Robert F. Gragg

Small-slope scattering from rough elastic ocean floors: General theory and computational algorithm

In this article acoustic scattering by a random rough interface that separates a fluid incident medium from an underlying uniform scattering medium, either fluid or elastic solid, in cases for which the Bragg scale lies within the power-law tail of the roughness spectrum is dealt with. The physical foundation is an inherently reciprocity-preserving, local small-slope theory. A fully bistatic formulation is developed for the scattering strength, together with a robust numerical implementation that allows a wide range of spectral exponent values. The practical result for ocean acoustics is a significantly improved description of the interface component of sea floor scattering. Calculations are presented to demonstrate the advantage of this approach over perturbation theory, and to illustrate its dependence on frequency and environmental parameters as well as its operation in bistatic geometries.

Backscatter from a limestone seafloor at 2–3.5 kHz: Measurements and modeling

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 degrees to 75 degrees 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.

Small-slope simulation of acoustic backscatter from a physical model of an elastic ocean bottom

An underwater acoustic experiment with a two-dimensional rough interface, milled from a slab of PVC, was performed at a tank facility. The purpose was to verify the predictions of numerical models of acoustic rough surface scattering, using a manufactured physical model of an ocean bottom that featured shear effects, nonhomogeneous roughness statistics, and root-mean-square roughness amplitude on the order of the acoustic wavelength. Predictions of the received time series and interface scattering strength in the 100-300 kHz band were obtained from the Bottom Reverberation from Inhomogeneities and Surfaces-Small-Slope Approximation (BORIS-SSA) numerical scattering model. The predictions were made using direct measurements of scattering model inputs-specifically, the geoacoustic properties from laboratory analysis of material samples and the grid of surface heights from a touch-trigger probe. BORIS-SSA predictions for the amplitude of the received time series were shown to be accurate with a root-mean-square residual error of about 1 dB, while errors for the scattering strength prediction were higher (2-3.5 dB). The work is part of an ongoing effort to use physical models to examine a variety of acoustic scattering and propagation phenomena involving the ocean bottom.

Mathematical Modeling and Computer-Aided Manufacturing of Rough Surfaces for Experimental Study of Seafloor Scattering

Diverse aspects of stochastic processes, time-series analysis, fractal geometry, and manufacturing technology are brought together to provide a unified theoretical framework for the mathematical characterization and physical fabrication of scale-model representations of the rough ocean bottom. These scale models can be used to validate the predictions of interface-scattering theories, particularly those relating to the dependence of scattering strength on roughness parameters. In acoustical oceanography, rough surfaces are conventionally described in terms of power-spectral density (PSD) or fractal dimension. Here, recently developed concepts describing modified power-law PSD and approximately self-affine stochastic fractals are used to account for issues of finite sample size and finite manufacturing resolution. The large- and small- bandwidth restrictions that these issues inherently impose are related to their effects on the properties of surfaces as characterized both mathematically and by visual observation. This development provides the groundwork for numerical generation of surfaces, using spectral methods, and their physical manufacture, using computer-aided manufacturing (CAM) techniques. Effects of the spectral method of numerical generation on the statistics of generated topography are detailed. A manufacturing technique using a computer numerically controlled (CNC) milling machine is discussed and topography-specific guidelines for accurate manufacture of rough surfaces are prescribed.

Doppler sidebands in the cross-spectral density of narrow-band reverberation from a dynamic sea surface

Analytic methods are used to formulate the impact of a random dynamic sea surface on the space-frequency characteristics of bistatic reverberation. A narrow-band point source is positioned beneath the time-dependent surface of a range-independent ocean. The small-waveheight perturbative approximation is invoked, and attention is focused on the Doppler sideband contributions to the reverberation cross-spectral density for an arbitrarily placed receiver pair. The new expression that results is identified as an active scattering generalization of the van Cittert-Zernike theorem from classical partial coherence theory. This work is the first to explicitly predict the sideband structure in the cross-spectral density of the field scattered from a realistic moving sea surface. A numerical example is presented for a shallow source and shallow receivers in a homogeneous ocean.

One-way propagation in weakly nonuniform media

Abstract A series of energy-conserving generalized rotations is applied to the Helmholtz equation for a weakly nonuniform one-dimensional medium. These transformations decouple the wave field, to arbitrary order in the strength of the nonuniformity, into counter-propagating modes with separately conserved energies. The result is asymptotic propagation without backscatter at arbitrary order. Low-order results coincide with the WKB form of the solution. Higher orders provide WKB modifications applicable to long-range propagation, e.g. in long time-of-flight applications or very-long-baseline interferometry. The result is illustrated in a numerical example.

Evaluation of a fundamental integral in rough-surface scattering theory

An algorithm is presented for the numerical evaluation of a fundamental but intractable integral that occurs in the physical theory of scattering from random rough surfaces. It is based on a rational-function approximation to an integrand factor, augmented with techniques for excluding poles and zeros from the path of integration. Examples, complete with error analysis, are provided for cases relevant to acoustic sea-floor and sea-surface scattering.

The Dependence of Long‐Range Reverberation on Bottom Roughness

At long‐range, shallow‐water reverberation can be driven by sub‐critical‐angle scattering, i.e. by rough interrace scattering. The Naval Research Laboratory has recently developed a small‐slope model for elastic seafloors that provides physics‐based estimates of the dependence of scattering on the incident and scattered angles, and physical descriptors of the environment. In this paper, this incoherent model is used as kernels in reverberation models, which in turn are used to assess the sensitivity at 3.5 kHz of long‐range monostatic reverberation to the roughness of the water‐sediment interface. It is shown that when sub‐critical‐angle scattering dominates, the acoustic field could be quite sensitive to the parameter values of the roughness, thus arguing for the need for regional in‐situ methods for its estimation.

Calculations of acoustic scattering from an elastic ocean bottom

The problem of acoustic scattering from the rough interface between the ocean and an elastic bottom is examined. A recently developed reciprocal scattering formalism [D. Wurmser, J. Math. Phys. 37, 4434–4479 (1996)] allows existing numerical and operator expansion methods to be used to calculate scattering from rougher and/or higher‐dimensional surfaces than would otherwise be possible. Here, the method is used to generate new physically intuitive versions of perturbation theory and the small slope approximation for the bottom scattering problem. The results are compared to those for the analogous two‐fluid theory. The relative merits of the various approximations are discussed. Finally, the significance of the work to low‐angle scattering is discussed. [Work supported by ONR.]

Topography measurement of scale-model representations of the rough ocean bottom by touch-trigger probe and its implications for spectral characterization

Scale models of the ocean bottom exhibiting multiscale roughness having power-law form power-spectral density are useful for validation of deterministic and stochastic rough-surface scattering theories. Such scattering theories require accurate knowledge of the topography of the scale-model surface, which, at acoustic scales, can be measured on a two-dimensional grid using a kinematic-resistive touch-trigger probe and represented by a digital elevation model. Both the discrete representation and the physical measurement process introduce spectral artifacts. While the theoretical relationship describing spectral effects of the discrete representation is well known, this relationship is more complex for the physical measurement process. In the later case, spectral effects result from a combination of random measurement errors and fundamental limitations of probe measurement. Here, a numerical model of the physical measurement process is presented, which is used to simulate spectral effects of probe measurement. While random measurement errors can be controlled for and tend to introduce only an additive white-noise component into the measured power-spectral density, limitations of probe measurement are due to the finite size of the probe stylus and result in a systematic error in the measured power-spectral density for spatial wavenumbers above a critical wavenumber