Adiabatic Mode Theory Research Articles

Sound-speed profile inversion using a large aperture vertical line array

In this paper, the single vertical slice ocean acoustic tomography problem is investigated with a large aperture vertical line array and a narrow-band low-frequency acoustic source deployed at a known location. Within the framework of matched-field processing and adiabatic normal mode theory, the self-cohering technique of Bucker is shown to provide a smooth measure of sound-speed mismatch. Simple simulation results lead to the development of a minimization procedure to invert for the sound-speed profile using a good starting solution (e.g., based on historical data). The technique then is demonstrated with additional simulations for sound-speed structures characteristic of the NE Pacific.

Numerical simulation of modal ocean acoustic tomography

Numerical simulation of modal tomography has been conducted at low frequency (f=50 Hz) for the same vertical slices of the ‘‘sampled ocean model’’ for which ray tomography is simulated [by M. Jones and T. Georges, in this session]. Modal travel time perturbation data are synthesized using an adiabatic mode theory and also by the modal spectrum of the PE field (MOSPEF) to investigate mode coupling effects. The inversions given by the different methods are compared. [Work supported by NOAA.]

The measurement of modal evolution in a range-dependent, shallow-water waveguide.

A technique is described for the measurement of normal mode eigenvalues as a function of range in a laterally inhomogeneous, shallow-water waveguide. It is based on the notion, originating in adiabatic mode theory, that the local modes adapt to the local environment. The trajectories of the eigenvalues in range are therefore influenced by lateral variations in the geoacoustic parameters of the seabed as well as bathymetric changes and fluctuations in the acoustic properties of the water column. Knowledge of the modal evolution in a waveguide plays a critical role in characterizing sound propagation and its relationship to the environment in shallow water. The method consists of beamforming narrow-band pressure field data acquired on a horizontal array using a short sliding aperture in range and examining the trajectories of the modal peaks in the beamformed output. The modal resolution can be significantly enhanced by mode filtering with a vertical array of receivers prior to the beamforming operation. Applications to synthetic and experimental data are discussed. [Work supported by ONR.]

Nonreciprocity of long-range reverberation in a wedged continental shelf.

The bottom scattering strength at low frequency and grazing angle in shallow water can only be extracted from long-range reverbertion (with large R/N ratio). Due to multipath effects the conventional model of boundary reverberation, based on geometric ray theory in the deep sea, is no longer suitable for calculating the long-range reverberation in shallow water. Up to date little reliable information on bottom scattering at low frequency and small grazing angle has been given in shallow water. For a frequency range of 50–2000 Hz and at a grazing angle of 0.5°–10°, it is hard to confirm the mechanism of bottom scattering and its dependence on frequency, grazing angle, and bottom type. In such a range, even the angular dependence of bottom scattering (a most basic relationship) is questionable. In this talk, using the WKB approximation to adiabatic normal mode theory, an averaged intensity expression R(r) of long-range reverberation in wedged homogeneous shallow water is derived. For two identical sources directed at each other, the monostatic reverberation intensities, obtained at two terminations with a depth of h1 or h2, would not be reciprocal. It is shown that Rh1(r)/Rh2(r) = (h1/h2)n, where n is an angular index of bottom backscattering. Based on this result, the value of n at small grazing angle can be determined from at-sea long-range reverberation data. A simulation tank experiment is suggested. [Work supported by ONR and the IAAS.]

Perth–Bermuda sound propagation (1960): Adiabatic mode interpretation

A 1960 experiment is examined in which sound from three underwater explosions near Perth, Australia, was detected near Bermuda. A recent attempt [Munk et al., J. Phys. Ocean. 18, 1876 (1988)] to calculate propagation paths for this event included rotational flattening of the Earth and horizontal refraction determined from the vertical sound speed minimum. That calculation left Bermuda in a shadow zone. The current work invokes adiabatic mode theory to include refraction due to horizontal variations in the vertical mode structure. These results include separate horizontal rays for each of the first few vertical modes, using an archival data set of 230 ocean sound profiles to generate the modes numerically. Where appropriate, interaction with bathymetry is included. This solution possesses two eigenray groups: Group A passes just south of the Cape of Good Hope, at which point group B is almost 1000 km to the south. Intermediate rays are blocked by islands. Group A proceeds unimpeded to Bermuda for a total time of flight of 13 354±5 s, while group B interacts slightly with bathymetry off Brazil, arriving at 13 403±9 s, and suffering roughly 7–12 dB more bottom attenuation. The spread in these arrivals overlaps satisfactorily with experimental data (main arrival at 13 364±5 s; pulse train half-width 15 s; second arrival 30±5 s later roughly 10 dB below the first arrival).

Perth, Bermuda (1962): A shot heard round the world according to adiabatic mode theory

A 1962 experiment is examined in which sound from an underwater explosion near Perth, Australia, was detected near Bermuda. A recent attempt [Munk et al., J. Phys. Ocean. 18, 1876 (1988)] to calculate propagation paths for this event included rotational flattening of the Earth and horizontal refraction due to the mean equatorward temperature gradient. That calculation left Bermuda in a shadow zone. The current work invokes adiabatic mode theory to include refraction due to horizontal variations in the vertical mode structure. The results include separate horizontal rays for each of the first few vertical modes, using an archival data set of 230 ocean sound profiles to generate the modes numerically. Where appropriate, interaction with bathymetry was included (scattering and bottom losses). This solution possesses two eigenray groups: group A passes just south of the Cape of Good Hope, at which point group B is almost 1000 km to the south. Intermediate rays are blocked by islands. Group A proceeds unimpeded to Bermuda for a total time-of-flight of 13 360 s, while group B interacts slightly with bathymetry off Brazil, arriving 37 to 45 s behind group A, and suffering 8 to 10 dB more bottom attenuation. These numbers are within experimental uncertainties (main arrival at 13 3644±5 s; pulse train half-width ∼15 s; second arrival ∼30±5 s later). [Work supported by ONR.]

Determination of compressional wave speed profiles using modal inverse techniques in a range-dependent environment in Nantucket Sound

An inverse method for determining geoacoustic properties in a horizontally stratified, shallow-water waveguide is extended to the case of a weakly range-dependent environment [Frisk et al., J. Acoust. Soc. Am. 86, 1928–1939 (1989)]. The technique consists of estimating the local modal eigenvalues from the beam-formed output of a horizontal array and using these data as input to modal inverse methods for obtaining the local bottom parameters. Specifically, the approach is applied to data at 140 and 220 Hz obtained in a shallow-water environment with a known abrupt change in bathymetry. First, a range-independent medium is assumed and both iteration of forward models and perturbative inversion methods are applied to the modal data to obtain estimates of the bottom sound velocity profile. Although the perturbative inversion results are clearly superior, neither approach reproduces the full dependence with range of the observed pressure fields or the complete modal peak structure. In particular, the data exhibit an apparent splitting of the modal peaks that is interpreted, within the context of adiabatic mode theory, as the superposition of the modal contributions from the two segments of the waveguide with differing depths. When the assumption of a range-independent medium is relaxed and the perturbative inverse technique is applied to the eigenvalue data from each portion of the waveguide, distinctly different bottom profiles are obtained for each part. These result in significantly improved agreement between theory and experiment, thereby demonstrating the effectiveness of the range-dependent inversion procedure.

Acoustic modal features associated with mesoscale eddy

Acoustic modal features associated with oceanic mesoscale eddy are the basic measurements for remote sensing and inferring mesoscale eddy by using modal ocean acoustic tomography (MOAT) [Shang, J. Acoust. Soc. Am. 85, 1531–1537 (1989)]. In this talk, modal travel time perturbation caused by mesoscale eddy has been calculated by three different methods and comparison has been made. Since the linear inverse is one of the conventional inverse schemes, the nonlinearity of modal travel time perturbation has been investigated for different eddy parameters as well as different frequencies. In order to determine the validity of using the adiabatic mode theory to MOAT, some interesting results of mode coupling caused by mesoscale eddy will also be presented in this talk. [Work supported by NOAA and ONR.]

Nonlinear bias of travel time in modal ocean acoustic tomography

In a previous paper [E. C. Shang, J. Acoust. Soc. Am. 85, 1531–1537 (1989)], ocean acoustic tomography based on the adiabatic normal mode theory has been developed. Mesoscale structure can be inverted by using a linear relationship connecting the modal travel time perturbation and the sound‐speed perturbation. This linear relationship is only an approximation due to the following two facts: (1) neglecting the difference between perturbed modal eigenfunction and the background model eigenfunction; (2) taking the first‐order expansion of the “potential function” ΔK2 (z,r) to calculate the modal wavenumber perturbation. Comparing with ray tomography, item (1) is the term corresponding to the ray path perturbation. It has been found that item (1) is not significant and the problem caused by item (2) can be recovered by a “post‐correction” procedure proposed in this paper. [Work supported by NOAA and ONR.]

The normal mode theory with bottom elasticity and range dependence

The theory of normal modes has been extended to accommodate the effects of shear waves in the elastic ocean floor. The effect of absorption has also been included as the imaginary component of the shear and compressional wavenumbers. A semi‐infinite elastic bottom basement layer has been included and it was found that the absorption causes the wavenumber spectrum of the radiating modes to be inherently discrete, hence, the number of radiating modes is drastically reduced relative to the ones obtained with the deep false boundary. The range dependence of the acoustic properties and of the boundaries of the ocean has been included by a modified version of the adiabatic normal mode theory. Comparisons are made with experimental measurements of the transmission loss of underwater explosives and with results from Collins' HEPE mode. [Work supported by the Independent Research Board of the Naval Surface Warfare Center and by NORDA.]

Determination of compressional wave speed profiles using modal inverse techniques in a range-dependent environment in Nantucket Sound

An inverse method for determining geoacoustic properties in a horizontally stratified, shallow-water waveguide is extended to the case of a weakly range-dependent environment. The technique consists of estimating the local modal eigenvalues from the beam-formed output of a horizontal array and using these data as input to modal inverse methods for obtaining the local bottom parameters. Specifically, the approach is applied to data at 140 and 220 Hz obtained in a shallow-water environment with a known abrupt change in bathymetry. First, a range-independent medium is assumed, and both iteration of forward models and perturbative inversion methods are applied to the modal data to obtain estimates of the bottom sound velocity profile. Although the perturbative inversion results are clearly superior, neither approach reproduces the full dependence with range of the observed pressure fields or the complete modal peak structure. In particular, the data exhibit an apparent splitting of the modal peaks which is interpreted, within the context of adiabatic mode theory, as the superposition of the modal contributions from the two segments of the waveguide with differing depths. When the assumption of a range-independent medium is relaxed, and the perturbative inverse technique is applied to the eigenvalue data from each portion of the waveguide, distinctly different bottom profiles are obtained for each part. These result in significantly improved agreement between theory and experiment, thereby demonstrating the effectiveness of the range-dependent inversion procedure.

Application of adiabatic mode theory to the calculation of horizontal refraction through a mesoscale ocean eddy

Adiabatic mode theory is used to calculate the path in the horizontal plane of vertical acoustic normal modes (at a frequency of 25 Hz) that have propagated through an idealized gulf-stream eddy of radius 125 km whose center is 625 km from the source. The arrival directions of the modes (which are calculated using a simple algorithm) and their relative amplitudes are used to calculate the apparent bearing of the source at positions beyond the eddy (the algorithm cannot be applied to positions within the inhomogeneity) on a line at 5° to the source-center axis. For source and receivers at a depth of 600 m, the source bearing deviation decreases from 0.5° at 740 km to 0.4° at 860 km (its asymptote, as range becomes infinite, is zero). The results for a transducer depth of 600 m are compared with a previously published calculation based on a parabolic equation method that used identical eddy parameters and transducer configurations. The two methods are found to predict broadly the same bearing deviation, except that the adiabatic mode-theory predictions are around 0.04° smaller than the average parabolic equation predictions. The effect of coupling from other modes on the amplitude of the important mode is found to be only 2% and hence negligible. For the eddy model considered, the degree of horizontal refraction of normal modes decreases as the mode number is increased. The bearing deviation therefore increases as the depth of the transducers is increased.

Mode coupling caused by a mesoscale oceanic eddy

From the full-wave theoretical point of view, there are two regimes for the interaction of acoustic waves with an oceanic mesoscale eddy: the weak coupling regime and the strong coupling regime. Understanding the limits of these two regimes is crucial for developing the new generation of ocean acoustic tomography systems based on the full-wave theoretical framework instead of ray theory. In the present paper, the mode coupling caused by an oceanic mesoscale eddy is investigated by numerical simulating on CYBER 205. The eddy is modeled as a Gaussian eddy with parameters: DR (horizontal scale), DZ (vertical scale), ZE (center depth), and DC (strength). By incorporating a single-mode starting field into a 2-D PE (IFD) code, mode coupling coefficients are given by a mode filtering processing after passing through the eddy. Mode coupling coefficients are calculated for a frequency range of 10–100 Hz, and a mode number range of 1–10. It has been found that the mode coupling has a complex “resonant feature” with eddy parameters and also has a dependence on frequency as well as on mode number. However, the limit of the weak coupling regime, within which the adiabatic mode theory is valid, can be drawn out but it is by no means a simple criterion. [Work supported by NOAA.]

Ocean acoustic tomography based on adiabatic mode theory

A new method of ocean acoustic tomography is developed on the basis of adiabatic normal-mode theory. It is a full-wave method suitable for low frequency in a slowly varying range-dependent waveguide. The modal phase difference perturbations are proposed as data for inverting the sound-speed profile perturbation for a continuous-wave (cw) source. The modal travel time perturbations are proposed as data for a pulse source. It is shown that the ‘‘normalized depth structure,’’ as well as the ‘‘range-averaged strength parameter’’ (defined as the product of the effective horizontal scale and the maximum sound-speed perturbation), can be retrieved from the vertical slice modal tomography. A simulation example of inversion is presented to verify the basic ideas.

Acoustic propagation in inhomogeneous underwater acoustic waveguides

The propagation of sound in horizontally stratified media may be treated using a variety of techniques. Realistic shallow‐water environments may, however, exhibit significant departures from this ideal geometry. This paper reviews various methods, both deterministic and stochastic, for handling range‐dependent shallow‐water propagation. Stochastic coupled‐mode theory is applied to consider the effects of sea surface scattering. It is shown that for low enough frequencies, adiabatic normal mode theory provides a sufficient treatment, in accord with the findings of both laboratory and at sea measurements. This same technique is also applied to consider the effects of inhomogeneities within the seabed and water column on the propagated field and its spatial coherence. Implications for matched field processing are discussed, and methods of estimating performance degradation due to rough boundary interaction are presented.

Distribution of surface‐generated noise in a three‐dimensional environment

Modeling the distribution of noise generated at the ocean surface over a wide ocean area with a spatially variable source level distribution in a three‐dimensional (3D) environment requires the capability of efficiently computing the acoustic field from all points on the surface to all points in the ocean. A previously developed noise theory is modified [W. A. Kuperman and F. Ingenito J. Acoust. Soc. Am. 67, 1988–1996 (1985)] and is combined with an approximate numerical technique for computing transmission loss in a 3‐D environment based on adiabatic mode theory [M. P. Porter, W. A. Kuperman, F. Ingenito, and A. Piacsek, J. Acoust. Soc. Am. Suppl. 1 81, S9 (1987)]. The wave equation solution of the acoustic field (or its cross‐spectral density) from any point to any point (or to any pair of points) reduces to a “spreadsheet” type manipulation of precomputed modal solutions. This 3‐D range‐dependent numerical scheme is shown to reduce to analytic expressions for the distribution of noise in a horizontally...

A new method of ocean acoustic tomographym—“Phase tomography.”

The ray travel time has been used as the basic measurement for inversion of the sound speed structure in the conventional ocean acoustic tomography. There are cases where the ray theory is neither applicable nor convenient, especially for lower frequency and axial propagation in the ocean sound channel. In the present paper, a new method of ocean acoustic tomography has been developed; it is a full‐wave approach based on the adiabatic normal mode theory. Instead of the ray travel time, the modal phase difference perturbations are used for inversion—we call it “phase tomography.” The method is suitable for lower frequency, and it is also suitable for axial propagation; moreover, the range‐dependent sound‐speed structure can be obtained from the vertical slice inversions. The mathematical derivation will be presented and some possible ways of implementation will be discussed.

Determination of ocean bottom sound‐speed profiles using a modal inverse technique in a range‐dependent shallow water environment

A modal inverse technique for determining ocean bottom sound‐speed profiles is applied to experimental data obtained in a range‐dependent shallow water environment. The method consists of towing a cw source away from moored receivers to obtain a well‐sampled measurement of the complex pressure versus range. The Hankel transform of the measured field then yields the depth‐dependent Green's function versus horizontal wavenumber, which contains distinct peaks at the modal eigenvalues. The eigenvalues are used as input data to a perturbative inverse scheme that generates the sound‐speed profile of the bottom. Data at 140 and 220 Hz were obtained in May 1984 in Nantucket Sound, MA in an area where the depth changed abruptly at about the midpoint of the experimental track. An associated change in bottom properties was assumed, and the concept of local modal eigenvalues yielding local geoacoustic parameters was then used to determine separate models for the two regions. Excellent agreement between the pressure fields computed for this range‐dependent model using adiabatic mode theory and the experimentally measured fields is demonstrated.

Normal mode filtering for downslope propagation in a shallow water wedge

Experiments to measure downslope sound propagation in a shallow water wedge with a penetrable bottom have been conducted in an indoor tank. Mode filtering has been applied to the signals received on a vertical array to extract the waveforms of individual normal modes. Results show that the normal modes of the wedge propagate with wave fronts which are curved into arcs of circles centered on the wedge apex. These wedge modes propagate downslope without coupling and are therefore the true normal modes of the wedge. Conventional normal modes with vertical wave fronts show coupling effects for slopes of 2° and higher. The group velocities of the wedge modes are strongly dependent on the slope and water depth and are in agreement with adiabatic normal mode theory. Modes which are not trapped in the water depth at the source can become trapped when the water depth is sufficient to support them.

Downslope propagation of normal modes in a shallow water wedge

Experimental results are presented for the downslope propagation of acoustic normal modes in a shallow water wedge with a penetrable bottom and a slope of up to 9°. A 7-element line source was used to generate individual low-order normal modes. To obtain single modes in a wedge it was necessary to align the line source to form an arc of a circle centered on the wedge apex. The resulting normal modes propagate with wave fronts which are also curved into arcs of circles centered on the wedge apex. These ‘‘wedge modes’’ propagate without coupling. The amplitudes, mode functions, and group velocities of the wedge modes are described by simple adiabatic normal mode theory, but the wave front curvature is not.