Split-step Parabolic Equation Research Articles

A hybrid split-step/finite-difference PE algorithm for variable-density media

Although variations in the density, ρ, are naturally incorporated into finite-difference parabolic equation (PE) solvers, split-step PE algorithms traditionally account for density changes by adding to the refractive index terms containing derivatives of ρ. As a consequence, geoacoustic density profiles that contain step discontinuities at layer interfaces must be smoothed appropriately before these extra terms can be evaluated. In this paper, a new hybrid method is proposed for treating density inhomogeneities in the split-step PE. This approach involves splitting the differential operator into density-independent and density-dependent components. While the former terms may be evaluated with the split-step Fourier technique, the influence of density changes is handled through a finite-difference procedure. Such an algorithm can be easily implemented in recently proposed hybrid split-step/finite-difference and split-step/Lanczos PE solvers. [J. Acoust. Soc. Am. 96, 396–405 (1994)].

Comparison of Wait and norton Formulations with Parabolic Equation Method in Ground Wave Propagation

In this paper, ground wave propagation characteristics are discussed for a vertically polarized short electrical dipole over smooth spherical earth. Analytical solution which is suitable for numerical computations and valid over a broad frequency range as well as medium parameters have not yet appeared. Because of this, attention have been paid to analytical approximate and/or pure numerical solutions. Two main analytical approaches named as Norton and Wait formulations are taken into account together with numerical split step Parabolic Equation (SSPE) method. Analytical and numerical advantages and difficulties of the methods are discussed in detail and various comparison are given among them with their validity regions and accuracies. Based on these comparisons, two efficient computer algorithms, GRWAVE and SSPE are build. GRWAVE combines Norton and Wait formulations in a dynamic way and decides where to switch in between them depending on the initial parameter set. On the other hand, SSPE produces reference solutions in regions where applicable. Also a new interesting method for path loss calculations in surface wave propagations is introduced and comparison of the results with ground wave path loss curves in CCIR Recommendations (CCIR Rec. 368-6, 1990) is given. The study presented here, briefly outlines ground wave propagation, its physical characterics and gives computaional results of a broad parameter range for comparisons.

An improved impedance-boundary algorithm for Fourier split-step solutions of the parabolic wave equation

A new implementation of the previously published mixed Fourier transform (MFT) method for including impedance boundaries in split-step parabolic equation solutions is described and demonstrated. The new algorithm is formulated entirely in the discrete domain which results in extended applicability and increased computation speed. A brief review of the original MFT solution is followed by a detailed description of the discrete formulation. The performance of the new algorithm is then demonstrated with a few examples which rely heavily on the accuracy of the impedance boundary. These examples include 10 MHz surface wave propagation over smooth and rough sea surfaces and 10 GHz calculations utilizing an effective rough surface impedance.

Coupling scattering from the sea surface to a one-way marching propagation model via conformal mapping: Validation

Propagation models in underwater acoustics usually incorporate sea surface scattering effects in an ad hoc manner which in most cases requires making severe approximations. In particular, to include in a coherent manner in a marching acoustic propagation model the scattering that occurs at a rough sea surface poses a serious problem. Dozier [J. Acoust. Soc. Am. 75, 1415–1432 (1984)] introduced a rigorous approach in the framework of the split-step parabolic equation model, which used a sequence of conformal mappings to flatten segments of the sea surface locally. Each conformal mapping preserved the elliptic form of the wave equation. In each transformed space the parabolic approximation is made and the solution advanced one range step. The method has the attractive feature of handling surface roughness within a propagation model in a mathematically consistent manner, including refraction and multiple surface interactions when and where they occur. In this work the technique developed by Dozier is implemented in the realm of the finite element parabolic equation model [Collins, J. Acoust. Soc. Am. 93, 1736–1742 (1993)] and applied to the problem of forward scattering from both periodic and single realizations of randomly rough surfaces. The validity of the technique is examined by comparison with an exact solution to the scattering problem obtained through the method of moments for both types of surfaces. It is shown that the current technique provides the correct energy distribution of the forward-scattered field. The performance of this hybrid model suggests that it might be well suited as the reference solution for comparing with propagation models that include sea surface scattering in a more approximate manner.

Split Step Parabolic Equation Analysis of Coupled Dielectric Waveguides

The propagation phenomena in parallel coupled slab waveguides is investigated in terms of the Split Step Parabolic Equation Method (SSPEM) and the computed eigenvalues and the coupling coefficients are compared with the exact solution. Conventionally, couplers are excited from one port and the power as it propagates along the structure is transferred periodically back and forth between the waveguides and the period L is referred to as the beat length. In weakly coupled structures the eigenvalues of the propagating surface waves appear in rather closely spaced pairs thereby severly reducing the accuracy in determining the beat length. An alternative use of SSPEM may then be utilized for improved accuracy, wherein one determines the initial field in such a way as to excite only two modes of the coupler and determines L utilizing the perfect imaging properties of the waveguide.

Hybrid methods for incorporating density variations into the split-step PE

Density variations are easily analyzed in the context of finite difference parabolic equation (PE) solvers by discretization of an appropriate differential operator. In split-step Fourier solution algorithms, however, variations in the density ρ are instead modeled by adding terms to the refractive index. Since these extra terms depend on derivatives of ρ, geoacoustic density profiles must be smoothed appropriately to remove any step discontinuities. In this paper, a new hybrid method is proposed for treating density inhomogeneities in the split-step PE. This approach involves splitting the differential operator into density-independent and density-dependent components. While the former component is propagated using the split-step Fourier technique, the influence of density changes is computed through a finite difference procedure. Such an algorithm is especially attractive as it may be transparently incorporated into the recently proposed hybrid split-step/finite-difference and split-step/Lanczos solvers [J. Acoust. Soc. Am. 96, 396–405 (1994)]. Both the hybrid and the standard finite-difference procedures are applied to a shallow-water test case involving jump discontinuities in both the sound speed and the density and are found to be in excellent agreement with reference solutions.

The influence of acoustic signals on a juvenile gray whale

In May 1994, a juvenile gray whale, Eschrichtiusglaucus, entered the Petaluma River, which empties into the north end of San Francisco Bay, CA. The Marine Mammal Center of Sausalito, CA, coordinated a rescue and asked us to lure the whale to deeper water with sound. The Petaluma River is muddy and brackish, 20 km long, and generally 75 m wide and 3 to 4 m deep. Recorded gray whale calls and synthetic signals in the range of 100–900 Hz were broadcast with a J-9 acoustic transducer providing a source level of 153 dB re: 1 μPa at 1 m. Over several hours, the whale, who surfaced for air every 140 s, seemed to be attracted to these sounds as we traveled at a few knots down river, our sound boat typically 50 m ahead of the whale. The whale appeared to lose interest in the sound boat much beyond 100 m. A split step parabolic equation model of acoustic propagation in the river suggests that the sound level was 123 dB near the river bottom at a range of 50 m and 120 dB at 200 m. On one occasion the whale approached to within 3 m of the active source. The sound level at this distance would be about 144 dB.

Marching out the parabolic equation model: A graphical approach

The parabolic equation (PE) acoustic model uses a Fourier spectral method of solving a second-order differential equation. The solution is ‘‘evolved’’ by solving a series of fast Fourier transforms whereby the source is convolved with the environment of the propagating energy. Each convolution marks a step of the march until the solution is marched out to the final range. This evolution has been described mathematically many times in previous published papers [Hardin and Tappert, SIAM Rev. 15, 423 (1973), Thomson and Chapman, J. Acoust. Soc. Am. 74, 1848–1854 (1983)]. A graphical evolution of the convolution of the source with the environment is important in providing a fresh perpective and further insight into the refinement and the understanding of the source function and the sensitivity to the reference sound speed. In addition, the observation of the geometry as it marches the propagation forward in wave space is an important graphic tool for understanding the analytical description of the Tappert split step PE approach.

Inclusion of rough surface scattering in a finite element parabolic equation propagation model

Propagation models in underwater acoustics usually incorporate surface scattering effects in an ad-hoc manner that in most cases requires making severe approximations. In particular, to include surface scattering in a marching algorithm for sound propagation in a coherent manner posses a difficult problem. Dozier [J. Acoust. Soc. Am. 75, 1415–1432 (1984)] introduced a novel and rigorous approach to include surface scattering in the framework of the split-step parabolic equation method. In this work the technique developed by Dozier is adapted and implemented in the realm of the finite element parabolic equation (FEPE) acoustic propagation model developed by Collins [J. Acoust. Soc. Am. 86, 1097 (1989)]. The technique is tested by applying it to scattering from a sinusoidal patch as well as from a rough surface patch defined by a Pierson–Moskowitz spectrum. These results are compared against highly accurate solutions to the surface scattering integral equation (method of moments). [Work supported by ONR.]

Comparisons and sensitivity analyses of numerical predictions of acoustic propagation in shallow-water and surface duct environments

Comparisons of commonly used numerical models of acoustic propagation are performed over a wide frequency band in selected shallow-water environments. The models include range-independent propagation loss codes, such as Colossus II and Generic Sonar model (GSM), as well as range-dependent codes, such as SNAP and a wide-angle split-step parabolic equation (PE) model. For both range-independent and range-dependent shallow-water environments, SNAP model predictions typically provide close approximations to measured propagation loss data. Alternatively, large differences and model biases are generally prevalent in comparisons between measured data and Colossus II or PE predictions. Comparisons between the PE and GSM models are presented for sound propagation in surface duct environments. Results pertinent to the sensitivity of shallow-water propagation loss predictions to the amount of range dependence in environmental quantities, such as sound speed, bathymetry, and bottom-interaction parameters, are provided. Advanced 3-D visualizations are shown that illustrate the sensitivity of the optimum frequency of acoustic propagation in shallow water regions to variations in source–receiver geometries and environmental quantities.

Surface loss, scattering, and reverberation with the split-step parabolic wave equation model

Moore-Head et al. [J. Acoust. Soc. Am. 86, 247–251 (1989)] have published a method to incorporate loss due to rough surface scattering into split-step parabolic wave equation (PE) codes which then has been extended by Dozier et al. [in Ocean Variability and Acoustic Propagation, edited by J. Potter and A. Warn-Varnas (Kluwer Academic, Dordrecht, 1991)] to include the incoherent scatter. A good agreement with other methods was reported in both cases. However, in the presence of sound-speed gradients this approach is an approximate solution and errors may occur. This will be discussed for some examples. Using the same procedure to estimate the angular acoustic spectrum at the sea-surface, the reverberation from the surface and/or bottom may be evaluated for any generic backscatter function. Two different approaches, based on either reciprocity or backpropagation, will be outlined, with both methods being applicable also to bistatic reverberation. The computationally simpler reciprocity approach has been programmed and results compare well with other models for range-independent environments. A comparison for range-dependent situations with a ray-based model gives good agreement on the main features and all remaining deviations can be explained.

Implementation of surface impedance boundary conditions in the split-step parabolic equation

The parabolic wave equation (PE) is a powerful numerical method for computing the full-wave complex acoustic pressure field in range-dependent environments. The split-step Fourier PE algorithm (SSFPE) of Hardin and Tappert provides a very efficient computational implementation of PE when the surface boundary condition is of the Dirichlet or Neumann form. This allows use of fast Fourier transform (FFT) methods to implement the SSFPE algorithm. In many cases, however, the surface boundary condition is of the mixed or impedance type which precludes use of simple FFTs. In this talk, generalizations of the SSFPE algorithm will be discussed that use a novel fast radiation transform (FRT). The FRT allows explicit incorporation of complex surface impedance boundary conditions into the SSFPE algorithm, while still retaining computational efficiency. Illustrative examples that use this new method will be shown.

A review of the Parabolic Equation Workshop II.

In Spring, 1981, the first Parabolic Equation (PE) Workshop was held [Davis, White, and Cavanagh, NORDA Technical Note 143 (1982)]. The purpose of that workshop was to provide a forum for theoretical and applied PE developers and to compare computer results for a set of ocean acoustic problems. Underwater acoustic modelers have benefited from that set of test cases and documented PE deficiencies for nearly a decade. The second PE workshop was recently held (in Spring, 1991) to document the impressive advances that have taken place in the development of PE models over that decade. PE developers from the ocean acoustics community were invited to provide computer solutions for a new set of ocean acoustic problems. The set of problems was designed to test the ability of PE models to handle (a) wide-angle propagation (nearly ±90° with respect to the axis of propagation), (b) shear waves and shear attentuations, (c) backscatter, (d) conservation of energy, and (e) complicated range-dependent environments. In addition, predictions from the PE models were compared with measured acoustic data taken in a region where the ocean environments (water column, sediment, ocean bottom, and subbottom) were well-known. Results from this PE Workshop II indicate that several significant advances have occurred in PE model development to the extent that some current PE models can include very wide-angle propagation, shear waves, and backscatter, while conserving energy. Additionally, an unexpected problem experienced by some wide-angle split-step PE models was also presented at the PE Workshop II. These and other results from the PE Workshop II will be presented and discussed. [Work supported by ONR AEAS and ONR Acoustic Reverberation SRP.]