Gary H. Brooke

Rational square-root approximations for parabolic equation algorithms

In this article, stable Pade approximations to the function 1+z are derived by choosing a branch cut in the negative half-plane. The Pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z<−1, I(z)=0 [I(z) denotes the imaginary part of z]. The characteristics of the corresponding square-root approximation are illustrated for various segments of the complex plane. In particular, for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case. An elastic parabolic equation algorithm is used to illustrate the application of the new Pade approximations to a realistic ocean environment, including elasticity in the ocean bottom.

One‐way wave equations for seismoacoustic propagation in elastic waveguides

One‐way or parabolic wave equations for time‐harmonic propagation in two‐dimensional elastic waveguides are considered. It is shown that the direct application of a rational linear approximation with real coefficients to the elastic wave propagation case results in exponential growth in the numerical solutions. Elementary analysis demonstrates that this kind of approximation does not treat properly the modes with complex wavenumber which can exist in elastic waveguides. A new bilinear square‐root approximation with complex coefficients is introduced that accommodates all mode types and leads to stable numerical solutions. In the case of thick elastic layers (such as sea‐bottom sediments), this new approximation gives accurate total field prediction. When thin elastic layers (such as ice on the sea surface) are present, however, the method introduces excessive damping to modes with wavenumbers significantly different from a reference wavenumber.

Linear and nonlinear measures of ocean acoustic environmental sensitivity

This letter defines linear, linearized, and nonlinear measures of environmental sensitivity for ocean acoustic propagation that account for realistic uncertainties in various environmental parameters (water-column sound-speed profile and seabed geoacoustic properties). Simple interpretations of sensitivity are based on the implicit assumption of a linear relationship between parameter sensitivity and parameter uncertainty. This assumption is examined by comparing the three sensitivity measures over a range of parameter uncertainties about the actual assumed environmental uncertainty. Sensitivity range and depth dependencies are illustrated for realistic geoacoustic uncertainties and oceanographic variability of the sound-speed profile.

Improving performance for matched field processing with a minimum variance beamformer

Matched field processing (MFP) employing reduced minimum variance beamforming (RMVB) has been described in the literature as being robust to signal phase errors for vertical line arrays in modal noise. These techniques are extended to horizontal line arrays and evaluated for both horizontal and vertical arrays. The evaluation of RMVB is also more general in that it is extended to white and modal noise fields in the presence of phase or amplitude errors. Two phase error models were employed, one corresponding to constant sensor‐position errors the other to short‐term fluctuations of mode phase during the covariance matrix estimation. For minimum variance beamforming (MVB), in the presence of both types of random phase errors, array gain, peak‐to‐sidelobe ratios, and peak‐to‐background ratios all decreased in a modal noise field. Performance was far less sensitive to phase errors in spatially white noise than in modal noise. MVB was modified by reducing the number of eigenvectors employed in the matching. F...

Non-local boundary conditions for high-order parabolic equation algorithms

Abstract Numerical parabolic equation (PE) solvers for underwater sound propagation usually approximate the subbottom radiation condition by appending an absorbing layer to the computational mesh. An alternative procedure uses a non-local boundary condition (NLBC) applied along (or below) the sea–bottom interface to transform the semi-infinite PE configuration exactly into an equivalent bounded domain. In this paper, we analyze NLBCs that are expressed in terms of plane-wave reflection coefficients and describe their implementation in the context of high-order Pade-based PE algorithms. In particular, we develop non-local boundary conditions to account for the coherent losses due to scattering by a statistically-rough sea surface. Several numerical examples are presented to illustrate the numerical implementation of NLBCs for high-order PEs.

Transient Scholte wave transmission along rough liquid–solid interfaces

Transmission characteristics of transient Scholte waves along periodic and random liquid–solid interfaces are demonstrated using laboratory ultrasonic models. These characteristics indicate that, for periodic interfaces, the transmitted Scholte wave is composed of an attenuated main pulse, which arrives first, followed by substantial energy at the Bragg frequency, which is delayed and spread over a wide time window. Random topographic variations in the liquid–solid interface eliminate the transmission of the late‐arriving components. The experiments are limited to a water ‘‘half‐space’’ over a high‐speed solid half‐space, in which case, most of the Scholte wave energy is in the water. The Scholte wavelength is comparable to the horizontal spatial wavelength in the surface but is much greater than the vertical scale of the spatial variations. Results from composite interfaces formed by placing thin parallel scatterers on a plane solid surface and from corrugated interfaces formed by directly machining groo...

Spatial field shifts in ocean acoustic environmental sensitivity analysis.

This article examines the effects of spatial field shifts in ocean acoustic environmental sensitivity analysis. Acoustic sensitivity studies are typically based on comparing acoustic fields computed for a reference environmental model and for a perturbed model in which one or more parameters have been changed. The perturbation to the acoustic field due to the perturbed environment generally includes a component representing a spatial shift of the field (i.e., local field structure remains coherent, but shifts in range and/or depth) and a component representing a change to the shifted field. Standard sensitivity measures based on acoustic perturbations at a fixed point can indicate high sensitivity in cases where the field structure changes very little, but is simply shifted by a small spatial offset; this can conflict with an intuitive understanding of sensitivity. This article defines and compares fixed-point and field-shift corrected sensitivity measures. The approaches are illustrated with examples of deterministic sensitivity (i.e., sensitivity to a specific environmental change) and stochastic sensitivity (sensitivity to environmental uncertainty) in range-independent and range-dependent environments.

Numerical implementation of a modal solution to a range‐dependent benchmark problem

Numerical results are presented for a modal representation of sound propagation in a range‐dependent waveguide with plane‐parallel, perfectly reflecting boundaries. Range dependence in the waveguide is embedded in the sound speed that varies sinusoidally with depth and exponentially with range. The modal expansion is based on a conformal mapping technique, whereby the solution to the Helmholtz equation in a range‐dependent waveguide is expressed in terms of a solution to a parabolic equation in a corresponding waveguide whose sound speed is constant in the mapped coordinates.

Shallow water acoustic studies using an air‐suspended water waveguide model

A novel experimental technique is introduced that allows multimode interference and coupling due to boundary topography and to the presence of scatterers in shallow water acoustic waveguides to be studied in the absence of radiation and elastic bottom effects. The air‐suspended water waveguide model consists of a water layer bounded above by air and below by a thin rubber membrane supported by pressurized air to counteract the weight of the water. Experimental data on the transmission of transient ultrasonic signals along the ‘‘perfect’’ waveguide are presented and compared with theory. Both experimental and numerical results are used to demonstrate that multiple nondispersed discrete pulses in a waveguide are equivalent to a summation of highly dispersed waveguide modes. Preliminary experimental results are also given demonstrating the propagation of transient acoustic waves in a free‐fluid layer waveguide with sloped or periodic boundaries.

Progress on the DRDC clutter model

The DRDC Clutter model provides bistatic reverberation and target echo predictions for a towed array in a range-dependent shallow water area. The sound speed and scattering properties are specified on a rectangular grid, and the computations-including the full 3-D beam patterns of the horizontal array-are performed along multiple radials from the receiver. The computational engine is written in Fortran 95, and uses adiabatic normal modes, so is quite efficient computationally. This paper provides an overview of the model, with model-data comparisons, and emphasis on progress made since 2011. Model-data comparisons are shown for towed array reverberation data from the Malta Plateau in the Mediterranean, and predictions are made for reverberation and target echoes from a near-shore area in the Gulf of Mexico where experiments are planned in 2013. In addition to the research version of the model, a Java and C interface developed by Brooke Numerical Services provides a GUI interface for user input, access to ...