Theoretical and numerical results are presented for modal characteristics of the seismo-acoustic wavefield in anisotropic range-independent media. General anisotropy affects the form of the elastic-stiffness tensor, particle-motion polarization, the frequency and angular dispersion curves, and introduces near-degenerate modes. Horizontally polarized particle motion (SH) cannot be ignored when anisotropy is present for low-frequency modes having significant bottom interaction. The seismo-acoustic wavefield has polarizations in all three coordinate directions even in the absence of any scattering or heterogeneity. Even weak anisotropy may have a significant impact on seismo-acoustic wave propagation. Unlike isotropic and transversely isotropic media with a vertical symmetry axis where acoustic signals comprise P-SV modes alone (in the absence of any scattering), tilted TI media allow both quasi-P-SV and quasi-SH modes to carry seismo-acoustic energy. Discrete modes for an anisotropic medium are best described as generalized P-SV-SH modes with polarizations in all three Cartesian directions. Conversion to SH is a loss that will mimic acoustic attenuation. An in-water explosion will excite quasi-SH.

Several problems of current interest involve elastic bottom range-dependent ocean environments with buried or earthquake-type sources, specifically oceanic T-wave propagation studies and interface wave related analyses. Additionally, observed deep shadow-zone arrivals are not predicted by ray theoretic methods, and attempts to model them with fluid-bottom parabolic equation solutions suggest that it may be necessary to account for elastic bottom interactions. In order to study energy conversion between elastic and acoustic waves, current elastic parabolic equation solutions must be modified to allow for seismic starting fields for underwater acoustic propagation environments. Two types of elastic self-starter are presented. An explosive-type source is implemented using a compressional self-starter and the resulting acoustic field is consistent with benchmark solutions. A shear wave self-starter is implemented and shown to generate transmission loss levels consistent with the explosive source. Source fields can be combined to generate starting fields for source types such as explosions, earthquakes, or pile driving. Examples demonstrate the use of source fields for shallow sources or deep ocean-bottom earthquake sources, where down slope conversion, a known T-wave generation mechanism, is modeled. Self-starters are interpreted in the context of the seismic moment tensor.

Recently, two-types of elastic self-starters have been incorporated into parabolic equation solutions for range-dependent elastic bottom underwater acoustic problems. These source fields generate parabolic equation solutions that can be used to study development of oceanic T-phases via the process of downslope conversion. More general range-dependence has also been shown to scatter elastic wave energy into water column acoustic modes which then propagate as T-phases. In certain circumstances, sources in the elastic bottom can also cause interface waves at the ocean bottom that contribute to the ocean acoustic field. Both types of waves can propagate long distances and could be source mechanisms for unexplained acoustic signals recorded near the sea floor and below the ray-theoretic turning point. Parabolic equation solutions will be used to study effects of parameters such as frequency, source location, and source type on T-phase and interface wave generation and propagation.

The shallow water environment may be highly variable, with both range dependence and anisotropy almost ubiquitous in the seafloor bottom/sub-bottom regions. Some common causes of range dependence include marine-sediment composition, non-planar boundaries, rough surfaces, strong density, or velocity contrasts, and variation in water-column depth and/or sediment-cover thickness. Common causes of elastic anisotropy are compositional layering or vertically aligned cracks. There is an apparent trade-off between anisotropy and range dependence, and difficult to separate the two effects in a propagating signal. If the symmetry axis of a compositionally layered sediment is not exactly normal to the seafloor, the seismo-acoustic wave field has particle-motion polarizations in all three coordinate directions. Even in a one-dimensional medium, an explosion source excites sediment particle motion with all three polarizations. Assuming sediment isotropy when it is not justified can cause errors in layer-thickness computations, elastic property-gradient determinations, acoustic attenuation, and predictions of long-range sound propagation. Employing a modal representation of the seismo-acoustic wavefield, we illustrate the effects of anisotropy and range dependence on the modes, such as mode-identity switching due to angular dispersion and coupling of longitudinal, vertical, and horizontal particle motion polarization.

This communication corrects errors and supplies missing parameter values for a previous publication by the authors (ibid., vol. 35, no. 3, pp. 603-617, Jul. 2010) regarding the geoacoustic bottom interaction model (GABIM).

The geoacoustic bottom interaction model (GABIM) has been developed for application over the low-frequency and midfrequency range (100 Hz to 10 kHz). It yields values for bottom backscattering strength and bottom loss for stratified seafloors. The model input parameters are first defined, after which the zeroth-order, nonrandom problem is discussed. Standard codes are used to obtain bottom loss, uncorrected for scattering, and as the first step in computation of scattering. The kernel for interface scattering employs a combination of the Kirchhoff approximation, first-order perturbation theory, and an empirical expression for very rough seafloors. The kernel for sediment volume scattering can be chosen as empirical or physical, the latter based on first-order perturbation theory. Examples are provided to illustrate the various scattering kernels and to show the behavior predicted by the full model for layered seafloors. Suggestions are made for improvements and generalizations of the model.

In continuous geoacoustic inversion, the resolution and uncertainty of the estimated ocean bottom are inherently linked by a tradeoff with each other. They also depend on the experiment geometry and the choice of formulation for the data and for the bottom model. For example, the resolution and variance will differ if the data are represented by an intensity envelope timeseries or a tau-p timeseries, or if wave slowness or bottom impedance is estimated. Previous work by the authors investigated variations in resolution and uncertainty with geometry and problem formulation based on linearization of the geoacoustic problem at a given solution. However, the problem is further complicated by the fact that the solution point itself can also depend on the geometry and data and model formulation so that the resolution and uncertainty vary for that reason also. This aspect of the nonlinear geoacoustic inverse problem is explored here.

Elastic anisotropy is a nearly ubiquitous feature of marine sediments. The simplest type of sediment anisotropy is transverse isotropy, characterized by five elastic constants, and results from layered deposition. A modal scattering theory for volume perturbations of the sediment elastic moduli is presented. The scattering theory is based on the coupled mode formulation for propagation in range dependent fluid-elastic media. The Born approximation is employed to derive a modal scattering matrix. Although the perturbations of the elastic moduli are random, they may not be arbitrary in the sense that certain symmetry and energy constraints among the moduli must be respected. Mode–mode coupling matrices are computed for quasi-P-SV-SH seismo-acoustic modes, which show mode mixing and the importance of non-nearest neighbor interactions. The effects of volume scattering can be combined with rough surface scattering and also incorporated into mode coupling caused by deterministic range dependence of the material properties. This work has implications for acoustic loss estimates for low-frequency shallow water acoustic propagation.

Modeling acoustic propagation in the ocean requires a representation of the ocean sound speed. For a 1-D, range independent ocean, probably the most common representation comprises constant sound speed layers and/or gradient layers. Gradients of the form 1/c²(z), admitting an exact solution for the acoustic pressure field in terms of Airy functions, are commonly employed. A disadvantage of using the Airy functions is that they are specific to that profile. For more general profiles, the best representation may be a stack of constant sound speed layers for which the pressure field is expressible in terms of simple exponentials.

As an example, computing the backscattering strength from a stack of layers containing volume heterogeneities requires evaluation of an integral proportional to the 4th power of the pressure field. Integrating the simple exponential solutions for constant sound speed layers is trivial. Integrating a product of four Airy functions, either analytically or numerically, is not. If the medium is represented only by constant sound sound speed layers, what is the best way to choose the layers, and how many layers are needed? These questions are investigated by employing an algorithm originally due to Fuchs (1968).

The resolution operator for a linear inverse problem indicates how much smearing exists in the map between the true model and the estimated model. The trace of the resolution operator provides an estimate of the number of model parameters model that are resolved. In a series representation of the resolution operator for a nonlinear problem, the higher-order terms indicates how much spurious nonlinear leakage there is from the true model to the estimated model.

In previous work, the solution of a simple nonlinear ocean acoustic inverse problem as a perturbation series in the horizontal wavenumber was constructed, and the linear data kernels were presented. This linear problem permitted the quantification of the magnitude of the perturbation and indicated when nonlinear effects must be taken into consideration. In this extension of previous work, the second order resolution kernel is constructed, which describes how effectively nonlinear effects can be removed from the reconstructed model.

The resolution operator for a nonlinear inverse problem is the product of the estimated inverse of the forward model operator and the forward model operator itself. If the inverse of the forward model operator were exact, then the resolution operator would be the identity, but in general the resolution operator describes the (noninvertible) transfer function between the unknown true environmental model and the limited-resolution version, which can be estimated from measurements. The nonlinear model resolution operator can be computed iteratively from the Neumann series representation of an estimate of the inverse of the forward model operator with the assumption that both the data functional and the model perturbation functional possess regular perturbation expansions. An example of a problem that fits these criteria is normal mode acoustic propagation with "slow enough" perturbations such that the modes adjust adiabatically to the perturbations, and the mode eigenvalues are "far" from cutoff. We examine the effects of the nonlinear components of the model and the higher order components of the model resolution on the reconstruction of the model estimate for a simple ocean acoustic propagation problem.

Inversion of ocean bottom geoacoustic properties from acoustic receptions in the water column is a nonlinear inverse problem that is inherently unstable and nonunique. One common approach to stabilizing this problem is to assume that the ocean bottom is made up of a small number of layers. The solution from this approach does allow one to reproduce the scattered sound field if all the other experiment parameters such as frequency and geometry are also reproduced. However without extensive prior information about that ocean bottom, this approach yields only one of many equivalent nonunique solutions and may not accurately describe the actual ocean bottom itself. An alternate approach, which may allow one to reuse the results later with a different frequency or geometry, is to use the tools of geophysical continuum inversion to specify the degree of nonuniqueness by quantifying both the uncertainty and limited resolution of the continuum bottom solution. This work compares inversion uncertainty and resolution results for different formulations of the data (e.g., matched field versus matched modes versus waveform), different geometries, and different formulations of the uncertainty (e.g., normally distributed versus including some higher-order moments).

An inverse problem is the estimation of a continuous model function given a finite set of data points. In most geophysical problems, due to data noise and lack of complete geometric coverage, this problem is intractable analytically so we parameterize the continuous model and solve as a regularized parameter estimation problem. Coarsening the discretization of an ocean bottom model is one way to stabilize the otherwise ill-conditioned problem; this is sometimes called regularization by discretization. But by setting the resolution of the model estimate prior to the inversion in this way, one also affects the model estimate's statistics, due to the inherent tradeoff between model resolution and uncertainty. This relationship is well understood for linear problems, but resolution and uncertainty in nonlinear inverse problems remains an active area of research. In nonlinear problems the probability distributions of the model parameters are not Gaussian, and so with Monte Carlo methods we explore the relationship between the magnitude of non-Gaussian features in the model probability, the discretization of the model, and the aliasing of its parameters.

The intrinsically non-Gaussian statistics of nonlinear inverse problems, including ocean geoacoustic problems, is explored via analytic rather than numerical means. While Monte Carlo Bayesian methods do address the non-Gaussian statistics in nonlinear inverse problems, they can be very slow, and intuitive interpretation of the results are at times problematic. There is great theoretical overlap between recursive filters/smoothers, such as the extended Kalman filter, and methods of linear and nonlinear geophysical inversion. The use of recursive filters in inversion is not in itself new, but our interest is in adapting statistical developments from one to the other. Classic analytic methods in both filtering theory and inverse theory assume Gaussian probability distributions, but newer nonlinear filters do not all make this assumption and are explored for their potential application to nonlinear inverse problems. The similarities and differences between the frameworks of filtering theory and inverse theory are laid out in a series of geoacoustic inverse problem examples. Recent work in nonlinear filters handles non-Gaussian probability densities that are constrained to a particular form, and also derives analytic expressions for higher order moments of these density functions. The application of these developments to geoacoustic inverse problems is addressed.

For a strictly linear inverse problem the model resolution matrix is a simple product of the linear system matrix and its generalized inverse. How much the model resolution differs from the identity is an indication of bias in the inverted model. Nonlinear inverse problems lead to higher order resolution operators, which allow one to quantify the effect of ignoring the nonlinearity in the inverse problem. In addition, the trade off between resolution and variance is examined for the nonlinear inverse problem. Ocean acoustic inverse problems for ocean and/or bottom structure are intrinsically underdetermined, because some continuous quantity such as sound speed or temperature is estimated from a finite number of data values, and the model is generally discretely parametrized to be numerically tractable. Discretizing the model has the positive effect of regularizing an intrinsically ill-posed problem. However, discretizing an intrinsically nonlinear problem extracts a penalty. Higher order resolution operators may contribute even when they should be identically zero. (For example, in a quadratically nonlinear model, cubic and higher order resolution operators should be zero, but won't be zero as a result of the discretization.) Modal propagation in an ocean acoustic waveguide is used to illustrate the issues.

The goal of solving geophysical inverse problems isn't just to find a model fitting the data. We can always fit a (N-1)th order polynomial to N data points, but it doesn't tell us much about the solution's uniqueness, or limits placed on resolution in the presence of noise. The real goal is to develop more information than just a model that fits the data. For truly linear problems, the statistics are Gaussian, constraints on the model information and uniqueness are characterized by the the null space of the operator which maps the model to the data, and the resolution matrices have well defined meanings. However, many inverse problems of interest in geophysics are nonlinear. Global solution methods such as simulated annealing or genetic algorithms tell us nothing about the statistics and little about the uniqueness of our model solution. Monte Carlo analyses are exhaustive and do provide information about uniqueness and a posteriori statistics but are numerically intensive. As an alternative we have examined a nonlinear filter that is an extension of the Kalman and Extended Kalman filters. We address the question: Can geophysically interesting problems be recast as nonlinear filtering problems, and if so what can they tell us about the evolution of the statistics, uniqueness, and model resolution?

Nonlinear inverse problems including the ocean acoustic problem have been solved by Monte Carlo, locally-linear, and filter based techniques such as the Extended Kalman Filter (EKF). While these techniques do provide statistical information about the solution (e.g., mean and variance), each suffers from inherent limitations in their approach to nonlinear problems. Monte Carlo techniques are expensive to compute and do not contribute to intuitive interpretation of a problem, and locally-linear techniques (including the EKF) are limited by the multimodal objective landscape of nonlinear problems. A truly nonlinear filter, based on recent work in nonlinear tracking, estimates state information for a nonlinear problem in continual measurement updates and is adapted to solving nonlinear inverse problems. Additional terms derived from the system's state PDF are added to the mean and covariance of the solution to address the nonlinearities of the problem, and overall the technique offers improved performance in nonlinear inversion.

The goal of solving ocean acoustic inverse problems is not just to find a model fitting the data. We can always fit an N-1th order polynomial to N data points, but it does not tell us much about the uniqueness, or limits placed on resolution in the presence of noise. The real goal is to develop more information than just a model that fits the data. For truly linear problems, constraints on the model information and uniqueness are characterized by the null space of the operator which maps the model to the data, the statistics are Gaussian, and the resolution matrices have well-defined meanings. Unfortunately most inverse problems of interest in ocean acoustics are nonlinear. We are faced with methods such as simulated annealing or genetic algorithms, which tell us nothing about the statistics and little about the uniqueness of our model solution, or exhaustive but numerically intensive Monte Carlo analyses. As an alternative we have examined a nonlinear filter, that is an extension of the Kalman and extended Kalman filters. We discuss the background of the ocean acoustic inverse problem for bottom properties, and how it can be addressed by employing an exact nonlinear filter.

The propagator for coupled-mode elastic waves can be cast into a number of different representations, which emphasize particular aspects of the wave propagation in a laterally heterogeneous medium. One representation has the form of a generalized scattering operator and contains a quantity that can be interpreted as the lateral impedance. Another representation reduces naturally to the JWKB approximation for smoothly varying media with no mode coupling. The propagator solution for the fields in a laterally heterogeneous elastic medium with weak random boundary fluctuations leads naturally to the application of Feynman diagram techniques for the derivation of Dyson's equation and the Bethe–Salpeter equation for the propagator mean and covariance, respectively. The diagram techniques are reviewed and their utility for solution of random media elastic wave problems is demonstrated.

If the Earth were a plane-layered semi-infinite half-space or a radially symmetric sphere oceanic, T-waves could not exist. This is apparent because the source depth of T-wave producing earthquakes is greater than the depth at which the low order modes comprising the T-waves have any significant amplitude. Bottom roughness provides a mechanism by which energy from high order source modes can be scattered into the lower order modes. The efficiency of this scattering is improved when there is a layer of sediments overlying the higher speed upper ocean crust. Some of the source modes develop a large anti-node, reminiscent of a Scholte wave, at the water sediment interface. Bottom roughness serves as a sheet of secondary sources placed directly on the anti-node of these modes, and contributes a significant amount of energy to the T-waves. However, a significant fraction of the total T-wave energy is also provided by small scattered contributions from many higher modes of the continuum spectrum, which is modeled using the locked mode approximation. Source mechanism modeling of T-wave excitation shows that normal fault earthquakes are inefficient generators of T-waves.

Physics-based interface scattering models for the seafloor [H.-H. Essen, J. Acoust. Soc. Am. 95, 1299-1310 (1994); Gragg et al., ibid. 110, 2878-2901 (2001)] exhibit features in their predicted grazing angle dependence. These features have a strong dependence on the assumed composition and roughness of the bottom. Verifying such predictions requires data that cover a wide range of grazing angles and involve minimal sub-bottom penetration. Such measurements were performed in the frequency band 2–3.5 kHz over an exposed limestone bottom off the Carolina coast during the second Littoral Warfare Advanced Development Focused Technology Experiment of 1996 (LWAD FTE 96-2). Direct-path bottom scattering strengths were obtained in shallow water (198–310 m deep) for grazing angles from 8° to 75° using data fusion from multiple experimental geometries coupled with careful signal processing. The processing included corrections for the surface-reflected path, other multipaths, and characteristics of the reverberation decay observed over the pulse duration at higher grazing angles. The resulting frequency and grazing-angle dependences exhibit trends consistent with theoretical predictions, and geoacoustic parameters obtained by inversion are consistent with values expected for limestone.

For any inverse problem, finding a model fitting the data is only half the problem. Most inverse problems of interest in ocean acoustics yield nonunique model solutions, and involve inevitable trade-offs between model and data resolution and variance. Problems of uniqueness and resolution and variance trade-offs can be addressed by examining the Frechet derivatives of the model–data functional with respect to the model variables. Tarantola [Inverse Problem Theory (Elsevier, Amsterdam, 1987), p. 613] published analytical formulas for the basic derivatives, e.g., derivatives of pressure with respect to elastic moduli and density. Other derivatives of interest, such as the derivative of transmission loss with respect to attenuation, can be easily constructed using the chain rule. For a range independent medium the analytical formulas involve only the Green's function and the vertical derivative of the Green's function for the medium. A crucial advantage of the analytical formulas for the Frechet derivatives over numerical differencing is that they can be computed with a single pass of any program which supplies the Green's function. Various derivatives of interest in shallow water ocean acoustics are presented and illustrated by an application to the sensitivity of measured pressure to shallow water sediment properties.

Bathymetry plays a strong role in the excitation of T-waves by breaking strict mode orthogonality and permitting energy from higher order modes to couple to the lower order modes comprising the T-phase. Observationally (Dziak, 2001) earthquakes with a strong strike–slip component are more efficient at generating T-waves than normal fault mechanisms with the same moment magnitude. It is shown that fault type and orientation correlates strongly with T-wave excitation efficiency. For shallow sources, the discrete modes contribute to the majority of the seismic source field, which is then scattered into the acoustic modes by irregular bathymetry. However, the deeper the earthquake source, the more important the continuum component of the spectrum becomes for the excitation. Deterministic bathymetry and random roughness enter the modal scattering theory as separate terms, and allow the relative contributions from the slope conversion mechanism and bottom roughness to be directly compared.

The excitation mechanism of oceanic T-waves has been a puzzle for almost fifty years, with refraction from a sloping seafloor and seafloor scattering as two of the most commonly invoked mechanisms. By representing the earthquake source field as a normal mode sum, it can be seen that both mechanisms are very closely related. Strict modal orthogonality prohibits the existence of T-waves in a laterally homogeneous semi-infinite half-space or radially symmetric sphere, as energy cannot be transferred from one mode to another in an homogeneous medium. Deterministic non-planar bathymetry, random boundary roughness, upper crustal heterogeneity, or a combination of these provides a physical mechanism to break the strict orthogonality. We show that modal scattering from the rough seabottom in the epicentral region converts energy from the directly excited ocean crustal/water column modes to the propagating acoustic modes comprising the oceanic T-wave. Submarine earthquake fault orientation also appears to be reflected in the T-wave excitation.

The amplitudes of the propagating acoustic modes associated with T-waves decay exponentially below their ray equivalent turning points, and cannot be excited directly by an earthquake. The modal decomposition for a T-wave producing earthquake that occurred near the western tip of the Blanco TFZ has been computed. The directly excited higher order modes are characterized by relatively large amplitudes in the ocean crust, significant water-borne components, and often strong interface components at the ocean–bottom boundary. Employing the modal scattering theory of Park and Odom (1999), it is found that energy has been transferred from higher order modes to the Stoneley fundamental and the lower order modes have significant amplitude at the water–bottom interface. Scattering from irregular ocean bottom bathymetry is the mechanism for the energy transfer. The lowest order acoustic modes, modes 1 and 2, are only very weakly excited because they have very small amplitudes at the bottom. This is consistent with the interpretation of de Groot-Hedlin and Orcutt (1999).