Random variability in the water column will affect the operation of a horizontal aperture sonar. Two sources of variability in shallow water are turbulence and internal waves. In a modeling study, the effects of turbulence and internal waves on a shallow-water imaging system are compared. The operational principles of a large aperture imaging system are first reviewed. A shallow-water internal wave model is developed by modifying the Garrett-Munk model. The internal waves are assumed to dissipate and drive the small-scale turbulence. The two phenomena are predicted to have markedly different effects on a system. Turbulence has short spatial correlation scales whose primary effects will be manifested in the variance of the acoustic phase. By contrast, internal waves will have much larger scattering but also a longer correlation scale. The primary acoustic quantity of interest for internal waves is shown to be the curvature of the phase as observed along the aperture. Properties of shallow-water internal waves are shown to preclude the use of standard acoustic calculations based on the Markov approximation. Using archival environmental data, sample calculations are presented for the site of a planned August 1996 experiment.

A model for the joint probability density function is proposed for the quadrature components of waves propagating in random media (WPRM). The model is based on a further generalization of the model for the intensity distribution for WPRM proposed by Ewart [J. Acoust. Soc. Am. 86, 1490–1498 (1989)]. Both this distribution model and the intensity distribution model apply to the full range of scattering strengths and ranges. That is, from short ranges where the intensity probability density function (pdf) is lognormal, through the region of the medium focus, where the intensity moments 〈I^q〉 can be well above q!, to the saturation regime, where the moments approach exponential. Simulation of the complex fields used to test the model was accomplished by parabolic wave equation marching of the field through a medium with a fourth‐order power law transverse spectrum (usually termed the modified exponential). The fourth moment of the fields are accurately predicted using parabolic fourth moment theory—giving some confidence in the higher moments. The resulting joint pdf's are parametrized in terms of the scattering strength and the scaled range that parametrize the simulations. This work provides a necessary first step in the important quest to include highly non‐Gaussian quadrature component statistics in signal processing formulations.

In this paper numerical results for the solution of the fourth moment equation describing intensity fluctuations in the ocean are described. The work of previous authors who used adaptive grid methods to deal with the propagation of scalar waves through idealized random media with simple correlation functions is followed. Similar methods are applied to the analysis and prediction of the acoustic intensities measured in the mid‐ocean acoustic transmission experiment (MATE). Here extensive measurements provide the parameters needed to define the statistical ocean medium completely, using existing models for internal waves and fine structure that are well‐documented in the literature. When such models are used in the numerical solution of the fourth moment equation it is found that there are some discrepancies when comparisons are made with acoustic measurements of intensity fluctuations. However, if the fine structure model is modified in a simple way, then very good agreement is obtained with the experimental data. Most results are presented for the acoustic fluctuations of a propagating acoustic signal of fixed frequency, but some predictions for the cross correlation between two signals of different frequencies are also given.

The potential difficulties associated with modeling acoustic propagation in shallow‐water environments are well documented. Larger scale deterministic features combine with random fluctuations in the water column, sediment, sea surface, and water–sediment interface to produce an extremely complicated propagation regime. The extent to which each of these factors needs to be included in realistic propagation modeling remains to be quantified. Toward this end, results from a series of detailed simulations generated using the parabolic equation method are presented. Beginning with a deterministic downward refracting sound‐speed profile and a known sloping bottom, realizations of the random features are sequentially added to the simulation. Wind‐driven surface gravity waves and power‐law bottom roughness are considered. The individual and cumulative effects of these scattering mechanisms on the acoustic wavefront are quantified. Successive interactions with the random interfaces are studied. A modal decomposition reveals evidence of significant mode coupling. Temporal fluctuations in the acoustic wave are related to the time evolution of the sea surface.

The performance of matched‐field processing is degraded because of randomness of the propagation medium. Simulations and theory specialized to the Bartlett processor are used to study this type of environmental mismatch. The purpose is to quantify the effects of deep‐water internal waves on matched‐field processing. This is done at three different levels of approximation. First, the case of an ocean waveguide with a quadratic average sound‐speed profile and vertically stationary sound‐speed fluctuation statistics with Gaussian correlation is examined. Next, approximations are introduced so that a relatively simple analytic model can be abstracted from the theory. Finally, vertically nonstationary fluctuations obtained from a realistic, dynamic internal wave model are considered. In this case, a realistic average deep‐water sound‐speed profile, consistent with the internal wave field, is used. The results are found to agree with the predictions of the analytic model when the effects of multipaths are unimportant.

The oceanography of the near‐surface zone of the temperate oceans is characterized by a mixed layer of varying depth and a sharp thermocline with large sound velocity gradients. Most studies of surface scattering have ignored the effect of the coupling between volume scattering and scattering from the rough surface. Recent Navy emphasis on ocean surface scattering measurements has also ignored the volume effects. The gradients at the bottom of a mixed layer can be large enough to cause wide variations in the transmission and reflection coefficients (including total internal reflection) for the shallow angles of long‐range propagation. The wave packets that are found on the sharp thermocline can have large displacements and space‐time scales near those of surface waves, but with differing dispersion relations. A wide‐angle PE code has been used to scatter sound from an ocean surface with a Pierson‐Moskowitz spectrum, and the effects of the randomness on the scattered field with and without thermocline waves have been studied. The PE code used was developed in a joint project with Eric Thorsos (APL‐UW). The internal waves were generated to be similar in scale to observations with a towed chain (Marmorino et al., J. Geophys. Res. 92 C12, 13049–13062). The results indicate that volume scattering cannot be ignored when modeling shallow‐angle surface scattering.

Ewart and Percival [J. Acoust. Soc. Am. 80, 1745 (1986)] have shown that the generalized gamma distribution effectively models intensity probability distributions of temporal fluctuations observed in a field experiment and transverse spatial fluctuations simulated in numerical experiments. In both cases the fluctuations are due to wave propagation through a medium with a random index of refraction. Here, the transverse spatial intensity fluctuations of a wave propagating through a medium with a power‐law autocorrelation function of wave speed are modeled over a regime that spans 108 in scattering strength and 106 in scaled range (range divided by the Fresnel length). This scattering parameter regime transforms to ranges between 100 m and 100 km and to frequencies between 100 Hz and 100 kHz when normalizations typical of observed ocean internal wave fluctuations are used. Contour plots of the variance, skewness, and kurtosis of the intensity distribution are presented for the range/frequency plane. It is shown that the region of saturation, i.e., exponential intensity distribution, cannot be attained except for very large source strengths. Also, the lognormal intensity distribution, which is assumed for scintillation indices near zero, can be applied only in the region of vanishingly small intensity fluctuations. This work, while based on plane‐wave propagation and a fourth‐order power‐law transverse spectrum of the medium, retains the essential character of the intensity fluctuations and provides a prescription for modeling the intensity distribution for any medium where the random index of refraction process can be assumed stationary.

A three-dimensional diffraction tomography algorithm based on image projections is implemented. For each view, the measured scattered field is directly backpropagated onto a single plane in the image space. The backpropagated field evaluated on the plane is defined as the image projection because it closely approximates the straight line projection of the object. The object is then reconstructed by parallel slices using conventional straight ray tomographic techniques. This approach permits practical three-dimensional reconstruction using a limited number of views. The reconstructions made with image projections are of comparable quality to ideal diffraction-limited images. By backpropagating the field prior to filtering, curved or misaligned recording surfaces can be used. The limits on the image projection technique for multiple object systems are explored. A diffuse structure is reconstructed.

An acoustic wave propagating in a medium with an index of refraction that is random in space and time acquires intensity modulations that can be modeled in terms of a space–time autocorrelation function, a scattering strength parameter γ, and a scaled range X. Over a wide range of γ, X, and medium autocorrelation functions, the probability distributions of intensity vary from lognormal at small X to exponential at large X. The generalized gamma distribution [E. W. Stacy, Ann. Math. Stat. 33, 1187–1192 (1962)] is characterized by three parameters, and reduces to many well‐known distributions. It varies smoothly from lognormal to exponential as the parameters change. It is proposed that this distribution is a general analytic form that represents the probability distribution of intensity as a function of range, depth, and time in forward scattering. This proposition is tested with the measured temporal intensity fluctuations from the Mid‐Ocean Acoustic Transmission Experiment, MATE. It is also tested wtih depth‐range results from Monte Carlo simulations of wave propagation in a random medium with a power law autocorrelation function. The fitted generalized gamma distributions for the data sets chosen lie within the 95% Kolmogorov confidence bands for the true unknown probability distributions. In past treatments of this subject, the intensity moments have been used almost exclusively in modeling the probability distributions. The benefits of using distribution modeling rather than moment methods are described. Also discussed are the anomalies encountered for a medium with Gaussian autocorrelation.

A transmitted signal can arrive at a receiver via several refracted Fermat paths. If the paths are independent in the Fresnel sense, then the received signal can be modeled as the sum of amplitude scaled and time shifted copies of a predetermined replica plus white noise. We present an algorithm that uses the replica to determine the time shifts and amplitudes for each path. It is referred to as an n-dimensional matched filter algorithm by analogy with the well-known matched filter algorithm. The cross correlation between the received signal and the replica oscillates near the center frequency of the transmitted signal. This causes the n-dimensional matched filter output to have many local maxima that are not globally optimal. The time shifts and amplitude scalings for the Fermat paths are determined by maximizing the output of the n-dimensional matched filter. The algorithm is more robust and efficient than others currently available. Simulated realizations of received signals were generated with multipath and noise characteristics similar to an ocean acoustic transmission case. These realizations were then separated into arrival times and corresponding amplitudes by the algorithm. The results of these tests and the general limitations of the algorithm are discussed.

In the Mid‐Ocean Acoustic Transmission Experiment (MATE), measurements were made of the fluctuating phase and amplitude of signals propagated 18.1 km at four frequencies (2,4,8, and 12.5 kHz) over a wholly refracted Fermat path. Extensive oceanographic instrumentation of that transmission path permits evaluation of the space–time correlations of the index of refraction. The scattering regimes span the important region from well before the medium focus to beyond it, but do not extend to the very far field. In the past two years, solutions for the propagation of the fourth moment of the complex amplitude of a wave propagating in a random field have become available; these solutions are applicable at all ranges. Ewart and Reynolds discuss MATE and compare the acoustic results with theory for the intensity autospectra [J. Acoust. Soc. Am. 75, 785–802 (1984)]. The theoretical solutions have recently been extended to include acoustic frequency dependence. In this paper predictions from the new theory, based on space–time medium correlations where both internal waves and low wavenumber finestructure have been considered, are compared with the measured acoustic results from MATE. These comparisons include the cross correlations and cross spectra of intensity as a function of acoustic frequency separation. The agreement between theory and experiment is shown to be somewhat poorer for the cross spectra than for the autospectra. It is postulated that the lack of agreement arises either from the sensitivity of the cross spectral predictions to the precise form of the medium correlation function or from the fact that the first order correction to the evaluation of the fundamental solution (which has not been evaluated for the two frequency case) is more significant for the cross spectra of intensity than for the autospectra.

Probability distributions of intensity fluctuations from the MATE, AFAR, and S. W. Bermuda underwater acoustics experiments are compared with recently derived theoretical expressions. The limitations and strengths of these expressions are discussed. In particular, it is found that the work of Furutsu [J. Math. Phys. 17, 1252–1263 (1976)] gives a good description of the probability distribution function of intensity or log intensity, requiring only a knowledge of the second‐ and third‐order intensity moments. Furutsu's description is not asymptotically correct at large range, so a modified form is proposed for the moments of intensity that reduce analytically to the log‐normal distribution at short range and to the exponential distribution at large range. This new form also predicts the higher moments well but cannot be inverted analytically. A numerical inversion is used, and the ensuing distribution agrees well with the analytical result of Furutsu. It is expected that the new expression will be applicable at all ranges.

A straightforward numerical algorithm for simulating the propagation of a wavefield in a two-dimensional randomly varying medium is described. Both a finite-difference and a fast Fourier transform method are described. These methods are well known, but there is a novel treatment of the random scattering term in the wave equation that allows accurate answers to be economically obtained. The methods are then tested by comparison with known approximate solutions for the fourth moment of a propagating wavefield. The two approaches show good agreement, thus confirming the usefulness of the analytic results and at the same time indicating that the simulation process should be a powerful tool for investigating the higher-order statistics of the field. The agreement should hold in situations encountered in optical and acoustic scattering experiments.

An experiment to measure phase (travel time) and intensity fluctuations in sound pulses transmitted at 2, 4, 8, and 13 kHz over an 18.1 km wholly refracted Fermat path is discussed. Simultaneously with the acoustic monitoring the index of refraction fluctuations were measured in space and time with sufficient resolution to determine the correlation function of the medium. The site was the Cobb Seamount in the northeast Pacific (46°46'N, 130°47'W), and the time period was 30 days in June–July, 1977. In terms of both the quality and quantity of acoustic and oceanographic measurements, this experiment represents a significant improvement over an earlier experiment in the same location [J. Acoust. Soc. Am. 60, 46–59 (1976)]. The acoustic measurements cover a wider range of acoustic frequencies and more closely represent measurements from a single Fermat path. Approximately 25% of the acoustic data are discussed here; the representations of the correlation function of the index of refraction are based on all of the oceanographic data. The physical processes responsible for the fluctuations in the index of refraction are those due to the tides, internal waves, and finestructure. The effects of internal waves are treated in detail. The moments of the observed intensity fluctuations are discussed, as are the spectral distributions of the second moments of phase and intensity. The observations are compared with theoretical predictions based on the Rytov approximation and on a multiple scatter formulation (approximate solution to the fourth moment equation).

This paper deals with the problem of the intensity fluctuations arising in a wave when it propagates through a medium that is randomly inhomogeneous in space and time. It is assumed that multiple scattering can occur and that the intensity fluctuations can become large. The parabolic moment equation for the fourth moment of the wave field is solved for a monochromatic point source immersed in the medium. Approximate expressions are obtained for the space–time spectrum of intensity fluctuations at any distance in the medium. The solution of the fourth moment equation is compared with results of the Rytov method of smooth perturbations, and the limitations of the latter are discussed.

The intensity fluctuations measured at 4 and 8 kHz in the Cobb experiment have been available for nearly 10 years and in that time have not successfully been predicted theoretically. We show that multiple scatter effects must be considered, and that neither the Born nor the Rytov approximation to the scattering formulation is appropriate. A companion paper [J. Acoust. Soc. Am. 74, 1474–14832  (1983)] provides the theoretical background for this work by presenting a general form of the analytical solution to the fourth moment equation in two dimensions for a point source transmission. We use the parameters of the Garrett–Munk model of the internal wave field appropriate to the Cobb experiment oceanographic regime to obtain the correlation functions of the acoustic refractive index field, and then predict the intensity fluctuations. We discuss corrections to the predicted spectrum that are due to fine structure effects and tidal motions. We include a discussion of the scattering parameters Γ and X, which are the scattering and range scaling parameters of the moment equation formulation, and the parameters of the path integral formulation, Γ and φ. We present a discussion of the measured correlation functions taken during a more recent experiment at Cobb seamount and indicate the directions we need to take to bring the scattering predictions based on internal waves and finestructure into closer agreement with experiment.

An experiment to measure the dispersion of a passive tracer injected at 300 and 1000 m depth in the Pacific 600 miles southwest of San Diego is described. Rhodamine dye was released as a (very nearly) point source, and the subsequent growth of the patch was mapped at various times, up to 66.2 hours after release, using a self‐propelled underwater research vehicle (SPURV), which carried temperature, conductivity, pressure, and dye sensors on a depth cycling trajectory through the patch. The analysis shows that the horizontal diffusion rates decrease with depth, that it is difficult to resolve the mixing in terms of a constant diffusion parameter based on any one of a number of models, that the 300 m results agree with those obtained by Schuert (1970), and that the patch growth began to develop eddy behavior at 66.2 hours when the rms second moment of the dye distribution was (r²)^l/2 = 210 m. For the purpose of predicting horizontal mixing, the results indicate that eddy diffusivities of 1150 and 360 cm²/s at 300 and 1000 m, respectively, are the best fit values for spatial scales <200 m horizontally. Although we have focused on horizontal (isopycnal) diffusion, a few aspects of the vertical (diapycnal) characteristics of the patch are also discussed.

A simple numerical simulation model is presented which includes the effects of oceanic finestructure together with previously developed theoretical treatments of the effects of internal waves on acoustic transmission. The wave equation solution to particular layer geometries developed by Brekhovskikh [Waves in Layered Media (Academic, New York, 1960)] is combined with the theoretical model of acoustic scattering based on internal waves developed by Desaubies [J. Acoust. Soc. Am. 64, 1460–1469 (1978)]. In this treatment it is assumed that a single finestructure layer is passively advected by internal waves. The layer modulates the effects of internal waves. Experiments carried out at Cobb Seamount over a fixed, horizontal, wholly refracted path indicate that the power spectrum of the phase fluctuations is fit almost exactly by models based on internal waves only, while the power spectrum of the log amplitude fluctuations is not. In the present treatment the predictions of the phase fluctuations remain unaffected by the presence of finestructure, and the predicted power spectrum of the log amplitude is in considerably better agreement with observation.

FM slide and pulsed‐tone signals at 2, 4, 8, and 16 kHz were transmitted from a source to a receiver at the same depth 1100 m distant. Three wholly refracted paths were observed in the return signals, and a discussion of the six‐hour time series of the arrival times and amplitudes for each path is presented. Evidence is given of acoustic frequency‐dependent scattering, and scattering from oceanic fine structure is suggested as the most likely physical mechanism to explain the observations.

A maximum‐likelihood procedure for estimating the amplitudes and arrival times of individual pulses in a multipath signal is derived. Computer simulation results are presented that compare this procedure with the more traditional matched‐filter and inverse‐filter techniques. The maximum‐likelihood technique is shown to be significantly more accurate than either the matched or inverse filter. A computer algorithm for implementing the maximum‐likelihood estimation procedure is described.

Amplitude and phase (transit time) fluctuations in pulses sent between a fixed transmitter and a fixed receiver over a wholly refracted 17.2‐km path were recorded for 144.5 h. The sites were the southwest flank of Cobb Seamount and a lesser peak 17.2 km away, both at 1000 m depth. Eight‐cycle pulses at 4166 Hz and 16‐cycle pulses at 8333 Hz were sent alternately every 15.7 sec and received at three receivers located at 0, 5, and 15 m along a horizontal arm located perpendicular to the transmission path. Power spectra have been computed from the time series of phase and amplitude at a single receiver and the phase difference between two receivers. Extreme care was used in the analysis of the data to ensure that the time series obtained represented a single path. The power spectra of the phase data exhibit dominant tidal peaks at 24, 12.4, and 6.2 h; the power spectra of the amplitudes show less evidence of the tides. Between the inertial frequency and the Väisälä frequency, the power spectra of the phase, amplitude and phase difference fall off at approximately ω^-3, ω^-1, and ω^-1/2, respectively. The power spectra of the phase differences for the 15‐, 10‐, and 5‐m spacings scale according to a plane wave arrival. The phase‐difference variance is below theoretical predictions, and the amplitude variance is above theoretical predictions. The high‐amplitude variance is tentatively identified as "micro" multipath interference. Evidence is presented to show that the oceanographic regime of the Cobb Seamount area is typical of the open ocean environment.

Estimates of the spatial spectrum of ocean temperature fluctuations at six different depths between 500 and 2500 m are presented. Measurements were made with a depth-stable, self-propelled instrument carrier. Temperature samples were taken at 0.2 m intervals over a track typically 10 km long, corresponding to a spectral bandwidth of 1–2500 cycles per kilometer (cpkm). The observed spectrum falls into three distinct bands; these are tentatively identified with the effects of internal waves (W), fine-scale layering (L), and turbulent mixing (T). Each band has a characteristic wavenumber dependence which is invariant with depth. The intensity in each band scales with depth in characteristic fashion.