an investigation of wave propagation over irregular ... · wave solution for the ﬁeld in the...

An Investigation of Wave Propagation over Irregular Terrain andUrban Streets using Finite Elements

KAMRAN ARSHADMiddlesex University

School of Computing ScienceLondon, NW4 4BTUnited Kingdom

FERDINAND KATSRIKUMiddlesex University


ABOUBAKER LASEBAEMiddlesex University


Abstract:A method to model electromagnetic wave propagation in troposphere on irregular terrain in the presenceof height dependant refractivity is presented using finite element analysis. In this work, the helmholtz equationapplied on radiowave propagation properly manipulated and simplified using pade approximation is solved usingfinite element method. Paraxial form of helmholtz equation is also called wide angle formulation of parabolicequation (WPEM) and is used because of its accuracy and behavior on large propagation angles. By using thismethod, horizontal and vertical tropospheric characteristics are assigned to every element, and different refractivityand terrain profiles can be entered at different stages. We also consider wave propagation on an urban street todemonstrate the effectiveness of our method.

Key–Words:Troposphere, finite element method, parabolic equation method.

1 Introduction

Radio coverage in the troposphere and urban areas hasbeen a challenging problem for many years [1]. Re-searchers in wave propagation area have been search-ing for efficient mathematical models for describingthe problem of electromagnetic wave propagation introposphere. Numerous methods are available for pre-dicting electromagnetic wave propagation in the at-mosphere [2]. However, the presence of vertical re-fractivity stratification in the atmosphere complicatesthe application of some methods. To model refractiv-ity variations in the horizontal as well as vertical di-rection, geometric optics, coupled-mode analysis, orhybrid methods have been employed [3].

In the past, emphasis was given to geometricaloptics (GO) techniques and modal analysis. GO pro-vide a general geometrical description of ray fami-lies, propagating through the atmosphere. Ray trac-ing methods present many disadvantages; for exam-ple the radiowave frequency is not accounted for andit is not always clear whether the ray is trapped by thespecific duct structure [4]. An alternative approachfor tropospheric propagation modelling was devel-oped by Baumgartner [5] and later extended [6], nor-mally known as Waveguide Model or Coupled ModeTechnique. The main disadvantage of coupled modetechnique lie in the complexity of the root findingalgorithms and large computational demands, espe-cially when higher frequencies and complicated duct-

ing profiles are involved.The solution of electromagnetic propagation

problems in the presence of an irregular terrain isa complicated matter. Three-dimensional variationsin refraction and terrain make the full vector prob-lem extremely difficult to solve in a reasonable time.Many existing prediction models are based on simpli-fied Deygout solution for multiple knife-edge diffrac-tion [7]. An important class of propagation modelsover irregular terrain is based on integral equation for-mulation which in general can be simplified by usingparaxial approximation [8]. If one chooses to simplifythe problem by assuming symmetry in one or more ofthe coordinate directions, the vector problem can bedecoupled into scalar problems.However the solutionof two dimensional scalar problem is still difficult forrealistic environments. Some approximations and nu-merical schemes for the solution are used to reducethe solution of the full two-way equation to one-wayequation. Benefits of one-way propagation are thesimple numerical implementation of range dependen-cies in the medium and the avoidance of prohibitivenumerical aspects of solving elliptic equations associ-ated with implementing two range-dependant bound-ary conditions.

One of the most reliable and widely used tech-nique in literature is parabolic equation (PE) method,initially developed for the study of underwater acous-tic problems and later on extended to tropospheric

Proceedings of the 6th WSEAS Int. Conference on TELECOMMUNICATIONS and INFORMATICS, Dallas, Texas, USA, March 22-24, 2007 105

propagation ones [9]. The PE is based on the solutionof the two dimensional differential parabolic equa-tion, fitted by homogenous or inhomogeneous refrac-tive profiles. Models based on the parabolic approx-imation of the wave equation have been used exten-sively for modelling refractive effects on troposphericpropagation [10, 11], in last decade. The biggest ad-vantage to using the PE method is that it gives a full-wave solution for the field in the presence of range-dependent environments. Solution of PE in complexenvironments require some numerical method like fi-nite element method.

In this work wide angle parabolic equation is usedto model radio wave propagation over irregular terrainin troposphere and on urban streets using finite ele-ment method (FEM). Parabolic equation in wide an-gle formulation is used to make sure that every singleray should be accounted in the solution. The main ad-vantage of using WPEM over irregular terrain is that itcombines the effects of both terrain diffraction and at-mospheric refraction while remaining straightforwardto implement [12]. The FEM has been applied previ-ously to model wave propagation over smooth ground[13]. In the present work, we extend this approachto the solution of wide angle parabolic equation inthe presence of irregular terrain and on urban streets.In this paper, helmholtz equation is properly manipu-lated and simplified using pade approximations [14],the solution of which is achieved by using finite ele-ment method. Vertical tropospheric profile character-istics are assigned to every mesh element, while solu-tion advances in small variable range steps, each ex-cited by solution of the previous step.

The remainder of the paper is organized as fol-lows. In section 2, wave propagation modelling us-ing paraxial approximation of wave equation is de-scribed along with boundary conditions and initialfield. Section 3 describes finite element formulationof the problem. Results and discussions are given insection 4 and section 5 concludes the paper.

2 Wave propagation modelling byparaxial form of helmholtz equa-tion

The paraxial form of Helmholtz equation is:

∂2u(x, z)∂x2

+ j2k0∂u(x, z)

∂x+

∂2u(x, z)∂z2

+ k20

(n2(x, z)− 1

)u(x, z) = 0

(1)

wherek0 is the free space wave number,u(x, z) isthe unknown electric or magnetic field depending on

polarization,n(x, z) is the refractive index of tropo-sphere,x is the propagation direction andz is thetransverse direction. Equation (1) is also called wideangel representation of parabolic equation method.In long-distance propagation scenarios the effect ofearth’s curvature must be considered. In earth flat-tening transformation refractive index is replaced withmodified refractive indexm(x, z) [2]:

m(x, z) = n(x, z)(1 +

z

R

)(2)

whereR is the radius of earth. After substituting equa-tion (2) in (1), it becomes,

∂2u(x, z)∂x2

+ j2k0∂u(x, z)

∂x+

∂2u(x, z)∂z2

+

k20

(n2(x, z)− 1 +

2z

R

)u(x, z) = 0

(3)

Wave propagation model based on WPEM as definedin equation (3) is subject to terrain boundary condi-tion, which represents the relationship that must holdbetween the fieldu(x, z) and terrain. Note from equa-tion (3) that the earth curvature enters only through the2zR term; if this term is ignored, the equation describespropagation over a flat earth.

2.1 Boundary and Initial ConditionsIn two dimensional wave propagation problem, thereare two boundaries, one at the starting height,z =zmin which in fact is the earth surface at some heightdepending on terrain profile, and at the maximum al-titude considered,z = zmax. An impedance typeboundary condition can be used at lower heights toaccount for the finite conductivity and permittivity ofthe surface of the earth. The entrance boundary con-ditions are expressed by the equation [9]:

[∂u(x, z)

∂z+ jk0qu(x, z)

]

z=zmax

= 0 (4)

where,

q = qv =jk0√

εr − j60σλ(5)

q = qh = jk0

√εr − j60σλ (6)

for vertical and polarization respectively.εr is thecomplex relative permittivity andσ is the conductivityof ground whereasλ is the wavelength of radio wavesin meters.

When numerical propagation simulations are per-formed, infinite propagation domains cannot be re-alized and the size of the propagation domain must


be truncated. This is accomplished numerically byimplementing absorbing boundary conditions or Per-fectly Matched Layer (PML) on the upper boundaryatz = zmax.

For initial field a normalised gaussian antennapattern is used:

u(0, z) = [A(z −H0)− A(z + H0)] (7)

whereH0 represents the antenna height and bar de-notes complex conjugate. The Fourier transform ofA(z) is given as,

a(ζ) = e−ζ2w2/4 (8)

wherew is defined as,

w =

√2 ln 2

k0 sin θbw2

(9)

wheresin θbw2 is the sine of3dB beamwidth.

3 Finite element formulation prob-lem

In this section finite element formulation of the modelis developed. Apply the finite element method overthe domainzmin ≤ z ≤ zmax, choose an interpola-tion function and than by using weak formulation ofGalerkin method, matrix equation for the system canbe defined as,

[M ]∂2{u}∂x2

+ j2k0[M ]∂{u}∂x

+ [K]{u}− k2

0[M ]{u} = {0}(10)

where matrices[M ] and[K] are defined as,

[M ] =∑

e

∫

z{N}T {N} dz (11)

and,

[K] =∑

e

∫

zk2

0

[n2 +

2z

R

]dz −

∑e

∫

z{N}T

{N} dz −∑

e

∫

z{N}T {N} dz (12)

and{N} is the shape function matrix,{N} = ∂{N}∂z ,

superscriptT denotes transpose function and{0} isthe zero matrix. Now, rewrite equation (10) as,

j2k0[M ]∂{u}∂x

= − [K]− k20[M ]

1 + 1j2k0

∂∂x

{u} (13)

Using Pade approximation [14],

∂

∂x≈ 1

j2k0

{[K]− k2

0[M ]}

(14)

Substituting equation (14) in (13) and after simplifica-tion it gives,

−j2k0

([M ]− [K]− k2

0[M ]4k2

0

)∂{u}∂x

=([K]− k2

0[M ]){u}

(15)

Define,

[M ] = [M ]− [K]− k20[M ]

4k20

equation (15) will becomes,

−j2k0[M ]∂{u}∂x

= [K]{u} − k20[M ]{u} (16)

So at range stepx + ∆x2 equation (16) can be written

as,

−j2k0[M ]∂{u}∂x

∣∣∣x+∆x2

={[K]− k2

0[M ]} {u}x+∆x

2

(17)Using crank nicolson approximation scheme,

− j2k0[M ]{u}x+∆x − {u}x

∆x

=([K]− k2

0[M ])[{u}x+∆x + {u}x

2

] (18)

yields final propagation algorithm as,{−j4k0[M ]−∆x

{[K]− k2

0[M ]}}{u}x+∆x

={−j4k0[M ] + ∆x

{[K]− k2

0[M ]}}{u}x

(19)

4 Results and DiscussionsIn this section, results for wave propagation in tropo-sphere, over irregular terrain and urban streets will bepresented. Coverage diagrams at different frequen-cies and under different environment conditions willbe presented. Initially wave propagation in tropo-sphere over smooth terrain is described along withsome ducting conditions and than results of propa-gation over irregular terrain will be shown. Accuratemodelling of radio waves over irregular terrain and es-pecially in urban environment is crucial for the plan-ning of cellular communications in cities.

To demonstrate the efficacy of the present ap-proach a number of simulations were carried out withvarious frequencies and media profiles. In all of the


simulations the following assumptions were made un-til stated otherwise. A transmitting antenna of heightH0 = 150m above sea level with a3dB-beam widthθbw = 10 and vertical polarization of the propagatingwave is assume. Boundary conditions and initial fieldgeneration is described in section?? and??. Simu-lation parameters use in this paper are summarized intable 1.

Parameter Value(s)

Tx Antenna Height 150mTx Antenna Gain 43.5dBRx Antenna Height 150mAntenna 3dB bandwidth 10◦

Ground Conductivity 0.01mho/mGround Permittivity 15PML zone 300mTropospheric Duct Height 300Antenna Polarization HPOL or VPOLFrequency of Waves 100MHz, 1GHz

Table 1: Simulation Parameters used for wave propa-gation in troposphere

For all simulations finite elements of lengthλ/10has been choosen. Different values of range step∆xsomewhere in between25 and180m has taken. If asmaller∆x is use the computation takes longer andmore computational resources will be needed. Forlarge ∆x, the computation is accelerated, but somesignificant atmosphere changes may miss and moreerror will incur in simulation results. Therefore,∆xshould be optimally choosen accordingly for eachproblem. However, it is possible to choose∆x andelement size having variable lengths in one simula-tion.

Finite element formulation of parabolic equationmethod using different terrain profiles is simple andstraightforward. As in the case of refractivity pro-files, at each range step∆x program requires a ter-rain profile as a function of height. However if ter-rain or refractivity profile is not changing along rangethan a single profile is sufficient. Terrain profile is en-tered in much the same way as refractivity profiles.All what is required is the series of data points cor-responding to height versus range to describe terrain.The simplest and most effective technique models ter-rain as a sequence of horizontal steps, and called stair-case method. In this paper we use staircase methodto model terrain, however other approaches are dis-cussed in detail in [2].

Atmospheric conditions can be entered into thesimulation program as profiles of refractivity-height

data where refractivityN is related with refractive in-dexn asN = (n− 1)× 106 [2]. In cases where hori-zontally inhomogeneous conditions needs to be mod-elled, different refractivity profiles can be entered atseveral ranges, and program can perform linear inter-polation in range and height for use at intermediatecalculation position.

Modelling ground wave propagation for upwardpropagated waves is a challenging signal processingtask. As we discussed earlier in section?? the upperboundary should be truncated for proper terminationof grid or domain. This can be accomplished numeri-cally by implementing an absorbing boundary condi-tion on the upper boundary or using perfectly matchedlayer (PML) as described in section??. Effective ab-sorption within PML block depends on number ofPML layers and artificial PML medium parameters.Figure 1 and 2 presents the coverage diagram of trans-mitting antenna at100MHz for standard atmosphericconditions without and with PML. As can be seenfrom figure 1 without the implementation of PML thewaves get reflected from the upper boundary. In fig-ure 2 when a perfectly matched layer (PML) boundaryis implemented these reflections are eliminated.

Figure 1: Coverage diagram with out PML at100MHz

It can be seen easily that waves propagates undis-turbed through the tropospheric medium under normalatmospheric conditions. In figures 3, a bilinear sur-face ducting profile is included at 100MHz, startingfrom the sea level to an altitude of300m. Standard at-mospheric conditions over this altitude were also as-sumed. Figures 4 and 5 shows trapping mechanismat 1GHz and3GHz respectively. From the results,it is clear that a duct of sufficient intensity is capa-ble of capturing whole energy at3GHz. At 100MHzduct is unable to divert waves but at 3GHz waves arealmost completely trapped between the sea level andduct height.


Figure 2: Coverage diagram with PML at 100MHz

Figure 3: Coverage Diagrams at different duct inten-sity profiles at 100MHz

Figure 4: Coverage Diagrams at low duct intensityprofiles at 1GHz

In generating results for terrain modelling, twodifferent scenarios are choosen. In first scenario (fig-ure 6), an irregular terrain is assumed while in secondcase (7), street in an urban area is considered. Three

Figure 5: Coverage Diagrams at low duct intensityprofiles at 3GHz

Figure 6: Coverage Diagrams at 100MHz on an irreg-ular terrain

Figure 7: Path Loss in an urban street at 100MHz

buildings of arbitrarily heights are placed on the streetand span of street is assumed to be 50m. Figure 6shows coverage diagram in the presence of irregularterrain generated by usingrand function in matlab.


Figure 8: Received Power in an urban street at100MHz

Figure 7 shows paath loss contour in an urban street.For urban environment radiating antenna is assumedto radiate at25m while receiving antenna is at10m.Three buildings of different heights and widths are as-sumed along the street. For the sake of simplicity ithas been assumed that the building surfaces are per-fectly conducting.

Figure 8 shows the received power at receiverheight in dBm. Diffraction of waves along the edgesof building is quite clear in figure 7. Buildings of vari-able heights are choosen to show the effect of shad-owing as seen in figure 7. Behind the large build-ing strength of signal is very low and obviously insidebuildings received power is very low (ideally zero) asshown in figure 8. These results clearly shows thatthe proposed method works well for propagation mod-elling over irregular terrain as well as for an urban en-vironment.

5 Conclusion

In this paper, the wide angle form of parabolic equa-tion for radio wave propagation modelling under dif-ferent profiles was introduced using finite elementapproach and simulation examples were presented.Finite element method formulation can easily pro-cess complex refractivity profiles of any kind and canhandle complex geometries without much computa-tional burden. Moreover, the refractive index and ter-rain variations being independent between consecu-tive range steps, giving the ability to include inho-mogeneous tropospheric profiles and different terrainvariations.

References:

[1] M. P. M. Hall, L. W. Barclay and M. T. He-witt, Propagation of radiowaves,The Institutionof Electrical Engineers, 1996.

[2] M. Levy, Parabolic Equation Methods for Elec-tromagnetic Wave Propagation,IEE Publisher,2005.

[3] H. V. Hitney, Refractive effects from VHF toEHF, Part A: Propagation mechanism,AGARD-LS-196, 4A-1-4A-13, 1994.

[4] P. L. Slingsby, Modeling tropospheric ductingeffects on VHF/UHF propagation,IEEE Trans-actions on Broadcasting, 37-2, 1991, pp. 25-34.

[5] G. B. Baumgartner, H. V. Hitney and R. A. Pap-pert, Duct propagation modeling for theintegrated-refractive-effects prediction program(IREPS),IEE Proc., Pt. F.Vol. 130, No. 7, 1983,pp. 630-642.

[6] C. H. Shellman, A new version of MODESRCHusing interpolated values of the magnetoionicreflection coefficients,Interim technical reportNOSC/TR-1143, Naval Ocean Systems Center,1986, ADA 179094.

[7] J. Deygout, Correction factor for multiple knife-edge diffraction,IEEE Trans. Antennas Propa-gation39, 1991, pp. 1256-1258.

[8] J. T. Hviid and J. Bach Anderson and J. Toftgardand J. Bøger, Terrain-based propagation modelfor rural area - an integral equation approach,IEEE Trans. Anteena Propagat., Vol. 43, 1995,pp. 41-46.

[9] G. D. Dockery, Modeling electromagneticwave propagation in the troposphere usingthe parabolic equation,IEEE Trans. AntennasPropag., Vol. 36, 1988, pp. 1464-1470.

[10] K. H. Craig, Propagation modelling in the tro-posphere: Parabolic equation method,IEE Elec-tronics Letters, vol. 24, 1988, pp. 1136-1139.

[11] K. H. Craig and M. F. Levy, Parabolic equationmodelling of the effects of multipath and ductingon radar systems,IEEE Proc. F, Vol. 138, No. 2,1991, pp. 153-162.

[12] A. Holstad and I. Lie, Radar Path Loss Computa-tions over Irregular Terrain,Storm Weather Cen-ter, 2005.

[13] K. Arshad and F. A. Katsriku and A. Lasebae,An investigation of Tropospheric Radio WavePropagation using Finite Elements,WSEASTransactions on Communications, 4-11, 2005,pp. 1186-1192.

[14] M. D. Collings, A split-step Pade solution for theparabolic equation method,J. Acoust. Soc. Am.,1993, pp. 1736-1742.


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