An exact spectral representation of the wave equation for propagation over a terrain

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HAL Id: hal-01022309

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Submitted on 22 Sep 2014

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An exact spectral representation of the wave equation for propagation over a terrain

Alexandre Chabory, Christophe Morlaas, Rémi Douvenot, Bernard Souny

To cite this version:

Alexandre Chabory, Christophe Morlaas, Rémi Douvenot, Bernard Souny. An exact spectral representation of the wave equation for propagation over a terrain. ICEAA 2012, International Conference on Electromagnetics in Advanced Applications, Sep 2012, Cape Town, South Africa. pp 717-720,

�10.1109/ICEAA.2012.6328722�. �hal-01022309�

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Equation for Propagation Over a Terrain

A. Chabory,C. Morlaas, R.Douvenot, and B. Souny

∗

Abstrat Anexatspetralrepresentationofthe

waveequationaboveadieletrigroundisproposed.

The formulation is based on the diagonalisation

of the vertial operator, takes into aount the

angle-dependane of the reexion oeient, and

does not inludeany paraxialapproximation. The

expressions of the spetrum omprise two parts:

a ontinuous part and a disrete part. The latter

orresponds to a possible surfae wave. The use

of this result in split-step algorithms to simulate

wave propagation requires a disretization of the

spetrum. To render the disretization onsistent,

an alternative disrete spetral representation is

proposed that intrinsialy inludes the trunation

oftheomputationdomainatanitehigh.

1 INTRODUCTION

To model the propagation of eletromagneti

waves over the ground at large distanes, one

an rely on split-step methods based on the

paraboliapproximationof the waveequation [1℄.

The omputation is performed step by step at

inreasing distanes, going bak and forth from a

spatial to a spetral representation of the wave.

Suh methods angenerallytakeinto aountthe

terrain prole, a possible ground wave, and the

eletrialharateristisoftheatmosphere.

To model a ground haraterized by a onstant

surfae impedane, the ontinuous mixed Fourier

transform has been proposed [2℄. This trans-

form mathes the spetral representation to an

impedane boundary ondition via a hange

of variable. A disretized ounterpart of this

transform, the disrete mixed Fourier transform

(DMFT),hasbeendevelopedtorenderthesheme

selfonsistentandavoidnumerialinstabilities.

In many appliations, e.g. for rough surfaes, a

onstantimpedanemaynotbesuienttomodel

theboundaryonditionatthegroundlevelbeause

the reexion oeient depends on the angle of

inidene. Todealwiththisdiulty, Kuttlerand

Dokery have proposed to keep the impedane

onstant at a given range, but to extrat the

impedane value from the dominant propagation

diretion [2℄. Janaswamy has proposed a more

rigourous solution to model the propagation over

∗

ENAC/Teleom Lab/EMA, 7 av E. Belin, 31055

Toulouse,Frane,e-mail:haboryreherhe.ena.f r,tel.:

+33562174325,fax:+33562174270.

a nononstant immittane plane [3℄, i.e. with an

angle-dependantreexionoeient.

Inthisartile,wetheoretiallydevelopanexat

spetral representation of the vertial operator

taking into aount the angle dependane of

the reexion oeient. We diretly start from

Maxwell equations, i.e. without the paraboli

approximation. The demonstration is diretly

founded on the diagonalisation of the vertial

operator[4℄. Besides, inorderto avoidthenumer-

ialdiulties that may arise when disretizinga

ontinuous spetral representation, we introdue

an alternative formulation for a domain of nite

high. Thus,thisformulationinludestheneessary

trunation of the omputation domain at a nite

altitude. We show that this leads to a disrete

spetrum.

This artile is organized as follows. In Setion

2we present theonguration. The spetral rep-

resentationis demonstratedin Setion 3. Next,in

Setion4wemodifythisrepresentationtoaount

for a domain of nite high before presenting the

appliationofthemethodin Setion5.

2 CONFIGURATION

We onsider a time-harmoni onguration where

the elds are transverse magneti (TM) with re-

spet to the vertial axis

z

^. ^Note ^that ^the ^trans-

verseeletriaseanbeanalyzedinasimilarway.

Let

(r, φ, z )

^be^the^loal^ylindrial^oordinate^sys-

tem with unit vetors

(ˆ r, φ, ˆ ˆ z)

^. ^We ^assume ^a ^ro-

tationnal symmetry about the vertial axis. The

ground/atmosphereareharaterizedbyaonstant

permeability

µ 0

^and^by spae-varyingpermittivity

ε r (r, z)

^and ^ondutivity

σ(r, z)

^The ^problem ^an

beformulatedfromavetorpotential

Π

^e ^suh^that

E = Π

^e

− ∇ 1

k ² ∇ · Π

^e

, H = 1

−jωµ 0

∇ × Π e .

(1)

where

k = −jωµ 0 (σ + jωε)

^is^thewavenumber.

(3)

3 HOMOGENEOUS ATMOSPHERE

ABOVE A DIELECTRIC GROUND

3.1 Formulation

Foranhomogeneousatmosphereandground,upon

replaing

Π

^e ^by

ψˆ z

^,^the^waveêquationân^beâst

inylindrialoordinatesas

− 1 r

∂

∂r

r ∂

∂r Ψ

− ∂ ²

∂z ² Ψ − k ² (z)Ψ = 0,

⁽²⁾

with

r ∈ ]0, ∞ [

^,

z ∈

^R. ^F^urthermore

k(z) =

( k a z > 0, k g z < 0,

(3)

with

k a

^and

k g

^thewavenumbersintheatmosphere andground,respetively. Atinnity,

Ψ

^is^subjeted

to radiating boundary onditions. At the ground

level

z = 0

^,^the^boundaryônditionsîsîmposed^by

theground/atmosphereinterfae.

3.2 Priniple of the determination of the

vertial spetral representation

Thevariables

r

^and

z

^are^separate ⁱⁿ ^(2). ^Conse-

quently,weintrodueanoperator

L z

^ating^on^the

vertialoordinate

z

^suh^that

L z Ψ = −

∂ ²

∂z ² + k ² (z)

Ψ.

⁽⁴⁾

This operator an be assoiated with a Sturm-

Liouville problem of the third kind. The spetral

representationisintroduedviathediagonalisation

ofthisoperator.

Todoso,weemploythemethoddevelopedin[4℄,

thatanbedividedin3steps:

•

^We^determine^the ^Green's^funtion

G(z, z ^′ , λ)

of the operator

L z − λI

^, ^with

λ ∈

^C,

I

^the

identity operator, and

(z ^′ , z)

^the ^position ^of

thesoureand observation,respetively.

•

^We^write^the^following^identity,demonstrated in[4℄,

1 2jπ lim

R→∞

I

C R

G(z, z ^′ , λ)dλ = − δ(z − z ^′ ),

⁽⁵⁾

where

C R

îs ^the îrle êntered ât

0

^of ^radii

R

ⁱⁿ ^the^omplex

λ

^-plane. ^Then, ^we^evaluate

expliitlytheintegralin (5)takingareofthe

ontributionsofbranhutsandpoles.

•

^Finally,^we^write

Ψ(r, z) =

Z ∞

0 δ(z − z ^′ )Ψ(r, z ^′ )dz ^′ .

⁽⁶⁾

Using (5), wean substitute

δ(z − z ^′ )

^by ^the

expliit ontributions of the branh uts and

poles. The expression that we obtain is the

spretralrepresentationoftheoperator.

3.3 Green'sfuntion

TheGreen'sfuntionof

L z − λI

^is^the^solution^of

− ∂

∂z ² G(z, z ^′ , λ)−(k ² + λ)G(z, z ^′ , λ) = δ(z−z ^′ ),

⁽⁷⁾

for

(z, z ^′ ) ∈

^R

²

^. ^Besides, ^at

z → + ∞

^,

G

^is ^sub-

jetedtoradiationboundaryonditions. At

z = 0

^,

areexion oeient is introdued to restrit the

omputation to thedomain

z ≥ 0

^and

z ^′ ≥ 0

^, ^be-

ause in this artile we are only interested by the

eld in theatmosphere. Using alassialmethod

[4℄ for the determination of Green's funtions, we

obtain

G(z, z ^′ , λ) = e ^−jk ^za ^|z−z ^′ ^| + Γe ^−jk ^za ^(z+z ^′ ⁾ 2jk za

,

⁽⁸⁾

for

z ≥ 0

^and

z ^′ ≥ 0

^, ^with

k za = p k a ² + λ

^.

The suitable determination of the square root is

the one that respets the radiation ondition, i.e.

Im

(k za ) ≤ 0

^. ^The^reexion ^oeient^is^given^by

Γ = Z a − Z g

Z a + Z g

,

⁽⁹⁾

with

Z a = jk za

σ a + jωε a

, Z g = jk zg

σ g + jωε g

.

⁽¹⁰⁾

Notethattheonditionat

z = 0

^orrespondsⁱⁿ^the

spetraldomaintothefollowingimpedanebound-

aryondition

∂

∂z G(z, z ^′ , λ) − jk za

1 − Γ

1 + Γ G(z, z ^′ , λ) = 0,

⁽¹¹⁾

insidewhihtheimpedanedependsonthespetral

variable

k za

^. ^This ôndition ân ^be ôbtained ^by

solving the equation in the omplete domain

z ∈

R. Thus,it isanexatformulationfor theground

interfae.

3.4 Integrationin theomplex

λ

^-plane

Wenowhavetoevaluatetheintegralin(5). Inthe

omplex

λ

^-plane,^the^square^rootⁱⁿ

k za

^introdues

a branh ut. As indiated in Fig. 1, we dene

a new ontour

C a

^to ^add ^the ontribution of the branhuttotheintegral.

Wenotethatonthelowerpartof

C a

^,

k za

^is^real

and positive, while on the upper part

k za

^is ^real

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Figure 1: Contours of integration in the omplex

λ

^-plane.

andnegative. Ahangeofvariable

λ ↔ k za

ⁱⁿ^the

integralof

G(z, z ^′ , λ)

^over^both^parts ^of

C a

^yields

Z

C a

G(z, z ^′ , λ)dλ = Z +∞

0 e ^jk ^za ^z + Γe ^−jk ^za ^z .

e ^jk ^za ^z ^′ + Γe ^−jk ^za ^z ^′ j Γ dk za .

(12)

Toevaluatetheintegralin(5),wealsohavetoon-

siderthepossiblepreseneof poles inside theon-

tour

C _R

^. ^Note ^that

G

^is ^regular^when

λ → k a ²

^de-

spitethefator

1/k za

ⁱⁿ ^(8). ^Thus, ^the^only^poles

wehaveto onsiderin

G

^are^the^poles^of

Γ

^. ^F^rom

(9),if

Γ

^hasâ^pole,îtsêxpressionîs^given^by

λ ^p = k _a ²

1 + ǫ ,

⁽¹³⁾

where

ǫ = σ a + jωǫ a

σ b + jωǫ b

.

⁽¹⁴⁾

Theresidueofthis pole orrespondsto

Res

(G, λ ^p ) = −2jk ^p _za

1 − ǫ ² e ^−jk ^p ^za ^(z+z ^′ ⁾

⁽¹⁵⁾

with

k ^p _za = r

k ² _a ǫ

1 + ǫ .

⁽¹⁶⁾

Finally,iftheontributionofthebranhut(12)

and the pole (15)are taken into aount, the ex-

pression(5) expliitlyresultsin

δ(z−z ^′ ) = 2jk ^p _za

1 − ǫ ² e ^−jk ^za ^p ^(z+z ^′ ⁾ + 1

2π Z +∞

0 1 Γ

e ^jk ^za ^z ^′ + Γe ^−jk ^za ^z ^′ . e ^jk ^za ^z + Γe ^−jk ^za ^z

dk za .

(17)

Substituting (17)in(6), andinterhangingtheor-

derofintegration,weendupwiththespetralrep-

resentation

Ψ(r, z) = ˜ Ψ ^p (r)e ^−jk ^za ^p ^z + Z +∞

0 Ψ(r, k ˜ z ) e ^jk ^z ^z + Γ(k z )e ^−jk ^z ^z dk z

(18)

with

Ψ ˜ ^p (r) = 2jk ^p _za 1 − ǫ ²

Z +∞

0 Ψ(r, z)e ^−jk ^p ^za ^z dz,

Ψ(r, k ˜ za ) = 1 2π

Z +∞

0 Ψ(r, z) Γ(k za )

. e ^jk ^za ^z + Γ(k za )e ^−jk ^za ^z dz.

(19)

Therst termof (18)orrepondsto the ontribu-

tionofthepole,and anbeassoiatedwithapos-

sibleground/surfaewave. Theseondtermisthe

ontinuous spetrum that represents plane waves

andtheirreexionovertheground.

The spetral representation has been obtained

hereforTM elds and foragroundmodeledbya

dieletriinterfae. Note thattheformulationan

beextended to otheraseswhere thereexion o-

eient

Γ(k za )

ândîtsâssoiated^polesâre^known.

Thismayinlude,forexample,groundsharater-

izedbyaroughsurfaeorbyamultilayerdieletri

slab.

4 HOMOGENEOUS ATMOSPHERE OF

FINITE HIGH

In the previous spetral representation, the on-

tinuous spetrum omes from the fat that the

domainisnotbounded, i.e.

z ∈ [0, + ∞ [

^. ^However,

for obvious numerial reasons, the omputation

domain must be bounded. We introdue in this

Setion a spetral representation for a domain of

nite high. We showthat this leadsto a disrete

spetrum.

Themethodissimilartothepreviousoneexept

that there now exists a boundary ondition at a

nitehigh

z = h

^given^by

∂

∂z G(z, z ^′ , λ) − jk za

1 − Γ

1 + Γ G(z, z ^′ , λ) = 0.

⁽²⁰⁾

TheevaluationoftheGreen'sfuntion yields

G(z, z ^′ ,λ) = −1

4k za Γ(k za ) sin(k za h) .

e ^jk ^za ^(z ^′ ^−h) + Γ(k za )e ^−jk ^za ^(z ^′ ^−h) . e ^jk ^za ^z + Γ(k za )e ^−jk ^za ^z

(21)

(5)

for

0 < z < z ^′

^,^and

G(z, z ^′ ,λ) = − 1

4k za Γ(k za ) sin(k za h) .

e ^jk ^za ^z ^′ + Γ(k za )e ^−jk ^za ^z ^′ .

e ^jk ^za ^(z−h) + Γ(k za )e ^−jk ^za ^(z−h)

(22)

for

z ^′ < z < h

^. ^Tâking âre ôf ^the ontributions of thepolesand branh uts, theintegration of

G

over

C R

ⁱⁿ ⁽⁵⁾ ^leads ^to ^a ^disrete ^spetrum. ^Its

expressionisgivenby

Ψ(r, z) = ˜ Ψ ^p (r)e ^−jk ^p ^za ^z +

+∞

X

n=1

Ψ ˜ n (r) e ^j ^nπ ^h ^z + Γ n e ^−j ^nπ ^h ^z

⁽²³⁾

with

Γ n = Γ(nπ/h)

^. ^The^spetral^omponents^an

beexpressedas

Ψ ˜ ^p (r) = 2jk _za ^p 1 −e ^−2jk ^za ^p ^h

1 1 − ǫ ²

Z h

0 Ψ(r, z)e ^−jk ^p ^za ^z dz Ψ ˜ n (r) =

Z h

0 Ψ(r, z) 2hΓ n

e ^j ^nπz ^h + Γ n e ^−j ^nπz ^h dz.

(24)

5 APPLICATION TO PROPAGATION

SIMULATIONS

5.1 Homogeneous ground/atmosphere

Thepreviousspetralrepresentationanbeusedto

model the propagation in an homogeneous atmo-

sphereaboveanhomogeneousground. Weassume

that

Ψ(r 0 , z)

^is^known. ^To^obtain

Ψ

^at

r > r 0

^, ^we

determine bothomponents of thespetral repre-

sentation

Ψ ˜ ^p (r 0 )

^and

Ψ(r ˜ 0 , k za )

^using ^the ^expres-

sionsin(24). Then,for

r > r 0

^,

Ψ ˜ ^p (r)

^and

Ψ(r, k ˜ za )

aresolutionsof

1 r

∂

∂r

r ∂

∂r Ψ ˜

+ (k a ² − k ² za ) ˜ Ψ = 0

⁽²⁵⁾

If theeld is propagating towards

r > r 0

^, ^we ^an

solvethisequation. Weendupwith

Ψ(r, k ˜ z ) = ˜ Ψ(r 0 , z) H ₀ ⁽²⁾ (k r r)

H ₀ ⁽²⁾ (k r r 0 ) ,

⁽²⁶⁾

with

H ₀ ⁽²⁾

^the ^Hankel ^funtion ^of ^the ^seond ^kind

and of order

0

^, ^and

k r = p

k ² _a − k ² _za

^where

Im

(k r ) ≤ 0

^. ^Finally, ûsing ^(18), ^we ân ôbtain

Ψ(r, z)

^from

Ψ(r, k ˜ z )

^. ^Note^that ^for

r − r 0

^greater

thanfew wavelength, the Hankelfuntions anbe

simplied,andtheexpressionisreduedto

Ψ(r, k ˜ z ) = ˜ Ψ(r 0 , z)e ^−jk ^r ^(r−r ⁰ ⁾ .

⁽²⁷⁾

5.2 General ase

Tomodel thepropagation,wehavegonebakand

forth from a spatial to a spetral representation.

Thus, thereexists an analogywith lassialmeth-

odsbasedontheparaboliequation(PWE).How-

everherenohypothesisontheparaxialityismade.

Besides the variation of the reexion oeient

with

k za

^(and ^thus ^with ^inidene) ^is intrinsialy modeled. As withPWE,itisbepossibletoderive

amore-generalsplit-stepalgorithm thattakesinto

aount the terrain prole, and variations in the

eletrial harateristis. Details of suh an algo-

rithmarepresentedin [5℄.

6 CONCLUSION

WehaveproposedaspetralrepresentationofTM

elds in an homogeneous atmosphere abovea di-

eletri groundthat takesinto aount the angle-

dependane of the reexion oeient. The spe-

tralrepresentationinludesaontinuousandadis-

retepart. Theformerorrespondsto planewaves

andtheirreexionovertheground. Thelatteror-

responds toapossibleground wave. Thismethod

anbe extended to TE ongurationsand to any

ground for whih the reexion oeient and its

polesareknown.

An alternative disrete formulation has been pro-

posedthat inludes atrunation of thedomain at

nite high. This formulation is suitable for be-

ingemployedinsplit-stepalgorithms,forwhihthe

problemmustbeboundedinspaeanddisretized.

Referenes

[1℄ M.Levy,ParaboliEquationMethodsforEle-

tromagneti Wave Propagation, ser. IEE ele-

tromagnetiwavesseries45,IET,Ed.,2000.

[2℄ D. G. Dokery and J. R. Kuttler, An

improved impedane-boundary algorithm for

Fourier split-step solutions of the paraboli

waveequation,IEEEtrans.AntennasPropag.,

vol.44,no.12, pp.1592-1599,1996.

[3℄ R. Janaswamy, Radio wave propagation over

a nononstant immitttaneplane, RadioSi.,

vol.36,no.3,pp.387-405,2001.

[4℄ D. G. Dudley, Mathematial Foundations for

Eletromagneti Theory, IEEE press, New-

York,1994.

[5℄ R. Douvenot, C. Morlaas, A. Chabory, B.

Souny, Matrix split-step resolution for propa-

gationbasedonanexatspetralformulation,

in ICEAA,2012.

An exact spectral representation of the wave equation for propagation over a terrain

HAL Id: hal-01022309

https://hal-enac.archives-ouvertes.fr/hal-01022309

Submitted on 22 Sep 2014

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.