tracing is used in these situations, but has its own limitations (e. g., shadow zones). We suggest using curved wave spectral estimation (CWS) that yields reliable results for any refractivity profile, in contrast to plane wave spectral estimation. CWS is used to derive multiple grazing angle clutter, a model for ocean surface clutter in the microwave region that depends on all incident angles at each range and their corresponding powers.

Gerstoft, P, Menon R, Hodgkiss WS, Mecklenbrauker CF.  2012.  Eigenvalues of the sample covariance matrix for a towed array. Journal of the Acoustical Society of America. 132:2388-2396.   10.1121/1.4746024   AbstractWebsite

It is well known that observations of the spatial sample covariance matrix (SCM, also called the cross-spectral matrix) reveal that the ordered noise eigenvalues of the SCM decay steadily, but common models predict equal noise eigenvalues. Random matrix theory (RMT) is used to derive and discuss properties of the eigenvalue spectrum of the data SCM for linear arrays, with an application to ocean acoustic data. Noise on the array is considered either incoherent or propagating acoustic noise that is coherent across the array. Using conventional three-dimensional or two-dimensional isotropic noise models with full or snapshot-deficient observations, realizations of the SCM eigenvalues are explained using RMT. Deep-water towed-array data are analyzed and it is shown that the eigenvalues of the SCM compare well with theory. It is demonstrated how RMT can be applied to study eigenvalue spectrum estimation as dependent on array properties (element spacing to wavelength ratio) and data sampling (snapshots). Apart from explaining the observed noise eigenvalue spectrum, the improved model of the eigenvalue spectrum has important applications in array signal processing. (C) 2012 Acoustical Society of America. []

Menon, R, Gerstoft P, Hodgkiss WS.  2012.  Cross-correlations of diffuse noise in an ocean environment using eigenvalue based statistical inference. Journal of the Acoustical Society of America. 132:3213-3224.   10.1121/1.4754558   AbstractWebsite

Cross-correlations of diffuse noise fields can be used to extract environmental information. The influence of directional sources (usually ships) often results in a bias of the travel time estimates obtained from the cross-correlations. Using an array of sensors, insights from random matrix theory on the behavior of the eigenvalues of the sample covariance matrix (SCM) in an isotropic noise field are used to isolate the diffuse noise component from the directional sources. A sequential hypothesis testing of the eigenvalues of the SCM reveals eigenvalues dominated by loud sources that are statistical outliers for the assumed diffuse noise model. Travel times obtained from cross-correlations using only the diffuse noise component (i.e., by discarding or attenuating the outliers) converge to the predicted travel times based on the known array sensor spacing and measured sound speed at the site and are stable temporally (i.e., unbiased estimates). Data from the Shallow Water 2006 experiment demonstrates the effectiveness of this approach and that the signal-to-noise ratio builds up as the square root of time, as predicted by theory. (C) 2012 Acoustical Society of America. []

Menon, R, Gerstoft P, Hodgkiss WS.  2012.  Asymptotic eigenvalue density of noise covariance matrices. IEEE Transactions on Signal Processing. 60:3415-3424.   10.1109/tsp.2012.2193573   AbstractWebsite

The asymptotic eigenvalues are derived for the true noise covariance matrix (CM) and the noise sample covariance matrix (SCM) for a line array with equidistant sensors in an isotropic noise field. In this case, the CM in the frequency domain is a symmetric Toeplitz sinc matrix which has at most two distinct eigenvalues in the asymptotic limit of an infinite number of sensors. Interestingly, for line arrays with interelement spacing less than half a wavelength, the CM turns out to be rank deficient. The asymptotic eigenvalue density of the SCM is derived using random matrix theory (RMT) for all ratios of the interelement spacing to the wavelength. When the CM has two distinct eigenvalues, the eigenvalue density of the SCM separates into two distinct lobes as the number of snapshots is increased. These lobes are centered at the two distinct eigenvalues of the CM. The asymptotic results agree well with analytic solutions and simulations for arrays with a small number of sensors.

Cho, SE, Song HC, Hodgkiss WS.  2012.  Asynchronous multiuser underwater acoustic communications (L). Journal of the Acoustical Society of America. 132:5-8.   10.1121/1.4726029   AbstractWebsite

An asynchronous multiuser system is proposed to support multiple-access underwater communications without the use of code-division multiple-access or a feedback channel. The rich multipath channels experienced by spatially separated users will be sufficient to ensure separation of collided packets at the base station. The iterative receiver will employ a combination of adaptive time-reversal processing, matching pursuit, and successive interference cancellation in a block-wise fashion to achieve multiuser separability. Data collected during the KAM11 experiment are used to illustrate the system's capability in a dynamic, time-varying environment. (C) 2012 Acoustical Society of America. []