A mos. Meas. Tech., 18, 4857–4870, 2025
h ps://doi.o g/10.5194/am -18-4857-2025
© Au ho (s) 2025. This wo k is dis ibu ed unde
he C ea i e Commons A ibu ion 4.0 License.
Simula ions o spec al pola ime ic a iables measu ed in
ain a W-band
Ioanna Tsikoudi1,2, Alessand o Ba aglia3,4, Ch is ine Unal5,6, and Eleni Ma inou1
1Ins i u e o As onomy, As ophysics, Space Applica ions and Remo e Sensing, Na ional Obse a o y o A hens, G eece
2Depa men o Physics, Na ional and Kapodis ian Uni e si y o A hens, G eece
3Depa men o En i onmen , Land and In as uc u e Enginee ing, Poli ecnico o To ino, Tu in, I aly
4Depa men o Physics and As onomy, Uni e si y o Leices e , Leices e , UK
5Geoscience and Remo e Sensing, Del Uni e si y o Technology, Del , he Ne he lands
6Clima e Ins i u e, Del Uni e si y o Technology, Del , he Ne he lands
Co espondence: Ioanna Tsikoudi ([email p o ec ed])
Recei ed: 14 Oc obe 2024 – Discussion s a ed: 2 Janua y 2025
Re ised: 16 July 2025 – Accep ed: 17 July 2025 – Published: 29 Sep embe 2025
Abs ac . In his wo k, he T-ma ix app oach is exploi ed
o p oduce simula ions o spec al pola ime ic a iables
(spec al di e en ial e lec i i y, sZDR, spec al di e en ial
sca e ing phase, sδHV, and spec al co ela ion coe icien ,
sρHV) o obse a ions o ain acqui ed om slan -looking
W-band cloud ada . The spec al pola ime ic a iables a e
simula ed wi h wo di e en me hodologies, aking in o ac-
coun ins umen noise and he s ochas ic mo emen o he
aind ops, in oduced by aind op oscilla ions and by u -
bulence. The simula ed esul s a e hen compa ed wi h ain
Dopple spec a obse a ions om W-band ada o mode -
a e ain a e condi ions. Two cases, di e ing in le els o u -
bulence, a e conside ed. While he compa ison o he simula-
ions wi h he measu emen s p esen s a easonable ag eemen
o equi- olume diame e s less han 2.25 mm, la ge disc ep-
ancies a e ound in he ampli ude (bu no he posi ion) o
he maxima and minima o sZDR and, mo e mildly, o sδHV.
This pinpoin s a gene al weakness in app oxima ing aind op
as sphe oids o simula e ada backsca e ing p ope ies a he
W-band.
1 In oduc ion
Cloud ada obse a ions a e c ucial o unde s anding cloud
mic ophysics, as p oposed in he g oundwo k laid by ada
pionee s (A las e al., 1973; Lhe mi e, 1990). In he las
25 yea s, his has been co obo a ed by an abundance o
s udies based on e ically poin ing spec al Dopple cloud
ada obse a ions in mul i- equency con igu a ions and/o
in syne gy wi h lida and adiome e s o be e cha ac e iz-
ing d izzle (e.g., O’Conno e al., 2005; Kollias e al., 2011;
Luke and Kollias, 2013), ain (Kollias e al., 2001, 2002; T i-
don e al., 2013; T idon and Ba aglia, 2015; Cou ie e al.,
2022), ice (Kalesse e al., 2016; Knei el e al., 2016; Li e al.,
2021; Luke e al., 2021), mixed-phase clouds (Luke e al.,
2010), and mel ing pa icles (e.g., Li and Moissee , 2019;
M óz e al., 2021). Pola ime ic a iables p o ide addi ional
cons ain s on hyd ome eo shape and o ien a ion and a e
ou inely measu ed by g ound-based p ecipi a ion ada ne -
wo ks using low-ele a ion scanning s a egies (Chand aseka
e al., 2023, and e e ences he ein). Howe e , e ically
poin ing cloud ada s miss mos o he pola ime ic in o -
ma ion o hyd ome eo s (wi h he sole excep ion o he lin-
ea depola iza ion a io; M óz e al., 2021), since hyd ome e-
o s end o all wi h hei maximum dimensions ho izon ally
aligned. In o de o o e come his limi a ion, mo e ecen ly, a
ew si es s a ed ope a ing cloud ada s wi h Dopple and po-
la ime ic capabili ies in slan obse a ion mode (Myagko
e al., 2020; Unal and an den B ule, 2024; Mak and Unal,
2025). This con igu a ion has he c i ical ad an age ha pa -
icles wi h di e en sizes a e sepa a ed in he spec al domain
(because hey ha e di e en sedimen a ion eloci ies), which
allows he con ibu ions o di e en pa icle ypes o be dis-
en angled. While e ically poin ing ada s can also achie e
his sepa a ion, ada s in slan pola iza ion mode addi ion-
Published by Cope nicus Publica ions on behal o he Eu opean Geosciences Union.
4858 I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables
ally exploi pola ime ic measu emen s. A highe equen-
cies like he W-band, whe e mul iple esonances occu ac oss
he pa icle size dis ibu ion (PSD), he pola ime ic a iables
– esul ing om in eg a ion o e he en i e PSD – end o a -
e age ou he cha ac e is ic ea u es o single-pa icle sca e -
ing, o en balancing posi i e and nega i e con ibu ions (Kol-
lias e al., 2011). This is especially e iden in he simula ions
o di e en ial e lec i i y (ZDR), whe e his pa ame e ex-
hibi s e y low alues and sensi i i y o PSD a ia ions (Unal
and an den B ule, 2024). Fu he , he pola ime ic a iables
e lec bo h sca e ing and p opaga ion e ec s. A way o mi -
iga e hese challenges a millime e wa eleng hs is o analyze
pola ime ic a iables in he spec al domain.
Fo Ka- and W-band obse a ions o ain a a 45° ele a ion
angle, Unal and an den B ule (2024) ha e demons a ed ha
using he Rayleigh pla eau, as p oposed in he li e a u e (T i-
don e al., 2013; Myagko e al., 2020), allows o he sep-
a a ion o p opaga ion and backsca e con ibu ions in he
spec al domain o pola ime ic a iables, speci ically he
di e en ial phase shi and di e en ial e lec i i y. The di -
e en ial phase a backsca e ing can hen be u ilized o in e
he cha ac e is ic d ople diame e o he d op size dis ibu-
ion (DSD). Inciden ally, W-band pola ime ic ada obse -
a ions a slan angles ha e also been p oposed in he ame-
wo k o he ESA spacebo ne WIVERN mission (Illingwo h
e al., 2018; Ba aglia e al., 2022), which aims o measu e
in-cloud winds by using he pola iza ion di e si y echnique
wi h an an enna scanning conically a an incidence angle o
41.6°. Al hough in he WIVERN case no spec al measu e-
men s a e en isaged, his mission will p o ide an unp ece-
den ed abundance o inciden al cloud ada pola ime ic ob-
se a ions globally.
Spec al pola ime ic obse a ions, u ilizing ei he slan
o ho izon al p o iling, e ec i ely dis inguish hyd ome e-
o s om clu e (Bachmann and Z ni´
c, 2007; Moissee and
Chand aseka , 2009; Unal, 2009; Chen e al., 2022) and also
enable he cha ac e iza ion o a ious hyd ome eo s (Spek
e al., 2008; P i zenmaie e al., 2018; Wang e al., 2019;
Lakshmi e al., 2024). In he case o ain, Moissee e al.
(2006) de i ed he shape–size ela ionship, while Yano sky
(2011) explo ed he e ec s o u bulence on spec al ZDR.
These s udies we e conduc ed a cen ime e -wa eleng h e-
quencies.
In o de o build quan i a i e e ie al algo i hms based
on spec al pola ime ic obse a ions, o wa d model simula-
o s o he pola ime ic spec a hemsel es a e needed. Sim-
ula ions o Dopple spec a obse ed by g ound-based e i-
cally poin ing ada ha e been pionee ed by Z ni´
c (1975) and
ha e been applied o di e en hyd ome eo s and o millime-
e ada by di e en au ho s (e.g., Kollias e al., 2011; T idon
and Ba aglia, 2015; Cou ie e al., 2024), including u bu-
lence e ec s and aind op ine ia (Zhu e al., 2023). The sim-
ula ion o pola ime ic spec a (Myagko e al., 2020; Unal
and an den B ule, 2024) has been explo ed only ma ginally
because slan obse a ions a e no so common.
Elec omagne ic sca e ing p ope ies o ain ha e been
his o ically compu ed by assuming sphe oid o Chebyshe
shapes (bo h o a ionally symme ic) ia he T-ma ix me hod
(Mishchenko e al., 2000). Such models ha e been ound sa -
is ac o y o explain ada and adiome ic measu emen s in
he S, C, X, Ku, and Kabands (Ba aglia e al., 2010; Kumjian
e al., 2019; Teng e al., 2018) bu hey ha e also been
used o simula e highe ada equencies (Aydin and Lu e,
1991; Knei el e al., 2020; Unal and an den B ule, 2024).
Howe e , aind ops gene ally change due o oscilla ions,
which cause depa u e om o a ionally symme ic shapes.
The T-ma ix me hod can, in p inciple, simula e sca e ing
om non- o a ionally symme ic pa icles (gi en nume ical
con e gence; W ied , 2002), bu such implemen a ions a e
compu a ionally demanding and no widely a ailable. As a
esul , mos T-ma ix applica ions ely on he assump ion
o o a ionally symme ic pa icles. Di e en s udies ha e
highligh ed he s ong impac o he shape assump ions in
modi ying he pola ime ic a iables (e.g., compa ed sphe e,
sphe oids, and equilib ium/Chebyshe d ops; Ekelund e al.,
2020), pa icula ly when conside ing pa icles in he eso-
nance egions (Thu ai e al., 2007) ( ha occu in he 5.5–
7 mm diame e egion a he C band and a smalle sizes and
in mul iple anges wi h inc eased equency). Such s udies,
howe e , a e based on a s udy o he DSD-in eg a ed pola i-
me ic a iables and he e o e do no ully cap u e he impac
o he shape o each single pa icle. Combining Dopple and
pola ime ic measu emen s, spec al pola ime y has he po-
en ial o es hyd ome eo shape models and hei associa ed
sca e ing p ope ies in g ea de ail.
The e o e, he i s goal o his s udy is o explo e how di -
e en assump ions ha a e ela ed o a mosphe ic condi ions
( u bulence) and whi e noise o a eal ada spec um impac
he simula ed spec al pola ime ic a iables. The second ob-
jec i e is o p esen a no el compa ison be ween simula ed
and obse ed da a.
The pape is s uc u ed as ollows. Fi s we de ail he
me hodology o simula ing he cloud ada spec a and po-
la ime ic a iables (Sec . 2); hen we p esen he esul s o
ou simula ions, desc ibe he obse a ional da ase , compa e
simula ions and obse a ions, and discuss he implica ions o
ou indings.
2 Me hodology o simula ions
2.1 Rain sca e ing p ope ies simula ed by T-ma ix
The simula ions a e gene a ed by using a Py hon package
o compu e he elec omagne ic sca e ing p ope ies o non-
sphe ical pa icles using he T-ma ix me hod (Leinonen,
2014).
In his s udy, he ain sca e ing p ope ies a e exclusi ely
a ge ed. The backsca e ing ampli ude ma ix, S, and he
phase ma ix, Z(Mishchenko e al., 2000, Chap e 16), a e
A mos. Meas. Tech., 18, 4857–4870, 2025 h ps://doi.o g/10.5194/am -18-4857-2025
I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables 4859
calcula ed o d ops o di e en diame e , D, wi h axis a ios
pa ame e ized acco ding o Keenan e al. (2001), Andsage
e al. (1999), and Bea d and Chuang (1987). The ollowing
equa ion is employed o desc ibe he aind op axis a io:
a
b(D) =
1/(0.9939 +0.00736 ·D−0.018485 ·D2
+0.001456 ·D3),
D < 0.89mm
1/(1.0048 +5.7×10−4D−2.628 ×10−2D2
+3.682 ×10−3D3−1.677 ×10−4D4),
D≥0.89mm,
(1)
whe e a/b deno es he a io o he majo o mino axes
o he obla e sphe oid. The use o wo di e en o mula-
ions e lec s he physical di e ences in aind op de o ma-
ion egimes. Fo small aind ops, he axis a io ollows he
pa ame e iza ion by Keenan e al. (2001), while o la ge
d ops, he i o Andsage e al. (1999) o he model o Bea d
and Chuang (1987) is used.
The b own line in Fig. 1 ep esen s he axis a io pa-
ame e iza ion used in his s udy and is plo ed agains he
equi alen ela ionship o Thu ai e al. (2008) (dashed g een
line) and he axis a io o sphe es (do ed pu ple line). The
i s wo lines p esen g ea ag eemen o pa icles wi h
equi- olume diame e s up o 3mm. Ve y small d ople s a e
concei ed as pe ec sphe es (axis a io ≈1). As hei size
inc eases, d ops a e modeled as sphe oid pa icles and an
obla e shape is assumed (axis a io >1). The sca e ing ge-
ome y o he simula ion co esponds o a ada poin ing
a a 45° ele a ion angle. Raind ops a e assumed o be pa -
ially aligned wi h hei maximum dimension p e e en ially
on he ho izon al plane: sca e ing p ope ies a e a e aged
o e Gaussian dis ibu ions o can ing angles wi h di e -
en s anda d de ia ions. The aind ops a e assumed o be a
10 °C; he complex ela i e pe mi i i y o wa e a his em-
pe a u e is 3.2−1.8ıa 94 GHz (Lhe mi e, 1990).
2.1.1 Compu a ion o single-pa icle pola ime ic
a iables
The phase ma ix Zdesc ibes how an elec omagne ic wa e
is sca e ed by a pa icle and how he sca e ing a ec s i s po-
la iza ion s a e (Mishchenko e al., 2000). I is a 4×4 ma ix
ha ans o ms he S okes ec o o an inciden elec omag-
ne ic wa e o he S okes ec o o he sca e ed wa e. F om
he elemen s Zij (D) o his ma ix, he ollowing backsca -
e ing quan i ies can be compu ed.
–Backsca e ing c oss sec ions o V-pola ized and H-
pola ized adia ion:
σVV(D) =2π(Z11 +Z12 +Z21 +Z22)[mm2],
σHH(D) =2π(Z11 −Z12 −Z21 +Z22)[mm2].(2)
Figu e 1. Axis a io (majo o mino axis) pa ame e iza ion as a
unc ion o equi- olume diame e s. The b own line is used in his
s udy and is calcula ed acco ding o Keenan e al. (2001) and And-
sage e al. (1999). The dashed g een line is he pa ame e iza ion o
Thu ai e al. (2008) and he do ed pu ple line is he axis a io o
sphe es.
–Di e en ial e lec i i y:
ZDR(D) =10log10
σHH(D)
σVV(D) [dB].(3)
–Copola co ela ion coe icien :
ρHV(D) =
p(Z33 +Z44)2+(Z43 −Z34)2
√(Z11 −Z12 −Z21 +Z22)(Z11 +Z12 +Z21 +Z22).(4)
–Di e en ial phase:
δHV(D) =a c anZ43 −Z34
Z33 +Z44 [°].(5)
The no malized backsca e ing c oss sec ion o an obla e
sphe oid aind op is shown in Fig. 2 wi h b own colo . The
axis a io o his compu a ion is he same as he b own line
o Fig. 1. The dashed g een line ep esen s he same quan-
i y bu compu ed by using he axis a io pa ame e iza ion
o Thu ai e al. (2008) (g een line in Fig. 1). The same ap-
plies o he pu ple do ed line, which is p oduced by us-
ing he sphe es’ axis a io. The pa ame e iza ions o he
wo di e en sphe oids esul in nea ly iden ical cu es, in-
dica ing ha he choice o axis a io o obla e shapes does
no signi ican ly a ec he backsca e ing c oss sec ion be-
ha io . In con as , he sphe ical pa ame e iza ion shi s he
Mie no ches sligh ly o he le , due o he di e en geome-
y o he sca e e s. The posi ions o he i s , second, and
hi d Mie no ches a e indica ed by he dashed blue lines
a D=1.68 mm, D=2.88 mm, and D=4.13 mm, espec-
i ely.
h ps://doi.o g/10.5194/am -18-4857-2025 A mos. Meas. Tech., 18, 4857–4870, 2025
4860 I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables
Figu e 2. No malized backsca e ing c oss sec ion o obla e
sphe oid model aind ops when poin ing a 45° ele a ion, as a unc-
ion o he sphe e equi- olume diame e D. The dashed ligh blue
lines indica e he i s (D=1.68 mm), second (D=2.88 mm), and
hi d (D=4.13 mm) Mie no ches.
Some T-ma ix esul s o he pola ime ic a iables a e
displayed in Figs. 3 and 4: di e en d op o ien a ion con-
di ions and aind op axis a ios a e conside ed. The dashed
black lines and he blue lines a e calcula ed by assuming pe -
ec ly o ien ed aind ops wi h axis a io pa ame e iza ion, as
p oposed by Thu ai e al. (2008) and acco ding o Eq. (1),
espec i ely. In Fig. 3, hose wo lines a e almos iden ical up
o app oxima ely 3 mm diame e bu hey di e ge a e wa ds.
No ably, o la ge aind ops, he dashed black line aligns
closely wi h he ligh blue line, which ep esen s a wobbling
aind op wi h a 5° can ing angle on a e age. This sugges s
ha he same ampli udes o he maxima and minima in he
spec al pola ime ic a iables can be achie ed by di e en
combina ions o axis a io pa ame e iza ions and a ying de-
g ees o wobbling. The e o e, in he ollowing, he pa am-
e e iza ion o Eq. (1) is used in combina ion wi h di e en
deg ees o wobbling.
The di e en ial phase (δHV) e e s o he phase shi in-
oduced a backsca e ing be ween he ho izon ally and e -
ically pola ized componen s o he ecei ed ada signal.
This pa ame e depends on he size o he hyd ome eo s and
p o ides in o ma ion abou hei shape and o ien a ion. In
Fig. 3b, δHV emains nea 0 o small d op diame e s, con-
sis en wi h Rayleigh sca e ing. As he diame e inc eases,
δHV depa s om 0 and exhibi s oscilla o y beha io , a -
ibu ed o esonance e ec s and he ansi ion om sphe -
ical o obla e shapes. These luc ua ions become mo e p o-
nounced a la ge diame e s. Va iabili y in d op o ien a ion
wi hin he ada sampling olume, desc ibed by he can -
ing angle dis ibu ion, u he con ibu es o he obse ed
a ia ions in δHV. The b oade he wid h o he can ing an-
gle dis ibu ion, he lowe he magni ude o he pola ime ic
a iables. When pa icles a e andomly o ien ed ( ed line in
Fig. 3), hei o ien a ions a e dis ibu ed uni o mly in all di-
ec ions. In his case, he ensemble-a e aged esponse o e
all possible o ien a ions leads o cancella ion e ec s in he
di e en ial phase (δHV =0, Fig. 3b) and in he di e en ial
e lec i i y (ZDR =0 dB, Fig. 3a). The cancella ion occu s
because, o a medium ha is a mix u e o andomly o ien ed
pa icles, he o -diagonal elemen s Z12,Z21,Z34,Z43 o he
phase ma ix become 0 (as shown in Mishchenko e al., 2000,
Chap e 3, Table II), hus leading o ZDR =0 and δHV =0
(see Eqs. 3–5). The dashed blue lines o Figs. 3 and 4 indi-
ca e he posi ions o he Mie no ches, as depic ed in Fig. 2.
The i s wo minima o δHV coincide wi h he Mie no ches,
while ZDR is app oxima ely 0 a hese poin s. Mo eo e , he
diame e s o he minima (D1,D3,D5) and maxima (D2,D4,
D6) a e demons a ed o ZDR.
The copola co ela ion coe icien (ρHV) quan i ies he
co ela ion be ween he ho izon ally and e ically pola ized
componen s o he ada signal. In Fig. 4, pe ec ly o ien ed
d ops (solid blue and dashed black lines) ha e ρHV =1. Con-
e sely, aind ops wi h a ia ions in he o ien a ion o il
o he d op axis ela i e o he di ec ion o mo ion (can -
ing) ha e ρHV sligh ly lowe han 1, showing a minimum
loss o co ela ion be ween he wo di e en pola iza ion
s a es. A b oade dis ibu ion o can ing angles would lead o
u he deco ela ion. E en when conside ing andomly o i-
en ed aind ops, ρHV ne e alls sho o 0.986. Realis ic al-
ues o can ing gene ally do no exceed 10° (Mishchenko e
al., 2000). Howe e , nei he an enna pa e n e ec s, no an-
enna coupling o he quasi-bis a ic ada con igu a ion, no
mul iple sca e ing, no noise, was included in he calcula-
ions o ρHV a his s age. One, o a combina ion, o hese
e ec s may d i e ρHV below 0.986.
2.1.2 D op size dis ibu ion and aind op eloci ies
The gamma dis ibu ion is a ma hema ical shape ypically
used o ep esen he a iabili y o a na u al ain all d op size
dis ibu ion (DSD) (Ulb ich, 1983):
N(D) =N0Dµexp(−3D) [mm−1m−3],(6)
whe e D[mm] is he sphe e equi- olume diame e , µis
he dimensionless shape pa ame e , N0[mm−1−µm−3] is
he numbe concen a ion pa ame e , and 3[mm−1] is he
slope pa ame e . The h ee pa ame e s (N0,µ, and 3) o
he gamma dis ibu ion enable a wide ange o ain all si -
ua ions o be desc ibed. The pa ame e 3can be de i ed
om 3=(4+µ)/Dm, whe e Dm[mm] is he mass-weigh ed
mean diame e (Ulb ich and A las, 2007; Tes ud e al., 2001).
Impo an ly o Dopple applica ions, he la ge he d ops,
he as e he e minal all speed, T. The ela ionship be-
ween he d op diame e s and he co esponding eloci ies is
pa ame e ized in SI uni s ollowing F isch e al. (1995) and
A mos. Meas. Tech., 18, 4857–4870, 2025 h ps://doi.o g/10.5194/am -18-4857-2025
I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables 4861
Figu e 3. Simula ions o (a) di e en ial e lec i i y, ZDR, and (b) di e en ial phase, δHV, as a unc ion o sphe e equi- olume diame e ,
o 94 GHz ada poin ing a 45°. Pe ec o ien a ion (PO) and andom o ien a ion (RO) a e ep esen ed by he da k blue and ed lines,
espec i ely, de i ed wi h axis a io pa ame e iza ion acco ding o Eq. (1). The dashed black line also co esponds o pe ec ly o ien ed
aind ops wi h axis a io pa ame e iza ion as p oposed by Thu ai e al. (2008). The emaining lines ep esen di e en deg ees o aind op
wobbling, wi h a Gaussian dis ibu ion a ound he ho izon al wi h s anda d de ia ions o 5° (ligh blue), 10° (g een), 20° (o ange), and 40°
(pink).
Figu e 4. As Fig. 3 bu o he copola co ela ion coe icien , ρHV,
as a unc ion o sphe e equi- olume diame e , o 94 GHz ada
poin ing a an ele a ion o 45°.
A las e al. (1973):
T(D) =
cloud =1.2×108·D
22,
D < 0.11 ×10−3m
d izzle =8333 ·D
2−0.0833,
0.11 ×10−3≤D≤0.86 ×10−3m
ain =9.65 −10.3·e−0.6×103·D,
D > 0.86 ×10−3m
.(7)
A ac o o (ρ0/ρ)0.4, wi h ρ0being he densi y a sea le el,
applies o di e en ai densi ies.
In Fig. 5, aind op e minal eloci ies a e plo ed agains
he diame e s acco ding o Eq. (7) and he pa ame e iza-
Figu e 5. Te minal all speed Tas a unc ion o he sphe e equi-
olume diame e , D, o Eq. (7), wi h hick b own line, and o Thu-
ai and B ingi (2005), wi h dashed black line.
ion om Thu ai and B ingi (2005) (solid b own and dashed
black lines, espec i ely). The ela i e di e ence be ween he
wo eloci y pa ame e iza ions ne e exceeds 2 %. The e-
o e, when mapping e minal eloci ies o diame e s, his
ansla es in o simila ela i e unce ain ies in he de e mi-
na ion o diame e s o any gi en eloci y. Fo ins ance, he
posi ion o he i s (second) Mie no ch is expec ed o occu a
e minal eloci ies o 5.89±0.11 ms−1(7.82 ±0.15 ms−1).
2.2 Simula ion o spec al pola ime ic a iables
Two me hodologies o simula ing spec al pola ime ic a i-
ables, as obse ed om W-band cloud ada , will be p e-
sen ed in his pape . The i s was de eloped based on Yu
e al. (2012) and Z ni´
c (1975), while he second is based
on Thu ai e al. (2008) and Chand aseka (1986). No ably,
h ps://doi.o g/10.5194/am -18-4857-2025 A mos. Meas. Tech., 18, 4857–4870, 2025
4862 I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables
bo h me hods show e y good ag eemen ; hey a e desc ibed
in de ail in Sec s. 2.2.1 and 2.2.2, espec i ely. The use o
bo h app oaches ensu es ha he in oduced s ochas ic pe -
u ba ions espec he physical ela ionships be ween sca e -
ing elemen s. Thei ag eemen inc eases con idence in he
simula ed u bulence s uc u e and suppo s he inding ha
obse ed disc epancies a e no a i ac s o he simula ion
me hod. Some p elimina y p ocessing is needed o bo h
me hodologies, as discussed nex .
Fi s ly, an ideal copola spec um SVV o he V channel is
independen ly gene a ed o each diame e (Unal, 2015):
SVV( LoS)=λ4
π5|K|2N(D)σVV(D) 1
sinθel
dD
d T(D) ,(8)
whe e λis he ada wa eleng h, |K2|is de i ed om he di-
elec ic ac o o wa e , N(D) is he DSD (see Sec . 2.1.2),
σVV is he backsca e ing c oss sec ion o he V chan-
nel (Sec . 2.1.1), LoS(D) =sinθel T(D)+wLoS deno es he
line-o -sigh (LoS) Dopple eloci ies o he d ops a he
gi en ele a ion angle θel, and LoS is he sum o he compo-
nen s o he aind op e minal eloci y and o he wind speed
along he LoS. Equa ion (8) is o mula ed o ele a ion an-
gles θel signi ican ly g ea e han 0, wi hou accoun ing o
he con ibu ion o u bulence. The spec um is mapped o
he eloci y domain ia Eq. (7) and sampled in co espon-
dence wi h he eloci y poin s j, wi h j=1,2,...,NFFT,
whe e NFFT is he numbe o FFT poin s, as dic a ed by he
Dopple eloci y esolu ion and Nyquis in e al en isaged
o any gi en ada sys em. The samples a e indica ed as
SVV( j). Simila ly, he H channel spec um can also be p o-
duced a each eloci y bin by eplacing σVV(D) wi h σHH(D)
in Eq. (8).
The c oss spec um, deno ed SHV(D), is de i ed as
SHV( LoS)=λ4
π5|K|2N(D)pσVV(D)σHH(D)
×1
sinθel
dD
d T(D)ρHV(D)eıδHV(D) ,(9)
whe e ı=√−1, ρHV(D) is he co ela ion coe icien be-
ween he V and H channels, and δHV(D) is he phase di -
e ence be ween he V and H channel signals, as desc ibed
in Eqs. (4) and (5). The spec um is sampled simila ly o
he V channel spec um a eloci y poin s jwi h j=
1,2,...,NFFT, and he samples a e deno ed SHV( j). No e
ha each Dopple eloci y spec um can be con e ed o he
equency domain by using he ela ionship D=2 LoS/λ
be ween equency Dopple shi , D, and LoS.
Gene ally, spec a a e de i ed a any gi en ange om he
FFT o he ime se ies o ada sampled ol age signals, he
so-called I(in-phase) and Q(quad a u e) signals collec ed a
he same ange dis ance (Do iak and Z ni´
c, 1993). In he ol-
lowing, complex ol ages will be iden i ied wi h callig aphic
s yle le e s (e.g., V,N). Also, such ol ages will always be
exp essed in he eloci y domain, as indica ed by hei unc-
ional a gumen . They co espond o he FFT o he ol ages
exp essed in he ime domain.
2.2.1 Me hodology I: di ec compu a ion o Iand Qin
he equency domain
This me hod allows Dopple spec a o be simula ed by wo k-
ing only in he eloci y ( equency) domain. Following Yu e
al. (2012), he ime se ies o complex ol age signals in he
V channel in he eloci y domain can be w i en as
V[1]
V( j,k) =q−SVV( j)lnu[1]
jk eıθ[1]
jk ,
j=1,2,...,NFFT;k=1,2,...,K,
(10)
whe e u[1]and θ[1]a e independen , iden ically dis ibu ed,
andom a iables wi h uni o m dis ibu ion be ween 0 and 1
and be ween −πand π, espec i ely. This p ocess can be e-
pea ed k=1,2,...,K imes, in o de o gene a e Kindepen-
den s ochas ic ealiza ions o he same spec um. Simila ly,
o he H channel in he eloci y domain:
VH( j,k) =qsZDR( j)hsρHV( j)V[1]
V( j,k)
+q1−sρHV2( j)V[2]
V( j,k)ieı sδHV( j),
j=1,2,...,NFFT,k =1,2,...,K,
(11)
whe e he spec al a iables sρHV,sδHV, and sZDR a e gen-
e a ed as desc ibed in Sec . 2.1 o each eloci y bin j, bu
also hold he p e ix sin he no a ion o di e en ia e hem
om he commonly used in eg al pola ime ic a iables. The
se ies V[2]
V( j,k) is gene a ed acco ding o Eq. (10), wi h he
same model spec um SVV( ) bu wi h a second independen
sequence o andom numbe s (u[2]and θ[2]). This p ocess
is epea ed o each eloci y bin o a o al o NFFT spec-
al poin s wi hin he Nyquis in e al. The in e se Fou ie
ans o ms o VV( j)and VH( j), wi h j=1,2,...,NFFT,
ep esen simula ed ime se ies o complex signals o he V
and H channels. Fo he implemen a ion o whi e noise, an
app oach simila o Eq. (10) is used:
NV( j,k) =q−NVlnu[3]
jk eıθ[3]
jk ,
NH( j,k) =q−NHlnu[4]
jk eıθ[4]
jk ,
j=1,2,...,NFFT,k =1,2,...,K,
(12)
whe e NVand NHa e he noise powe le els o he V and
H channels co esponding o he p esc ibed alues o signal-
o-noise a io (SNR), and u[3],θ[3],u[4], and θ[4]a e again
gene a ed independen ly.
The complex numbe s ha ep esen he simula ion o he
noisy Iand Qin he equency domain o he V and H chan-
A mos. Meas. Tech., 18, 4857–4870, 2025 h ps://doi.o g/10.5194/am -18-4857-2025
I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables 4863
nels a e calcula ed om
SV( j,k) =VV( j,k) +NV( j,k),
SH( j,k) =VH( j,k) +NH( j,k),
j=1,2,...,NFFT,k =1,2,...,K.
(13)
2.2.2 Me hodology II: co ela ion ma ix
Al e na i ely, he Iand Qgene a ion can be pe o med us-
ing he me hodology p oposed by Unal and Moissee (2004),
based on he co ela ion ma ix. Fi s , he co ela ion ma ix
Ris buil wi h he Dopple powe spec a in he diagonal
e ms and he c oss-pola spec um in he an idiagonal ele-
men s, as
R( j)=SVV( j)+NVSHV( j)
S?
HV( j) SHH( j)+NH,
j=1,2,...,NFFT ,
(14)
wi h all e ms gi en by Eqs. (8) and (9). Noise has also been
included bu wi h no copola co ela ion. Because Ris He -
mi ian and posi i e de ini e, i may be w i en as R=T†T
ia Cholesky decomposi ion, whe e †deno es he He mi-
ian anspose. Gi en 2NFFT ze o-mean independen s an-
da d ci cula Gaussian andom a iables, y1,y2,...,y2NFFT
(i.e., yj=1/√2(ξj+ıηj), whe e ξjand ηja e no mally dis-
ibu ed wi h mean equal o 0 and s anda d de ia ion equal o
1), he complex numbe s
SV( 1)
SH( 1)
SV( 2)
SH( 2)
.
.
.
SV( FFT)
SH( FFT)
=T†
y1
y2
y3
y4
.
.
.
y2NFFT−1
y2NFFT
(15)
ha e componen s dis ibu ed as no mally dis ibu ed a i-
ables wi h ze o mean and wi h co ela ion p o ided by R.
The p ocedu e can be epea ed K imes o simula e Kdi e -
en spec a.
2.2.3 Compu a ion o pola ime ic a iables om I
and Q
Once Iand Qha e been ob ained wi h ei he o he wo
me hodologies, hen noisy Dopple spec a can be compu ed
as a spec al a e age o Kspec a:
SVV( j)=h|SV( j)|2i= 1
K
K
X
k=1SV( j,k)
2,(16)
SHH( j)=h|SH( j)|2i= 1
K
K
X
k=1SH( j,k)
2.(17)
The spec al pola ime ic a iables sρHV( ) and sδHV( )
a e calcula ed acco ding o Mishchenko e al. (2000):
sρHV( j)eısδHV( j)=hSH( j)S?
V( j)i
qh|SH( j)|2ih|SV( j)|2i
,(18)
whe e hSH( j)S?
V( j)iis he a e age,
1
K
K
X
k=1
SH( j,k)S?
V( j,k) .
2.2.4 Inclusion o u bulence in he simula ions
Unde s anding he e ec s o u bulence on he Dopple spec-
um is c ucial o imp o ing he accu acy o ada obse a-
ions and hei in e p e a ion. A mosphe ic u bulence causes
andom luc ua ions in he eloci y o hyd ome eo s, hus
b oadening he Dopple spec um. All d ople s a e he e as-
sumed o ha e no ine ial e ec s and he e o e ac like pe ec
ace s. Thus, o in oduce he u bulen mo ions o d ops in
he simula ions, he Dopple spec a mus be con ol ed wi h
a u bulence e m Sai :
S u b
VV ( LoS)=(SVV ∗Sai )( LoS)
=
∞
Z
−∞
SVV( LoS −ξ)Sai (ξ)dξ , (19)
whe e he symbol ∗deno es con olu ion, ξis he con olu ion
a iable, and Sai accoun s o he u bulen mo ions wi hin
he a mosphe e:
Sai ( ) =1
√2πσ
e− 2
2σ2
,(20)
wi h σ exp essing he u bulence b oadening o he Dopple
spec um. Equa ions simila o Eq. (19) can be used o
compu e he u bulence-b oadened spec a S u b
HH ( ) o H-
pola ized adia ion, as well as o S u b
HV ( ). Then he b oad-
ened sZ u b
DR ( ) can be compu ed as he a io o S u b
HH ( ) o
S u b
VV ( ), whe eas he u bulen -b oadened pa ame e s sρ u b
HV
and sδ u b
HV a e hen calcula ed as, espec i ely, he ampli ude
and he phase o he a iable:
sρ u b
HV ( )eısδ u b
HV ( ) =S u b
HV ( )
qS u b
HH ( )S u b
VV ( )
.(21)
Fo he gene a ion o Iand Q:
h ps://doi.o g/10.5194/am -18-4857-2025 A mos. Meas. Tech., 18, 4857–4870, 2025
4864 I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables
–Fo me hodology 1 (Sec . 2.2.1), he simula ed spec al
pola ime ic a iables sZ u b
DR ( ),sδ u b
HV ( ), and sρ u b
HV ( )
will eplace he ideal quan i ies in Eq. (11).
–Fo me hodology 2 (Sec . 2.2.2), S u b
VV ,S u b
HH , and S u b
HV
a e used di ec ly in he de ini ion o he co ela ion ma-
ix in Eq. (14).
2.2.5 Ra ionale o simula ion based on I /Q
The eason we chose o gene a e noisy spec a using I/Q
componen s, ins ead o wo king wi h a e age spec a wi h
added noise powe , is o explici ly in es iga e whe he he
use o andom indi idual noisy spec a can help explain o
ep oduce he a iabili y and deg ada ion o en obse ed in
measu ed spec al pola ime ic a iables, pa icula ly in a i-
ables ha ely on c oss-channel co ela ions, like SHV, a low
SNR and low co ela ions whe e app oxima ed o mulas, as
demons a ed in Myagko and O i (2022), end o ail.
By simula ing he noisy spec a om I/Q componen s, we
aimed o es whe he noise cha ac e is ics con ibu e o he
spec al a iabili y seen in obse a ions. In his sense, ou
wo k seeks o ill a gap in he li e a u e and o e an al e -
na i e angle o unde s anding he ole o noise in ada po-
la ime y.
3 Compa isons wi h measu emen s
To assess he accu acy o he cloud ada simula ion me h-
ods, we compa e he measu emen s and he simula ed da a.
This compa ison aims o alida e he pe o mance o he
simula ions and iden i y any disc epancies ha may a ise
om model assump ions o pa ame e se ings. The cloud
ada measu emen s we e ob ained using an RPG equency-
modula ed con inuous wa e (FMCW) dual pola iza ion W-
band cloud Dopple ada sys em, ope a ing a 94 GHz in
a simul aneous ansmission–simul aneous ecep ion (STSR)
mode. The ada sys em was con igu ed o in es iga e po-
la ime ic and spec al pola ime ic measu emen s o clouds
and p ecipi a ion in he oposphe e o a pe iod o 4 mon hs
(Janua y–Ap il 2021). The models desc ibed in Sec s. 2.2.1
and 2.2.2 we e ini ialized based on he cha ac e is ics (SNR;
pulse epe i ion equency, PRF; FFT bins) o he eal mea-
su emen s o gene a e simula ed ada da a, o compa ison
wi h he eal da a.
Two case s udies om 3 Feb ua y 2021 a e p esen ed, bo h
cha ac e ized by mode a e ain all, wi h ain a es app ox-
ima ely be ween 6 and 7 mmh−1. The i s one ocuses on
a spec um acqui ed a an al i ude o 105m abo e g ound
le el, while he second one a ge s a spec um a 484m. The
cases di e p ima ily in he le el o a mosphe ic u bulence
obse ed a speci ic heigh s. Excluding cases o s ong wind
shea (e.g., je s eams) and deep con ec i e sys ems (e.g.,
hunde s o ms), highe al i udes a e gene ally cha ac e ized
by signi ican ly less u bulence han lowe le els, as u bu-
lence is mos ly gene a ed by su ace hea ing and ic ion. The
measu ed spec og am on he e ical channel, SV, and he
pola ime ic a iables, sZDR,sδHV, and sρHV, a e p esen ed
in Fig. 6. The xaxis ep esen s he Dopple eloci y, LoS,
co esponding o he un olded measu ed Dopple eloci y.
The spec al signa u es associa ed wi h small aind ops ap-
pea on he le side o he spec a. As aind op sizes in-
c ease and become compa able o he ada wa eleng h, non-
Rayleigh sca e ing occu s, leading o esonance ea u es, ob-
se ed on he igh side o he spec a.
To acili a e he compa ison be ween simula ions and ob-
se a ional da a, he e minal eloci y, T, was selec ed o
he eloci y axis in Sec s. 3.1 and 3.2. Acco dingly, he
Dopple eloci ies shown in Fig. 6 we e i s adjus ed along
he eloci y axis o emo e he con ibu ion o he adial
wind, wLoS. This co ec ion was achie ed by iden i ying he
i s Mie sca e ing minimum (Kollias e al., 2002). A an ele-
a ion angle o θel =45°, he i s Mie minimum co esponds
o a eloci y o 5.89sinθel =4.16 ms−1. The esul ing co -
ec ed Dopple eloci ies, LoS −wLoS, we e hen di ided by
sinθel, yielding an es ima e o he e minal eloci ies o he
obse a ions.
A compa ison be ween measu ed and simula ed sρHV is
challenging. The measu emen o sρHV is subjec ed o bi-
ases (pa icula ly a low signal- o-noise le els, Touzi e al.,
1999) and is a ec ed by ada -speci ic cha ac e is ics (e.g.,
an enna- ela ed), which a e di icul o quan i y and accoun
o (Myagko e al., 2025). The e o e sρHV is no u he an-
alyzed in his pape .
3.1 Case s udy 1: mode a e u bulence condi ions
The Dopple spec um measu ed a a heigh o 105 m is p e-
sen ed in Fig. 7 wi h a black line. The p esence o u bu-
lence is depic ed as he b oadening e ec o he spec um
and he no ches a e smoo hed ou . To accu a ely ma ch he
measu ed ada spec um, a a ie y o gamma d op size dis-
ibu ions (DSDs) we e p oduced by adjus ing he pa ame-
e s desc ibed in Sec . 2.1.2, aiming o ind he DSD ha
bes i s he obse ed spec um (blue line). Di e en com-
bina ions o µ,N0,Dm( om Eq. 6), and σ ( om Eq. 20)
a e es ed o be e ep esen he eal measu emen . To iden-
i y he op imal i , he leas squa es me hod was employed.
This me hod minimizes he sum o he squa ed di e ences
be ween he measu ed and simula ed spec a, ensu ing ha
he bes - i ing gamma DSD is selec ed. The spec a a e com-
pa ed in loga i hmic scale a he han in linea uni s o be e
cap u e he wide dynamic ange o ada e lec i i y. In his
way, bo h high and low e lec i i y alues a e app op ia ely
weigh ed, a oiding he dominance by la ge alues ha oc-
cu s in linea compa isons. In o de o a oid o e i ing he
ails o he spec um (and de e io a ing he i s o he high
SNR pa o he spec um, e.g., in co espondence o he Mie
no ch), only he pa o he spec um abo e he dashed pu ple
line a −8 dBZ(ms−1)−1is i ed. Tha emphasizes he es-
A mos. Meas. Tech., 18, 4857–4870, 2025 h ps://doi.o g/10.5194/am -18-4857-2025
I. Tsikoudi e al.: Simula ions o spec al pola ime ic a iables 4865
Figu e 6. E en o 3 Feb ua y 2021, 12:40 UTC, wi h e ical p o iles o (a) e lec i i y, (b) di e en ial phase shi , (c) di e en ial e lec i -
i y, and (d) co ela ion coe icien spec a. The wo le els ha a e used o case s udies a e ma ked by he solid (105m) and dashed (484 m)
ec angles.
Figu e 7. 3 Feb ua y 2021, 12:40UTC, 105 m: measu ed Dopple
spec um (black line) and op imum- i ed gamma DSD (blue line).
The dashed pu ple line indica es he h eshold o applying he
leas squa es me hod in o de o ind he op imum i . The pa ame-
e s ha cha ac e ize he i ed spec um a e µ=0, Dm=1.8 mm,
N0=987 mm−1−µm−3, and σ =0.5 ms−1.
onance no ches – whe he sha p o smoo hed – p o iding a
mo e obus indica ion o he magni ude o σ . This h eshold
is an empi ical ule o humb de i ed om his s udy, which
p ima ily ocused on cases wi h ain a es o 5–9 mmh−1.
In Fig. 8, he black lines ep esen he measu ed spec al
pola ime ic a iables sZDR (le ) and sδHV ( igh ), while he
blue and ed lines a e he esul s o he wo simula ion me h-
ods, ob ained by using he a o emen ioned op imum- i ed
Dopple spec um (see Fig. 7). Nex o he ada ele a ion
angle, he p ima y physical ac o s in luencing he spec al
pola ime ic a iables a e he axis a io–diame e ela ion-
ship and he can ing angle dis ibu ion (Unal and an den
B ule, 2024), as well as he a iabili y in ai mo ion, cha ac-
e ized by σ . The alues o sZDR and sδHV do no depend
on he aind op size dis ibu ion (Unal and an den B ule,
2024). Howe e , wha may a y in Fig. 8 is he e minal e-
loci y ange – o example, unde low u bulence condi ions,
he eloci y ange na ows when Dmis small, as in he case
o ligh ain.
In o de o p o ide a consis en e e ence o sphe i-
cal aind ops, he measu ed sZDR and sδHV we e adjus ed
along he yaxis o 0 dB and 0°, espec i ely. The adjus -
men was de e mined based on he measu ed alues o he
smalles pa icles, which a e expec ed o be nea ly sphe i-
cal. This co ec ion accoun s o p opaga ion e ec s and in-
s umen miscalib a ions o he pola ime ic a iables. The
spec al pola ime ic a iables a e analyzed ou side he g ay-
shaded egions, whe e he Dopple spec al powe exceeds
−8 dBZ(ms−1)−1, o ensu e a su icien ly high signal- o-
noise a io.
As expec ed, he e is excellen ag eemen (wi hin he
s ochas ic noisiness) be ween he wo me hods used o gen-
e a ing he simula ions (blue and ed lines) o he wo a i-
ables. The use o bo h me hods desc ibed in Sec . 2.2 is o
ensu e ha he s ochas ic pe u ba ions espec he physical
ela ionships be ween he sca e ing elemen s. The ac ha
h ps://doi.o g/10.5194/am -18-4857-2025 A mos. Meas. Tech., 18, 4857–4870, 2025
