Frédéric Sturm

Numerical study of broadband sound pulse propagation in three-dimensional oceanic waveguides

In this paper, the propagation of a broadband sound pulse in three-dimensional (3D) shallow water waveguides is investigated numerically. Two cases are examined: (i) the 3D ASA benchmark wedge, and (ii) the 3D Gaussian canyon. The numerical method used to solve the four-dimensional acoustic problem is based on a Fourier synthesis technique. The frequency-domain calculations are carried out using the fully 3D parabolic equation based model 3DWAPE, recently modified to include a wide-angle paraxial approximation for the azimuthal component. A broadband sound pulse with a central frequency of 25 Hz and a bandwith of 40 Hz is considered. For both test cases, 3D results corresponding to a 25 Hz cw point source are first presented and compared with predictions from a 3D adiabatic modal model. Then, the acoustic problem is solved considering the broadband source pulse. The modal structure of the received signals is analyzed and exhibits multiple mode arrivals of the propagating signal.

On the use of higher-order azimuthal schemes in 3-D PE modeling.

In this paper, the issue of using higher-order finite difference schemes to handle the azimuthal derivative term in a three-dimensional parabolic equation based model is addressed. The three-dimensional penetrable wedge benchmark problem is chosen to illustrate the accuracy and efficiency of the proposed schemes. Both point source and modal initializations of the pressure field are considered. For each higher-order finite difference scheme used in azimuth, the convergence of the numerical solution with respect to the azimuth is investigated and the CPU times are given. Some comparisons with solutions obtained from another 3-D model [J. A. Fawcett, J. Acoust. Soc. Am. 93, 2627-2632 (1993)] are presented. The numerical simulations show that the use of a higher-order scheme in azimuth allows one to reduce the required number of points in the azimuthal direction while still obtaining accurate solutions. The higher-order schemes have approximately the same efficiency as a FFT-based approach (in fact, may outperform it slightly); however, the finite difference approach has the advantage that it may be more flexible than the FFT approach for various PE approximations.

Numerical investigation of out-of-plane sound propagation in a shallow water experiment

In an experiment in the Florida Straits, broadband pulses were transmitted over a range of 10 km and received by a vertical hydrophone array. For pulses with center frequency below 400 Hz, the received signal consisted of a dominant arrival followed by a secondary one delayed by about 0.4 s. A hypothesis that the secondary arrival was caused by 3D out-of-plane propagation is investigated here numerically with a 3D parabolic equation model (3DWAPE) and a 3D ray model (MOC3D). Both models clearly predict a secondary arrival caused by 3D horizontal refraction from the sloping bottom in the shoreward direction.

Comparisons of laboratory scale measurements of three-dimensional acoustic propagation with solutions by a parabolic equation model

In this paper, laboratory scale measurements of long range across-slope acoustic propagation in a three-dimensional (3-D) wedge-like environment are compared to numerical solutions. In a previous work, it was shown that the experimental data contain strong 3-D effects like mode shadow zones and multiple mode arrivals, in qualitative agreement with theoretical and numerical predictions. In the present work, the experimental data are compared with numerical solutions obtained using a fully 3-D parabolic equation based model. A subspace inversion approach is used for the refinement of some of the parameters describing the model experiment. The inversion procedure is implemented in a Bayesian framework based on the exhaustive search over the parameter space. The comparisons are performed both in the time and in the frequency domain using the maximum a posteriori estimates of the refined parameters as input in the 3-D model. A very good quantitative agreement is achieved between the numerical predictions provided by the 3-D parabolic equation model and the experimental data.

Scaled model experiment of long-range across- slope pulse propagation in a penetrable wedge

In this paper, laboratory scale measurements of long-range across-slope propagation of broadband pulses in a shallow-water wedge-shaped environment with a sandy bottom are reported. The scaled model was designed to study the three-dimensional (3D) acoustic field in the presence of only a few propagating modes. The recorded time series exhibit prominent 3D effects such as mode shadow zones and multiple mode arrivals. Inspection of the spectral content of the time signals gives evidence of intra-mode interference and frequency dependence of the mode cut-off range in the across-slope direction.

An explicit analytical solution for sound propagation in a three-dimensional penetrable wedge with small apex anglea)

A problem of sound propagation in a shallow-water waveguide with a weakly sloping penetrable bottom is considered. The adiabatic mode parabolic equations are used to approximate the solution of the three-dimensional (3D) Helmholtz equation by modal decomposition of the acoustic pressure field. The mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of the 3D wedge. Using the analytical formula for modal amplitudes, an explicit and remarkably simple expression for the acoustic pressure in the wedge is obtained. The proposed solution is validated by the comparison with a solution of the 3D penetrable wedge problem obtained using a fully 3D parabolic equation that includes a leading-order cross term correction.

Leading-order cross term correction of three-dimensional parabolic equation models

The issue of handling a leading-order cross-multiplied term in three-dimensional (3D) parabolic equation (PE) based models is addressed. In particular, numerical results obtained incorporating a leading-order cross-term correction in an existing 3D PE model, written in cylindrical coordinates, based on higher-order Padé approximations in both depth and azimuth, and a splitting operator technique are reported. Note that the numerical algorithm proposed in this paper could be used in the future to update any 3D PE codes that neglect cross terms and use a splitting numerical technique. The 3D penetrable wedge benchmark problem is chosen to illustrate the accuracy of the now-fully wide-angle enhanced 3D PE model. The comparisons with a 3D reference solution based on the image source clearly show that handling the leading-order cross term in the 3D PE computation is sufficient to remove the phase errors inherent to any 3D PE models that neglect cross terms in their formulations.

Conservative initial‐boundary value problems for the Wide‐Angle PE in waveguides with variable bottoms

We consider the third‐order, Claerbout‐type Wide‐Angle Parabolic Equation (PE) in the context of Underwater Acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There are strong indications, that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition on B, may not be well‐posed, for example when B is downsloping. In previous work we proposed an additional bottom boundary condition that, together with the zero field condition on B, yields a well‐posed problem. In the present paper we continue our investigation of additional bottom boundary conditions that yield well‐posed, physically correct problems. Motivated by the fact that the solution of the wide‐angle PE in a domain with horizontal layers conserves its L2 norm in the absence of attenuation, we seek additional boundary conditions on a variable‐topography bottom, that yield L2‐ conservative solutions of the problem. We identify a family ...

Benchmarking two three‐dimensional parabolic equation methods

Results of acoustic propagation modeling using two different three‐dimensional parabolic equation methods are presented. One model [J. Fawcett, J. Acoust. Soc. Am. 93, 2627–2632 (1993)] uses standard splitting and operator approximations while the other method [F. Sturm, M. Pelissier, and D. Fattaccioli, Proc. of the Third European Conference on Underwater Acoustics, 243–248 (1996)] does not split the azimuthal operator and maps the bathymetry into an equivalent flat bottom problem. Examples of propagation over wedge and corrugated bathymetries are considered for different frequencies. Both point‐source and modal initial fields are used. The three‐dimensional solutions exhibit interesting three‐dimensional refractive effects. The convergence of the solutions with respect to azimuthal discretization is also discussed.

On the Feasibility of a Matched-Field Inversion in a Three-Dimensional Oceanic Environment Ignoring Out-of-Plane Propagation

Inverse problems in ocean acoustics are based on 2-D modeling of sound propagation, hence ignoring the effects of horizontal refraction, referred to as 3-D propagation effects. However, the acoustic propagation in shallow-water environments, such as the continental shelf, may be affected by 3-D effects requiring 3-D modeling to be accounted for. The aim of this work is to investigate the importance of the 3-D effects with respect to the performance and reliability of typical 2-D-model-based inversion procedures of ocean acoustics. The study is carried out on a well-established synthetic test case which exhibits well-known 3-D effects. A matched-field inversion procedure is implemented based on the exhaustive search over the parameter space. The feasibility and the limits of inverting low-frequency noisy 3-D synthetic data for some parameters describing the test case by matching replica from 2-D computations are explored. Both synthetic data and replica are generated using a parabolic-equation-based code. This approach highlights the relevance of using 2-D propagation models when inversions are performed at relatively short ranges from the source. On the other hand, important mismatch occurs when inverting at farther ranges, demonstrating that the use of fully 3-D forward models is required.