Doppler Passive Fathometry.

Passive fathometry is a technique whereby broadband ambient ocean noise received on an array of hydrophones is averaged and cross-correlated to produce a sub-bottom profile [Siderius, Harrison, and Porter (2006). J. Acoust. Soc. Am. 120, 1315-1323]. Here this technique is extended to determine the vertical velocity of the array, and compensate for it, without any prior knowledge, i.e., Doppler Passive Fathometry. For Fourier transform lengths beyond a certain limit, the differing Doppler between the direct and bottom reflected paths spoils the correlation match, however it is shown by using some experimental data, where the array was known to suffer from arbitrary but near periodic motion, that compensation is possible, enabling continuing time integration. In the process, the vertical velocity becomes known. Velocity, with peak value ∼±1 m/s, is plotted against time and shown to be 90° out of phase with depth, as expected for periodic motion. Since stationary targets have already been detected by noise correlation [Harrison (2008). J. Acoust. Soc. Am. 123, 1834-1837], it is implied that the range of moving targets can also be determined.

The ocean noise coherence matrix and its rank.

An expression for the cross-spectral density matrix of ocean noise naturally separates into a Toeplitz part and a Hankel part [Harrison (2017). J. Acoust. Soc. Am. 141, 2812-2820]. The Toeplitz part is shown to be substantially rank-deficient for all practical acoustic cases, which has implications for adaptive beam forming. The influence of the Hankel part on passive fathometry is investigated, and its effect on adaptive beam forming is shown to be weak or negligible. Numerical demonstrations of these findings including beam patterns and eigenvalue spectra derived via circulant matrices are given based on a simple half-space with a Rayleigh reflection coefficient. Two sets of experimental data are revisited in this context, deriving eigenvalue spectra, beam patterns, and passive fathometry impulse responses with conventional and adaptive processing and differing amounts of averaging. The solution to a long-standing puzzle of processing inconsistency is suggested.

Separation of measured noise coherence matrix into Toeplitz and Hankel parts.

The cross-spectral density of ocean ambient noise is usually estimated from the product of the complex hydrophone signals, each of which already corresponds to the summed responses of sources from all angles. The true coherence is the integral over all angles of the angle-dependent product. The influence of this distinction on necessary time integration in geoacoustic inversion and passive fathometry is explored, and a meaningful separation of the cross-spectral density matrix into Toeplitz and Hankel parts is proposed. Various processing techniques are applied to synthesized data and some experimental vertical array data in a passive fathometry context. Passive fathometry is only sensitive to the Hankel part of the matrix.

Efficient modeling of range-dependent ray convergence effects in propagation and reverberation.

In an earlier paper [Harrison (2013). J. Acoust. Soc. Am. 133, 3777-3789] the computationally efficient energy flux approach to modeling sound propagation was modified to include focusing, ray convergence, and caustic-like behavior. The derivation started with the coherent normal mode sum but retained only terms that interfered on a scale of a ray cycle distance. Here, by starting with the adiabatic mode sum, the formulation is extended to a slowly varying range-dependent environment and applied to the target-echo and reverberation model, Artemis. Some examples are given.

Ray convergence in a flux-like propagation formulation.

The energy flux formulation of waveguide propagation is closely related to the incoherent mode sum, and its simplicity has led to development of efficient computational algorithms for reverberation and target echo strength, but it lacks the effects of convergence or modal interference. By starting with the coherent mode sum and rejecting the most rapid interference but retaining beats on a scale of a ray cycle distance it is shown that convergence can be included in a hybrid formulation requiring minimal extra computation. Three solutions are offered by evaluating the modal intensity cross terms using Taylor expansions. In the most efficient approach the double summation of the cross terms is reduced to a single numerical sum by solving the other summation analytically. The other two solutions are a local range average and a local depth average. Favorable comparisons are made between these three solutions and the wave model Orca with, and without, spatial averaging in an upward refracting duct. As a by-product, it is shown that the running range average is very close to the mode solution excluding its fringes, given a relation between averaging window size and effective number of modes which, in turn, is related to the waveguide invariant.

Synthetic array processing of ocean ambient noise for higher resolution seabed bottom loss estimation.

Predicting transmission loss in the ocean often strongly depends on the bottom loss. Bottom loss can be estimated using ocean noise and vertical array beam-forming [Harrison and Simons, J. Acoust. Soc. Am. 112, 1377-1389 (2002)]. With finite length arrays, the bottom loss estimate using this method can be smoothed due to beam widths. This paper describes how the noise coherence function can be synthetically expanded, which is similar to extending the length of an array. A full wave ocean noise model is used to demonstrate, in simulation, how this leads to improvements in the resolution of bottom loss estimates.

A relation between multipath group velocity, mode number, and ray cycle distance.

Weston's ray invariant or "characteristic time" in a range-dependent environment is exactly equivalent to the Wentzel-Kramers-Brillouin phase integral for ducted normal modes. By considering a ray element it is shown that the ray invariant can also be written in terms of ray cycle distance and cycle time. This leads to a useful formula for group velocity in terms of cycle distance and mode number. Drawing a distinction between the ray and wave interpretation, the Airy phase (i.e., the existence of a group velocity minimum) can be included in this approach. Favorable comparisons are made with group velocities derived from a normal mode model. The relationship is valid for variable sound speed and variable bathymetry, and this is demonstrated numerically. The formula is applicable to active sonar, multipath pulse shape, target signatures, reverberation, tomography, and underwater communications.

Bayesian geoacoustic inversion using wind-driven ambient noise.

This paper applies Bayesian inversion to bottom-loss data derived from wind-driven ambient noise measurements from a vertical line array to quantify the information content constraining seabed geoacoustic parameters. The inversion utilizes a previously proposed ray-based representation of the ambient noise field as a forward model for fast computations of bottom loss data for a layered seabed. This model considers the effect of the array's finite aperture in the estimation of bottom loss and is extended to include the wind speed as the driving mechanism for the ambient noise field. The strength of this field relative to other unwanted noise mechanisms defines a signal-to-noise ratio, which is included in the inversion as a frequency-dependent parameter. The wind speed is found to have a strong impact on the resolution of seabed geoacoustic parameters as quantified by marginal probability distributions from Bayesian inversion of simulated data. The inversion method is also applied to experimental data collected at a moored vertical array during the MAPEX 2000 experiment, and the results are compared to those from previous active-source inversions and to core measurements at a nearby site.

Target time smearing with short transmissions and multipath propagation.

In active sonar the target echo level is often estimated with a propagation model that adds all multipath arrivals. If the (post-correlator) transmitted pulse is short compared to the multipath time spread then there is effectively an extra loss (which may be substantial) since only a few of the paths contribute to the target echo at any one instant. This well known "time-smearing" loss is treated in a self-consistent manner with previous calculations of reverberation [Harrison, J. Acoust. Soc. Am. 114, 2744-2756 (2003)] to estimate the target response and the signal-to-reverberation-ratio. Again isovelocity water, Lambert's law, and reflection loss proportional to angle are assumed. In this important short pulse regime the target response becomes independent of boundary reflection properties but proportional to transmitted pulse length. Thus the signal-to-reverberation-ratio becomes independent of pulse length. The effect on signal-to-ambient-noise is also investigated and the resulting formulas presented in a table.

The relation between the waveguide invariant, multipath impulse response, and ray cycles.

The waveguide invariant, ß, that manifests itself as interference fringes or "striations" in a plot of frequency vs source-receiver separation, is usually thought of as a modal phenomenon. This paper shows that striations can be explained simply through the variation of the eigenray arrival times with range, in short, the variation of the multipath impulse response. It is possible to calculate ß for a number of sound speed profiles analytically and to find what ß depends on, why it switches from one value to another, how it depends on source-receiver depth, how it depends on variable bathymetry, and how smooth the sound speed profile needs to be for clear fringes. The analytical findings are confirmed by calculating striation patterns numerically starting from eigenray travel times in various stratified environments. Most importantly the approach throws some light on what can be deduced from ß alone and the likelihood and utility of striations in reverberation.

Fixed time versus fixed range reverberation calculation: analytical solution.

Reverberation is commonly calculated by estimating the propagation loss to and from an elementary area, defined by transmitted pulse length and beam width, and treating the resulting backscatter from the area as a function of its range. In reality reverberation is strictly a function of time and contributions for a given time come from many ranges. Closed-form solutions are given for reverberation calculated both at fixed range and at fixed time isovelocity water and some variants of Lambert's law and linear reflection loss with an abrupt critical angle. These are derived by considering the shape of the two-way scattered multipath pulse envelope from a point scatterer. The ratio of these two solutions is shown to depend on the dominant propagation angle spread for the particular range or time. The ratio is largest at intermediate ranges (though typically less than 1 dB) and depends explicitly on the critical angle. At longer ranges mode-stripping reduces the propagation angle spread and the ratio reduces ultimately to unity. At short range the ratio is also close to unity although interpreting it requires care.

An approximate form of the Rayleigh reflection loss and its phase: application to reverberation calculation.

A useful approximation to the Rayleigh reflection coefficient for two half-spaces composed of water over sediment is derived. This exhibits dependence on angle that may deviate considerably from linear in the interval between grazing and critical. It shows that the non-linearity can be expressed as a separate function that multiplies the linear loss coefficient. This non-linearity term depends only on sediment density and does not depend on sediment sound speed or volume absorption. The non-linearity term tends to unity, i.e., the reflection loss becomes effectively linear, when the density ratio is about 1.27. The reflection phase in the same approximation leads to the well-known "effective depth" and "lateral shift." A class of closed-form reverberation (and signal-to-reverberation) expressions has already been developed [C. H. Harrison, J. Acoust. Soc. Am. 114, 2744-2756 (2003); C. H. Harrison, J. Comput. Acoust. 13, 317-340 (2005); C. H. Harrison, IEEE J. Ocean. Eng. 30, 660-675 (2005)]. The findings of this paper enable one to convert these reverberation expressions from simple linear loss to more general reflecting environments. Correction curves are calculated in terms of sediment density. These curves are applied to a test case taken from a recent ONR-funded Reverberation Workshop.

Anomalous signed passive fathometer impulse response when using adaptive beam forming (L).

The impulse response of the seabed can be extracted from sea surface ambient noise by cross-correlating the time series from an upward and a downward steered beam. When the steering for each beam is standard minimum variance adaptive beam forming it has been found that the impulse response for significant echoes appears to have the same amplitude but opposite sign. A mathematical explanation is offered for this strange phenomenon. Crucial contributing factors are that the cross-spectral density matrix for the vertical array typically consists of the sum of a Toeplitz matrix and a much weaker Hankel matrix and that it is ill-conditioned.

Bottom profiling by correlating beam-steered noise sequences.

It has already been established that by cross-correlating ambient noise time series received on the upward and downward steered beams of a drifting vertical array one can obtain a subbottom layer profile. Strictly, the time differential of the cross correlation is the impulse response of the seabed. Here it is shown theoretically and by simulation that completely uncorrelated surface noise results in a layer profile with predictable amplitudes proportional to those of an equivalent echo sounder at the same depth as the array. The phenomenon is simulated by representing the sound sources as multiple random time sequences emitted from random locations in a horizontal plane above a vertical array and then accounting for the travel times of the direct and bottom reflected paths. A well-defined correlation spike is seen at the depth corresponding to the bottom reflection despite the fact that the sound sources contain no structure whatsoever. The effects of using simultaneously steered upward and downward conical beams with a tilted or faceted seabed and multiple layers are also investigated by simulation. Experimental profiles are obtained using two different vertical arrays in smooth and rough bottom sites in the Mediterranean. Correlation peak amplitudes follow the theory and simulations closely.

Passive fathometer processing.

Ocean acoustic noise can be processed efficiently to extract Green's function information between two receivers. By using noise array-processing techniques, it has been demonstrated that a passive array can be used as a fathometer [Siderius, et al., J. Acoust. Soc. Am. 120, 1315-1323 (2006)]. Here, this approach is derived in both frequency and time domains and the output corresponds to the reflection sequence. From this reflection sequence, it is possible to extract seabed layering. In the ocean waveguide, most of the energy is horizontally propagating, whereas the bottom information is contained in the vertically propagating noise. Extracting the seabed information requires a dense array, since the resolution of the bottom layer is about half the array spacing. If velocity sensors are used instead of pressure sensors, the array spacing requirement can be relaxed and simulations show that just one vertical velocity sensor is sufficient.

Multipath pulse shapes in shallow water: theory and simulation.

In shallow water propagation the steeper ray angles are weakened most by boundary losses. Regarding the sound intensity as a continuous function of angle it can be converted into a function of travel time to reveal the multipath pulse shape received from a remote source (one-way path) or a target (two-way path). The closed-form isovelocity pulse shape is extended here to the case of upward or downward refraction. The envelope of the earliest arrivals is roughly trapezoidal with a delayed peak corresponding to the slowest, near horizontal refracted paths. The tail of the pulse falls off exponentially (linearly in decibels) with a decay constant that depends only on the bottom reflection properties and water depth, irrespective of travel time, a useful property for geoacoustic inversion and for sonar design. The nontrivial analytical problem of inverting explicit functions of angle into explicit functions of time is solved by numerical interpolation. Thus exact solutions can be calculated numerically. Explicit closed-form approximations are given for one-way paths. Two-way paths are calculated by numerical convolution. Using the wave model C-SNAP in several broadband cases of interest it is demonstrated that these solutions correspond roughly to a depth average of multipath arrivals.

Geoacoustic inversion using multipath pulse shape.

Experimental data, measured in a shallow water region of the Mediterranean Sea, are used to show that the variation of received intensity with time is well described by existing expressions [Harrison and Nielsen, J. Acoust. Soc. Am. 121, 1362-1373 (2007)]. These expressions indicate that the effect of the sea-water sound speed profile can be neglected for times greater than the peak intensity arrival. Beyond this time, intensity is shown to decay at a rate determined by the seabed acoustic properties in a manner very similar to that for an isovelocity water column. It is shown that a method of determining seabed acoustic properties, previously restricted to isovelocity water columns [Prior and Harrison, J. Acoust. Soc. Am. 116, 1341-1344 (2004)], can consequently be used in the presence of a sound-speed profile. The method relates the decay rate of smeared multipath arrivals to the angular derivative of seabed reflection loss. Two datasets are studied and the method is used to describe average seabed properties and to detect changes in seabed type. The seabed descriptions thus derived are used to predict total received intensity as a function of source-receiver separation. Agreement between the propagation measurements and predictions is shown to be within measurement uncertainties.