emo e sensing
A icle
Imp o ing GNSS Zeni h We Delay In e pola ion by
U ilizing T oposphe ic G adien s: Expe imen s wi h a
Dense S a ion Ne wo k in Cen al Eu ope in he
Wa m Season
Flo ian Zus 1,*, Jan Douša 2, Michal Kaˇcmaˇ ík3, Pa el Václa o ic 2, Ky iakos Balidakis 1,
Galina Dick 1and Jens Wicke 1,4
1GFZ Ge man Resea ch Cen e o Geosciences, 14473 Po sdam, Ge many; [email p o ec ed] (K.B.);
[email p o ec ed] (G.D.); [email p o ec ed] (J.W.)
2Geode ic Obse a o y Pecný, Resea ch Ins i u e o Geodesy, Topog aphy and Ca og aphy,
250 66 Zdiby, Czech Republic; [email protected] (J.D.); pa el. acla o [email protected] (P.V.)
3Depa men o Geoin o ma ics, VŠB – Technical Uni e si y o Os a a, 708 33 Os a a, Czech Republic;
[email p o ec ed]
4Ins i u e o Geodesy and Geoin o ma ion Science, Technical Uni e si y o Be lin, 10623 Be lin, Ge many
*Co espondence: [email p o ec ed]; Tel.: +49-331-288-1969
Recei ed: 1 Feb ua y 2019; Accep ed: 19 Ma ch 2019; Published: 21 Ma ch 2019


Abs ac :
The Benchma k da a se collec ed wi hin he Eu opean COST Ac ion ES1206 has aimed o
suppo he de elopmen and alida ion o ad anced Global Na iga ion Sa elli e Sys em (GNSS)
oposphe ic p oduc s, in pa icula high- esolu ion zeni h delays and oposphe ic g adien s. In his
wo k we u ilize his unique da a se o show ha he in e pola ion o GNSS Zeni h We Delays
(ZWDs) can be imp o ed by u ilizing oposphe ic g adien s. To do his we i s p o e he concep
wi h simula ed obse a ions, ha is, zeni h delays and oposphe ic g adien s de i ed om a
Nume ical Wea he Model. We show how oposphe ic g adien s can be con e ed o ZWD g adien s.
Then he ZWD g adien s oge he wi h he ZWDs a selec ed e e ence s a ions a e used in an in e se
dis ance weigh ing in e pola ion scheme o es ima e he ZWD a some a ge s a ion. Fo a s a ion
con igu a ion wi h an a e age s a ion dis ance o 50 km in Ge many and a pe iod o wo mon hs
(May and June 2013), we ind an imp o emen o 20% in in e pola ed ZWDs when oposphe ic
g adien s a e aken in o accoun . Nex , we eplace he simula ed by eal obse a ions, ha is, zeni h
delays and oposphe ic g adien s om a P ecise Poin Posi ioning (PPP) solu ion p o ided wi h he
G-Nu /Te nu analysis so wa e. He e we ind an imp o emen o 10% in in e pola ed ZWDs when
oposphe ic g adien s a e aken in o accoun .
Keywo ds:
GNSS; zeni h we delay; oposphe ic g adien ; nume ical wea he p edic ion
model; in e pola ion
1. In oduc ion
Ca ie phase (and code) obse a ions om a g ound-based Global Na iga ion Sa elli e Sys em
(GNSS) s a ion allow he es ima ion o he Zeni h To al Delay (ZTD) [
1
] and he oposphe ic
g adien [
2
]. F om he ZTD, he Zeni h We Delay (ZWD) can be e ie ed. The oposphe ic g adien
a he s a ion can be oughly ela ed o he ho izon al ZWD g adien a he s a ion, ha is, he pa ial
de i a i e o he ZWD wi h espec o he longi ude (la i ude). In o he wo ds, a single s a ion p o ides
in o ma ion on he ZWD and ho izon al ZWD g adien . Thus i is na u al o y o imp o e ZWD
in e pola ion by u ilizing oposphe ic g adien s. The ZWD can be con e ed o he In eg a ed Wa e
Remo e Sens. 2019,11, 674; doi:10.3390/ s11060674 www.mdpi.com/jou nal/ emo esensing
Remo e Sens. 2019,11, 674 2 o 15
Vapo (IWV), which is closely linked o p ecipi a ion. An imp o ed ZWD map implies an imp o ed
IWV map, which is aluable in me eo ological s udies [3,4].
A p e equisi e o an accu a e in e pola ion is he quali y o ZTDs and oposphe ic g adien s.
To da e mos a en ion has been paid o ZTDs (see, e.g., [
5
] and e e ences he ein). Li le a en ion has
been paid o oposphe ic g adien s. Ba -Se e e al. p esen ed in [
2
] p elimina y esul s p o iding
some e idence ha he oposphe ic g adien s con ain eal oposphe ic ea u es. To wha ex en
he oposphe ic g adien s con ain use ul oposphe ic in o ma ion was no exploi ed u he by
any means. A numbe o compa ison s udies ollowed. Fo example, Li e al. [
6
] showed ha an
imp o ed obse a ion geome y yields imp o ed oposphe ic g adien es ima es. Mo el e al. used
in [
7
] di e en so wa e packages and compa ed oposphe ic g adien s om a dozen o s a ions
loca ed on an island in he Medi e anean Sea (Co sica). Typically, such s udies a e es ic ed o a
ew s a ions. Dousa e al. p esen ed in [
8
] a compa ison o hund eds o s a ions including wo
GNSS es ima ion me hods (double- and ze o-di e encing) and compa ed hem wi h oposphe ic
g adien s de i ed om Nume ical Wea he Models (NWMs). The compa ison o oposphe ic g adien
maps, ha is, he isual inspec ion, p o ided clea e idence ha he GNSS oposphe ic g adien s
include eal oposphe ic ea u es. Al hough, he esul s we e encou aging some poin s needed
u he in es iga ion. Fo example, due o he low ho izon al esolu ion o he unde lying NWMs,
he magni ude o NWM oposphe ic g adien s was unde es ima ed wi h espec o he magni ude
o he GNSS oposphe ic g adien s. In addi ion, he ole o he GNSS da a p ocessing op ions and
pa ame e s emained unclea . Fo example, he impac o highe -o de ionosphe ic co ec ions on
es ima ed oposphe ic g adien s was analyzed in [
9
]. Clea a i icial signals in oposphe ic g adien s
we e obse ed du ing he 2nd EUREF ep ocessing [
10
] and iden i ied as an abso p ion o asymme ic
e ec caused by p oblems ela ed o low-ele a ion obse a ions p o ided by p og essi ely deg ading
ins umen a ion. Kacma ik e al. s udied in [
11
] he sensi i i y o oposphe ic g adien s o se e al
p ocessing op ions. Gene ally, all he solu ions un in he pos -p ocessing mode p o ided a obus
oposphe ic g adien es ima ion wi h a clea ela ion o eal wea he condi ions. The quali y o
oposphe ic g adien es ima es in eal- ime mode mainly depended on he ac ual quali y o he
eal- ime o bi s and clocks. In addi ion, [
11
] showed ha he highe ho izon al esolu ion o he NWMs
yields a be e ma ch be ween NWM and GNSS oposphe ic g adien maps. As a ma e o example,
Figu e 1shows he GNSS and NWM oposphe ic g adien map o an epoch whe e s ong oposphe ic
g adien s a e p esen (10 May 2013, 6 UTC). De ails on he GNSS and NWM oposphe ic pa ame e s
a e p o ided in Sec ion 2below. In o de o show ha oposphe ic g adien s a e o en accompanied by
se e e wea he , we o e laid he oposphe ic g adien maps wi h he ada image (ins an aneous ain)
p o ided by he Deu sche We e Diens (DWD). We also added he s a ion speci ic GNSS IWV alues
in o de o show ha he oposphe ic g adien s can be oughly ela ed o ho izon al IWV g adien s;
ypically, he oposphe ic g adien s poin om d y o mois a eas. This was also ecognized by [
12
]
s udying p elimina y signs o deep con ec ion. In ac , hey used ZTDs and oposphe ic g adien s o
de i e so-called pseudo-ZTDs in o de o imp o e IWV isualiza ion. Simila ly, Douša e al. [
13
] used
he pseudo-ZTDs o imp o ing a new wo-s age oposphe ic co ec ion model. I p ecise oposphe ic
g adien s a e es ima ed along wi h ZTDs, he co esponding pseudo-ZTDs could be used o imp o e
spa ial in e pola ion.
Remo e Sens. 2019,11, 674 3 o 15
Remo e Sens. 2018, 10, x FOR PEER REVIEW 3 o 15
Figu e 1. We choose an epoch whe e s ong oposphe ic g adien s a e p esen (10 May 2013, 6 UTC).
The le panel shows he Global Na iga ion Sa elli e Sys em (GNSS) oposphe ic g adien map. The
middle panel shows he Nume ical Wea he Models (NWM) oposphe ic g adien map. The
oposphe ic g adien maps a e o e laid wi h ada p ecipi a ion p o ided by he Deu sche We e
Diens (DWD). The ada image shows ins an aneous ain in mm/h using a colo scale o
yellow-g een-blue-pu ple- ed, whe e a da ke one o a speci ic colo means a highe ain all
in ensi y. The igh panel shows he s a ion speci ic GNSS In eg a ed Wa e Vapo (IWV).
The pu pose o his s udy is o show ha ZWD in e pola ion can be imp o ed by u ilizing
oposphe ic g adien s. The algo i hm de eloped in his s udy can be ega ded an imp o ed
oposphe ic co ec ion model. Such a oposphe ic co ec ion model can be used o p o iding
oposphe ic augmen a ion co ec ions in posi ioning applica ions (see, e.g., [13] and e e ences
he ein). The in e pola ed ZWD can be con e ed o IWV, which makes he oposphe ic co ec ion
model aluable in me eo ological applica ions as well.
The s uc u e o his wo k is as ollows. In Sec ion 2 we in oduce ZTDs and oposphe ic
g adien s and desc ibe he da a se s used in his s udy. In Sec ion 3 we show how oposphe ic
g adien s can be con e ed o ho izon al ZWD g adien s and we desc ibe he in e pola ion me hod.
In Sec ion 4 we p esen he esul s. A i s , we make use o simula ed obse a ions be o e we make
use o eal obse a ions. The pu pose o u ilizing simula ed obse a ions is o p o e he concep in
ad ance. In Sec ion 5 we discuss he esul s. The conclusion is gi en in Sec ion 6.
2. ZTDs and T oposphe ic G adien s
In he GNSS analysis, he oposphe ic delay is pa ame e ized u ilizing Mapping Func ions
(MF). Speci ically, o he ele a ion angle e and azimu h angle a, he pa ame e ized oposphe ic
delay T eads as
𝑇󰇛𝑒,𝑎󰇜=𝑚
󰇛𝑒󰇜⋅𝑍𝐻𝐷+𝑚󰇛𝑒󰇜⋅𝑍𝑊𝐷+𝑚
󰇛𝑒󰇜󰇟𝑐𝑜𝑠󰇛𝑎󰇜⋅𝑁+𝑠𝑖𝑛󰇛𝑎󰇜⋅𝐸󰇠 (1)
whe e ZHD deno es he Zeni h Hyd os a ic Delay, ZWD deno es he Zeni h We Delay, N
deno es he no h-g adien componen , E deno es he eas -g adien componen , mh deno es he
hyd os a ic MF, mw deno es he we MF, and mg deno es he g adien MF [14]. The ZTD is gi en by
𝑍𝑇𝐷 = 𝑍𝐻𝐷+𝑍𝑊𝐷 (2)
Simila ly o he ZTD, he oposphe ic g adien can be w i en as he sum o a hyd os a ic and
we con ibu ion. In essence,
𝐸=𝐸
+𝐸
 (3)
𝑁=𝑁
+𝑁

whe e Nh (Nw) deno es he hyd os a ic (we ) no h-g adien componen and Eh (Ew) deno es he
hyd os a ic (we ) eas -g adien componen . I is impo an o no e ha he GNSS analysis (see below)
p o ides accu a e ZTDs and ( o al) oposphe ic g adien s. The GNSS analysis does no p o ide
ZWDs and we g adien s. In his s udy, he GNSS ZWDs and we g adien s a e ob ained om he
GNSS ZTDs and oposphe ic g adien s by sub ac ing NWM ZHDs and hyd os a ic g adien s.
Figu e 1.
We choose an epoch whe e s ong oposphe ic g adien s a e p esen (10 May 2013,
6 UTC). The le panel shows he Global Na iga ion Sa elli e Sys em (GNSS) oposphe ic g adien
map. The middle panel shows he Nume ical Wea he Models (NWM) oposphe ic g adien map.
The oposphe ic g adien maps a e o e laid wi h ada p ecipi a ion p o ided by he Deu sche
We e Diens (DWD). The ada image shows ins an aneous ain in mm/h using a colo scale o
yellow-g een-blue-pu ple- ed, whe e a da ke one o a speci ic colo means a highe ain all in ensi y.
The igh panel shows he s a ion speci ic GNSS In eg a ed Wa e Vapo (IWV).
The pu pose o his s udy is o show ha ZWD in e pola ion can be imp o ed by u ilizing
oposphe ic g adien s. The algo i hm de eloped in his s udy can be ega ded an imp o ed
oposphe ic co ec ion model. Such a oposphe ic co ec ion model can be used o p o iding
oposphe ic augmen a ion co ec ions in posi ioning applica ions (see, e.g., [
13
] and e e ences
he ein). The in e pola ed ZWD can be con e ed o IWV, which makes he oposphe ic co ec ion
model aluable in me eo ological applica ions as well.
The s uc u e o his wo k is as ollows. In Sec ion 2we in oduce ZTDs and oposphe ic g adien s
and desc ibe he da a se s used in his s udy. In Sec ion 3we show how oposphe ic g adien s can
be con e ed o ho izon al ZWD g adien s and we desc ibe he in e pola ion me hod. In Sec ion 4
we p esen he esul s. A i s , we make use o simula ed obse a ions be o e we make use o eal
obse a ions. The pu pose o u ilizing simula ed obse a ions is o p o e he concep in ad ance.
In Sec ion 5we discuss he esul s. The conclusion is gi en in Sec ion 6.
2. ZTDs and T oposphe ic G adien s
In he GNSS analysis, he oposphe ic delay is pa ame e ized u ilizing Mapping Func ions (MF).
Speci ically, o he ele a ion angle eand azimu h angle a, he pa ame e ized oposphe ic delay T
eads as
T(e,a)=mh(e)·ZHD +mw(e)·ZWD +mg(e)[cos(a)·N+sin(a)·E](1)
whe e ZHD deno es he Zeni h Hyd os a ic Delay, ZWD deno es he Zeni h We Delay, Ndeno es he
no h-g adien componen , Edeno es he eas -g adien componen , m
h
deno es he hyd os a ic MF, m
w
deno es he we MF, and mgdeno es he g adien MF [14]. The ZTD is gi en by
ZTD =ZHD +ZWD (2)
Simila ly o he ZTD, he oposphe ic g adien can be w i en as he sum o a hyd os a ic and
we con ibu ion. In essence,
E=Eh+Ew
N=Nh+Nw
(3)
whe e N
h
(N
w
) deno es he hyd os a ic (we ) no h-g adien componen and E
h
(E
w
) deno es he
hyd os a ic (we ) eas -g adien componen . I is impo an o no e ha he GNSS analysis (see below)
p o ides accu a e ZTDs and ( o al) oposphe ic g adien s. The GNSS analysis does no p o ide ZWDs
Remo e Sens. 2019,11, 674 4 o 15
and we g adien s. In his s udy, he GNSS ZWDs and we g adien s a e ob ained om he GNSS
ZTDs and oposphe ic g adien s by sub ac ing NWM ZHDs and hyd os a ic g adien s.
2.1. Benchma k Da a Se
We base ou s udy on he Benchma k da a se which was collec ed wi hin he Eu opean COST
Ac ion ES1206 GNSS4SWEC (Ad anced GNSS oposphe ic p oduc s o moni o ing se e e wea he
and clima e). Among a b oad se o a ious me eo ological obse a ions and p oduc s, i con ains da a
om 430 GNSS e e ence s a ions in he a ea o cen al Eu ope co e ing Ge many, he Czech Republic,
and pa s o Poland and Aus ia. The a e age s a ion dis ance is abou 50 km. The s a ion loca ions
a e shown in Figu e 1. All he da a and some GNSS and NWM e e ence oposphe ic p oduc s a e
a ailable o he pe iod o May and June 2013, when some signi ican ain all episodes o a ious o igin
leading o ex ensi e loods occu ed in he selec ed a ea. Fo comp ehensi e and de ailed in o ma ion
abou he Benchma k campaign, he eade is e e ed o [8].
2.2. GNSS ZTDs and T oposphe ic G adien s
We used he G-Nu /Te nu so wa e [
15
] o p ocessing GNSS obse a ions om GPS and
GLONASS cons ella ions. The so wa e employed he P ecise Poin Posi ioning (PPP) me hod [
16
]
and we u ilized p ecise p oduc s o sa elli e o bi s and clocks p o ided by he Eu opean Space
Agency, ESA (h p://na iga ion-o ice.esa.in /GNSS_based_p oduc s.h ml). We used pseudo ange
and ca ie -phase obse a ions wi h he 3
◦
ele a ion angle cu -o and we applied an obse a ion
weigh ing based on he unc ion 1/sin
2
(e). The es ima ed si e-speci ic s a e ec o comp ised
ecei e coo dina es, ecei e clock co ec ions, ZTD, oposphe ic g adien , loa ini ial ambigui ies
o ca ie -phase ionosphe e- ee linea combina ion, and In e Sys em Bias (ISB) o synch onizing
di e en ime ames o GPS and GLONASS. Ex ended Kalman il e suppo ed by he backwa d
smoo he was applied o he s a e ec o es ima ion [
17
]. We modelled GNSS obse a ions acco ding
o he IERS 2010 con en ions [
18
], and we used he ile igs14_2013.ATX o de i ing an enna phase
cen e o se s and a ia ions [
19
], which a e consis en wi h he IGS2014 e e ence ame [
20
] used o
p ecise sa elli e posi ions.
The s a ion coo dina es we e es ima ed along wi h o he pa ame e s, conside ed cons an o e
a day, ha is, igh ly cons ained. The sigma o a p io coo dina es we e 100 m. The ambigui ies
we e es ima ed as loa alues and hus no esol ed o in ege alues. The s a ion clock co ec ions
we e es ima ed wi hou any dependency be ween consecu i e epochs. We es ima ed he oposphe ic
pa ame e s a a 5-min sampling a e as a andom walk s ochas ic p ocess wi h a noise o 6 mm/
√
h
and 0.6 mm/
√
h o ZTD and oposphe ic g adien s, espec i ely. The ZHD, conside ed as he a
p io i alue o ZTD, was calcula ed om he Global P essu e and Tempe a u e model, GPT [
21
] and
in oduced in o he GNSS adjus men model wi h an a p io i sigma o 10 cm. Bo h hyd os a ic and
we pa s o ZTD we e mapped om he zeni h o pa icula sa elli e di ec ion using hyd os a ic
and we ac o s de i ed om he Global Mapping Func ion, GMF [
22
]. T oposphe ic g adien s we e
ini ialized by ze o alues, and Chen and He ing g adien MF [
14
] was employed o de i ing slan
delay con ibu ion.
2.3. NWM ZTDs and T oposphe ic G adien s
We u ilized he Wea he Resea ch and Fo ecas ing (WRF) model [
23
] o simula e he e ac i i y
ield o he a mosphe e. The ini ial and bounda y condi ions o he limi ed a ea model we e a Global
Fo ecas Sys em (GFS) analysis o he Na ional Cen e s o En i onmen al P edic ion (NCEP). The 24-h
ee o ecas s s a e e y day a 0 UTC. The ollowing physics was applied: he Thompson scheme [
24
]
o he mic ophysics, he Kain–F i sch scheme [
25
] o he cumulus pa ame e iza ion, he Yonsei
Uni e si y scheme [
26
] o he plane a y bounda y laye , he RRTMG Sho and Longwa e scheme [
27
]
o he adia ion, he Uni ied Noah Land Su ace Model [
28
] o he land su ace, and he e ised MM5
scheme [
29
] o he su ace laye . The e ac i i y, which was calcula ed om he p essu e, empe a u e,
Remo e Sens. 2019,11, 674 5 o 15
and humidi y [
30
], was a ailable e e y hou wi h a ho izon al esolu ion o 10 km on 50 e ical model
le els (up o 50 hPa). The e ac i i y a an a bi a y poin was ob ained by in e pola ion [31].
Fo a gi en s a ion loca ion, he oposphe ic g adien s we e de i ed om ay- aced oposphe ic
delays [
32
]. Fi s ly, a se o oposphe ic delays was compu ed (i consis s o 120 oposphe ic delays
whe e he ele a ion angles a e 3
◦
, 5
◦
, 7
◦
, 10
◦
, 15
◦
, 20
◦
, 30
◦
, 50
◦
, 70
◦
, and 90
◦
and he spacing in azimu h
is 30
◦
). Secondly, we compu ed azimu h independen oposphe ic delays unde he assump ion o a
sphe ically laye ed oposphe e. Finally, he di e ences be ween he azimu h dependen oposphe ic
delays and he azimu h independen oposphe ic delays we e compu ed and hen he g adien
componen s we e de e mined by a leas -squa e i [
33
]. The Chen and He ing g adien MF [
14
] was
used. The hyd os a ic delays we e compu ed sepa a ely in o de o calcula e hyd os a ic g adien s.
The we g adien s we e ob ained om he oposphe ic g adien s by sub ac ing he hyd os a ic
g adien s. The ZTD was he oposphe ic delay in he zeni h di ec ion. The ZHD was he hyd os a ic
delay in he zeni h di ec ion. The ZWD was ob ained by sub ac ing he ZHD om he ZTD.
We es ima ed he accu acy o he NWM ZHD and hyd os a ic g adien by compa ing wi h he
ZHD and hyd os a ic g adien de i ed om he a mosphe ic eanalysis ERA5 [
34
]. The ERA5 is
p oduced a he Eu opean Cen e o Medium Range Wea he Fo ecas s (ECMWF). The e ac i i y
ields a e a ailable hou ly wi h a ho izon al esolu ion o abou 31 km (T639 spec al iangula
unca ion). We conside ed he hou ly da a om he ull benchma k pe iod. On a e age (mean o e
all s a ions) he Roo Mean Squa e De ia ion (RMSD) was 3 mm o he ZHD and below 0.1 mm o
bo h hyd os a ic g adien componen s. Fo compa ison, he RMSD was 12 mm o he ZWD and abou
0.4 mm o bo h we g adien componen s. In essence, he WRF model allows an accu a e (sho ange)
p edic ion o he ZHD and he hyd os a ic g adien componen s. Hence, as a as one is conce ned
wi h he ZHD and he hyd os a ic g adien componen s, he WRF model can be po en ially used in
eal ime applica ions as well.
3. Me hod
3.1. Con e T oposphe ic G adien s o Ho izon al ZWD G adien s
The ollowing ela ions a e app oxima ely ue, see, o example Re e ence [35]
Ew∼C∂ZWD
∂x=C
R∂ZWD
∂λ 1
cosφ
Nw∼C∂ZWD
∂y=C
R∂ZWD
∂φ
(4)
He e
λ
and
φ
deno e he s a ion longi ude and la i ude, Rdeno es Ea h’s adius, and he ac o C
is ela ed o he scale heigh o he we e ac i i y g adien . Hence, i we know he we g adien a he
s a ion hen we know app oxima ely he ho izon al ZWD g adien a ha s a ion and ice e sa [
36
].
In o de o demons a e his, we chose a s a ion and selec ed neighbo ing s a ions wi hin a adius o 1
◦
.
We collec ed he ZWDs om hese s a ions, assumed ha he ZWD can be expanded in a Taylo -se ies
and ob ained by a leas squa e i he ho izon al ZWD g adien a he chosen s a ion. This ho izon al
ZWD g adien a he s a ion was oughly p opo ional o he we g adien a he s a ion. We epea ed
his p ocedu e o each s a ion in he benchma k da a se . Fo some s a ions he complex e ain may
play a ole, howe e , so we le his in es iga ion o u u e esea ch. We chose an epoch whe e s ong
oposphe ic g adien s we e p esen (31 May 2013, 18 UTC). We expe imen ed wi h he ac o Cand
ound ha an app op ia e choice was 4 km. We u ilized NWM da a (simila esul s a e ob ained i we
u ilize GNSS da a). The le panels o Figu e 2show he we eas and no h-g adien componen and
he igh panels o Figu e 2show he app oxima ion o he we eas and no h-g adien componen .
The g adien componen s do no ma ch pe ec ly bu he alidi y o he app oxima ion is ob ious.
To some ex en he di e ences be ween he le and igh panel a e due o he ac ha Cis chosen
o be 4 km o any loca ion. Ne e heless, i is clea ha we ha e a ool wi h which we can a leas
app oxima ely con e oposphe ic g adien s o ho izon al ZWD g adien s.

Remo e Sens. 2019,11, 674 6 o 15
Remo e Sens. 2018, 10, x FOR PEER REVIEW 6 o 15
Figu e 2. We chose an epoch whe e la ge oposphe ic g adien s a e p esen (31 May 2013, 18 UTC).
The le panel shows he we oposphe ic g adien s de i ed om ay- aced oposphe ic delays.
The igh panel shows he we oposphe ic g adien s app oxima ed by ho izon al Zeni h We Delay
(ZWD) g adien s. The ho izon al ZWD g adien s we e ob ained by a leas squa e i . Fo de ails e e
o he ex .
3.2. In e pola ion Algo i hm
The ZWD in e pola ion algo i hm consis s o wo s eps. A i s , he ZWD and he we g adien
componen s a he e e ence s a ions a e compu ed acco ding o
𝑍𝑊𝐷 = 𝑍𝑇𝐷 −𝑍𝐻𝐷 (5)
𝐸=𝐸−𝐸

𝑁=𝑁−𝑁

He e he ZHD and he hyd os a ic g adien componen s a e de i ed om he NWM. Second,
he ZWD a some a ge s a ion (loca ion), deno ed 𝑍𝑊𝐷, is ob ained h ough an in e se dis ance
weigh ing in e pola ion echnique
𝑍𝑊𝐷=∑𝑤⋅󰇡𝑍𝑊𝐷+
 𝛥𝜆+
 𝛥𝜙󰇢⋅𝑒𝑥𝑝󰇡−
󰇢 (6)
whe e w deno es he weigh s o he in e se dis ance weigh ing scheme, ∆λ deno es he
longi ude di e ence, ∆φ deno es he la i ude di e ence, and ∆h deno es he heigh di e ence
be ween he a ge and he e e ence s a ion espec i ely. The index i indica es he i h e e ence
s a ion. Following Equa ion 4, he pa ial de i a i e o he ZWD wi h espec o he longi ude and
la i ude can be app oxima ed u ilizing he we g adien componen s, which leads o
𝑍𝑊𝐷=∑𝑤⋅󰇡𝑍𝑊𝐷+

𝑅𝑐𝑜𝑠󰇛𝜙󰇜𝛥𝜆+

𝑅𝛥𝜙󰇢⋅𝑒𝑥𝑝󰇡−
󰇢 (7)
Figu e 2.
We chose an epoch whe e la ge oposphe ic g adien s a e p esen (31 May 2013, 18 UTC).
The le panel shows he we oposphe ic g adien s de i ed om ay- aced oposphe ic delays.
The igh panel shows he we oposphe ic g adien s app oxima ed by ho izon al Zeni h We Delay
(ZWD) g adien s. The ho izon al ZWD g adien s we e ob ained by a leas squa e i . Fo de ails e e
o he ex .
3.2. In e pola ion Algo i hm
The ZWD in e pola ion algo i hm consis s o wo s eps. A i s , he ZWD and he we g adien
componen s a he e e ence s a ions a e compu ed acco ding o
ZWD =ZTD −ZHD
Ew=E−Eh
Nw=N−Nh
(5)
He e he ZHD and he hyd os a ic g adien componen s a e de i ed om he NWM. Second,
he ZWD a some a ge s a ion (loca ion), deno ed
ZWD
, is ob ained h ough an in e se dis ance
weigh ing in e pola ion echnique
ZWD =∑wi·ZWDi+∂ZWDi
∂λ ∆λi+∂ZWDi
∂φ ∆φi·exp−∆hi
H(6)
whe e wdeno es he weigh s o he in e se dis ance weigh ing scheme,
∆λ
deno es he longi ude
di e ence,
∆φ
deno es he la i ude di e ence, and
∆
hdeno es he heigh di e ence be ween he a ge
and he e e ence s a ion espec i ely. The index iindica es he i h e e ence s a ion. Following
Remo e Sens. 2019,11, 674 7 o 15
Equa ion (4), he pa ial de i a i e o he ZWD wi h espec o he longi ude and la i ude can be
app oxima ed u ilizing he we g adien componen s, which leads o
ZWD =∑wi·ZWDi+Ei
w
CRcosφi∆λi+Ni
w
CR∆φi·exp−∆hi
H(7)
Fo any a ge s a ion we conside e e ence s a ions wi hin a adius o 1
◦
. The ac o Cis chosen
o be 4 km. The scale heigh o he ZWD, deno ed H, is chosen o be 3 km. A mo e sophis ica ed
e ical adjus men u ilizing loca ion and ime dependen scale heigh s (see, e.g., [
37
]) emains o be
implemen ed. We no e ha he in e pola ion algo i hm abo e can be w i en as
ZWD =∑wi·ZWDi·exp−∆hi
H+∑wi·Ei
w
CRcosφi∆λi+Ni
w
CR∆φi·exp−∆hi
H(8)
whe e he i s sum can be ega ded as he s anda d in e pola ion algo i hm and he second sum can
be ega ded as he co ec ion o i .
Some end use s a e in e es ed in he IWV and no he ZWD. The IWV a he a ge loca ion,
deno ed IWV , is ob ained h ough
IWV =κ(Tm)·ZWD
whe e he con e sion ac o
κ
depends on he weigh ed mean empe a u e Tm a he a ge loca ion.
The weigh ed mean empe a u e is ela ed o he su ace empe a u e [
1
]. We made use o he NWM o
ob ain he su ace empe a u e a he a ge loca ion.
We no e ha o he mo e sophis ica ed in e pola ion schemes ( o example k iging) a e possible.
Howe e , we lea e such in es iga ions o u u e s udies. The ad an age o he chosen in e pola ion
scheme is he ease o implemen a ion.
4. Resul s
Each s a ion in he benchma k da a se was ea ed as a a ge s a ion. The ZWD a he a ge
s a ion was ega ded he ue ZWD. Da a om he su ounding s a ions, he e e ence s a ions, we e
used o es ima e ZWD a he a ge s a ion. This es ima ed ZWD a he a ge s a ion was hen
compa ed wi h he ue ZWD a he a ge s a ion. I was hus s aigh o wa d o es ima e he e o o
he in e pola ion s a egy. We compa ed wo in e pola ion s a egies: (1) only ZWDs om he e e ence
s a ions a e used and (2) ZWDs and oposphe ic g adien s om he e e ence s a ions a e used.
He eina e he wo me hods a e e e ed o as he i s and second in e pola ion s a egy espec i ely.
In he i s subsec ion we make use o simula ed obse a ions, ha is, NWM ZTDs and oposphe ic
g adien s. In he second subsec ion we eplace he simula ed obse a ions by eal obse a ions, ha is,
GNSS ZTDs and oposphe ic g adien s.
4.1. Expe imen wi h NWM Da a
Di e ences be ween he wo in e pola ion s a egies can be expec ed o la ge oposphe ic
g adien s. The e o e we i s conside a single epoch whe e la ge oposphe ic g adien s a e p esen .
The sca e plo in Figu e 3shows he s a ion speci ic ZWD di e ences be ween he in e pola ion
s a egies, ha is, he co ec ion o he s anda d in e pola ion algo i hm, on 31 May 2013 a 18 UTC.
The ZWD di e ences each up o 25 mm. The ZWD di e ences show a speci ic pa e n. A quick look a
Figu e 2explains why ZWD di e ences a e la ge in some a eas han in o he a eas. The oposphe ic
g adien s show a speci ic pa e n and he ZWD di e ences ollow his pa e n. Figu e 3does no
p o ide in o ma ion on how well he in e pola ion s a egies pe o m. In Figu e 4 he uppe (lowe )
panel shows he ZWD e o o he i s (second) in e pola ion s a egy. On a e age (mean o e
all s a ions) he Roo Mean Squa e E o (RMSE) is educed by mo e han 30% when oposphe ic
Remo e Sens. 2019,11, 674 8 o 15
g adien s a e aken in o accoun . Nex , ins ead o a single epoch, we conside he hou ly da a om he
ull benchma k pe iod o wo mon hs. In Figu e 5, he uppe (lowe ) panel shows he s a ion speci ic
ZWD RMSE o he i s (second) in e pola ion s a egy. On a e age he RMSE is educed by abou 20%
when oposphe ic g adien s a e aken in o accoun .
Remo e Sens. 2018, 10, x FOR PEER REVIEW 8 o 15
Figu e 3. The sca e plo shows he s a ion speci ic ZWD di e ences be ween he wo in e pola ion
s a egies on 31 May 2013, 18 UTC.
Figu e 4. The uppe panel shows he ZWD in e pola ion e o when only ZWDs a e used. The lowe
panel shows he ZWD in e pola ion e o when bo h ZWDs and oposphe ic g adien s a e used.
The ed numbe s p o ide he Roo Mean Squa e E o (RMSE) (a e age o e all s a ions). Resul s a e
alid on 31 May 2013, 18 UTC.
Figu e 3.
The sca e plo shows he s a ion speci ic ZWD di e ences be ween he wo in e pola ion
s a egies on 31 May 2013, 18 UTC.
Remo e Sens. 2018, 10, x FOR PEER REVIEW 8 o 15
Figu e 3. The sca e plo shows he s a ion speci ic ZWD di e ences be ween he wo in e pola ion
s a egies on 31 May 2013, 18 UTC.
Figu e 4. The uppe panel shows he ZWD in e pola ion e o when only ZWDs a e used. The lowe
panel shows he ZWD in e pola ion e o when bo h ZWDs and oposphe ic g adien s a e used.
The ed numbe s p o ide he Roo Mean Squa e E o (RMSE) (a e age o e all s a ions). Resul s a e
alid on 31 May 2013, 18 UTC.
Figu e 4.
The uppe panel shows he ZWD in e pola ion e o when only ZWDs a e used. The lowe
panel shows he ZWD in e pola ion e o when bo h ZWDs and oposphe ic g adien s a e used.
The ed numbe s p o ide he Roo Mean Squa e E o (RMSE) (a e age o e all s a ions). Resul s a e
alid on 31 May 2013, 18 UTC.
Remo e Sens. 2019,11, 674 9 o 15
Remo e Sens. 2018, 10, x FOR PEER REVIEW 9 o 15
Figu e 5. We conside he hou ly da a a ailable o a pe iod o wo mon hs (May and June 2013). The
uppe panel shows he s a ion speci ic ZWD RMSE when only ZWDs a e used in he in e pola ion.
The lowe panel shows he s a ion speci ic ZWD RMSE when bo h ZWDs and oposphe ic g adien s
a e used in he in e pola ion. The ed numbe s p o ide he a e age RMSE (mean o e all s a ions).
4.2. Expe imen wi h GNSS Da a
Nex we used GNSS ZTDs and oposphe ic g adien s. The sca e plo in Figu e 6 shows he
s a ion speci ic ZWD di e ences be ween he in e pola ion s a egies on 31 May 2013 a 18 UTC.
Again, he ZWD di e ences each up o 25 mm. The pa e n in Figu e 6 closely ollows he pa e n in
Figu e 3 due o he ac ha he GNSS oposphe ic g adien s closely ollow he NWM oposphe ic
g adien s. In Figu e 7 he uppe (lowe ) panel shows he in e pola ion e o o he i s (second)
s a egy. On a e age (mean o e all s a ions) he RMSE is educed by abou 20% when oposphe ic
g adien s a e aken in o accoun . Figu e 7 can be compa ed wi h Figu e 4 p o ided abo e. The
imp o emen when u ilizing eal obse a ions is smalle compa ed o he imp o emen when
u ilizing simula ed obse a ions. Again, ins ead o a single epoch, we conside he hou ly da a om
he benchma k pe iod o wo mon hs. In Figu e 8 he uppe (lowe ) panel shows he s a ion speci ic
ZWD RMSE o he i s (second) in e pola ion s a egy. On a e age he RMSE is educed by abou
10% when oposphe ic g adien s a e aken in o accoun . Figu e 8 can be compa ed o Figu e 5
p o ided abo e. Clea ly, he RMSE is educed when oposphe ic g adien s a e u ilized, howe e ,
he imp o emen when u ilizing eal obse a ions is smalle compa ed o he imp o emen when
u ilizing simula ed obse a ions.
Figu e 5.
We conside he hou ly da a a ailable o a pe iod o wo mon hs (May and June 2013).
The uppe panel shows he s a ion speci ic ZWD RMSE when only ZWDs a e used in he in e pola ion.
The lowe panel shows he s a ion speci ic ZWD RMSE when bo h ZWDs and oposphe ic g adien s
a e used in he in e pola ion. The ed numbe s p o ide he a e age RMSE (mean o e all s a ions).
4.2. Expe imen wi h GNSS Da a
Nex we used GNSS ZTDs and oposphe ic g adien s. The sca e plo in Figu e 6shows he
s a ion speci ic ZWD di e ences be ween he in e pola ion s a egies on 31 May 2013 a 18 UTC.
Again, he ZWD di e ences each up o 25 mm. The pa e n in Figu e 6closely ollows he pa e n in
Figu e 3due o he ac ha he GNSS oposphe ic g adien s closely ollow he NWM oposphe ic
g adien s. In Figu e 7 he uppe (lowe ) panel shows he in e pola ion e o o he i s (second) s a egy.
On a e age (mean o e all s a ions) he RMSE is educed by abou 20% when oposphe ic g adien s
a e aken in o accoun . Figu e 7can be compa ed wi h Figu e 4p o ided abo e. The imp o emen
when u ilizing eal obse a ions is smalle compa ed o he imp o emen when u ilizing simula ed
obse a ions. Again, ins ead o a single epoch, we conside he hou ly da a om he benchma k pe iod
o wo mon hs. In Figu e 8 he uppe (lowe ) panel shows he s a ion speci ic ZWD RMSE o he i s
(second) in e pola ion s a egy. On a e age he RMSE is educed by abou 10% when oposphe ic
g adien s a e aken in o accoun . Figu e 8can be compa ed o Figu e 5p o ided abo e. Clea ly, he
RMSE is educed when oposphe ic g adien s a e u ilized, howe e , he imp o emen when u ilizing
eal obse a ions is smalle compa ed o he imp o emen when u ilizing simula ed obse a ions.
The inclusion o he oposphe ic g adien s in he in e pola ion only makes a di e ence du ing
se e e wea he . The oposphe ic g adien s a e smalle du ing calm wea he which is somewha ue
o mos o he days. As a esul , he a e age impac is na u ally small. I is hus ecommended o ake
a close look on speci ic epochs. As ano he case s udy we choose 10 May 2013, 6 UTC, he epoch used
o illus a ion in he in oduc ion ( e e o Figu e 1). Howe e , ins ead o he in e pola ed ZWD we
look a he in e pola ed IWV. The sca e plo in Figu e 9shows he s a ion speci ic IWV di e ences
be ween he in e pola ion s a egies on 10 May 2013 a 6 UTC. The oposphe ic g adien s shown
in Figu e 1explain why IWV di e ences a e la ge in some a eas han in o he a eas. In sou h-wes
Ge many, IWV di e ences each up o 3 kg/m
2
. In Figu e 10 he uppe (lowe ) panel shows he IWV
e o o he i s (second) in e pola ion s a egy. On a e age (mean o e all s a ions) he Roo Mean
Squa e E o (RMSE) is educed by abou 10% when oposphe ic g adien s a e aken in o accoun .