Eu opean Cong ess on Compu a ional Me hods in Applied Sciences and Enginee ing
ECCOMAS 2000
Ba celona, 11-14 Sep embe 2000
 ECCOMAS
1
METHOD OF MOMENTS APPLIED TO THE ANALYSIS OF ROUGH
SURFACES MODELLED BY FRACTALS
Me cè Vall-llosse a, Nu ia Du o, Ad iano Camps, Ignasi Co bella, Ja ie Ba á, F ancisco
To es, Miquel Guillamon
Depa men o Signal Theo y and Communica ion
Campus No d UPC Edi ici D-3
Jo di Gi ona, 1-3, 08034 Ba celona, Spain
E-mail: m[email p o ec ed], el: 34-934017261, Fax: 34-4017232
Key wo ds: Me hod o Momen s, Sca e ing, Rough su aces, F ac als.
Abs ac . The Sca e ing and Emissi i y o ough su aces in ol e solu ions o non-linea
di e en ial equa ions. Di e en app oaches ha e been used in he li e a u e o ob ain
app oxima e solu ions unde some hypo hesis. Fo example Ki chho solu ion is used when
he oughness is gen le on he scale o he wa eleng h.
In his pape he Me hod o Momen s is used o analyze he sca e ing o a bi a y su aces.
No app oxima ion abou he scale oughness is necessa y. Bo h Gaussian and F ac al
su aces ha e been modeled and compa ed. The in oduc ion o ac al geome y p o ides a
new ool o desc ibe na u al ough su aces. A i s inside o he p ope ies and pa ame e s
ha desc ibe ac al geome y has been done in o de o cha ac e ize hem s a is ically. I has
been demons a ed ha geome ical and sca e ing cha ac e is ics a e con olled by F ac al
desc ip o s, including ac al dimension1.
As a i s s ep, ou simula ions e e o a ( opological) one-dimensional (1-D) p o ile
embedded in a wo-dimensional (2-D) space. Physically, his co esponds o assume ha bo h
he elec omagne ic ield and he su ace heigh a e cons an along a ixed di ec ion.
Ex ension o he case o a 2-D su ace embedded in a h ee-dimensional (3-D) space is no
concep ually di icul , bu any simula ion un equi es a much longe compu a ional ime.
Fu he mo e, sca e ing esul s ob ained o 1-D p o iles gi e also a good indica ion o
sca e ing dependence on 2-D su ace pa ame e s.
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
2
1 INTRODUCTION
The p oblem o sol ing elec omagne ic wa e sca e ing and emission om andomly
ough su aces is o g ea in e es in emo e sensing and elecommunica ion applica ions. Bo h
analy ical and nume ical me hods ha e been de eloped o i s e alua ion. Among he a ious
heo ies, he mos well known ones a e he Small Pe u ba ion Me hod (SPM), Ki chho
app oxima ion ( angen plane app oxima ion) and Geome ic Op ics (GO). The alidi y o
hese analy ical me hods depends on he app oxima ion made pe aining o he physical
condi ions such as he su ace oughness and equency. Fo su aces wi h oughness scale
small compa ed wi h he wa eleng h, he me hod o Ki chho is used, while he adi ional
pe u ba ion me hod applies when ms heigh and ms slope o he ough su ace a e small
ela i e o he wa eleng h. In ei he case, addi ional assump ions mus be done o ob ain
ma hema ically ac able solu ions. Recen ly, new app oaches o he sca e ing p oblem ha e
eme ged in o de o ex end he applicabili y o hese models in o he in e media e equency
ange. The In eg al Equa ion Me hod (IEM) seeks a co ec ion e m o he su ace cu en in
addi ion o he Ki chho app oxima ion and he Phase Pe u ba ion Me hod pe u bs only he
ield ampli ude.
The e i ica ion o heo ies equi es expe imen al measu emen s o su ace sca e ing
coe icien s unde con olled labo a o y. Nowadays, compu e simula ion o e s an e icien
al e na i e. Axline and Fung2 simula ed he wa e sca e ing om a pe ec ly conduc ing
andom su ace by calcula ing he su ace cu en densi y induced by a impinging plane wa e
by he Me hod o Momen s (MoM). Chen and Bay3 ex ended his simula ion echnique o
include backsca e ing om dielec ic su aces.
On he o he hand, he cha ac e iza ion o wa e in e ac ion wi h ough su aces, such as sea
su aces, ocean bo oms and ough e ain needs o ma hema ical models o such ough
su aces. The in oduc ion o ac al geome y p o ides a new ool o desc ibe na u ally
occu ing ough s uc u es, since ac als hold in balance long- ange o de and sho - ange
diso de and can be used o desc ibe bo h de e minis ic and andom s uc u es o an
app op ia e blend. In his pape we use ac al unc ion o model ough su aces. Fi s o all,
we p esen a sui able ac al model p oposed by Jagga d and Sun1. Rela ions be ween i s
de ining pa ame e s o ac al desc ip o s and hose o adi ional andom model a e enclosed.
Sec ion 3 is de o ed o he analysis o andom su aces using MoM. Sec ion 4 demons a es
ha HH and VV sca e ing coe icien s ob ained om ac al su aces a e e y simila o hose
om gaussian su aces wi h he same s a is ics.
2 FRACTAL MODEL
In his i s s udy he ac al model p oposed by Jagga d and Sun1 is used. They p oposed a
ze o-mean, band-limi ed ac al unc ion, exp essed as a weigh ed sum o pe iodic unc ions:
()
∑
−
=
+−= 1
0
0)(1)(
N
n
n
n
nxbKsinDCx
φσ
(1)
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
3
Whe e, D is he ac al dimension in he unc ion and gi es a measu e o he su ace
oughness, anging om D=1 (smoo h pe iodic cu e) o D=2 ( ough, a ea illing cu e). K0
is he undamen al spa ial wa e numbe ( e
K
λπ
/·2
0=, in which λe is de ined as he
undamen al spa ial wa eleng h), b (>1) is he spa ial equency scaling pa ame e ,
φ
n a e
a bi a y phases and N is he numbe o ones. The ampli ude con ol ac o
()
[]
()
[]
[]
2/1
2
2/1
2
2
)1(1
)2(2
11
112










−−
−
=










−−
−−
=NN D
DD
D
D
C (2)
is chosen so ha he unc ion has a s anda d de ia ion ( ms heigh ) σ, while he alue o b
can be chosen such ha he ac al unc ion is almos pe iodic. Ob iously, he pe iodic
unc ions o inc easing equency in he summa ion o equa ion (1) p oduce he ine
s uc u es. Clea ly, o he pe iodic unc ions could be used in (1) o eplace he sine unc ion i
i was desi ed.
This unc ion has a ini e band o spa ial equency and exhibi s sel -simila i y o e he
co esponding ini e ange o esolu ion.
2.1 Rela ions be ween ac al and adi ional pa ame e s
As we can see om ela ion (1), he s uc u al p o ile o he ough su ace is de e mined by
he pa ame e s σ ( ms heigh ), D ( ac al dimension), b ( equency scaling), K0 ( undamen al
wa e numbe ), and N (numbe o ones). The adi ional pa ame e s used in andom su ace
modeling a e σ ( ms heigh ), Γ (co ela ion leng h), and σs ( ms slope). In o de o compa e
gaussian su aces and ac al ones wi h he same andom pa ame e s i is necessa y o ela e
hese wo se s o pa ame e s. The only common pa ame e is σ ( ms heigh ). The ms slope o
his kind o ac al su aces can be ound by de i ing he ms alue o he i s de i a i e o
unc ion (1), which esul s in:
()
[]
()
[]
[]
[]
2/1
22
22
2
2
0)1(1
)1(1
11
11










−−
−−
−−
−−
=
Db
Db
D
D
K
NN
N
s
σσ
(3)
No e he special case
σ
s=K0 ·
σ
when ei he D=1 o N=1. Figu e 1 g aphically shows he
dependence o
σ
s wi h he ac al pa ame e s (D, N, b and
λ
e). I is demons a ed ha
inc easing D inc eases always he ms slope, and his inc emen becomes much as e o D
g ea e han 1.4. On he o he hand, plo s in igu e 1 make e iden ha highe alues o b,
σ
,
and N ob ain highe alues o ms slope; meanwhile con a y beha io is obse ed a ying
λ
e
(see igu e 1c)).
The co ela ion leng h o his model can be ound wi h he aid o he au oco ela ion
coe icien
ρ
(
τ
) o he ac al unc ion gi en by:
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
4
()
[
]
()
[]
()
∑
−
=
−
−−
−−
=
+
=1
0
0
2
2
2
)cos(1
11
11
)(),(
)(),(
)(
N
n
n
n
N
bKD
D
D
x x
x x
τ
τ
τρ
(4)
The angle b acke s deno e he ensemble a e age. The co ela ion leng h
Γ
is de ined as he
i s oo o
ρ
(
τ
)=1/e when
τ
inc eases om ze o. Figu e 2 shows he a ia ion o he ms
slope ( ig. 2 a)) and he co ela ion leng h ( ig. 2b)) espec o he ac al dimension. These
plo s ha e been ob ained o =5GHz,
λ
e =2·
λ
0=12 cm, N=6 and
σ
=0.35. I is obse ed ha
he equi alen heigh de ia ion o he ac al su ace (
σ
=0.35) equals o he gaussian one o
D=1. Fu he mo e, igu e 2a) shows a maximum o D=1.4 ( he same alue whe e he
beha io o he ms espec o D changes). Finally, igu e 2b) demons a es ha he equi alen
co ela ion leng h
Γ
dec eases as D inc eases, which con i ms ha he ac al dimension is a
measu e o he oughness. Bu no clea dependence is obse ed wi h b.
σ s = (N, b, σ, λe)
N
= 6
σ = 0.15 cm
λe = 3 λ0
λ0 = 3 cm
σ s = (N, b, σ, λe)
N = 6
b= 2e/3
λe = 3 λ0
λ0 = 3 cm
σ = 0.10
σ = 0.15
σ = 0.20
σ = 0.30
σ s = (N, b, σ, λe)
N = 6
b= 2e/3
σ = 0.20
λ0 = 3 cm
λe = 8 λ0
λe = 5 λ0
λe = 3 λ0
λe = λ0
σ s = (N, b, σ, λe)
b= 2e/3
σ = 0.20
λe = 4 λ0
λ0 = 3 cm
a)
c) d)
b)
Figu e 1: ms slope espec o ac al dimension a ying o he ac al pa ame e s: a) Di e en alues b,
b) Di e en alues o
σ
, c) Di e en alues o
λ
e , and d) Di e en alues o N .
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
5
Figu e 2: a)
σ
espec o he ac al dimension (D), b)
Γ
espec o ac al dimension (D).
3 METHOD OF MOMENTS (MOM)
Figu e 3: The geome y o ough su ace sca e ing in wo-dimensional space.
The p oblem o sol ing elec omagne ic wa e sca e ing om andomly ough su aces
p esen ed in igu e 3 has been sol ed by he Me hod o Momen s. A e sion o he algo i hm
p esen ed by Chen and Bay3 o compu ing he backsca e ing and bis a ic sca e ing om
ough dielec ic su aces has been implemen ed. As he algo i hm o mula ion is de ailed in
he Chen and Bay3 pape we a e no including i .
I has al eady been men ioned in he in oduc ion ha his s udy e e s o a ( opologically)
one-dimensional p o ile embedded in a wo dimensional space. Physically, his co esponds o
assume ha bo h he elec omagne ic ield and he su ace heigh a e cons an along a ixed
di ec ion. The eason o sol ing his p oblem ins ead o a 2-D su ace embedded in a h ee-
dimensional space is ha any simula ion un equi es a much longe compu a ional ime. To
pe o m compu e simula ion a andomly ough cu e wi h p esc ibed su ace heigh densi y
dis ibu ion and au oco ela ion unc ion has o be gene a ed, as i is shown in igu e 3.
Gaussian cu es ha e been gene a ed using he me hod p oposed by Fung and Chen4,
a) b)
x
zEi
Hi
θ
Ei
Hi
θ
HH VV
L L
∆x

M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
6
meanwhile he ac al ones ha e been gene a ed as i is desc ibed in his pape a sec ion 2.
In p ac ice, a Gaussian ape unc ion o he o m exp(-g-2x2cos2
θ
) is applied o he inciden
ield o imp o e he con inui y a he edge o he illumina ed a ea. Owing o ini e compu e
s o age and p ac ical es ic ions on he ma ix size, he illumina ed leng h (L in igu e 3),
mus be ini e. Repea ed calcula ions o Ns segmen s (each o leng h L) mus be pe o med o
achie e meaning ul es ima es o he sca e ing coe icien s. Axline and Fung2 make a
discussion a ound he app op ia e alues o L, Ns and g pa ame e s.
4 SIMULATIONS RESULTS
The i s s ep was o assess ha MoM ag ees wi h Ki chho solu ion unde he hypo hesis
o alidi y o his las analy ical me hod. I was c ea ed a gaussian co ela ed su ace wi h
oughness pa ame e s K·
σ
=1.256, K·l= 8.38,
σ
s=0.21 and dielec ic cons an
ε
=80. A
segmen o ha su ace is p esen ed in igu e 4a). Besides igu e 4b) compa es plo s o he
HH and VV backsca e ing coe icien s ob ained using MoM wi h esul s ob ained using
Ki chho app oxima ion. In o de o assu e enough accu acy in using MoM 75 segmen s
whe e c ea ed and analyzed. E e yone had a size wel e imes he co ela ion leng h.
A e ha , simula ions compa ing ac al and gaussian su aces whe e ca ied ou . Random
ac al su aces using he o mula ion p esen ed in sec ion 2 ha e been ob ained and analyzed
and he sca e ing coe icien s ha e been compa ed wi h he ones ob ained om gaussian
su aces wi h he same s a is ical pa ame e s. These simula ions demons a ed ha adi ional
pa ame e s o andom su aces could be ob ained om ac al pa ame e s in he way
desc ibed in sec ion 2. Figu e 5a) plo s wo su ace samples one gaussian ( ed dashed line)
and ano he ac al (blue con inuous line). Bo h wi h he same adi ional pa ame e s:
a) b)
Backsca e ing coe icien (dB)
Figu a 4: Figu e 4: a) Gaussian p o ile wi h oughness pa ame e s K·
σ
=1.256, K·l= 8.38,
σ
s=0.21 and
dielec ic cons an
ε
=80. b) Compa ison o he backsca e ing coe icien s be ween MoM simula ion and
Ki chho solu ion.
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
7
kσ=0.29, kl=3.64 and σs=0.11. The ac al p o ile has been c ea ed applying equa ion 1 wi h
he ollowing pa ame e s: N=6, b=e/1.8, D=1.6, λe=4λ0. The oughness o hose su aces is
e y simila . In he simula ion i has been used λ0=3cm and ε =16. HH and VV Backsca e ing
coe icien o hese wo p o iles a e compa ed in igu e 5. Good ag eemen is ob ained. In
o de o achie e meaning ul es ima es o he sca e ing coe icien s 85 segmen s ha e been
analyzed using MoM in bo h kind o su aces.
Figu a 5:a) F ac al p o ile (blue con inuous line) wi h ac al pa ame e s: N=6, b=e/1.8, D=1.6, λe=4λ0,
σ=0.15cm, σs=0.12 compa ed wi h gaussian p o ile whose s a is ical pa ame e s a e kσ=0.29, kl=3.64 and
σs=0.11. I has been used: λ0=3cm and ε =16. in bo h cases b) Compa ison be ween HH and VV
backsca e ing coe icien s o a ac al su ace and a gaussian one wi h he same adi ional pa ame e
coe icien s. A sample o hese su aces is shown in 5a)
a) b)
Backsca e ing coe icien (dB)
Figu a 6: Compa ison be ween HH and VV backsca eing coe icien s o a gaussian su ace when
scala Ki chho app oxima ion (VV blue line wi h poin s and HH dashed ed line wi h c osses) is
applied wi h MoM solu ions (con inuous ed and g een lines)
Ki chho
MoM HH
VV
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
8
Figu e 6 compa es HH and VV backsca e ing coe icien s ob ained by scala ki chho
app oxima ion wi h he esul s ob ained by MoM o he same gaussian p o ile. One sample o
his p o ile has been shown in igu e 4. I can be seen ha hey ag ee e y well o inciden
angles closed o nadi di ec ion. Bu he cu es inc ease sepa a ion wi h he angle o
incidence. This disag eemen appea s because we a e no in he alid ange o alues o he
p oduc k·l o applying ki chho app oxima ion, because Scala Ki chho app oach gi es
good esul s o : k·l>6, k·
σ
<2 and
σ
s<0.25.
5 CONCLUSIONS
This pape demons a es ha ac al su aces a e a good ool o designing andom ough
su aces. He e we ha e p esen ed a e y easy echnique o gene a ing band limi ed ac al
s uc u es. We ha e s udied he a ia ion o he oughness wi h he ac al pa ame e s. On he
o he hand ela ions be ween ac al pa ame e s and adi ional s a is ical pa ame e s ha e
been p esen ed. Fu he wo k is going o be held s udying o he ac al s uc u es ha can
model mo e ealis ic su aces.
On he o he hand, i is demons a ed ha MoM is a sui able echnique o ob aining he
sca e ing coe icien s o andom su aces. I s mo e impo an ad an age, in on o analy ical
echniques, is ha no hypo hesis is conside ed. Consequen ly, sampling adequa ely he
andom su ace, i can be applied o any kind o su ace, wi h no es ic ion o oughness and
a any ange o equency.
The big d awback o MoM is ha he compu a ion ime inc eases wi h he size compa ed
o he wa eleng h and wi h he numbe o segmen s o be analyzed. Then i exis s a
comp omise be ween accu acy and ime consuming.
The nex s ep in he use o MoM is o apply a e y op imized algo i hm in a 2-D su ace
design embedded in a 3-D p oblem.
ACKNOWLEDGEMENTS
This wo k has been suppo ed by he Spanish Comision In e minis e ial de Ciencia y
Tecnología (CICYT TIC 99-1050-C03-01).
REFERENCES
[1] D.L. Jagga d and X. Sun, “Sca e ing om ac ally co uga ed su aces”, Jou . Op . Soc.
Am., Vol. 7, No. 6 (June 1990).
[2] R.M. Axline and M.F. Fung, “Nume ical compu a ion o sca e ing om a pe ec ly
conduc ing andom su ace”, IEEE T ans. An ennas P opaga ion, Vol. AP-26, pp.482-
488 (1978).
[3] M.F. Chen and S.Y. Bai, “Compu e Simula ion o Wa e Sca e ing om a Dielec ic
Random Su ace in Two Dimensions-Cylind ical Case”, Jou . O Elec omagne ic Wa e
M. Vall-llosse a, N. Du o, A. Camps, I. Co bella, J. Ba á, F. To es, M. Guillamon
9
and Applica ions, Vol. 4, No. 10, 963-982 (1990).
[4] A.K. Fung and M.F. Chen, “Nume ical simula ion o sca e ing om simple and
composi e andom su aces”, Jou . Op . Soc. Am., A/Vol. 2, No. 12 (Decembe 1985).