An approximate formula for the waveguide invariant in the shallow-water Pekeris model, analytical insights and experimental evidence.

The "invariant parameter" called "beta" is often useful for describing the acoustic interference pattern in a waveguide. For some shallow water waveguides, the measured acoustic intensity might contain contributions from several propagating acoustic modes. For each pair of these modes, a different value for the waveguide invariant might apply. If the acoustic intensity is measured over some distributed aperture and finite bandwidth, it may become difficult to assign a single value to beta. In the present work, the waveguide invariant is treated as a distribution. An algorithm for estimating this distribution for a general measurement geometry is developed. The algorithm is exercised for different classes of shallow water waveguides. When the propagation is dominated by modes interacting with the sea surface, the distribution can be sharply peaked. For cases where the sound speed profile creates a duct, the distribution is more diffuse. The effects of source/receiver depth, range, bandwidth and bottom attenuation are quantified.

Waveguide invariant analysis is a useful tool for understanding spectral interference patterns from broadband sources in shallow water waveguides. These interference patterns (in the form of intensity striations) were observed in time-frequency plots of surface ships passing a horizontal line array located along the continental shelf off southeast Florida during a 2007 acoustical experiment. Previous work has shown that results from the Range Dependent Waveguide Invariant Distribution (RaDWID) method match well with simulations from parabolic equation acoustical models and require significantly less computation time; however, work to understand the processor's ability to recreate real data was incomplete. We will discuss how RaDWID processing can be applied to ship-radiated broadband noise to model spectral interference patterns. The implementation of the processor on these data including handling of environmental parameter uncertainty, broadband source power spectrum, and environmental features will also be discussed. [Work supported by ONR Undersea Signal Processing.]

Passive acoustics is a versatile tool for maritime situational awareness, enabling applications such as source detection and localization, marine mammal tracking, and geoacoustic inversion. This study focuses on estimating the range between an acoustic receiver and a transiting ship in an acoustically range-independent shallow water environment. Here, acoustic propagation can be modeled by a set of modes that are determined by the shallow water waveguide and seabed characteristics. These modes are dispersive, with phase and group velocities varying with frequency, and their interference produces striation patterns that depend on range and frequency in single-hydrophone spectrograms. These striation patterns can often be characterized by the waveguide invariant (WI), a single parameter describing the waveguide's properties. This paper presents a statistical model and corresponding WI-based range estimation approach using a single hydrophone, leveraging broadband and tonal sounds from a transiting ship. Using data from the Seabed Characterization Experiment 2017 (SBCEX17), the method was evaluated on two commercial ships under different environmental conditions and frequency bands. Range estimation errors remained below ±4% up to 62 km in the best case, with robust performance demonstrated in the 40-60 Hz band.

Active sonar signal processing in shallow water has proven to be a challenging problem due to the strong interaction of sound with the boundaries of the channel and the dependence on typically unknown environmental parameters. This has motivated research on properties of acoustic propagation that are not sensitive to those factors, such as the waveguide invariance. The invariance principle has found application in passive sonar signal processing by relating the frequency content of a broadband source to the range between source and receiver. More recently, experimental evidence has suggested that a similar structure exists for active sonar. This structure provides additional information about the location of a target, and information can be exploited in sonar processing algorithms such as target tracking. Data from several sea experiments have been analyzed to determine the behavior of the active invariance, and tank experiments have been designed to confirm the presence of the range‐frequency structure in signals reflected by a moving target. This presentation provides an overview of the active invariance phenomena and describes the performance of a target tracking formulation that incorporates invariance structure into the state space representation for improved performance.

A technique for characterizing range‐dependent shallow water waveguides is described. The method consists of determining the beamformed output of a horizontal array over short apertures for signals due to a cw point source. By modeling the acoustic field locally as a sum of damped normal modes and using Prony's method to perform the beamforming, the local modal structure of the waveguide can be resolved. As a result, the modal composition of the waveguide as a function of range can be determined and interpreted in terms of range‐dependent mode theories (e.g., adiabatic mode theory). In addition to identifying important propagation characteristics such as mode cutoff, the method can be used to determine range‐dependent acoustic properties of the bottom. Examples of the application of the technique to the case of propagation in a wedge‐shaped ocean are presented. [Work supported by ONR.]

A survey is presented of recent related work (comprising numerical modeling and field experiments) on the determination of seafloor geoacoustic properties in shallow water from monochromatic pressure field measurements obtained using synthetic aperture and towed arrays. Two related techniques are discussed, their common foundation lying in the Hankel transform inversion of the pressure versus range data to obtain a depth-dependent Green's function versus horizontal wave number. Characteristic features of the Green's function, such as the peaks associated with normal mode propagation in the shallow water waveguide, are used to extract the geoacoustic properties (compressional and shear) as a function of depth in the bottom. INTRODUCTION Two related methods for determining geoacoustic properties in shallow water waveguides have recently been under development (Frisk and Lynch, 1984; Frisk et al., 1985; Kuperman et al., 1985; Werby and Tango, 1985). In both cases, the required input data are the magnitude and phase versus range of the pressure field due to a monochromatic (CW) point source. These data are numerically Hankel transformed to obtain the depth-dependent Green's function versus horizontal wave number. In the context of normal mode theory, the Green's function contains information about the nature of the discrete and continuous modal spectra as well as the plane-wave reflection coefficients of the waveguide boundaries. Specifically, the Green's function contains prominent peaks at horizontal wave numbers corresponding to the eigen-values for any trapped and virtual modes excited in the waveguide. The positions and magnitudes of these modal peaks are sensitive to the acoustic properties of the bottom, and this sensitivity is the basis of the two inverse methods for determining geoacoustic models in shallow water. The WHOI method utilizes an exact Hankel transform inversion of synthetic aperture array data obtained by towing a CW source away from two moored receivers. A geoacoustic model is obtained by computing the theoretical Green's function for various values of the bottom parameters and determining the parameter set which provides the best agreement with the experimental Green's function, particularly in the positions and relative magnitudes of the modal peaks. The NORDA technique involves an approximate Hankel transform inversion in which the beam formed output of a towed array due to the self-noise of the tow ship is computed. Layer sound velocity estimates are obtained from the identification of modal critical angles in the variable-offset beam former response. THEORY Consider a water column of thickness h, characterized by a constant density and a sound speed c(z), and bounded at the surface and bottom by horizontally stratified media as shown in Fig. 1. Then the spatial part of the acoustic pressure p, due to a point source at r =.0 and z = Zo with harmonic time satisfies the inhomogeneous Helmholtz equation(Mathematical equation available in full paper) The Green's function satisfies along with impedance boundary conditions at the surface and bottom. These boundary conditions can be ex pressed in terms of the plane-wave reflection coefficients RS(kr) and RB(kr) of the surface and bottom respectively (Frisk et al., 1980).

Green’s function extrapolation provides an efficient framework for predicting acoustic fields at adjacent locations from a reference observation. Conventional formulations of this approach have been developed primarily under the assumption that the waveguide invariant [Formula: see text] equals unity, which is valid for range-independent waveguides with flat bathymetry. This study demonstrates the applicability of Green’s function extrapolation in more general ocean waveguides where [Formula: see text] deviates from unity due to either bathymetry-induced range dependence or sound-speed-induced effects. The extrapolation is formulated as a two-step procedure consisting of a geometric time delay followed by a Doppler-like resampling operation, with the latter governing the dilation or compression of the arrival-time structure. Numerical simulations show that accurate extrapolation cannot be achieved by considering the time delay alone; instead, proper scaling of the arrival structure must be incorporated through the appropriate value of [Formula: see text]. Results are presented for range-independent shallow-water waveguides with flat bathymetry, range-dependent waveguides with linearly sloping and bow-shaped bathymetry, and a deep-water Arctic waveguide characterized by an [Formula: see text]-linear sound speed profile (SSP). While the assumption [Formula: see text] fails in environments where [Formula: see text], close agreement is recovered when the appropriate value of [Formula: see text] is employed. These findings establish the central role of [Formula: see text] in Green’s function extrapolation and extend its applicability to a broad class of realistic ocean waveguides.

Bayesian hierarchical modelling is a well-established branch of Bayesian inference. In this letter, we derive and study the estimation performance for the Bayesian hierarchical linear model (BHLM). Specifically, we consider a linear model with hierarchical priors for the involved amplitude and noise vectors. We provide closed-form expressions of the Bayesian Cramer-Rao bound (BCRB) for the following settings: (i) an arbitrary prior and hyperprior and (ii) a Gaussian- $\mathcal{Y}$ prior for the amplitudes, while the prior of noise is a Gaussian- $\mathcal{X}$ in both cases. Gaussian- $\mathcal{X}$ means that the conditional prior given the hyperparameter is Gaussian and $\mathcal{X}$ is the hyperprior. For the hierarchical distribution associated with spherical invariant random variables, the BCRB has a compact closed-form expression and enjoys several interesting properties that are discussed. Finally, we provide a theoretical analysis of the statistical efficiency of the linear minimum mean square error (MMSE) estimator in the low- and high-noise variance regimes when the hyperparameters are stochastically dominant.

Abstract Electrohydrodynamic instabilities have been investigated in cholesteric liquid crystals with negative dielectric anisotropy. We have used a room-temperature mixture of nematic p-methoxy benzilidene p-n butyl aniline (MBBA) and cholesteryl nonanoate(CN). The ac electric field is applied parallel to the helical axis of the planar texture. The onset of a two-dimensional periodic pattern is optically observed at threshold. According to the excitation frequency, two regimes must be distinguished, namely a low frequency (conduction) regime and a high frequency (dielectric) regime. For both regimes, we give a full experimental analysis of the threshold voltage and of the period of the instability as a function of the various parameters (frequency, sample conductivity, cholesteric pitch, sample thickness). The dependence of the threshold conditions vs. frequency can be quantitatively accounted for by Hurault's theory in the low frequency regime. As far as the high frequency regime is concerned, no theory is available. It appears nevertheless that part of our results can be interpreted along the lines of the model developed by Dubois-Violette et al. for nematics. Finally, the behavior above threshold is briefly described.

Attenuation of sound in the seabed plays an important role in predicting transmission loss in shallow water waveguides. Methods to invert the attenuation from low-frequency acoustic field data include time-frequency techniques that make use of modal dispersion. Since modal separation improves as a sound signal that propagates to longer ranges, most of the inversion methods based on modal dispersion were carried out with long range data. Recently a time-warping signal processing technique was introduced that enables high resolution of modes at relatively short ranges. Time-warping involves an axis transformation that transforms the original time-frequency relationship of the modes to a new domain in which the modes are approximately tonal and are well resolved. This paper shows that the inversion can be carried out directly in the time-warped domain, and extends the work to estimate low-frequency seabed attenuation.

There is a lack of conclusive experimental evidence, in terms of animal production, to substantiate some of the generally accepted “basic principles” of veld management in the Republic of South Africa. It is suggested that this may be due to the failure to recognise and/or consider the large number of possible variables associated with the comparison of grazing management systems. Due to the low carrying capacity of the veld, its resilience, and a large number of variables, experimentation in this field is expensive, laborious, of long duration and occupies large areas. Specially selected and adapted experimental designs may however facilitate this research.

Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cramer–Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this letter is to derive this expression directly from the Slepian–Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are given.

A tractable and compact closed-form expression on the channel-averaged signal-to-interference-plus-noise ratio (SINR) is derived for M-ary orthogonal coded/balanced transmitted-reference (BTR) systems, taking into consideration both inter-pulse interference (IPI) and multiple-access interference (MAI) in dense multipath ultra-wideband (UWB) channels. The UWB channel here is a realistic and standard one with lognormal channel gain distribution and double independent Poisson arrival distribution of cluster and ray. Hence, the analytical framework developed here can be applied to typical UWB channel models, especially considering the channel sparseness and cluster overlapping observed in realistic UWB channels. Based on the channel-averaged SINR, the effect of inter-pulse distance (between the reference and data pulses in BTR) on the multiuser performance is fully investigated. A proper selection of user-specific inter-pulse distances in multiuser scenario is then determined to maximize the user capacity for a given UWB channel.

The field inversion and machine learning (FIML) approach is leveraged to obtain a closed-form correction for the Spalart–Allmaras turbulence model to improve predictions of separated flows. Based on field inversion results obtained using the first-generation FIML Classic approach, a simple and compact closed-form expression is chosen to be used as correction model. The thus obtained correction model is optimized using the second-generation FIML Direct approach. Training and validation cases consist of a selection of airfoils in a wide range of flow conditions as well as the flat plate. The correction model and results for the training and validation cases obtained with the augmented turbulence model are presented, demonstrating the improved flow predictions.

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages.