Douglas A. Abraham

Novel physical interpretations of K-distributed reverberation

Interest in describing and modeling envelope distributions of sea-floor backscatter has increased recently, particularly with regard to high-resolution active sonar systems. Sea-floor scattering that results in heavy-tailed-matched-filter-envelope probability distribution functions (i.e., non-Rayleigh distributions exemplified by the K, Weibull, Rayleigh mixture, or log-normal distributions) is often the limiting factor in the performance of these types of sonar systems and in this context is referred to as reverberation or acoustic clutter analogous to radar clutter. Modeling of reverberation has traditionally entailed fitting various candidate distributions to time samples of the envelope of the scattered sonar (or radar) returns. This type of descriptive analysis and the asymptotic (infinite number of scatterers) analysis defining the K-distribution yield little insight into the environmental mechanisms responsible for heavy-tailed distributions (e.g., distributions and, clustering of discrete scatterers, patchiness in geo-acoustic properties, scattering strength of scatterers, etc.) and do not allow evaluation of the effect of changing sonar system parameters such as bandwidth and beamwidth. In contrast, we derive the envelope distribution for the scattered returns starting from simple physical descriptions of the environment with a finite number of scatterers. It is shown that plausible descriptions of the environment can lead to K-distributed reverberation. This result explains, at least partially, the success of the K-distribution in the modeling of radar clutter and sonar reverberation at a variety of frequencies and scales. The finite-number-of-scatterers model is then used to predict how the shape parameter of the K-distribution will change as the beamwidth of a towed-array receiver is varied. Analysis of reverberation data from a low-frequency (450-700 Hz) active sonar system illustrates that, within our ability to estimate it, the shape parameter of the K-distribution is proportional to the beamwidth of the towed-array receiver, a result important for sonar simulation and performance prediction models. These results should prove useful in developing methods for modeling, predicting and mitigating reverberation on high-resolution sonar systems.

Statistical characterization of high-frequency shallow-water seafloor backscatter

Knowledge of the background reverberation environment is a prerequisite for the design of any target detection scheme. While the problem of understanding and predicting high-frequency background seafloor reverberation level or mean energy scattered per unit area of the seabed has received considerable attention, studies of high-frequency reverberation amplitude statistics are relatively scarce. Of these studies, many have dealt with scattering from more or less homogeneous seafloors in terms of bottom type, whereas most shallow-water areas will not be homogeneous but will have patchiness in space and time which will have a strong influence on scattered amplitude statistics. In this work, a comparison is presented between 80-kHz seafloor backscatter statistics obtained at shallow-water sites around Sardinia and Sicily. The data include measurements from several distinct bottom provinces, including sites with Posidonia Oceanica sea grass and sites covered with live shellfish. Results of the analysis are cast both in terms of mean power level or backscattering coefficient as well as of the amplitude statistics. The reverberation statistics did not always exhibit a Rayleigh probability distribution function (PDF), but exhibited statistical distributions with heavier tails. Several more appropriate models of reverberation PDF were examined in order to better describe the measured amplitude distributions. The Rayleigh mixture and the K models were found to be the most robust in describing the observed data.

Simulation of non-Rayleigh reverberation and clutter

The simulation of active sonar reverberation time series has traditionally been done using either a computationally intensive point-scatterer model or a Rayleigh-distributed reverberation-envelope model with a time-varying power level. Although adequate in scenarios where reverberation arises from a multitude of scatterers, the Rayleigh model is not representative of the target-like non-Rayleigh reverberation or clutter commonly observed with modern high-resolution sonar systems operating in shallow-water environments. In this paper, techniques for simulating non-Rayleigh reverberation are developed within the context of the finite-number-of-scatterers representation of K-distributed reverberation, which allows control of the reverberation-envelope statistics as a function of system (beamwidth and bandwidth) and environmental (scatterer density and size) parameters. To avoid the high computational effort of the point-scatterer model, reverberation is simulated at the output of the matched filter and is generated using efficient approximate methods for forming K-distributed random variables. Finite impulse response filters are used to introduce the effects of multipath propagation and the shape of the reverberation power spectrum, the latter of which requires the development of a prewarping of the K distribution parameters to control the reverberation-envelope statistics. The simulation methods presented in this paper will be useful in the testing and evaluation of active sonar signal processing algorithms, as well as for simulation-based research on the effects of the sonar system and environment on the reverberation-envelope probability density function.

Using McDaniel's model to represent non-Rayleigh reverberation

Reverberation in low-frequency active sonar systems operating in shallow water has often been observed to follow non-Rayleigh statistical distributions. McDaniels model, generalized to allow noninteger valued parameters, has shown promise as being capable of accurately representing real data with a minimal parameterization. This paper first derives an exact analytical expression for the cumulative distribution function (CDF) of the generalized McDaniel model and then compares it with numerical inversion of the characteristic function. Both methods are seen to provide adequate and equivalent precision; however the characteristic function inversion method is significantly faster. The latter CDF evaluation technique is then applied to the analysis of simulated and real data to show that, when minimal data are available, McDaniels model can more accurately represent a wide variety of non-Rayleigh reverberation than the K or Rayleigh mixture models. This result arises from the generality of McDaniels model with respect to the K-distribution (i.e., the K-distribution P/sub fa/ estimate can be dominated by model mismatch error) and to its compact parameterization with respect to the Rayleigh mixture (i.e., the Rayleigh mixture model P/sub fa/ estimate is usually dominated by parameter estimation error).

Reverberation envelope statistics and their dependence on sonar bandwidth and scattering patch size

Increasing transmit waveform bandwidth in an active sonar-system results in an increase in the signal-to-reverberation power ratio in reverberation-limited environments, but also changes the probability density function of the reverberation envelope. A recent model that relates a description of the sonar system and environment to the parameters of the K-distribution predicts that the shape parameter (/spl alpha/) is proportional to range, beamwidth, and the density of scattering patches on the sea floor and is inversely proportional to bandwidth. In this paper, the bandwidth relationship is examined with real data from a low-frequency active sonar system with a towed array receiver. The inverse proportionality is observed at low bandwidths as a trend away from a Rayleigh-distributed envelope (decreasing /spl alpha/) as bandwidth increases. However, a trend back toward Rayleigh reverberation (increasing /spl alpha/) is observed as bandwidth continues to increase. Hypothesizing that the increase in /spl alpha/ arises from over-resolving scattering patches in range and not in angle, the model of is extended to account for patch size relative to that of the sonar-resolution cell. The shape parameter of a moment-matched K-distribution derived from the extended model is then seen to provide a good fit to that estimated from the data.

Reliable Methods for Estimating the

Parameter estimation for the K-distribution is an essential part of the statistical analysis of non-Rayleigh sonar reverberation or clutter for performance prediction, estimation of scattering properties, and for use in signal and information processing algorithms. Computational issues associated with maximum-likelihood (ML) estimation techniques for K-distribution parameters often force the use of the method of moments(MoM). However, as often as half the time, MoM techniques will fail owing to a noninvertible equation relating the shape parameter (α) to a particular moment ratio, which is equivalent to the detection index (D) of the matched-filter envelope. In this paper, a Bayesian approach is taken in developing a MoM-based estimator for D, and therefore a, that reliably provides a solution and is less computationally demanding than the ML techniques. Analytical-approximation (AA) and bootstrap-based (BB) approaches are considered for approximating the likelihood function of D and forming a posterior mean estimator, which is compared with the standard MoM and ML techniques. Computational complexity (in the form of execution time) for the Bayes-MoM-AA estimator is on the order of the standard MoM estimator while the Bayes-MoM-BB estimator can be 1-2 orders of magnitude greater, although still less than ML techniques. Performance is seen to be better than the standard MoM approach and the ML techniques, except for very small α (<; 3) where the ML techniques remain superior. Advantages of the Bayesian approach are illustrated through the use of alternative priors, the formation of Bayesian confidence intervals, and a technique for combining estimates from multiple experiments.

K

The detection-threshold (DT) term in the sonar equation describes the signal-to-noise ratio (SNR) required to achieve a specified probability of detection (Pd) for a given probability of false alarm (Pfa). Direct evaluation of DT requires obtaining the detector threshold (ft.) as a function of Pfa and then using h while inverting the often complicated relationship between SNR and Pd. However, easily evaluated approximations to DT exist when the background additive noise or reverberation is Gaussian (i.e., has a Rayleigh-distributed envelope). While these approximations are extremely accurate for Gaussian backgrounds, they are erroneously low when the background has a heavy-tailed probability density function. In this paper, it is shown that by obtaining h appropriately from the non-Gaussian background while approximating Pd for a target in the non-Gaussian background by that for a Gaussian background, the easily evaluated approximations to DT extend to non-Gaussian backgrounds with minimal loss in accuracy. Both fluctuating targets (FTs) and nonfluctuating targets (NFTs) are considered in Weibulland K-distributed backgrounds. While the Pd approximation for FTs is very accurate, it is coarser for NFTs, necessitating a correction factor to the DT approximations.

-Distribution Shape Parameter

Active sonar systems have recently been developed using larger arrays and broad-band sources to counter the detrimental effects of reverberation in shallow-water operational areas. Increasing array size and transmit waveform bandwidth improve the signal-to-noise ratio-and-reverberation power ratio (SNR) after matched filtering and beamforming by reducing the size of the range-bearing resolution cell and, thus, decreasing reverberation power levels. This can also have the adverse effect of increasing the tails of the probability density function (pdf) of the reverberation envelope, resulting in an increase in the probability of a false alarm. Using a recently developed model relating the number of scatterers in a resolution cell to a K-distributed reverberation envelope, the effect of increasing bandwidth (i.e., reducing the resolution cell size) on detection performance is examined for additive nonfluctuating and fluctuating target models. The probability of detection for the two target models is seen to be well approximated by that for a shifted gamma variate with matching moments. The approximations are then used to obtain the SNR required to meet a probability of detection and false-alarm performance specification (i.e., the detection threshold). The required SNR is then used to determine that, as long as the target and scatterers are not over-resolved, decreasing the size of the resolution cell always results in an improvement in performance. Thus, the increase in SNR obtained by increasing bandwidth outweighs the accompanying increase in false alarms resulting from heavier reverberation distribution tails for K-distributed reverberation. The amount of improvement is then quantified by the signal excess, which is seen to be as low as one decibel per doubling of bandwidth when the reverberation is severely non-Rayleigh, as opposed to the expected 3-dB gain when the reverberation is Rayleigh distributed.

Detection-Threshold Approximation for Non-Gaussian Backgrounds

Active sonar systems operating in shallow-water environments are often faced with high numbers of false alarms, generically referred to as clutter, arising from among other sources bottom scattering that results in heavy tails in the matched filter envelope probability density function compared with the Rayleigh distribution. In this paper, the effect of multipath propagation on the envelope statistics (i.e., the disparity from the Rayleigh distribution) is modeled through the use of the -distribution where the shape and scale parameters are formed from the autocorrelation function of the transmit waveform, the multipath structure, and the strength and spatial density of the bottom scatterers. Use of the -distribution is justified by showing that it is the limiting distribution of the sum of independent but not identically distributed -distributed random variables, which is representative of multipath when the bottom produces -distributed backscatter. The shape parameter, which drives the clutter statistics, is seen to be inversely proportional to bandwidth at bandwidths low enough that the multipath is not resolved and again at bandwidths high enough that all of the paths are resolved. As has been previously reported by LePage [IEEE J. Ocean. Eng., vol. 29, no. 2, pp. 330-346, 2004], multipath is shown to make clutter statistics more Rayleigh-like, which in this analysis equates to an increase in the -distribution shape parameter. The model is used to evaluate the effect on clutter statistics of varying environmental characterizations and system configurations where it is seen that, for a constant sound-speed profile, increasing the vertical aperture of the sonar, the center frequency, or surface roughness can lead to less multipath and, therefore, a reduction in the -distribution shape parameter and an increase in the probability of false alarm.

Signal excess in K-distributed reverberation

False alarms in active sonar systems arising from physical objects in the ocean (e.g., rocks, fish, or seaweed) are often called clutter. A variety of statistical models have been proposed for representing the sonar probability of false alarm (Pfa) in the presence of clutter, including the log-normal, generalized-Pareto, Weibull, and K distributions. However, owing to the potential sparseness of the clutter echoes within the analysis window, a mixture distribution comprising one of the clutter distributions and a Rayleigh-distributed envelope (i.e., an exponentially distributed intensity) to represent diffuse background scattering and noise is proposed. Parameter-estimation techniques based on the expectation-maximization (EM) algorithm are developed for mixtures containing the aforementioned clutter distributions. While the standard EM algorithm handles the mixture containing log-normal clutter, the EM-gradient algorithm, which combines the EM algorithm with a one-step Newton optimization, is necessary for the generalized-Pareto and Weibull cases. The mixture containing K -distributed clutter requires development of a variant of the EM algorithm exploiting method-of-moments parameter estimation. Evaluation of three midfrequency active-sonar data examples, spanning mildly to very heavy-tailed Pfa, illustrates that the mixture models provide a better fit than single-component models. As might be expected, inference on clutter-source scattering based on the shape parameter of the clutter distribution is shown to be less biased using the mixture model compared with a single-component distribution when the data contain both clutter echoes and diffuse background scattering or noise.