Split-step Parabolic Equation Research Articles

Comparative analysis of finite‐difference and split‐step based parabolic equation methods for tunnel propagation modelling

AbstractRadio wave propagation modelling in railway environments is of fundamental importance in designing reliable train communication systems. Parabolic equation (PE) methods have been widely applied to the modelling of wave propagation in tunnels due to their high computational efficiency and fidelity. The finite‐difference parabolic equation (FDPE) and the split‐step parabolic equation (SSPE) methods are two commonly used approaches to solve PE numerically. However, the relevant literature is still missing a comprehensive study of their performance, including the selection of parameters such as discretisation steps and the tradeoffs involved in terms of their accuracy and efficiency, especially as current wireless systems shift to high frequencies. In this study, a systematic analysis of the error and computational complexity of the FDPE and SSPE methods for radio wave propagation modelling in tunnels is provided. Guidelines for the choice of their parameters are provided, and their performance is demonstrated through both numerical examples and experimental measurements in actual tunnel cases.

A hybrid numerical model for long-range electromagnetic wave propagation

A hybrid numerical model is presented for solving long range electromagnetic wave propagation problems involving objects on or above the ground surface by hybridizing the two-way split-step parabolic equation (2W-SSPE) method with the method of moments (MoM). The advantages of the proposed model are twofold: (i) It reduces the staircasing error in irregular terrain modeling, which usually occurs when the standard SSPE method is used alone. This is achieved by employing the MoM to more accurately obtain the scattered fields from slanted/curved surfaces. (ii) It enables the SSPE method to handle the problems involving objects above the Earth's surface, which cannot be easily modeled by the standard SSPE method due to difficulty in imposing boundary conditions. The accuracy of the hybrid method is numerically verified by comparing the numerical results with those of the 2W-SSPE and the GO+UTD (geometric optic + uniform theory of diffraction) methods in some representative propagation problems.

Prediction method of SSPE propagation based on improved terrain reconstruction

In view of the difficulty of wave propagation prediction and analysis in complex environment, a propagation prediction method of split-step parabolic equation (SSPE) based on terrain reconstruction is proposed. This method uses a two-way SSPE solution with a wide-angle propagation factor and variable step size, and uses the SSPE propagation prediction method based on improved terrain reconstruction to simulate and test the radio wave propagation characteristics of real terrain. Compared with the existing radio wave propagation prediction method, this method combines real terrain, which improves the terrain modeling effect by improving the reconstruction algorithm, and uses variable step size two-way SSPE solution to predict the propagation loss. Simulation test results show that the prediction method is within 6 dB. The accuracy meets the requirements of a high-precision radio wave propagation model, indicating that the method has practical significance and popularization value.

UTD-PO Formulation for the Analysis of Multiple-Plateau Diffraction When Considering Illumination From a Low Source

A hybrid uniform theory of diffraction-physical optics (UTD-PO) formulation for the analysis of the multiple-diffraction (MD) that takes place over rectangular-shaped obstacles (plateaus) when the latter are illuminated from a low source is presented. The solution, which is based on Babinet's principle, has been validated with measurements taken at 60 GHz on a scaled model of the environment under study and with additional measurements performed by other authors. Moreover, the proposed approach has been compared with the wide-angle Fourier split-step parabolic equation method, with the UTD-PO formulation showing better accuracy against the experimental data, especially when the distance between the transmitter and the receiver is increased. Furthermore, the main advantage of the proposed solution is that due to recursion, the calculations imply only single diffractions, thereby avoiding the need of higher order diffraction terms in the diffraction coefficients. The results and findings of this communication are particularly applicable to the planning of microcellular mobile communication systems when the transmitting antenna is located below the height of the surrounding buildings.

PETOOL v2.0: Parabolic Equation Toolbox with evaporation duct models and real environment data

A new version of PETOOL (Parabolic Equation Toolbox) is introduced with various additional capabilities. PETOOL is an open-source and MATLAB-based software tool with a user-friendly graphical user interface (GUI) for the analysis and visualization of electromagnetic wave propagation over variable terrain and through arbitrary atmosphere. Four novel features of the second version are as follows: (i) Several evaporation duct models have been developed. (ii) Real atmosphere data have been included in the form of “Binary Universal Form for Representation (BUFR)” data developed by “World Meteorological Organization (WMO)”. (iii) Real terrain data have been incorporated into the toolbox in the form of “Digital Terrain Elevation Data (DTED)” developed by “National Imagery and Mapping Agency (NIMA)”. (iv) A special add-on has been developed to generate a 3D coverage map of propagation factor/loss on real terrain data. The toolbox can be used for research and/or educational purposes to analyze more realistic propagation scenarios in an easier and flexible manner. Program summaryProgram title: PETOOL v2.0 (Parabolic Equation Toolbox v2.0)CPC Library link to program files:http://dx.doi.org/10.17632/v8f42rn2zs.1Licensing provisions: GNU General Public License 3Programming language: MATLAB (MathWorks Inc.) R2019a. Partial Differential Toolbox, Curve Fitting Toolbox and Mapping Toolbox required.Journal Reference of previous version: O. Ozgun, G. Apaydin, M. Kuzuoglu, and L. Sevgi, PETOOL: MATLAB-based one-way and two-way split-step parabolic equation tool for radiowave propagation over variable terrain, Computer Physics Communications 182 (2011) 2638–2654.Does the new version supersede the previous version?: YesReasons for the new version: The new version of the toolbox has been enriched with several add-ons which allow the toolbox to make more realistic analyses with real terrain and atmospheric data. The toolbox has the ability to read real atmospheric data in the form of BUFR data and real terrain data in the form of DTED data. A wide range of evaporation duct models has been incorporated into the toolbox. A special add-on has been developed to generate a 3D coverage map of propagation factor/loss on real terrain data. Hence, the toolbox can be used to analyze more realistic propagation scenarios in an easier and flexible manner.Summary of revisions: (i) Several evaporation duct models have been included with real atmospheric data in BUFR format. (ii) Real terrain data in DTED format have been incorporated into the toolbox. (iii) A special add-on has been developed to generate a 3D coverage map of propagation factor/loss on real terrain data.Nature of problem: This program is designed with a user-friendly GUI for the analysis and visualization of radio-wave propagation over variable terrain on the Earth’s surface, and through homogeneous and inhomogeneous atmosphere by using real atmosphere and terrain data. It can easily model both horizontally- and vertically-varying atmospheric refraction (especially ducting) and multipath effects.Solution method: The program employs one-way and two-way Split-Step Parabolic Equation (SSPE) algorithms with a wide-angle propagator. The SSPE is an initial-value problem starting from a reference range (typically from an antenna), and marching out in range by obtaining the field along the vertical direction at each range step, through the use of step-by-step Fourier transformations. The two-way algorithm incorporates the backward-propagating waves into the standard one-way SSPE by utilizing an iterative forward–backward scheme for modeling multipath effects over a staircase-approximated terrain.

A Groundwave Propagation Model Using a Fast Far-Field Approximation

A fast far-field approximation (FAFFA), which is simple to use, is applied to groundwave propagation modeling from a nonpenetrable surface with both soft and hard boundaries. The results are validated against available reference models as well as compared to other numerical methods such as split step parabolic equation model and the method of moments.

Study on the maximum calculation height and the maximum propagation angle of the troposcatter wide‐angle parabolic equation method

In this study, a method to choose the maximum calculation height and the maximum propagation angle of the split-step troposcatter wide-angle parabolic equation (TWAPE) method is presented. The method is implemented with the height and the scatter angle of the troposcatter common volume. The performance of this method is then demonstrated with a few examples which rely heavily on the choosing of the maximum calculation height and the maximum propagation angle. These examples include the comparison of calculated results of TWAPE for choosing different maximum calculated height and maximum propagation angle at frequencies of 3 and 1 GHz, path lengths of 300 and 500 km, respectively. Finally, the seasonal variation mechanism of the troposcatter propagation loss is discussed based on the seasonal variation of the refractive-index structure parameter.

Development of a Finite-Difference Time Domain (FDTD) Model for Propagation of Transient Sounds in Very Shallow Water.

This finite-difference time domain (FDTD) model for sound propagation in very shallow water uses pressure and velocity grids with both 3-dimensional Cartesian and 2-dimensional cylindrical implementations. Parameters, including water and sediment properties, can vary in each dimension. Steady-state and transient signals from discrete and distributed sources, such as the surface of a vibrating pile, can be used. The cylindrical implementation uses less computation but requires axial symmetry. The Cartesian implementation allows asymmetry. FDTD calculations compare well with those of a split-step parabolic equation. Applications include modeling the propagation of individual fish sounds, fish aggregation sounds, and distributed sources.

Sensitivity analysis of three-dimensional sound pressure fields in complex underwater environments

A sensitivity kernel for sound pressure variability due to variations of index of refraction is derived from a higher-order three-dimensional (3D) split-step parabolic-equation (PE) solution of the Helmholtz equation. In this study, the kernel is used to compute the acoustic sensitivity field between a source and a receiver in a 3D underwater environment, and to quantify how much of the medium change can cause significant consequence on received acoustic signals. Using the chain rule, the dynamics of sensitivity fields can be connected to the dynamics of ocean processes. This talk will present numerical examples of sound propagation in submarine canyons and continental slopes, where the ocean dynamics cause strong spatial and temporal variability in sound pressure. Using the sensitivity kernel technique, we can analyze the spatial distribution and the temporal evolution of the acoustic sensitivity fields in these geologically and topographically complex environments. The paper will also discuss other applications of this sound pressure sensitivity kernel, including uncertainty quantification of transmission loss prediction and adjoint models for 3D acoustic inversions. [Work supported by the ONR.]

Comments on “Propagation Modeling Over Irregular Terrain by the Improved Two-Way Parabolic Equation Method”

The above paper [1] is about the two-way split-step parabolic equation method (2W-SSPE) over irregular terrain, and claims that they have developed the “improved” version of the 2W-SSPE approach that has been proposed and validated by us in literature [2]-[11]. The paper [1] claims to derive the 2W-PE directly from 2D Helmholtz equation. They do this by listing and explaining equations (1)-(9). The authors needed to state that this is quite different from the formulation given in Levy's book (reference [9] in [1]). Actually, factorization before or after does not mean anything; their equations (8) and (9) are exactly the same. As a result listing equations (1)-(9) does not mean derivation of the new two-way PE.

Reply to “Comments on 'Propagation modeling over irregular terrain by the improved two-way parabolic equation method'”

In this paper, we focused on deriving and refining the two-way parabolic equation (2WPE) formulations and developing the improved two-way split-step parabolic equation (2W-SSPE) method to model both forward- and backward-propagating waves over various lossy surfaces with impedance boundary conditions (IBC) in urban and tropospheric scenarios.

Performance Analysis of Techniques Used for Determining Land Mines

Today, remote sensing is used for different methods and different purposes. In all of the detection methods, some considerations such as low energy consumption, low cost, insensitivity to environmental changes, high accuracy, high reliability and robustness become important. Taking into account these facts, remote sensing methods are used in applications such as geological and archeological research, engineering areas, health services, preserving and controlling natural life, determination of underground sources, controlling air, sea and road traffic, military applications, etc. The method to be used is based on the object type to be detected, material to be made, and location to be found. The remote sensing methods from the past up to today can be listed as acoustic and seismic, ground penetration radar (GPR) detection, electromagnetic induction, infrared (IR) imaging, neutron quadrupole resonance (NQR), thermal neutron activation (TNA), neutron back scattering, X-ray back scattering, and magnetic anomaly detection. In these methods, detected raw images have to be processed, filtered and enhanced. In order to achieve these operations, some algorithms are needed to be developed. In this study, the methods used in detecting land mines remotely and their performance analysis have been given. In this way, the last situation on the advantages and disadvantages of methods used, application areas and detection accuracies are determined. Furthermore, the algorithms such as transmission line matrix (TLM), finite difference time-domain (FDTD), the method of moment (MoM), split step parabolic equation (SSPE) and image processing and intelligent algorithms are presented in detail.

Propagation Modeling Over Irregular Terrain by the Improved Two-Way Parabolic Equation Method

The two-way split-step parabolic equation (2W-SSPE) method is improved to be able to model both forward- and backward-waves propagation over irregular perfectly electrical conducting (PEC) surfaces and irregular lossy surfaces with impedance boundary conditions (IBC). The discrete mixed Fourier transform (DMFT) is applied in both forward- and backward- direction to deal with IBC, and the boundary shift (BS) method is used to dispose of the irregular terrain effects. Numerical results obtained by the improved 2W-SSPE algorithm for several typical scenarios with different ground types are compared with those got by the previous 2W-SSPE method and the one-way parabolic equation method.

An Angle-Dependent Impedance Boundary Condition for the Split-Step Parabolic Equation Method

A formulation of the impedance boundary condition (IBC) is derived, which can be used in split-step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties. The approach relies on ensuring that the plane-wave decomposition of the field satisfies the appropriate IBC for each spectral component. Numerical experiments illustrating the robustness of the approach for low-contrast interfaces and angles-of-incidence near the Brewster angle are presented.

A Novel Two-Way Finite-Element Parabolic Equation Groundwave Propagation Tool: Tests With Canonical Structures and Calibration

A novel two-way finite-element parabolic equation (PE) (2W-FEMPE) propagation model which handles both forward and backward scattering effects of the groundwave propagation above the Earth's surface over irregular terrain paths through inhomogeneous atmosphere is introduced. A Matlab-based propagation tool for 2W-FEMPE is developed and tested against mathematical exact and asymptotic solutions as well as the recently introduced two-way split-step PE model through a canonical validation, verification, and calibration process for the first time in literature.

PETOOL: MATLAB-based one-way and two-way split-step parabolic equation tool for radiowave propagation over variable terrain

A MATLAB-based one-way and two-way split-step parabolic equation software tool (PETOOL) has been developed with a user-friendly graphical user interface (GUI) for the analysis and visualization of radio-wave propagation over variable terrain and through homogeneous and inhomogeneous atmosphere. The tool has a unique feature over existing one-way parabolic equation (PE)-based codes, because it utilizes the two-way split-step parabolic equation (SSPE) approach with wide-angle propagator, which is a recursive forward–backward algorithm to incorporate both forward and backward waves into the solution in the presence of variable terrain. First, the formulation of the classical one-way SSPE and the relatively-novel two-way SSPE is presented, with particular emphasis on their capabilities and the limitations. Next, the structure and the GUI capabilities of the PETOOL software tool are discussed in detail. The calibration of PETOOL is performed and demonstrated via analytical comparisons and/or representative canonical tests performed against the Geometric Optic (GO) + Uniform Theory of Diffraction (UTD). The tool can be used for research and/or educational purposes to investigate the effects of a variety of user-defined terrain and range-dependent refractivity profiles in electromagnetic wave propagation. Program summary Program title: PETOOL (Parabolic Equation Toolbox) Catalogue identifier: AEJS_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJS_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 143 349 No. of bytes in distributed program, including test data, etc.: 23 280 251 Distribution format: tar.gz Programming language: MATLAB (MathWorks Inc.) 2010a. Partial Differential Toolbox and Curve Fitting Toolbox required Computer: PC Operating system: Windows XP and Vista Classification: 10 Nature of problem: Simulation of radio-wave propagation over variable terrain on the Earthʼs surface, and through homogeneous and inhomogeneous atmosphere. Solution method: The program implements one-way and two-way Split-Step Parabolic Equation (SSPE) algorithm, with wide-angle propagator. The SSPE is, in general, an initial-value problem starting from a reference range (typically from an antenna), and marching out in range by obtaining the field along the vertical direction at each range step, through the use of step-by-step Fourier transformations. The two-way algorithm incorporates the backward-propagating waves into the standard one-way SSPE by utilizing an iterative forward–backward scheme for modeling multipath effects over a staircase-approximated terrain. Unusual features: This is the first software package implementing a recursive forward–backward SSPE algorithm to account for the multipath effects during radio-wave propagation, and enabling the user to easily analyze and visualize the results of the two-way propagation with GUI capabilities. Running time: Problem dependent. Typically, it is about 1.5 ms (for conducting ground) and 4 ms (for lossy ground) per range step for a vertical field profile of vector length 1500, on Intel Core 2 Duo 1.6 GHz with 2 GB RAM under Windows Vista.

A Novel Split-Step Parabolic-Equation Package for Surface-Wave Propagation Prediction Along Multiple Mixed Irregular-Terrain Paths

A novel MATLAB™-based split-step parabolic-equation (SSPE) propagation package is introduced for the prediction of electromagnetic surface-wave attenuation along multiple mixed irregular and lossy paths. The impedance boundary condition is implemented via the discrete mixed Fourier transform (DMFT). Irregular terrain modeling is handled via coordinate transformation. Tests are performed along sea paths including smooth and hilly islands. Comparisons with ITU Recommendations and a Finite-Element-Method (FEM) based numerical propagator are given.

A curvilinear coordinate-based split-step parabolic equation method for propagation predictions over terrain

Propagation of radiowaves over irregular terrain and in an inhomogeneous atmosphere is solved by the parabolic equation method using the split-step Fourier algorithm on a terrain-conformal mesh. A piecewise continuous coordinate system is generated by the specification of: (1) the terrain profile shape at discrete points and (2) an upper height. The resulting mesh is conformal to the terrain at the lower boundary and gradually flattens off at the maximum height. In addition to preserving the number of points on any vertical line between the terrain and the maximum height from one range step to another, the coordinate transformation used in the paper produces a correction term in the refractive index whose gradient diminishes with height. As a result, the sampling requirements over steep terrain are relaxed when compared to the Beilis-Tappert transformation. Formulation and results are given both for the horizontal and vertical polarizations.

A hybrid (parabolic equation)-(Gaussian beam) algorithm for wave propagation through large inhomogeneous regions

The wide-angle split-step parabolic equation (PE) algorithm is used to model electromagnetic wave propagation over general inhomogeneous terrain up to a height h. The PE-computed fields at h are then projected onto a complete Gabor basis from which we effect Gaussian beam propagation at altitudes greater than h. The Gaussian beams can be propagated through general inhomogeneous media, devoid of failures at caustics and shadow boundaries (as befalls ray tracing). The accuracy of the Gaussian beam algorithm is demonstrated via two realistic examples: (1) low-frequency (HF) ionospheric propagation with application to over-the-horizon radar and (2) near-grazing high-frequency propagation for communication or surveillance applications. In the context of these examples, we discuss relevant numerical issues associated with the hybrid algorithm from which general advantages and disadvantages are addressed.

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)].