Normal Mode Model Research Articles

Three-dimensional matched-field tracking in shallow water using SWellEx-96 Data

A three-dimensional version of matched-field tracking (MFT), a generalization of matched-field processing (MFP), is used to locate and track a towed source using a vertical array during SWellEx-96—an experiment conducted off the San Diego coast in May 1996. Archival and in situ environmental data are used as input into a forward propagating adiabatic normal mode (ANM) model and a reciprocal propagating parabolic equation (PE) model to create the three-dimensional replica fields used to compute ambiguity volumes (the three-dimensional generalization of the ubiquitous two-dimensional ambiguity surface). Interaction with variable bathymetry caused by a downward refracting sound-speed profile and shallow water is not sufficient to provide uniqueness of the data and replica fields, thus creating many ambiguities for any given data within the fiducial volume. These ambiguities can be minimized via MFT, which uses source motion to reduce the number and relative magnitude of false ambiguities by averaging the ambiguity functions corresponding to trial source tracks defined by an initial position and velocity vector. As increasing numbers of ambiguity functions are averaged, steady movement within a fluctuating background can be detected.

Data-model comparisons of reverberation at three shallow-water sites

Reverberation measurements made by the SACLANT Undersea Research Centre at three shallow-water sites (130-190-m depth) are compared with each other and with estimates from the DREA normal-mode reverberation model OGOPOGO. The experiments over silt-clay and sand seabeds were conducted at slightly bistatic geometries (0.7-6.0-km source-receiver separation), using explosive sources detonated at mid-water depths. The signals were received on hydrophones of either a vertical or horizontal array and analyzed in one-tenth-decade frequency bands from 25 to 1000 Hz. The data are compared with each other to investigate the site differences and frequency dependencies, and with the estimates from the reverberation model OGOPOGO to interpret the data and to obtain a qualitative measure of the scattering. For modeling purposes, geoacoustic models of the seabed were assumed, and the reverberation data were fitted by adjusting the Lambert bottom scattering coefficients. Good model agreement was obtained with both individual hydrophone and data. Though somewhat sensitive to the geoacoustic the Lambert coefficients give a measure of the frequency dependence of the scattering. For the silt-clay bottom, the scattering is weak but is independent of frequency; for the sand bottoms, the scattering is stronger and increases with frequency. These results are compared with estimates from other experiments.

Wind effects on sub-tidal currents in Puget Sound

Wind effects on sub-tidal currents are studied using current meter records obtained at six moorings across the main basin of Puget Sound. High correlations between wind speeds and currents are found near the surface and at mid-depths of about 100 m. Empirical Orthogonal Function analysis applied to the axial currents in 1984 and 1985 shows that mode 1, containing over 60% of the variance, is highly correlated with wind speed even without any near surface current records. When near surface stratification is strong, direct wind effects are limited to the upper 30 m with counter currents in the lower layer indicating a baroclinic response. The transport in the lower layer almost balances the transport in the upper layer. When near surface stratification is weak, direct wind effects on currents can be detected to about 100 m. In this case, there is no clear and consistent depth at which one can separate the upper from the lower layer. Time series show that the acceleration in the surface layer initially increases in the same direction as the wind when the wind starts blowing, but it reaches a maximum, starts decreasing, and eventually changes to the opposite direction (decelerates) while the wind continues to blow in one direction. Results of a continuously stratified normal mode model and estimations from the observations suggest that friction at solid boundaries is a major cause of these phenomena. The model shows that modal currents of normal modes 2 and 3 are as important as mode 1, although the resultant vertical structure of total current shows a two-layer type pattern with only one zero crossing. The effect of the baroclinic pressure gradient is only apparent at low frequencies and among lower modes.

A LOCAL MODE APPROACH TO MOLECULAR VIBRATION ROTATION SPECTROSCOPY OF LOCAL MODE VIBRATIONAL STATES

We have developed a method to evaluate the effective vibration rotation parameter using the wavefunction from the local mode model, on the basis of the wavefunction equivalence of the two models describing the stretching overtone——the NMDD normal mode model and HCAO local mode model. By using this approach, an improved “α-relation” is derived, which exists among the effective vibration rotation parameters of local mode vibrational state at the local mode limit.

Ray theory in mode models: Demonstration of high-frequency collective-mode interference effects

A number of acoustic reflection and refraction effects, normally associated with ray theory, are used here as tests of a normal mode model in the high-frequency limit, where the modes become dense in the complex wave-number plane. The ray effects, which are associated with saddle points or branch line integrals in the wave-number integration method, do not appear as characteristics of any single mode in the modal solution. Rather, they emerge as collective effects from the coherent addition of large numbers of modes. The effects demonstrated include beam splitting at the Brewster angle for a fluid–fluid interface and at the Rayleigh angle for a fluid–solid interface, beam displacement near the critical angle for total reflection, and beam spreading due to a fluid–solid lateral wave. Gaussian beams are modeled efficiently using the ORCA normal mode model, by assigning complex values to the source coordinates. This technique extends, in a full mode solution, methods that have previously been implemented for Gaussian beams propagating in free space [Deschamps, Electr. Lett. 7, 684 (1971)] and reflecting from single interfaces [Ra et al., J. Appl. Math.24 , 396 (1973)]. [Work supported by the U.S. Navy Office of Naval Research.]

Poro-elastic and poro-acoustic propagation modeling

Until recently, sound propagation modelers have treated the ocean bottom as an acoustic or elastic medium. This paper will discuss some recent progress in poro-elastic propagation modeling. Biot’s theory of poro-elasticity has been incorporated into normal mode, wave-number integration, and parabolic equation models. Models have also been developed for poro-acoustic media, the limit of Biot’s theory in which shear waves are neglected. A series of benchmark problems that emphasize poro-elastic and poro-acoustic effects has been designed to test the accuracy of the models and their implementations. In addition to the propagation models, techniques have been developed for solving scattering problems and inversion problems involving poro-elastic layers. [Work supported by ONR.]

Model-based identification: an adaptive approach to ocean-acoustic processing

A model-based approach is developed to solve an adaptive ocean-acoustic signal-processing problem. Model-based signal processing is a well-defined methodology enabling the inclusion of propagation models, measurement models, and noise models into sophisticated processing algorithms. Here, we investigate the design of a so-called model-based identifier (MBID) for a general nonlinear state-space structure and apply it to a shallow water ocean-acoustic problem characterized by the normal-mode model. In this problem, we assume that the structure of the model is known and we show how this parameter-adaptive processor can be configured to jointly estimate both the modal functions and the horizontal wave numbers directly from the measured pressure-field and sound speed. We first design the model-based identifier using a model developed from a shallow-water ocean experiment and then apply it to a corresponding set of experimental data demonstrating the feasibility of this approach. It is also shown that one of the benefits of this adaptive approach is a solution to the so-called mismatch problem in matched-field processing (MFP).

A theoretical and simulation study of acoustic normal mode coupling effects due to the Barents Sea Polar Front, with applications to acoustic tomography and matched-field processing

Normal mode coupling due to a shallow water coastal front is considered, using oceanographic data from the 1992 Barents Sea Polar Front (BSPF) experiment as input to normal mode and parabolic equation (PE) acoustic propagation models. Criteria for the sensitivity of mode coupling to coastal front widths are derived, and applied to the BSPF as a representative example. The effects of coastal fronts on tomographic schemes are considered, particularly in terms of travel time-based tomography. The effects of frontal mode coupling on matched-field processing schemes are also considered, using both the maximum likelihood and variable coefficient likelihood methods, which show differing sensitivities to mismatch. Finally, directions for future research are discussed.

Analytical visco‐elastic relaxation models

In the past, relaxation processes employing PREM or 1066B‐stratified earth models have been solved numerically, either directly in the time domain (initial value models) or in the Laplace‐transformed domain (normal mode models). Solving N‐layer stratified models analytically has only been performed for small values of N. We present a brief outline of the analytical method of general radially stratified N‐layer, self‐gravitating, Maxwell rheological models. The analytical models allow for the relaxation times of the myriad of relaxation modes to be determined with very high accuracies, as the secular determinant from which they derive is an analytical function. We show by explicitly comparing results on a 5 and 30‐layer incompressible Maxwell model with a convex mantle viscosity profile, that the differences in the results between the two models are small when the 5‐layer model has the volume‐averaged structural and rheological properties of the 30‐layer model. The fact that in both models the fluid values of the Love numbers reach the same expected limits leaves no room for hypothetical so‐called non‐modal contributions. The application to compressible linear rheologies of this generalized analytical normal mode method is more involved, but straightforward.

Mode coupling and mode stripping in propagation across the range-dependent Atlantic Generating Station (AGS) site.

The complicated patterns of modal structure and attenuation observed in sound propagation through a range-dependent shallow-water environment are used to investigate and visualize the mechanisms of mode coupling and stripping. Broadband experiments have been carried out at the geologically known AGS site in the past few years. This presentation reports on a recent experiment which provides significantly more acoustic data along different propagation tracks and, consequently, permits high resolution of physical processes in range. A mode-filtering technique is used to construct modal structure from the data, while modal attenuation is obtained from spectral ratios. The evolution of modal structure is determined as a function of frequency and range. This is examined in view of environmental parameters along a selected propagation track. Numericalsimulations are carried out using interpolations of the known geoacoustic environment and employing the KRAKEN normal mode and RAM parabolic equation models. Results from these simulations are compared with the experimental data.

How to reconcile body-wave and normal-mode reference earth models

SUMMARY Reference earth models can be retrieved from either body waves or normal-mode eigenperiods. However, there is a large discrepancy between different reference earth models, which arises partly from the type of data set used in their construction and partly from differences in parametrization. Reference models derived from body-wave observations do not give access to density, attenuation factor and radial anisotropy. Conversely, reference models derived from normal modes cannot provide the correct locations for the depth of seismic discontinuities, nor the associated velocity jump. Eigenperiods derived from reference models constructed using body-wave data together with classifical attenuation models differ significantly from the observed eigenperiods. The body-wave and normal-mode approaches can be reconciled. The V’ and V, velocities given by body-wave models are considered as constraints, and an inversion is performed for parameters that cannot be extracted from body waves in the context of a radially anisotropic model, i.e. the density p, the quality factor Q, and the anisotropy parameters 5, (b and q. The influence of anelasticity is very large, although insufficient by itself to reconcile the two types of model. However, by including in the inversion procedure the density and the three anisotropic parameters, body-wave models can be brought into complete agreement with eigenperiod data. A number of reference models derived from body waves were tested and used as starting models: iasp91, sp6, and two new models ak303 and ak135. A number of robust features can be extracted from the inversions based on these different models. The quality factor Q,, is found to be much larger in the lower mantle than in previous models (e.g. prern). Anisotropy, in the form of transverse isotropy with a vertical symmetry axis, is significant in the whole upper mantle, but very small in the lower mantle except in the lower transition zone (between the 660 km discontinuity and 1000 km depth) and in the D”-layer. Compared with prem there is an increase of density in the D”-layer and a decrease in the lower transition zone. The attenuation estimates have been derived using velocity dispersion information, but are in agreement with available direct measurements of normal-mode attenuation. Such attenuation data are still of limited quality, and the present results emphasize the need for improved attenuation measurements.

Broadband frequency response modeling for inversion of a shallow-water waveguide with a range-dependent soft clay bottom (Yellow Shark ’95 experiment).

During the Yellow Shark ’95 experiment the broadband frequency response (200–800 Hz) of a shallow-water waveguide was measured for the purposes of performing multimodal inversion of its acoustic and geoacoustic properties. Measurements from vertical receiving arrays deployed every 8 km along a 40-km tomography transect showed that the frequency response was mostly flat with high transmission loss at selected frequencies. This differs from often observed frequency dependence in other shallow-water environments where there is some low frequency with minimum loss, suggesting an optimum frequency of propagation. Range-dependent geoacoustic and seismic data along the transect and water sound-speed data measured during the acoustic transmissions permitted the development of a realistic environmental model. Normal-mode modeling results with C-SNAP indicated that the presence of a soft clay top layer with a speed less than the water column was a decisive factor in the observed frequency dependence, since it acted as an alternative waveguide to lower-order modes. The modeled frequency dependence and absolute level of transmission loss closely agreed with the measurements obtained at the different ranges and depths, resulting in a validation of theoretical arguments involving selected frequencies of high transmission loss as a major feature of this type of environment.

Matched-field source localization in a range-dependent environment

In this paper source localization results obtained by matched-field (MF) processing of vertical line array (VLA) data from the PACIFIC ECHO III sea trial are presented. The experimental site was located in a region of significant bathymetric variation. An harmonic source was towed on a series of arcs, providing data for a sequence of different ranges and azimuths. In spite of the significant bathymetric variation, good MF localization results were obtained for a large fraction of the trial time. In the first part of this analysis it is assumed that the azimuth of the source is known. Substantial improvements in MF localization were obtained by using an adiabatic normal mode model to calculate replica acoustic fields for the range-dependent bathymetry. The improvement over the results obtained with flat-bottom waveguide replicas is demonstrated for data from the longer range arcs. In the second part of the analysis, the concept of environmental signal processing, an enhancement to MF processing that is obtained by making use of the complexity of the oceanic waveguide environment, is investigated. The azimuthal variation of bathymetry for the source tracks along the arcs was used to obtain estimates of the source bearing from the VLA data. In many instances, the bearing estimates were in reasonable agreement with the known source bearings.

Broadband frequency dispersion prediction using shallow-water data from Tanner Bank

Recent analyses of explosive charge reverberation and echo data [Davies etal. (1994)] showed a useful active broadband dispersion phenomena that was observed empirically in measured data for propagation paths within a mixed layer (ML). This Wilson Dispersion Phenomena (WDP) was originally noted in an environment which was characterized by a mixed layer. In April, 1994 an experiment [Holland etal. (1994)] was conducted under Navy sponsorship off the coast of California (Tanner Bank) in order to examine low-frequency active acoustic sources for use in characterizing bottom parameters in shallow water. This data set provided an ideal scenario for further examination of WDP. Upon preliminary examination of the Tanner Bank data set, the WDP was found to not only exist, but also to be predominant in environmental scenarios other than in the presence of a mixed layer. The phenomena proves to be highly dependent upon source–receiver geometry and environmental acoustic (EVA) characteristics such as sound-speed structure, bathymetry, bottom sediments, etc. It has been shown using the Tanner Bank data set that the occurrence of this phenomena can be modeled to a reasonable degree of accuracy using a time domain finite element parabolic equation model and a normal mode model. [Work supported by ONR.]

Selective high-resolution environmental parameter estimation using sector-focused matched-field processing

Sector-focusing (SF) has been shown to be effective at enhancing range-depth resolution while minimizing the instability caused by environmental mismatch. It has also been shown that a stable Bartlett type response could be obtained with SF using a small sector in range-depth space. This property is used to advantage when matched-field processing is used for source localization. By extending the sector focusing concept to include environmental parameter variation within the sector, high resolution may be selectively achieved. In this way greater resolution may be realized for a chosen set of search parameters by including in the sector, replicas derived from a set of ‘‘neighboring’’ environmental parameters. Conversely, by not including replicas derived from other ‘‘known’’ parameters, such as source range or depth, a broad Bartlett-like response may be obtained. Thus the estimate will simultaneously attain high resolution only in the selected search parameter while remaining insensitive to measurement errors in all other parameters, including errors in source location. Examples are presented for a simple parametrization scheme using the KRAKEN normal mode model to simulate a canonical shallow-water environment.

Rovibrational Spectroscopy of C 35Cl 4

High-resolution rotationally resolved infrared spectra of the spherical-top rotor C 35Cl 4 (in samples of natural isotopic composition) were obtained both in static cells by FTIR spectroscopy (ν 4, ν 1 + ν 4) and in a supersonically cooled expansion with tunable diode lasers (ν 3). For the ν 3 fundamental at 799 cm −1, quantum assignments were made for individual transitions from P(20) to R(24), and the partially resolved Q branch was matched with synthesized spectra. The bands ν 4 (316 cm −1) and ν 1 + ν 4 (774 cm −1) exhibit much reduced manifold splitting, and scalar fits were made to the R(24)- R(70) and P(38)- R(56) frequencies, respectively. The ν 3/(ν 1 + ν 4) Fermi dyad yields Coriolis constants ζ 3 = 0.621 and ζ′ 4 = −0.145 that satisfy the harmonic sum rule, but are perturbed by the Fermi resonance and are inconsistent with intramolecular force fields previously reported based on isotope shifts in matrix spectra. The true bending-mode Coriolis constant ζ 4 = −0.414, obtained from analysis of the isolated and unperturbed ν 4 band, agrees well with the isotope shifts. CCl 4 and other tetrahedral XY 4 molecules are shown to vibrate according to a simple vector-displacement normal-mode model for which ζ 3 ≈ 4/[3( m x / m y ) + 4].

Very broadband seismic analysis of the 1992 Flores, Indonesia, earthquake (Mw = 7.9)

The December 12, 1992, Flores Island earthquake (Mw = 7.9), which caused a destructive tsunami, is located in the back arc of the transition zone between the Sunda and Banda arcs, a region in active collision with the Australian continental margin. We study the earthquake using four seismological techniques spanning a period range of 1–1000 s and infer that the Flores event may be characterized as a shallow back arc thrust with fault plane strike 70°±30°, dip 28°±10° and rake 80° to 140° (linear trade‐off with strike), a rupture consisting of three discrete pulses with a total duration of 70 s (centroid time 26.3 s) and an estimated moment release between 7.5×1020 and 8.0×1020 N m. From the slip vector we infer that compressional forces play an important role in the deformation mechanism of the back arc north of Flores Island. Body waves constrain the hypocentral depth to be 16 km with rupture mainly propagating updip to shallow depth and bilaterally along strike with most subevents located toward the northeast. Evidence for rupture directivity from surface wave deconvolutions is inconclusive, but the overall source function is consistent with body wave results. Our body wave finite fault model is in rough agreement with two models proposed to explain tsunami run‐up observations. Normal mode modeling of W phase observations at periods longer than 70 s shows that slow slip is negligible. From this and the slip distribution determined from the body wave analysis we infer that the seismic source is an unlikely candidate to explain anomalously high tsunami run‐ups measured on northeastern Flores Island.

Model-based environmental inversion: A shallow water ocean application

A model-based approach to invert or estimate the sound-speed profile (SSP) from noisy pressure-field measurements or more succinctly a solution to the model-based inversion problem is discussed. Recall that model-based signal processing is a well-defined methodology enabling the inclusion of environmental (propagation) models, measurement (sensor arrays) models and noise (shipping, measurement) models into sophisticated processing algorithms. Here the design of a model-based processor (MBP) is investigated based on a Taylor series expansion of the SSP about the most current set of parameter estimates augmented into the state-space representation of a normal-mode propagation model to solve the environmental inversion problem. The resulting processor is adaptive in terms of simultaneously estimating the sound speed, modes, and pressure field at each iteration. Using data obtained from from the well-known Hudson Canyon experiment, a noisy shallow water ocean environment, the processor is designed and the results compared to those predicted using various propagation models and data. It is shown that the on-line, MBP not only predicts the sound speed quite well, but also is able to simultaneously provide enhanced estimates of both modal and pressure-field measurements which are useful for localization and rapid ocean environmental characterization.

A practical application of the Galerkin method to the broadband calculation of normal modes for underwater acoustics

A new application of the Galerkin technique to the calculation of the normal modes that are used to describe underwater acoustic propagation is presented. This expansion approximation method (EAM) is especially relevant for broadband normal mode calculations in which normal mode eigenvalues, depth functions, and attenuations are needed. The EAM is used to construct approximate normal mode eigenvalues and depth functions for some frequency from eigenvalues and depth functions previously calculated at another, higher frequency. The EAM approach for broadband underwater acoustic propagation calculations is evaluated in comparison with an exact normal mode model calculation of impulse responses and other broadband results.

Coriolis interaction parameters of the (2100; F 2) bands of SiH 4 and GeH 4. A test of local mode models

The high-resolution spectra of the (2100; F 2, N), N = 1 and 2, bands of SiH 4 and GeH 4 have been recorded and preliminary analyzed. The Coriolis interaction parameter 2 Bζ 3 is obtained; it provides a test of three widely used models: the harmonically coupled anharmonic oscillators model with two different kinds of variables, and the normal mode model with Darling-Dennison resonance included.