petool: matlab-based one-way and two-way split-step parabolic equation tool for radiowave...

Computer Physics Communications 182 (2011) 2638–2654

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Computer Physics Communications

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

Ozlem Ozgun a,∗, Gökhan Apaydin b, Mustafa Kuzuoglu c, Levent Sevgi d

a Department of Electrical and Electronics Engineering, Middle East Technical University, Northern Cyprus Campus, Guzelyurt, Mersin 10, Turkeyb Department of Electrical and Electronics Engineering, Zirve University, Gaziantep, Turkeyc Department of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkeyd Department of Electronics and Communications Engineering, Dogus University, Istanbul, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 April 2011Received in revised form 5 July 2011Accepted 9 July 2011Available online 27 July 2011

Keywords:PETOOLElectromagnetic propagationRefractivityDuctingSplit-step parabolic equationTerrain factorsMultipath effectsValidation, verification and calibrationMATLAB program

A MATLAB-based one-way and two-way split-step parabolic equation software tool (PETOOL) has beendeveloped 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 itutilizes the two-way split-step parabolic equation (SSPE) approach with wide-angle propagator, which is arecursive forward–backward algorithm to incorporate both forward and backward waves into the solutionin 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 calibrationof PETOOL is performed and demonstrated via analytical comparisons and/or representative canonicaltests performed against the Geometric Optic (GO) + Uniform Theory of Diffraction (UTD). The tool canbe used for research and/or educational purposes to investigate the effects of a variety of user-definedterrain and range-dependent refractivity profiles in electromagnetic wave propagation.

Program summary

Program title: PETOOL (Parabolic Equation Toolbox)Catalogue identifier: AEJS_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJS_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 143 349No. of bytes in distributed program, including test data, etc.: 23 280 251Distribution format: tar.gzProgramming language: MATLAB (MathWorks Inc.) 2010a. Partial Differential Toolbox and Curve FittingToolbox requiredComputer: PCOperating system: Windows XP and VistaClassification: 10Nature of problem: Simulation of radio-wave propagation over variable terrain on the Earth’s surface, andthrough 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 areference range (typically from an antenna), and marching out in range by obtaining the field along thevertical 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 utilizingan iterative forward–backward scheme for modeling multipath effects over a staircase-approximatedterrain.

✩ This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

* Corresponding author.E-mail address: [email protected] (O. Ozgun).

0010-4655/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.cpc.2011.07.017

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O. Ozgun et al. / Computer Physics Communications 182 (2011) 2638–2654 2639

Unusual features: This is the first software package implementing a recursive forward–backward SSPEalgorithm to account for the multipath effects during radio-wave propagation, and enabling the user toeasily 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 (forlossy ground) per range step for a vertical field profile of vector length 1500, on Intel Core 2 Duo 1.6 GHzwith 2 GB RAM under Windows Vista.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Radio-wave propagation over the Earth’s surface and in an in-homogeneous atmosphere is affected by several scattering phe-nomena, such as reflection, refraction, and diffraction. Understand-ing the effects of varying conditions on radio-wave propagation is,therefore, essential for designing reliable radar and communicationsystems. Especially, tropospheric waves may play a dominant rolein communications because they can propagate over the horizonand increase the coverage area, and hence, they may disrupt thecommunication links due to the interference that is not normallythere. Such waves are propagated by bending or refraction due tothe abrupt change in the refractive index in the troposphere, andcause so called ‘anomalous’ propagation. If the refractive gradientexceeds some certain limits, the radio waves may be trapped in a‘duct’ and guided over distances far greater than the normal range.In addition to tropospheric effects, irregular terrain surfaces haveconsiderable influence on radio-wave propagation because they re-flect and diffract the electromagnetic waves in a complex way.Hence, the design of an effective radar or communication systemcan be achieved by using a model that can properly incorporatethe refractivity and terrain factors.

The rigorous analytical and numerical modeling of radio-wavepropagation in such environments is a challenging task and hasattracted the attention of researchers for many decades. The dif-ficulty stems from the vast variability of the properties of themedium and also the surfaces and obstacles that re-direct thepropagating energy, making the radio propagation somewhat un-predictable. Initially, analytical techniques (such as ray tracingmethods, diffraction methods, and waveguide mode theory) havebeen employed to predict the radio propagation [1–6]. However,they require the geometry to be represented as a member of aset of some canonical geometries and suffer from the presence ofthe vertically-varying refractivity profile in the troposphere. Withthe advances in computers, some numerical techniques have beendevised to easily handle the above-mentioned difficulties. ParabolicEquation (PE) model has been widely used in propagation modelingto predict the wave behavior between a transmitter and a receiverover the two-dimensional (2D) Earth’s surface, because of its highcapability in modeling both horizontally- and vertically-varyingatmospheric refraction (especially ducting) effects. The standardPE is derived from Helmholtz’s equation in such a way that therapidly-varying phase term is discarded to obtain a reduced func-tion having slow variation in range for propagating angles closeto the paraxial direction. Helmholtz’s equation is approximated bytwo differential equations, corresponding to forward and backwardpropagating waves, each of which is in the form of a parabolic par-tial differential equation. The standard PE method takes only theforward part into account, namely, it is a one-way, forward scattermodel, valid in the paraxial region. Although the initial introduc-tion of the PE method has been credited to [7], its wide-spreadusage has become possible after the development of Fourier split-step algorithm by [8]. The split-step based PE method (SSPE) is, ingeneral, an initial-value problem starting from a reference range(typically from an antenna), and marching out in range by ob-taining the field along the vertical direction at each range step,through the use of step-by-step Fourier transformations [9–14].

Apart from the Fourier split-step algorithm, the solution of thePE has also been achieved by the finite difference (FD)-based [15–17] and finite element (FE)-based [18–22] algorithms. The Fouriersplit-step algorithm is more robust, since it provides the use oflarger range increments and a faster solution for long-range propa-gation scenarios. Apart from these studies, there exist several com-puter software programs, most of which have been developed formilitary purposes, for predicting radar coverage under the effect ofenvironmental factors that influence refractivity. These are IREPS(Integrated Refraction Effects Prediction System), EREPS (Engineer’sRefractive Effects Prediction System), TESS (Tactical Electronic Sup-port System), AREPS (Advanced Refractive Effects Prediction Sys-tem), TEMPER (Tropospheric Electromagnetic Parabolic EquationRoutine), and TPEM (Terrain Parabolic Equation Model). These pro-grams implement the ray optics or one-way PE techniques, or ahybrid model combining these methods.

In this paper, a novel software tool (PETOOL), which is devel-oped in MATLAB® with graphical user interface (GUI), is intro-duced for the analysis and visualization of radio-wave propagationthrough homogeneous and inhomogeneous atmosphere, by incor-porating variable terrain effects with the aid of the two-way split-step algorithm employing wide-angle propagator. Why yet developanother PE-based program for radio-wave propagation? The reasonis twofold: The first is that PETOOL is a free/open-source pro-gram, and has been designed with a user-friendly GUI, to serveas a research/educational tool for propagation engineers/instructorsto investigate the phenomenon in an illustrative manner, and/orto achieve the analysis/design/planning of reliable communicationsystems. It displays the propagation factor/loss on a range/altitudescale, and enables the user to easily visualize, enter and save allrelated input/output parameters. More importantly, the user caneasily create arbitrary terrain and refractivity profiles. The sec-ond reason is that PETOOL is indeed the first software packageimplementing both one- and two-way SSPE algorithms, the lat-ter of which incorporates the multipath effects into the PE solu-tion of the radio-wave propagation through a recursive forward–backward algorithm. The standard one-way PE method, in spite ofits wide-spread usage, suffers from two major drawbacks: (i) ThePE method handles only the forward-propagating waves, and ne-glects the backscattered waves. The forward waves provide almostaccurate results for typical long-range propagation problems, onlyif there does not exist obstacles that redirect the incoming wavein the form of reflections and diffractions. However, the accurateestimation of the multipath effects, occurring during propagationover terrain, requires the correct treatment of backward waves aswell. Moreover, the PE method takes the diffraction effects intoaccount within the paraxial approximation, degrading the accuracyof the approach in deep-shadow regions where the diffracted fieldsdominate. There are a number of studies in the literature to over-come such difficulties [23–30]. Recently, a two-way SSPE algorithmhas been proposed by the authors to incorporate the backward-propagating waves into the standard one-way SSPE by utilizingan iterative forward–backward scheme for modeling multipath ef-fects over a staircase-approximated terrain [31–33]. (ii) The seconddrawback is that the standard PE is a narrow-angle approximation,which consequently restricts the accuracy to propagation anglesup to 10◦–15◦ from the paraxial direction. A typical long-range

2640 O. Ozgun et al. / Computer Physics Communications 182 (2011) 2638–2654

propagation encounters propagation angles that are usually lessthan a few degrees, whereas the short range propagation prob-lems, as well as the problems involving multiple reflections anddiffractions because of hills and valleys with steep slopes, can onlybe solved by a PE model that is effective for larger propagationangles. To handle propagation angles up to 40◦–45◦ , wide-anglepropagators have been introduced [34–39]. PETOOL implementsthe two-way algorithm proposed in [31], with the exception thatnarrow-angle propagators are replaced by wide-angle propagatorsto handle larger propagation angles.

The validation, verification, and calibration (VV&C) process ofPETOOL is also demonstrated in this paper through several numer-ical tests and comparisons realized in some canonical and morecomplex scenarios. It is known that the VV&C process is one ofthe key issues for electromagnetic simulation codes because it isnecessary for the user to specify and master the validity domainof the simulation. Validation process is the process of determin-ing whether the right model is built (or solving the right equations).Verification assessment examines if the model is built right (or solv-ing the equations right). The verification of a code usually involvesthe error analysis, which intends to search for bugs, incorrect im-plementations, errors in the inputs or in other parts of the code,accuracy in the calculations, etc. Finally, the calibration processis “the process of adjusting numerical or physical modeling parame-ters in the computational model for the purpose of improving agreementwith experimental data (AIAA G-077-1998)”. The calibration is usu-ally done against exact solutions and/or other numerical methods,and can occur as a part of either validation or verification. The cal-ibration results of PETOOL are checked against the analytical data,the results of existing software (AREPS), as well as the GeometricOptic (GO) + Uniform Theory of Diffraction (UTD). The 3D imagevisualization of the GO + UTD fields in a propagation scenario in-volving multiple knife-edges is introduced and compared with thePE method.

This paper is organized as follows: Section 2 presents the brieftheoretical formulation of the one- and two-way SSPE methods, forthe sake of completeness. Section 3 discusses the structure and theGUI capabilities of PETOOL. Section 4 illustrates the VV&C tests viaseveral comparisons. Finally, Section 5 draws some conclusions.

Throughout the paper, the suppressed time-dependence of theform exp(−iωt) is assumed.

2. Formulation of one-way and two-way SSPE methods

2.1. One-way SSPE

The problem of our interest is the 2D electromagnetic problembounded by the ground/sea at the bottom, and unbounded at thetop, i.e. extending to infinity. The problem is governed by the scalarHelmholtz equation expressed as follows:

∂2ϕ

∂x2+ ∂2ϕ

∂z2+ k2n2ϕ = 0, (1)

where k = 2π/λ is the free-space wavenumber (λ is the wave-length), n(x, z) is the refractive index, and ϕ denotes the electricor magnetic field in horizontal or vertical polarization, respectively.Here, x and z represent range and altitude coordinates, respec-tively.

The PE is derived from Helmholtz equation by separating therapidly varying phase term to obtain a reduced (amplitude) func-tion that varies slowly in range for propagating angles close tothe paraxial direction (horizontal x-direction in this paper). By in-troducing the reduced function as u(x, z) = exp(−ikx)ϕ(x, z), thewave equation in terms of u is given as follows [12]:[

∂2

2+ 2ik

∂ + ∂2

2+ k2(n2 − 1

)]u(x, z) = 0. (2)

∂x ∂x ∂z

The differential operator in (2) can be factored in terms of twopseudo-differential operators and put into forward and backwardpropagating waves, as follows:

∂u

∂x=

{−ik(1 − Q )u forward,

−ik(1 + Q )u backward,(3)

where

Q = √1 + q and q = 1

k2

∂2

∂z2+ (

n2 − 1). (4)

The formal solution of the forward propagation part in (3) can beexpressed as

u(x + �x, z) = exp[−ik�x(1 − Q )

]u(x, z), (5)

which is amenable to numerical solution by marching-type algo-rithms along range. Note that backward propagating waves in (3)are omitted in the standard PE. The operator Q is approximatedby using the first-order Taylor expansion (i.e.,

√1 + q ≈ 1 + q/2),

yielding the standard PE as follows [12]:[∂2

∂z2+ 2ik

∂

∂x+ k2(n2 − 1

)]u(x, z) = 0. (6)

The accuracy of the standard PE is limited to propagation anglesless than 10◦–15◦ , and the error in the approximation increaseswith sin4 θ , where θ is the propagation angle from the horizon-tal, due to the first neglected term in Taylor’s expansion. Hence,the standard PE is known as the “narrow-angle” approximation tothe wave equation. As the propagation angles encountered in long-range propagation problems are usually less than a few degrees,the accuracy of the standard PE is adequate for numerical model-ing.

In problems involving large propagation angles (such as shortrange propagation problems or the problems exhibiting strongmultipath effects), a more accurate expansion of the operator Qis needed. In such cases, the use of higher-order polynomials forthe operator causes instabilities in the numerical results. There arevarious convenient methods proposed in the literature to handlelarge propagation angles, such as Claerbout equation [39] and split-step Padé parabolic equation method [35], both of which dependon the first-order Padé approximation. A very efficient approach,providing a good approximation to the wave equation for largeangles, has been proposed in [36], and re-visited in [37]. In thisapproach, the operator Q is re-written as Q = √

1 + A + B , and isapproximated as Q ≈ √

1 + A + √1 + B − 1 where A = ∂2/k2∂z2

and B = n2 − 1. By making use of the operator identity√

1 + A =1 + A[√1 + A + 1]−1, the wide-angle parabolic equation is givenby

∂u

∂x= [

ikA(√

1 + A + 1)−1 + ik(n − 1)

]u. (7)

In this paper, the wide-angle PE in (7) is preferred and imple-mented because it can be solved by the Fourier split-step algo-rithm as if it is a narrow-angle PE, which will be clear in thesequel.

The numerical solution of the PE is achieved by the Fouriersplit-step method, which is a widely-used and robust algorithm.The algorithm starts at a reference range (usually at an antenna),and marches the solution in range in a way such that it obtains thevertical field profile at a given range by using the field at the pre-vious range, with appropriate boundary conditions at the top andbottom boundaries of the domain (see Fig. 1). The split-step solu-tion of the narrow-angle parabolic equation in (6) is given by [12]

u(x + �x, z)

= exp

[ik

(n2 − 1

)�x]

F −1{

exp

[−ip2 �x

]F{

u(x, z)}}

, (8)
2 2k


Fig. 1. One-way SSPE framework with forward propagating waves. (White nodes represent zero fields.)

where F indicates the Fourier transform, and p = k sin θ is thetransform variable (θ is the propagation angle from the horizon-tal). The wide-angle split-step solution of (7) is computed by [37]

u(x + �x, z) = exp[ik(n − 1)�x

]× F −1

{exp

[− ip2�x

k

(√1 − p2

k2+ 1

)−1]

× F{

u(x, z)}}

. (9)

In radio-wave applications two parameters are of interest: prop-agation factor and propagation loss (usually called path loss). Aftercalculating the field inside the entire computational domain via(8) or (9), the propagation factor (field strength relative to its free-space value in dB) is computed by

PF = 20 log |u| + 10 log x + 10 logλ. (10)

The path loss, which is the ratio between the power radiated bythe transmitter antenna and the power available at a point inspace, can be determined by

PL = −20 log |u| + 20 log(4π) + 10 log x − 30 logλ. (11)

In the numerical realization of the split-step algorithm, thereare certain important issues that require special and careful treat-ment. These issues can be classified as (i) imposition of the bound-ary conditions on the upper and lower terminations, (ii) discretiza-tion (the determination of the altitude and range increments,�z and �x, respectively), (iii) definition of the initial vertical pro-file (source imposition), (iv) handling terrain factors, and (v) han-dling atmospheric refractivity variations. The subsequent para-graphs address these issues, which are the essential componentsof the validation, verification and calibration process.

(i) The propagation problem under consideration involves a ver-tically open region, and therefore, the condition u(x, z)|z→∞ = 0must be satisfied. Since the Fourier transform is realized by dis-cretization (such as by Fast Fourier Transform, FFT), the field istruncated abruptly along z, yielding non-physical reflections fromthe upper boundary. Such artificial reflections can be removedby extending the maximum height, and by decaying the fieldsmoothly in this extended region. This can be achieved by usingabsorbing regions or by applying windowing functions (such asHanning, Hamming), or by adding a small imaginary part to therefractive index (by making n complex).

The boundary condition (BC) that must be satisfied over theEarth’s surface is expressed as[α1

∂ + α2

]u(x, z) = 0, (12)

∂z

where α1 and α2 are constants. The cases where α1 = 0 andα2 = 0 refer to Dirichlet (horizontal polarization) and Neumann(vertical polarization) BCs, respectively, over the perfectly conduct-ing (PEC) surface. For lossy ground surface, the Cauchy-type BCis defined by setting α1 = 1, α2 = ik(εr + i60σλ)1/2 and α1 = 1,α2 = ik(εr + i60σλ)−1/2 for the horizontal and vertical polariza-tions, respectively. Here, σ and εr are the conductivity and therelative permittivity of the Earth’s surface, respectively. The stan-dard split-step PE method cannot handle the BCs automatically.The ground losses can be incorporated into the standard split-stepPE approach through the use of mixed Fourier transform [11]. Tosatisfy the BCs over PEC ground, the upper boundary is extendedfrom [0, zmax] to [−zmax, zmax], and then, in accordance with theimage theory, the odd and even symmetric field profiles of u areconstructed for Dirichlet and Neumann BCs, respectively, to be ableto apply the FFT. Another option, to avoid the height extension,is to reduce the Fourier transform to one-sided discrete sine orcosine transforms (DST or DCT), for Dirichlet and Neumann BCs,respectively.

(ii) As obvious from (8) and (9), the split-step algorithm oper-ates between z and p domains (namely, Fourier transform pairs)in a consecutive manner. In the numerical implementation, thedomains are truncated at zmax and pmax. The altitude and rangestep sizes �z and �x, respectively, and the maximum altitudezmax are determined according to the source/observation require-ments, as well as the sampling criterion to avoid aliasing effects.Once zmax is decided, pmax is obtained from the Nyquist criterionzmax × pmax = π N where N is the Fourier transform size. Note thatpmax = k sin θmax, where θmax is the maximum allowable propa-gation angle. Since �z = zmax/N , the altitude increment shouldsatisfy �z � λ/(2 sin θmax). Although the choice of �z is quite crit-ical in simulations, the selection of the range increment �x (takingthe refractivity gradients into account) is chosen by the user, andcan be much larger than the wavelength.

(iii) The initial vertical field at the starting range position (usu-ally x = 0) must be properly determined in accordance with theparameters of the antenna pattern being modeled. The initial fieldcan be computed by means of near-field/far-field transformationthat relates the aperture field and beam pattern, along with theutilization of fast Fourier transform. The height and elevation an-gle of the antenna can be included by using Fourier shift theorems.The antenna pattern is specified by three parameters: height za ,the 3 dB beamwidth θBW , and the tilt (or elevation) angle θtilt. Thefirst step is to specify the initial field in the p domain via

U (0, p) = f (p)exp(−ipza) − f ∗(−p)exp(ipza). (13)

The initial field in spatial z-domain is found by taking the In-verse Fast Fourier Transform (IFFT) of (13). The Gaussian antennapattern is often used in applications since it represents vari-


ous antenna types (such as parabolic antennas). The horizontally-polarized Gaussian antenna pattern can be defined as

f (p) = exp[−p2 w2/4

]where w =

√2 ln 2

k sin(θBW /2). (14)

The tilt angle is introduced by shifting the antenna pattern, i.e.,f (p) → f (p − k sin θtilt).

(iv) Terrain factors can be incorporated into the split-step PEalgorithm with different approaches. It is worthwhile mention-ing that, the majority of the PE approaches in the literature inmodeling radio-wave propagation over a variable terrain, charac-terizes only forward-propagating waves, and neglects backwardones. Although there are algorithms considering the terrain diffrac-tion, they still omit the backward waves. Recently, the two-wayalgorithm has been proposed to model multipath effects by mod-eling an arbitrary terrain by staircase approximations [31] (seeSection 2.2). In the context of forward propagation, the simplestand the effective approach is the staircase terrain modeling. In thisapproach, on each segment of the constant altitude, the verticalfield profile is calculated in the usual way, applying the desiredboundary conditions on the ground surface, and then, is simply setto zero on the vertical terrain facets. This approach provides sat-isfactory results in the approximate sense because the boundaryconditions on sloping facets are not properly taken into account,and also, the corner diffraction is ignored. There are more accurateapproaches in the literature to modeling of sloping facets by repre-senting the terrain as a sequence of piecewise linear functions, orby employing coordinate transformations [23–25], which achieveconformal mappings over terrain curvature.

(v) The lower troposphere affects radio-wave propagation innumerous ways. Especially, its non-uniform nature causes electro-magnetic waves to be bent or refracted. Refraction of electromag-netic waves is due to the variation of the velocity of propagationwith altitude. It is known that index of refraction (n) is the ra-tio of the velocity in free-space to the velocity in the medium ofinterest (atmosphere in our case); and is caused by pressure, tem-perature and water vapor variations in both space and time. It isconventional to define a new quantity, so called modified refractiv-ity, which takes the Earth’s curvature into account, as follows:

M = (n2 − 1 + 2z/ae

) × 106 (M-units), (15)

where ae is the Earth’s radius and z is the height above sur-face. Note that 2z/ae corresponds to the Earth’s curvature. Thevariations of the vertical gradient dM/dz of the modified refrac-tivity determine four types of atmospheric conditions: subrefrac-tion (dM/dz > 118 M-units/km), standard (dM/dz = 118 M-units/km), superrefraction (dM/dz < 118 M-units/km), and ducting(dM/dz < 0). Non-standard atmospheric conditions cause anoma-lous propagation because rays bend upwards in subrefraction, anddownwards to the Earth’s surface in superrefraction and ductingconditions, in a way different from the standard atmosphere. Espe-cially, atmospheric ducts (i.e., wave trapping layers) are of specialinterest, because the negative vertical gradient leads to the captureof energy within the duct, and the trapped energy can propagateto ranges beyond the normal horizon what would be expectedwith a standard atmosphere. Such conditions significantly affectthe radio communication links and radar performance. There arebasically four types of ducting conditions: surface duct, surface-based duct, elevated duct and evaporation duct (which is indeeda type of surface duct occurring over water due to water vaporevaporated from the sea) (see Fig. 5 in Section 3). These are theconsequences of several meteorological conditions, which are be-yond the scope of this study. For detailed discussion, the readersare referred to [40].

2.2. Two-way SSPE

As mentioned in Section 2.1, the standard PE method is a one-way, forward propagation model, because of ignoring the backwardpropagation term in (3). The standard PE model cannot reflect theeffect of the interaction between the forward and backward waves,especially if there are valleys or hills with steep slopes along thepropagation path. In the presence of such obstacles, the effectsof not only forward but also backward reflected, refracted, anddiffracted waves must be very well predicted to be able to get re-liable results.

The two-way SSPE algorithm is basically the iterative imple-mentation of the one-way SSPE by simply switching the directionof propagation back-and-forth to estimate the multiple-reflectioneffects [31–33]. As illustrated in Fig. 2, the algorithm can be ap-plied to a variable terrain by using staircase approximations. If thevertical field meets an obstacle, it is split into two componentspropagating in forward and backward directions. The forward fieldprofile continues in the usual way after setting it to zero on thevertical terrain facet. During forward propagation, the wide-anglesplit-step solution in (9) is employed to march the solution. How-ever, it is evident that, with regard to the physics of the problem,the field must be partially-reflected from the terrain facet. This isachieved in the two-way algorithm in such a way that, first the ini-tial field of the backward field is obtained by imposing the bound-ary conditions at the facet (e.g., the total tangential field must bezero on the PEC facet), and then this initial field is marched back inthe −x-direction by reversing the signs of k and �x in (9). Hence,the backward vertical field profile is found at range x − �x as fol-lows:

ub(x − �x, z) = exp[ik(n − 1)�x

]× F −1

{exp

[− ip2�x

k

(√1 − p2

k2+ 1

)−1]

× F{

u(x, z)}}

. (16)

Note that the same form of the equation is derived for the reducedfunction in backward propagation (as expected), but the originalfield is expressed as ϕb(x, z) = ub(x, z)exp(−ikx).

In this way, both forward and backward fields continue tomarch out in their own paraxial directions. At each time the wavehits a terrain facet, the field is again separated into forward andbackward components. The total field is then obtained by super-posing the backward- and forward-fields at each range step. Bear-ing in mind that the field contributions of the multiple reflectionsdecrease as the iterative calculations are performed (due to 1/

√r

term in 2D Green’s function), the convergence of the algorithm isachieved and checked against a certain threshold criterion com-paring the total fields at every iteration (i.e., ‖un − un−1‖/‖un−1‖where un and un−1 are the superposed fields at the nth and(n−1)st steps). The default value of the threshold is set to 0.025 inPETOOL, but is automatically decreased unless the maximum rangeis arrived. It is useful to note that, in the implementation of the it-erative two-way algorithm, the computation time and the memoryrequirements may increase depending on the number of forward–backward propagating waves that must be handled at once. SSPE isideally capable of handling infinite number of propagating waves,of course limited by the available memory, and therefore, it canbe used for any type of terrain in a flexible way. However, thebookkeeping process might place burden on the computationalresources, but can be eliminated by utilizing parallel processingtechniques. The parallel version of the software is currently beingdeveloped, and will be available as a freeware program in the fu-ture.


Fig. 2. Recursive implementation of two-way SSPE with forward (F)–backward (B) propagating waves for staircase-approximated terrain.

3. PETOOL software

PETOOL software package has been developed in MATLAB® ver-sion R2010a with a user-friendly GUI for the analysis and visual-ization of radio-wave propagation. The GUI has been designed soas to meet the following goals:

• The user should be able to easily visualize, load and save thepropagation factor/loss on a range/altitude scale in radio-wavepropagation problems over variable terrain and through homo-geneous and inhomogeneous atmosphere.

• The user should be able to define her/his own input param-eters, and to load/save them if desired. The user should bewarned if s/he enters inappropriate input values.

• The user should be able to easily define an arbitrarily-shapedterrain profile by just locating a number of points on thegraph through left-clicking the mouse. The user should beable to load/save the terrain parameters from/in a user-definedfile.

• The user should be able to easily specify range-dependent orindependent refractivity profiles by just selecting from a list ofvarious types of atmosphere profiles.

The main m-file to run the program is petool.m. An overview ofthe program structure, as a flowchart, is shown in Fig. 3. The mainwindow of the program is depicted in Fig. 4. The window is basi-cally divided into four panels. The left and right panels are locatedon a large gray background, whereas the top and bottom panelsare located on thin blue backgrounds. These panels are defined indetail below.

The top blue panel is reserved for five operational pushbuttons(load, save, exit, run, about). The load and save buttons are used forall input parameters of the simulation. Once clicked, a modal dialogbox is opened to select or specify a file the user wants to createor save. While exiting PETOOL, the user is also warned whether ornot s/he wants to save the parameters.

The bottom blue panel is used to show warning text messageswhenever needed, especially in case of inappropriate input en-trance.

On the left panel, there are six sub-panels (domain, analysis, an-tenna, surface, atmosphere, terrain), where the input parametersare defined by the user. The input parameters are summarized inTable 1.

• In specifying a refractivity profile, the user selects an atmo-sphere type from a menu list, and a new window is opened


Fig. 3. Flowchart of the program structure.


Fig. 4. PETOOL main window and 2D graphics windows obtained by right-clicking the mouse on the 3D map. (For interpretation of the references to color in this figure, thereader is referred to the web version of this article.)

accordingly, enabling the user to enter the modified refractiv-ity (M) values for the specified atmosphere type. The avail-able atmosphere profiles are standard atmosphere, surfaceduct, surface-based duct, elevated duct, evaporation duct, anduser-defined duct, whose windows are illustrated in Fig. 5. Ifthe range-dependent refractivity profile is to be defined, theabove-mentioned selection is performed for each range value.The profiles lying between two consecutive range values arecomputed automatically through linear interpolation. It is use-ful to note that the user can load/save the atmosphere pa-rameters separately. In addition, the user can easily modify ordelete the parameters in the profile and range lists, by meansof a special dialog box that is opened when the user clicks anitem from the list.

• In specifying a terrain profile, the user has three options:(i) s/he can locate a number of points on the top graphicsof the right panel by clicking “Locate Points” button, (ii) s/hecan define the terrain points manually by entering the val-ues into the range-height list boxes, or (iii) s/he can load auser-generated text file including the terrain parameters. Inall cases, the user can save/load/clear/plot the terrain profile.If the user prefers to create her/his own terrain by locat-ing points on the graphics, the values of the selected pointsare automatically placed into the range and height list boxes.Hence, it is possible to store the graphically-generated terrainprofiles in files, as well as to modify or delete the parametersin the list boxes. Once the terrain points are specified by usingone of the above-mentioned ways, the overall terrain profile is


Fig. 5. Windows for specifying atmosphere types: (a) standard atmosphere, (b) surface duct, (c) surface-based duct, (d) elevated duct, (e) evaporation duct, (f) user-definedduct.

created by performing a kind of interpolation (linear or cubic-spline) between two consecutive terrain points along range. Ifthe interpolation method is chosen to be ‘none’, the terrainprofile appears as a collection of knife-edges. After definingthe terrain profile, the user must click on the ‘run’ button tosee the analysis results of the new geometry. It is useful tonote that the program does not allow the terrain to extendbelow zero level.

On the right panel, there are two graphics (top graphics wherethe terrain is specified, and the bottom graphics showing the col-ored 3D map of the propagation factor/loss), together with fivesub-panels (plot type, current point, 2D graphics, colorbar, save re-sult) related to the visualization or storage of output parameters.After the user clicks on the ‘run’ button, the code performs theone- or two-way split-step algorithm, and then, plots the 3D prop-agation factor map on the bottom graphics. Although the defaultis propagation factor (PF), the user can switch to path loss (PL)map by clicking the appropriate button in the ‘plot type’ panel.Whenever the user moves the mouse over the bottom graphics, the

values (range, height, PF/PL) automatically appear in the ‘currentpoint’ panel. The user can plot the 2D graphics (PF/PL versus rangefor fixed altitude, or PF/PL versus altitude for fixed range) eitherby entering the values into the boxes in ‘2D graphics’ panel, or byright-clicking the mouse on the desired point of the 3D map (seeFig. 4). The ‘colorbar’ panel is used to adjust the colorbar scale ofthe 3D map for better visualization. Finally, the ‘save result’ panelis used to store the PF/PL maps in the form of a MATLAB file (.mat)or a picture file (.tif). We also note that 2D graphics in Fig. 4 canbe saved in text files by using the ‘save’ button.

4. Validation, verification and calibration tests

VV&C starts with the validation process, which is related towhether the model is correct. Referring to the above-mentioneddiscussions, the limitations of the standard split-step based para-bolic equation (SSPE) model are two-fold: (i) it is a one-way, for-ward propagation model, which neglects backward waves; (ii) bothnarrow- and wide-angle PE models are valid within the parax-ial region. For long-range propagation problems, these limitationsmay not be serious. However, they must be treated with special


Table 1Input parameters of PETOOL.

Domain parametersMaximum range (km)Maximum altitude (m)

Antenna parametersPolarization (horizontal or vertical)3 dB beamwidth (degree)Elevation angle (degree)Antenna height (m)Frequency (MHz) (must be > 30 MHz)

Analysis parametersOne-way or two-way (which algorithm is to be performed)Range step (m): horizontal step sizeAltitude step (m): vertical step size

Surface parametersPerfectly conducting or impedance surfaceIf impedance surface is chosen:Type: sea, fresh water, wet ground, medium dry ground, very dry ground,user-defined groundIf user-defined ground is chosen:Dielectric constantConductivity (S/m)

Atmosphere parametersRange-independent or range-dependent refractivityIf range-independent refractivity is chosen:Type: standard atmosphere, surface duct, surface-based duct, elevatedduct, evaporation duct, and user-defined ductIf range-dependent refractivity is chosen:Refractivity type for each range value defined in list-boxes

Terrain parametersNone (flat surface) or terrainIf terrain is chosen:Interpolation type (none, linear, cubicspline)Number of points: number of points to be placed on top graphics to definethe terrain profileRange and height values for terrain points defined in list-boxes

attention for waves propagating upwards/downwards with largertilts and/or for propagation paths having steep irregular terrainprofiles. The discussion of the former issue is left to Section 4.2in the context of the two-way SSPE. The latter issue in conjunctionwith the one-way SSPE is the main concern of Section 4.1.

4.1. Numerical simulations with one-way SSPE

A critical issue in the VV&C procedure is the construction of thereference (analytical or exact) solution. Two canonical problemsare taken into account to produce the reference solution: (i) 2Dpropagation inside a parallel-plate waveguide with PEC boundaries,(ii) 2D groundwave propagation through a surface duct. The wavefunction inside the PEC parallel-plate waveguide with Dirichlet BChaving the width of d and located longitudinally along x may berepresented in terms of modal superposition given as:

u(x, z) =√

2

d

N∑q=1

cq sin

(qπ

dz

)exp(iβqx),

βq =√

k2 − (qπ/d)2. (17)

Here, βq is the longitudinal propagation constant for the mode qand cq is the modal excitation coefficient, numerically computedfrom the given Gaussian source function g(z), using the orthonor-mality property as:

cq =d∫

g(z) sin

(qπ

dz

)dz,

0

g(z) = 1√2πσ 2

exp

[−(z − za)2

2σ 2

]. (18)

Here, za and σ are the height and the spatial width of the Gaus-sian source, respectively.

In the second canonical problem, the 2D surface duct over thePEC flat Earth is caused by a linearly decreasing vertical refractivityprofile (i.e., n2(z) = 1 − a0z, where a0 is a positive constant con-trolling the duct strength) and the exact solution is available interms of Airy functions for the range-independent vertical refrac-tive index. The analytical modal solution using N modes is givenby [41,42]

u(x, z) =N∑

q=1

cqψq(z)exp(iβqx) (19)

where cq is the normalization constant, βq is the longitudinalpropagation constant for the mode q, ψq(z) = Ai[(a0k2)1/3z −σq] isthe mode function satisfying the 1D wave equation in along z, Ai isAiry function of the first kind, and σq is the root of the equationsatisfying the boundary condition Ai′(−σq) + αAi(−σq) = 0. Thecases where Ai(−σq) = 0 and Ai′(−σq) = 0 denote the Dirichletand Neumann boundary conditions, respectively. Here, the primerefers to the derivative with respect to the vertical coordinate. Inthe numerical calculation, first the antenna pattern is determined.Then, the modal superposition in (17) or (19) is used together withthe orthonormality condition, and the number of modes and theirexcitation coefficients are computed for a given error boundary.

Assuming a horizontally polarized Gaussian antenna at 3 GHzwith 3 dB beamwidth of 30◦ , located at 0.5 m height and tilted45◦ downwards, inside a 1 m wide parallel-plate PEC waveguide,the performances of narrow- and wide-angle SSPE tools with re-spect to the analytical result are illustrated in Fig. 6 by meansof 3D field maps. Fig. 6(a) alone seems to be logical and phys-ical. A down-tilted Gaussian beam hits the lower boundary andreflects up causing interference, reaches the upper boundary andreflects down with the pattern shown there. The true 3D wavepatterns are shown in Figs. 6(b) and 6(c). Fig. 6(b) (the analyti-cal result) is produced with the superposition of 53 modes with amaximum error of 4 × 10−10. With reference to the exact result,the maximum and mean errors for the narrow-angle SSPE are 73%and 13% respectively, whereas for the wide-angle SSPE, the errorsare only 2.3 × 10−5 and 4.8 × 10−6 respectively. Hence, these re-sults validate the accuracy of the wide-angle SSPE, and show thatthe narrow-angle SSPE cannot handle large tilt angles. To validatethe narrow-angle SSPE, the angle must be constrained to be lessthan 10◦ while considering both beamwidth and tilt angle. Finally,Fig. 6(d) shows field strength vs. range computed from analyti-cal exact, narrow and wide angle SSPE models. As shown here,narrow-angle SSPE is far away from handling propagation charac-teristics of this scenario.

The performance of the narrow-angle SSPE in horizontal polar-ization can be visualized in Fig. 7, for the propagation scenarioinvolving two Gaussian antennas at 250 m and 500 m, with 3 dBbeamwidth of 1◦ , and tilt angle of 0.5◦ downwards and upwards,respectively. The frequency is 300 MHz; and a vertically decreas-ing refractivity profile with a slope of dM/dz = −600 M-units/km,corresponding to a strong anti-guiding atmosphere, is taken intoconsideration. The range and altitude increments chosen in thesimulation are �x = 25 m and �z = 0.5 m, respectively. In the an-alytical result, 350 modes are used with a maximum field error of10−8. With reference to the exact result, the maximum error forthe narrow-angle SSPE is obtained as 1.4%.

Note that mathematically exact representations do not neces-sarily yield reference solutions; unless numerical data is generatedwith a desired accuracy. Eqs. (17) and (19) are exact solutions in


Fig. 6. [Calibration of one-way SSPE.] Field maps for a Gaussian antenna at 0.5 m height with 3 dB beamwidth of 30◦ , tilt angle of 45◦ inside a parallel-plate waveguide at3 GHz: (a) narrow-angle SSPE, (b) analytical result, (c) wide-angle SSPE, (d) horizontal field profiles at z = 0.25 m and z = 0.5 m [�x = 5 mm, �z = 2.5 mm].

Fig. 7. [Calibration of one-way narrow-angle SSPE.] Two Gaussian antennas at 250 m and 500 m with 3 dB beamwidth of 1◦ , tilt angles of −0.5◦ and 0.5◦ at 300 MHz abovePEC ground, and inside a vertically decreasing refractivity with a slope of 600 M-units/km: (a) field map for analytical result, (b) field map for narrow-angle SSPE, (c) verticalfield profiles at two different ranges [�x = 25 m, �z = 0.5 m, 350 modes are used for the analytical result].

Table 2Number of modes as a function of tilt angle, for maximum initial field er-ror (< 10−8). (a) Parallel plate PEC waveguide ( f = 3 GHz, za = 0.5 m, θBW =30◦). (b) Surface duct ( f = 300 MHz, za = 250 m, θBW = 65◦ , dM/dz =−600 M-units/km).

(a) (b)

Tilt angle(degree)

Numberof modes

Tilt angle(degree)

Numberof modes

0 37 0 1910 40 2 19120 43 4 79530 46 6 209940 49 8 438045 50 10 7926

terms of modal summation, therefore may serve as reference if nu-merically computed with a specified accuracy. Although an exactmathematical solution is available at hand, it might be inefficientto perform numerical computation of the modal summation forlarger tilts, because the number of modes increases as the antennatilt increases. This is clarified in Table 2, which lists the numberof modes required to establish a given Gaussian antenna pattern

as a function of antenna tilt. Table 2(a) belongs to the parallel-plate waveguide with PEC boundaries, showing that the numberof modes (for the given parameters) would be much less than100 for tilts up to 45◦ (note that, although only trapped modeswith real propagation constants contribute fields beyond severalwavelength distances, all trapped and non-trapped modes are stillrequired to precisely reconstruct near fields). The number of modesincreases as the frequency increases and the beamwidth gets nar-rower, but all are numerically manageable. On the other hand,Table 2(b) shows the number of modes for the surface duct prob-lem. Tens of thousands of modes would not be sufficient to takethe tilt angles more than 10◦ into account (even for the twice widebeamwidth). Also the computations become a real challenge! Firstof all, modes are confined between the ground and their caustics[13] which go higher and higher as the mode number increases.The modal excitation coefficients are calculated numerically usingthe orthonormality principle, therefore a numerical integration isessential in the vertical domain for each mode. This means thatboth the integration step and the upper integral boundary shouldbe changed dynamically. Finally, all the modes are required in fieldcomputations at every range.


Hence, it might be difficult to produce numerical reference datafor the VV&C tests. A convenient way of testing narrow- and wide-angle SSPE models is to tilt up or down the pattern up to 40◦–45◦ . In the third scenario, five Gaussian antennas located at thesame place (1000 m height) with 3 dB beamwidth of 0.5◦ each,and 10◦–20◦–30◦–40◦–50◦ tilt angles illuminate the PEC ground ina standard atmosphere, assuming that the frequency is 1 GHz, the

Fig. 8. [Calibration of one-way SSPE.] Propagation Factor (PF) maps for five Gaussianantennas with different tilt-angles illuminating PEC ground: (Top) narrow-angle,(Bottom) wide-angle [specular reflection points are shown in Table 3].

polarization is horizontal, the range and the altitude incrementsare �x = 10 m and �z = 0.13 m, respectively. The 3D field maps,corresponding to narrow- and wide-angle SSPE are illustrated inFig. 8. The specular reflection points (ri) on the ground surfaceare tabulated in Table 3 for the tilt angles used in Fig. 8, togetherwith various tilt angles. Although there is an almost exact matchbetween the desired specular reflection points and those found bythe wide-angle SSPE, the results of the narrow-angle SSPE start todeteriorate as the tilt angle increases.

Table 3Specular reflection points and errors as a function of tilt angle.

Tilt angle(degree)

Exact and wide-angle reflectionpoint (km)

Narrow-anglereflection point(km)

Percentagedifference(%)

8 7.1 7.2 1.419 6.3 6.4 1.58

10 5.6 5.77 1.7611 5.14 5.25 2.1412 4.7 4.82 2.5513 4.33 4.45 2.7714 4.01 4.14 3.2415 3.73 3.87 3.7520 2.74 2.92 6.5725 2.14 2.36 10.2830 1.73 2 15.6135 1.43 1.74 21.6740 1.56 1.19 31.0945 1 1.42 42.0050 0.84 1.3 54.76

Fig. 9. [Calibration of one-way SSPE with AREPS.] A Gaussian antenna at 50 m height with 3 dB beamwidth of 3◦ , tilt angle of 0◦ inside a surface duct: (Upper-left) mainwindow with PF map, (Upper-right) modified refractivity profile for surface duct, (Lower-left) PF vs. range at 50 m altitude, (Lower-right) PF vs. altitude at 120 km range[ f = 3 GHz, �x = 200 m and �z = 0.29 m].


Fig. 10. [Calibration of two-way SSPE via GO + UTD.] A finite-height wall (50 mheight, 50 km range) in front of an infinite-height wall at 60 km illuminated by aline-source at 250 m: (a) PF maps, (b) PF vs. altitude at 55 km range, (c) PF vs.range at 50 m altitude [ f = 3 GHz, �x = 200 m and �z = 0.29 m].

Finally, the performance of the one-way SSPE is demonstratedagainst AREPS1 ver. 3.0, which simulates only the one-way prop-agation. The comparison is based on the narrow-angle SSPE be-cause this version of AREPS implements so. As shown in Fig. 9,a Gaussian antenna at an altitude of 50 m illuminates a variableterrain inside a surface duct, which is modeled by the modifiedrefractivity profile M . As illustrated by the propagation factor (PF)maps, a good agreement is observed between PETOOL and AREPSin narrow-angle case.

The good agreement among the results illustrates the successand completeness of the VV&C process for the one-way SSPE.

4.2. Numerical simulations with two-way SSPE

This section demonstrates the test results of the VV&C pro-cess of the two-way SSPE tool over different propagation scenar-ios. In the first two scenarios, the calibration is done against theGO + UTD results, assuming that the frequency is 3 GHz. It is use-ful to note that the calibration via GO + UTD is feasible only forsufficiently high frequencies such that the ray-optics interpretationis valid.

The first scenario involves two walls, one of which is finite (alsoknown as a knife-edge) and the other one is infinite in height.

1 http://areps.spawar.navy.mil/.

Fig. 11. [Calibration of two-way SSPE via GO + UTD.] Two finite-height walls (100 mhigh wall at 20 km range and 150 m high wall at 40 km range) illuminated by aline-source at 5 m: (Top) PF maps, (Bottom-left) PF vs. altitude at 15 km (in the in-terference region), (Bottom-right) PF vs. altitude at 35 km (in the shadow/diffractionregion) [ f = 3 GHz, �x = 200 m and �z = 0.29 m].

In Fig. 10, the finite-height (50 m) wall is located at 50 km; theinfinite-height wall is at 60 km; the source is at 250 m; and thepolarization is vertical. This is one of the classical structures inthe field of diffraction theory, and its approximate solution canbe computed by using ray-optic techniques, combined with specialdiffraction methods. In the implementation of the GO method, thereflected waves from the ground and the wall can simply be cal-culated by employing the principles of image theory that replacesthe original problem with the equivalent problem represented byimage sources with respect to the boundary conditions that mustbe satisfied on the boundaries depending on the polarization. Thetotal field is obtained by the sum of the direct ray, reflected raysemanating from image sources, and the diffracted rays from the tipof the wall, by also checking the line-of-sight (LOS) conditions be-

http://areps.spawar.navy.mil/


Fig. 12. [Calibration of two-way SSPE.] PF maps for two Gaussian antennas above ground with different tilt angles illuminating two opposite infinite-walls: (Left) narrow-angle, (Right) wide-angle [specular reflection points: z1 = 176 m, z2 = 577 m (wide-angle and exact), z1 = 174 m, z2 = 500 m (narrow-angle)] [ f = 3 GHz, �x = 5 m and�z = 0.06 m].

Fig. 13. [Calibration of two-way SSPE.] PF maps for a variable terrain illuminated by a Gaussian antenna at 50 m in an elevated-duct environment: (Left-top) one-waynarrow-angle, (Left-bottom) one-way wide-angle, (Right-top) two-way narrow-angle, (Right-bottom) two-way wide-angle [ f = 3 GHz, �x = 200 m and �z = 0.29 m].

tween the source(s) and the observation point. As shown in Fig. 10,multiple reflections (almost resonance behavior) occur especiallyin the region between the walls. In comparing the two-way SSPEwith the GO + UTD approach, the contribution of the waves hittingthe walls up to three times is superposed. To achieve fair compar-isons up to third degree of reflections, the GO+UTD code accountsfor 35 types of rays bouncing from the walls and the ground. Inaddition to reflected waves, the diffracted waves from the finite-height wall are also computed. However, the multiple bouncing ofthe diffracted fields from the walls and the ground is ignored duetheir negligible effects compared to strong reflections. The goodagreements among the results illustrate the success of the two-way SSPE with respect to GO + UTD approach.

The scenario presented in Fig. 11 involves two finite-heightwalls along the propagation path (100 m high wall at 20 km rangeand 150 m high wall at 40 km range) illuminated by a line-sourceat 5 m (i.e., multiple-wedge problem). The figures on the top showthe PF maps obtained with the one-way SSPE, two-way SSPE andGO + UTD models. The boundaries of both incident (i.e., LOS) andreflected fields are observed in both two-way SSPE and GO + UTDmaps. The artificial effects around these boundaries are also clearly

observed in the GO + UTD model, which are inherently not ob-served in the SSPE map. The two plots on the bottom belong to PFvs. height at two different ranges. As observed from two-way SSPEand GO + UTD results, a good agreement is obtained in the firstregion (at 15 km that is in the interference region), but a slight dis-crepancy is observed in the shadow region (at 35 km that is in theshadow/diffraction region). This discrepancy can be eliminated bytaking into account the slope diffraction coefficient in the GO+UTDmodel, whose accuracy decreases in deep-shadow regions. Further-more, this discrepancy might be due to ignoring some of the lesscontributing components (such as double-diffractions), and due tothe limitation of the SSPE within the paraxial regions; thereforefurther investigation is required in order to speculate about thisdiscrepancy.

The forthcoming scenario is designed to better grasp the dif-ferences between narrow- and wide-angle propagators in an en-vironment exhibiting strong reflections. Two Gaussian antennas,located just above ground, with 3 dB beamwidth of 0.3◦ each, and−10◦ , −30◦ tilt angles illuminate two opposite infinite-walls, asshown in Fig. 12. It is assumed that the polarization is horizon-tal, and the medium is free-space. The expected specular reflection


Fig. 14. [Miscellaneous PETOOL simulations.] PF maps for a variable perfectly-conducting terrain illuminated by a Gaussian antenna at 10 m assuming different refractivityprofiles: (a) surface duct (one-way), (b) surface duct (two-way), (c) surface-based duct (one-way), (d) surface-based duct (two-way) [ f = 3 GHz, �x = 200 m and �z =0.29 m].

Fig. 15. [Miscellaneous PETOOL simulations.] The same scenario in Fig. 14, except that the antenna is at 100 m height.


Fig. 16. [Miscellaneous PETOOL simulations.] PF maps for a variable terrain illuminated by a Gaussian antenna at 5 m assuming different ground surface parameters instandard atmosphere: (a) perfectly-conducting surface (one-way), (b) perfectly-conducting surface (two-way), (c) very dry ground (εr = 3, σ = 1e–4) (one-way), (d) very dryground (two-way) [ f = 100 MHz, �x = 100 m and �z = 10 m].

points of the beams on the walls are as follows: z1 = 176 m andz2 = 577 m. The wide-angle version of the two-way SSPE providesaccurate results for all tilt angles and specular reflection points.However, the narrow-angle SSPE is accurate for only −10◦ tilt an-gle (Narrow: z1 = 174 m and z2 = 500 m). The error increasesas the tilt angle increases in the narrow-angle SSPE, as expected.The new scenario in Fig. 13 illustrates the comparison betweenone-way and two-way SSPE models with narrow and wide anglepropagators over a variable terrain and in an elevated duct envi-ronment. A Gaussian antenna with 2◦ tilt angle is located at 50 mheight. The frequency is 3 GHz, and the polarization is horizontal.Referring to the descriptions in Fig. 5, the elevated duct is definedas follows: M0 = 300, M1 = 330, M2 = 310, M3 = 350, z1 = 100 m,z2 = 150 m, z2 = 300 m. These examples help us to visualize theimportance of the correct representation of large propagation an-gles, as well as the backward propagating waves.

As emphasized in the Introduction part, PETOOL has been de-signed for research/educational purposes. Hence, propagation en-gineers/instructors can simulate different propagation scenariosto investigate the radio-wave propagation and/or to design re-liable communication links. In order to demonstrate what en-gineer/instructors can do with PETOOL, we have shown variousscenarios in Figs. 14, 15 and 16. In Fig. 14, a Gaussian antenna,which is located at 10 m height, radiates into surface duct andsurface-based duct environments. Referring to the descriptions inFig. 5, the surface duct is defined as follows: M0 = 350, M1 = 300,M2 = 350, z1 = 200 m, z2 = 300 m. The surface-based duct isdefined as follows: M0 = 340, M1 = 356, M2 = 340, M3 = 358,z1 = 135 m, z2 = 150 m, z2 = 300 m. Polarization is horizontal,and the ground is perfectly conducting. Computations times are:4 s (one-way SSPE), 820 s (Fig. 14(b)), 643 s (Fig. 14(d)), on Intel

Core 2 Duo 1.6 GHz with 2 GB RAM PC. Note that the computationstimes are dependent on the number of iterations in the two-waySSPE, which in turn depend on the amount of the wave interac-tions (multiple reflections) between the hills. In Fig. 15, the samescenarios in Fig. 14 are performed, except that the antenna is lo-cated at 100 m. Computations times are: 4 s (one-way SSPE), 672 s(Fig. 15(b)), 963 s (Fig. 15(d)). In Fig. 16, different ground parame-ters are simulated assuming that a Gaussian antenna is located at5 m height and radiates into a standard atmosphere. In Fig. 16(a)and 16(b), the surface is perfectly conducting. In Fig. 16(c) and16(d), the surface is “very dry ground” with dielectric constantεr = 3 and conductivity σ = 1e–4 siemens/m. Polarization is ver-tical, and the frequency is 100 MHz. Computations times are: 8 s(Fig. 16(a)), 112 s (Fig. 16(b)), 8 s (Fig. 16(c)), 129 s (Fig. 16(d)).

5. Conclusions

A MATLAB-based one-way and two-way split-step based para-bolic equation program — called PETOOL — has been introducedand discussed systematically for a radio-wave propagation problemover an arbitrary ground profile through homogeneous and inho-mogeneous atmosphere. It has been concluded that the standardPE model has certain drawbacks in terms of handling backwardpropagation, as well as large propagation angles. These issues maybe predominantly observed in problems involving arbitrary obsta-cles on the propagation path or in short-range problems, or combi-nation of two (for example, in urban propagation scenarios). Theymay also be observed in propagation scenarios where abrupt, se-vere changes in atmospheric conditions occur. The wide-angle ver-sion of the two-way SSPE overcomes these limitations up to a greatextent, as demonstrated by several numerical calibration tests per-


formed with respect to reference results. The program structureand the GUI capabilities of PETOOL have been discussed. It is be-lieved that PETOOL serves as a user-friendly research/educationaltool for propagation engineers/instructors.

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petool: matlab-based one-way and two-way split-step parabolic equation tool for radiowave...

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