wa e
A icle
Op imal Calib a ion o E apo a ion Models agains
Penman–Mon ei h Equa ion
Dagma Dlouhá , Vik o Dubo ský * and Lukáš Pospíšil


Ci a ion: Dlouhá, D.; Dubo ský, V.;
Pospíšil, L. Op imal Calib a ion o
E apo a ion Models agains
Penman–Mon ei h Equa ion. Wa e
2021,13, 1484. h ps://doi.o g/
10.3390/w13111484
Academic Edi o : Guido D’U so
Recei ed: 26 Feb ua y 2021
Accep ed: 24 May 2021
Published: 25 May 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Depa men o Ma hema ics, Facul y o Ci il Enginee ing, VSB-TU Os a a, Lud íka Podéš ˇe 1875/17,
708 00 Os a a, Czech Republic; dagma [email p o ec ed] (D.D.); [email p o ec ed] (L.P.)
*Co espondence: ik o [email p o ec ed]
Abs ac :
We p esen an app oach o he calib a ion o simpli ied e apo a ion model pa ame e s
based on he op imiza ion o pa ame e s agains he mos complex model o e apo a ion es ima ion,
i.e., he Penman–Mon ei h equa ion. This model compu es he e apo a ion om se e al inpu
quan i ies, such as ai empe a u e, wind speed, hea s o age, ne adia ion e c. Howe e , some imes
all hese alues a e no a ailable, he e o e we mus use simpli ied models. Ou in e es in ee wa e
su ace e apo a ion is gi en by he need o ongoing hyd ic eclama ion o he o me Ležáky–Mos
qua y, i.e., he ongoing es o a ion o he land ha has been mined o a na u al and economically
usable s a e. Fo eme ging pi lakes, he p edic ion o e apo a ion and he le el o wa e plays
a c ucial ole. We examine he me hodology on se e al popula models and s anda d s a is ical
measu es. The p esen ed app oach can be applied in a gene al model calib a ion p ocess subjec o
any heo e ical o measu ed e apo a ion.
Keywo ds:
model calib a ion; e apo a ion; Penman–Mon ei h equa ion; op imiza ion; c oss- alida ion
1. In oduc ion
E apo a ion and e apo anspi a ion play a c ucial ole in wa e managemen in a
wide ange o human ac i i ies and hus he e is a s ong need o accu a e es ima es.
This need leads o a conside able numbe o pape s and s udies ha a e o e ing new
me hods o such es ima es o compa ison o me hods al eady used in hyd ologic engi-
nee ing applica ions. The esul s o es ima ion could be compa ed wi h e e ence e apo a-
ion/e apo anspi a ion calcula ed by FAO Penman–Mon ei h equa ion
EFAO
, which is
ecommended as he s anda d me hod [
1
,
2
]. This equa ion is conside ed o be an e alon
o which he esul s o he o he me hods can be ela ed and compa ed. The pape s
dealing wi h e apo a ion o e apo anspi a ion p esen he compa ison o he FAO Penman–
Mon ei h me hod esul s o o he me hods p oposing he ela ions be ween inpu da a less
complica ed and compu a ionally less demanding. Fo ins ance, such a p ocedu e was
pe o med in he s udy [
3
] o ind he bes es ima ion o wa e los om a co e ed ese oi .
In [
4
] i is s a ed ha
EFAO
p o ides good ag eemen wi h e apo a ion measu ed on
120ha
dam. In [
5
],
EFAO
is used no only as a e e ence me hod and bu also a ounda ion o new
nume ical models de i ed by mul iple linea eg ession and design o expe imen me hod,
ollowed by he simpli ied me hodology o he quan i ica ion o he e apo a ion a e o
a basin wi h a pho o ol aic sys em. Simila ly, he possibili y o educ ion o Lake Nasse
e apo a ion using a loa ing pho o ol aic sys em is desc ibed in [
6
]. Ou esea ch is no
ocused only on he a ea o Lake Mos , bu also on pi lakes ha a e only planned, he e o e
i is impossible o use limnological and ba hyme ic da a, such as empe a u e p o ile o
wa e dep h. Such a si ua ion is conside ed in [
7
], whe e au ho s used he
EFAO
o he
compu a ion o he open wa e e apo a ion es ima ion.
FAO Penman–Mon ei h me hod is cha ac e ized by a s ong likelihood o co ec ly
p edic ing e apo anspi a ion in a wide ange o loca ions and clima es wi h di e ing
Wa e 2021,13, 1484. h ps://doi.o g/10.3390/w13111484 h ps://www.mdpi.com/jou nal/wa e
Wa e 2021,13, 1484 2 o 23
local condi ions, e.g., sola adia ion, sunshine du a ion, wind speed, ai humidi y, ai
empe a u e [8–10].
Ou in e es in ee wa e su ace e apo a ion is gi en by he need o ongoing hyd ic
ecul i a ion o he o me Ležáky–Mos qua y (Czech Republic), i.e., Lake Mos , as well
as ano he planned hyd ic ecul i a ion in he egion. One o he key componen s o hyd ic
eclama ion planning is he secu i iza ion o long- e m sus ainabili y, which is based on he
capabili y o keeping he s able le el o a dimension o he inal wa e le el.
Hyd ic ecul i a ion was p oposed o be he bes way o deal wi h he esidual o open-
cu coal mines in he no h-wes e n egion o he Czech Republic. A e he mine is closed,
he oid could be illed by su ace wa e uno and g oundwa e . In he case ha hese
esou ces a e no s ong enough, he pi lake has o be illed a i icially and ha is he case o
he o me Mos -Ležáky mine and Lake Mos . The le el o he new lake has been p oposed
o be s able wi h a wa e le el a
199m
abo e sea le el assuming he abula ed alues o
p ecipi a ion and e apo a ion be ween
500mm
and
600mm
annually [
11
]. Howe e , his
assump ion ails o be ue [12].
The e olu ion o a i icial pi lakes is a ec ed by a wide ange o chemical, physical,
and namely hyd ological p ocesses such as sa u a ion o he coas al lines, leakage h ough
he bo om, and ee su ace e apo a ion. As he lake bo om was sealed be o e illing,
e apo a ion was supposed o be he main cause o he obse ed wa e loss.
Toge he wi h he p ecipi a ion, he open wa e e apo a ion and ege a ion e apo an-
spi a ion o m he main componen s o he wa e cycle in na u e, and i is said ha he
e apo a ion o e he land su ace amoun s o abou wo hi ds o he a e age p ecipi a ion,
see [
13
]. Howe e , hese es ima es di e om loca ion o loca ion and e apo a ion mea-
su emen o calcula ion p ocedu es a e complica ed and bu dened wi h a high deg ee o
unce ain y. This happens due o he complexi y o e apo a ion as a physical phenomenon
and se e al ac o s ha a ec his p ocess. The a e o e apo a ion could be measu ed o
calcula ed, howe e , because o he simpli ica ion o he e apo a ion p ocess desc ip ion,
bo h measu emen and compu a ional me hods p o ide only he app oxima ion o ac-
ual e apo a ion. Fo u he discussion abou he his o ical de elopmen o e apo a ion
and e apo anspi a ion, see [
14
] men ioning 166 models and equa ions ob ained du ing
he las h ee cen u ies. The desc ip ion and cha ac e iza ion o all physical p ocesses
a ec ing e apo a ion could be ound in a classical book by B u sae [
15
], o Maidmen [
16
]
( Shu lewo h’s chap e ).
Fo compu a ional me hods, he e a e wo ways o handle e apo a ion, desc ibed as
mass ans e o ene gy budge me hods, see [
13
]. Fu he mo e, he models could be iewed
as empe a u e-based, adia ion-based,mass ans e -based, and combined me hods based on he
inpu s used o calcula e he a e o e apo a ion, o addi ional de ails see [2,16–18].
Addi ionally, wi hin each g oup, he e a e se e al equa ions, which a e widely ci ed
in he echnical li e a u e. Du ing he s udy o a ious e apo a ion models in he li e a u e,
one can obse e se e al di icul ies. One o he main di icul ies is he inconsis encies in he
used physical uni s. Fo ins ance, acco ding o he ime and place o publica ion o a icles
o books, he same equa ion can be encoun e ed wi h he p essu e gi en in
kPa
,
mba
,
To
o millime e s o me cu y column. This a iabili y o uni s can cause di e en shapes o he
same equa ions in di e en sou ces, e en when he same uni s a e used. Fu he mo e,
di e en ypes o me hods equi e di e en ype o da a. Howe e , in p ac ical applica ions,
we a e no able o measu e all ypes o inpu pa ame e s, he e o e he choice o he model
depends no only on he modelling quali y bu mainly on he abili y o measu e he equi ed
inpu physical quan i ies.
This is one o he main easons why he de elopmen o new simpli ied models is s ill
ac i e. The FAO equa ion is a solid s anda d, bu some imes oo complex o be handled in
p ac ice. Fo ins ance, in he a icle [
19
], eigh een empe a u e-, adia ion-, mass ans e -based
and combined me hods a e s udied unde he condi ion o clima ic change in Ge many. The
FAO equa ion is compa ed wi h 31 me hods unde he humid clima e condi ion in I an
Wa e 2021,13, 1484 3 o 23
in he pape [
20
]. In [
21
], i e empe a u e-based me hods and h ee adia ion me hods
a e conside ed.
In his pape , we a e looking o he simpli ica ion o he FAO equa ion in e ms o
he numbe o inpu quan i ies. Ou goal is o use less complex models o model he
e apo a ion in he a ea o Lake Mos by calib a ing he pa ame e s o he models in he
i ing op imiza ion p ocess agains he e apo a ion es ima ion by FAO using selec ed
s a is ical measu es. The mo i a ion came om he lack o measu ing de ices in he di ec
a ea o he lake. In his pape , we conside models which equi e only he ai empe a u e,
wind speed, and ela i e humidi y. Besides he Lake Mos , we a e in e es ed also in he
planned pi lakes. The e o e, he ypes o models a e limi ed o hose which equi es only
hese basic me eo ological da a. In his case, i is no possible o measu e, o ins ance, he
empe a u e o he wa e .
The pe o mance o he conside ed me hods can be e alua ed s a is ically, and se e al
s a is ical measu es could be used. The mos commonly used a e he Roo Mean Squa e
E o (RMSE), Mean absolu e e o (MAE), and he Mean Bias E o (MBE), see, o in-
s ance, [
22
]. Addi ionally, he RMSE and MBE could be combined o calcula e he so-called
-s a is ic es , which exp esses he le el o con idence be ween he models, see [
23
]. A
qui e unusual measu e could be ound in [
24
], i is mean a io
MR
, which is compu ed as
he a e age o a ios be ween he p edic ed and obse ed alues. Ano he way o compa e
wo gi en models is o compu e Pea son’s co ela ion coe icien (PCC)
o o compu e
R2
coe icien o de e mina ion. These wo measu es a e closely connec ed since R2is squa e
o
. The simples measu e o he p edic ion quali y is he pe cen age exp ession o he
di e ence be ween models. This is a s a is ical measu e ha is well known as pe cen
bias (PBIAS). The las class o s a is ical measu es ha desc ibe he co espondence o
obse ed and simula ed da a a e ag eemen indices, such as Nash–Su cli e e iciency (NSE)
and Willmo ’s ag eemen index. These a e used, o example, in [
21
,
25
,
26
]. Fo u he
discussion o s a is ical model e alua ions, see [
27
]. In his pape , we compa e se e al se-
lec ed s a is ical measu es, namely
NSE
,
RMSE
,
MAE
, and
PBIAS
. Howe e , we suppose
ha ou me hodology can be applied o any chosen dis ance unc ion. The applicabili y
depends on he abili y o sol e he co esponding eg ession p oblem.
The choice o empe a u e-based me hods in he case o he Lake Mos s udy is no due
o he cu en lack o me eo ological da a since Kopis y me eos a ion is loca ed only
1km
om he lake. The simpli ica ion o he equa ion is mo i a ed by he u he planned hyd ic
ecul i a ions in he egion. Planned pi lakes a e mo e dis an om Kopis y and he e o e
he me eo ological da a p o ided o he models om Kopis y would no be su icien . The
da a p o ided o he models on new lakes will be measu ed di ec ly on he a ea o new
lakes. In he case o Lake Mos , we can iden i y he app op ia e simpli ied model because
o ad an ageous loca ion o Kopis y wi h espec o Lake Mos . The e o e, we a e in e es ed
in he iden i ica ion o he simples sui able model wi h he low demand on inpu da a.
The cons uc ion o he new wea he s a ion in he a ea close o he new lake does no
make any sense om he inancial poin o iew (because o he p esence o Kopis y wea he
s a ion). On he o he hand, o p o ide be e es ima ions, we should measu e he inpu
da a as close as possible o he a ea o in e es . The measu emen o , o ins ance, he
empe a u e is ela i ely cheap. The only ques ion is i he empe a u e is a su icien
amoun o inpu me eo ological da a o p o iding a su icien es ima ion. The accu acy o
he wa e loss due o e apo a ion is c ucial o hose planned a i icial lakes. To p o ide he
bes possible es ima e, he expe iences om he Lake Mos will be used. In his pape , we
compa e se e al simpli ied models wi h espec o di e en s a is ical measu es, namely
NSE,RMSE,MAE, and PBIAS.
Addi ionally, o a oid he o e i ing o he calib a ed model, we adop he c oss-
alida ion me hodology [
28
]. We andomly spli he da a in o calib a ion and alida ion
pa s. The pa ame e s o he model a e op imized on he calib a ion se and es ed on he
alida ion pa . Resul s om he alida ion pa a e u he analyzed and he bes model is
chosen conce ning esul s om all c oss- alida ion spli ings.
Wa e 2021,13, 1484 4 o 23
The pape is o ganized as ollows. Sec ion 2.4 in oduces he me hods and ma e ials
used in ou compu a ion. To be mo e speci ic, we s a wi h he p esen a ion o he Lake
Mos in Sec ion 2.1, and he da a p o ided o he models in Sec ion 2.2. A e wa ds, we
e iew he FAO equa ion in Sec ion 2.3 and he simpli ied models in Sec ion 2.4. Du ing
he calib a ion p ocess, we use he s a is ical measu es p esen ed in Sec ion 2.5. The whole
me hodology is implemen ed in R p og amming language, see Sec ion 2.6 o de ails. This
sec ion also includes he desc ip ion o he used c oss- alida ion p ocess. The esul s a e
p esen ed in Sec ion 3and discussed in Sec ion 4. Finally, Sec ion 5concludes he pape .
The pape can be conside ed o be an ex ension o ou p e iously published wo k [
29
].
2. Ma e ials and Me hods
2.1. S udy A ea
The Lake Mos is si ua ed in he No h o he Czech Republic nea he ci y o Mos
50
◦
C310
0
N, 13
◦
C360
0
E, see Figu e 1. I was c ea ed by he hyd ic ecul i a ion o he
Mos –Ležáky qua y in he cen al pa o he No h Bohemian b own coal basin. The o me
mine hea ily a ec ed he a ea o
1254ha
and he pi lake, as a pa o i s e i aliza ion, was
planned o ha e a su ace a ea o abou
300ha
. The p ojec o he e i aliza ion is secu ed
by he s a e en e p ise Pali o ý kombiná Ús í (PKU) [30].
Figu e 1. Lake Mos and he su ounding a ea (sou ce: www.pku.cz, Google Ea h).
Be o e he looding, i was necessa y o ake echnical a angemen s such as sealing he
bo om o he u u e lake, cons uc ion o an unde g ound sealing wall, and s eng hening
he sho eline. All hese a angemen s allow iewing he Lake Mos as a closed sys em
wi hou na u al in low o ou low. Due o he absen na u al in low, he esidual pi o
he lake was illed h ough an a i icial eede du ing he pe iod om 2008 o 2014. In he
inal phase o lake illing, i.e., in he yea 2014, he su ace le el ose om
197.74m
o he
equi ed le el o 199m abo e sea le el.
A e inishing he illing p ocess, Lake Mos has an ac ual su ace a ea o
309.4ha
,
a coas al line leng h o
8.9km
, a o al wa e olume o
70.5millionm3
, and a maximum
dep h o
75m
. Th oughou he illing o he lake, bo h ope a ional and basic me eo ological
da a we e moni o ed. The ope a ional da a con ain da a on he achie ed al i ude o he
lake le el, i s su ace a ea, and especially on he olume o wa e admi ed. The illing o
he lake has been inished in 2014 achie ing he equi ed su ace le el o 199m.
Wa e 2021,13, 1484 5 o 23
2.2. Da a and Da a Sou ces
In ou esea ch, we a e using he me eo ological da a collec ed du ing he yea s 2015–
2019. The collec ion includes all da a necessa y o he calcula ion o he Penman–Mon ei h
equa ion (see Sec ion 2.3). These me eo ological measu emen s we e pe o med a he
Kopis y wea he s a ion si ua ed app oxima ely
1km
om he lake. The s a ion is ope a ed
by CHMI—Czech Hyd ome eo ological Ins i u e and he da a a e eco ded a en-minu e
in e als. The da ase ob ained om CHMI was s a is ically p ocessed o be used in he
equa ions o model he e apo a ion. We p esen he da a basic s a is ics in Figu e 2. In
Kopis y wea he s a ion, he wind speed is measu ed a
10m
abo e he g ound o a oid
he in luence o he g ound. The ai empe a u e and humidi y a e measu ed a
2m
abo e
he g ound.
We also included p ecipi a ion equency o he demons a ion o he hyd ological
balance in he a ea o in e es . In compa ison wi h he a e age empe a u e and p ecip-
i a ion in he Czech Republic, he a ea o he planned hyd ic eclama ion is in he a ea
wi h he empe a u e s ongly abo e he a e age and p ecipi a ion s ongly below no mal
p ecipi a ion, and wi h he numbe o hou s o sunshine below he ypical alue in he
Czech Republic [11].
Figu e 2.
The basic s a is ics o he daily measu emen s pe o med a he Kopis y wea he s a ion du ing he yea s 2015–2019:
a e age day empe a u e
Ta
, a mosphe ic p essu e
P
, dayligh hou s pe day
n
, ela i e humidi y
RH
, wind speed
u2
, and
p ecipi a ion p . These da a a e used in he equa ions o modelling he e apo a ion.
2.3. Penman–Mon ei h Equa ion
The EFAO equa ion is o he o m
EFAO =0.408 ∆(Rn−G)+γ900
Ta+273 u2(es−ea)
∆+γ(1+0.34 u2). (1)
Please see Sec ion Abb e ia ions a he end o his pape o he desc ip ion and
physical uni s o he used a iables.
Acco ding o Linac e pape [
31
] o daily es ima es o he e apo a ion a e o ee wa e
le el, he e m Gcan be neglec ed, i.e., we se G=0.

Wa e 2021,13, 1484 6 o 23
The e m
(es−ea)
in
[kPa]
, is he di e ence o sa u a ion apo p essu e and ac ual
apo p essu e. The alues esand eaa e gi en by
es=1
20.6108 e 17.27 Tmax
Tmax+237.3 +0.6108 e 17.27 Tmin
Tmin+237.3 ,ea=RH
100 es. (2)
The psych ome ic cons an
γ
depends on he a mosphe ic p essu e
P
in
[kPa]
and
on abo e-men ioned cons an s
Cp=1013Jkg−1°C−1
,
λ=2.45MJkg−1
and
ε=
0.622
[−]
,
wi h
ε
being a io molecula weigh o wa e apo o d y ai . To compu e i s alue, he
ollowing o mula is used
γ=CpP
ε λ =0.665 ×10−3P. (3)
Using his o mula, he compu ed alue o
γ
depends only on one measu ed quan i y
and ha is a mosphe ic p essu e Pand is gi en in kPa°C−1.
The slope
∆
desc ibes he ela ionship be ween sa u a ion apo p essu e and empe -
a u e. Fo a gi en empe a u e Ta, he co esponding ∆is gi en by
∆=
40980.6108e
17.27 Ta
Ta+237.3 
(Ta+237.3)2. (4)
The esul ing uni o ∆is kPa °C−1.
The ne adia ion a he su ace
Rn
in
hMJm−2day−1i
is, by [
1
], gi en as he di e ence
incoming ne sho wa e adia ion Rns and ou going ne long wa e adia ion Rnl, i.e.,
Rn=Rns −Rnl.
To e alua e
Rnl
, he knowledge o sola adia ion
Rs
and global ex a e es ial adia ion,
Rais equi ed.
Ex a e es ial adia ion is he amoun o adia ion inciden on a uni o he ho izon al
su ace a he ou e bounda y o he a mosphe e. Fo places o simila la i ude, i is
app oxima ely he same, changing only du ing he yea . The e is no in luence o cloud
u bidi y o ai pollu ion o e he Ea h’s a mosphe e, and he e o e, he dose o sola
ene gy is he highes a any gi en ime. In addi ion o he sola cons an , he angle o
incidence o he sun’s ays a a gi en loca ion o he a mosphe e bounda y mus also be
aken in o accoun . The e o e, he alue o
Ra
is exp essed depending on hese quan i ies as
Ra=24 ×60
πGsc d (ωssin ϕsin δ+sin ωscos ϕcos δ). (5)
The e ms included in Equa ion (5) a e
Gsc =0.082 MJ m−2min−1, global sola cons an ,
d =1+0.33 cos 2π
365 JD, Ea h-Sun ela i e dis ance [−],
δ=0.409 sin 2π
365 JD −1.39, sola declina ion [ ad],
ϕla i ude o he si e o in e es [ ad],
ωs=a ccos (− an ϕ an δ), sunse hou angle [ ad],
JD numbe o Julian day.
Wa e 2021,13, 1484 7 o 23
The e ms Rsand Rns could be compu ed by
Rs=as+bsn
NRa, (6)
Rns =(1−α)Rs, (7)
whe e
as
,
bs
a e Angs öm coe icien s,
n[h]
and
N[h]
a e ac ual and maximum possible du a-
ion o dayligh , espec i ely. Finally, αdeno es albedo, i.e., he coe icien o e lec ion.
As he Angs öm coe icien s a e no calcula ed based on he ac ual sola adia ion mea-
su emen s he e, he FAO pape [
1
] ecommenda ion
as=
0.25 and
bs=
0.5 in Equa ion
(6)
is used. Fu he mo e, he ee wa e su ace albedo is se as α=0.08 based on [1].
The maximum dayligh du a ion Nis compu ed as
N=24
πωs. (8)
To de e mine ne longwa e adia ion Rnl, he ollowing o mula is used
Rnl =σ (Tmax +273.16)4+(Tmin +273.16)4
2! 0.34−0.14√ea!1.35 Rs
Rso −0.35. (9)
The abo e o mula
(9)
uses he S e an-Bol zmann cons an
σ=4.903 ×10−9MJK−4m−2day−1
and
Rso
in
hMJm−2day−1i
, which is he clea -sky
adia ion.
The alue o he clea -sky adia ion Rso is calcula ed as
Rso =0.75 +2×10−5zRa,
whe e zis he si e al i ude in [m]abo e he sea le el.
Howe e , despi e i s complexi y o he FAO Penman–Mon ei h Equa ion
(1)
, i is
no possible o conside
EFAO
esul s o be accu a e, since he numbe o inpu da a o
be measu ed o calcula ed by empi ical o mulae based on measu ed inpu da a. Such
an es ima ion p ocess is a ec ed by measu emen and calcula ion e o s. Fo example,
in [
32
,
33
], one could ind sensi i i y analysis o he FAO Penman–Mon ei h equa ion in
di e en clima e condi ions.
I should be poin ed ou ha Equa ion
(1)
was de i ed as a me hod o de e mine he
e e ence a e o e apo anspi a ion, i.e., he e e ence a e o e apo a ion om g owing plan s
wi h he cha ac e is ics o hypo he ical e e ence c ops such as heigh ,ae odynamic esis ance o
hei su ace, and albedo. Fo eal c ops, he a e o e apo anspi a ion is de e mined om
EFAO by mul iplying he c op- ela ed coe icien Kc
Ec op =KcEFAO.
Wi h he p ope coe icien , he
EFAO
o mula could be used o es ima e open wa e
e apo a ion. The alues o coe icien
Kc
, i.e.,
Kc,mid
and
Kc,end
o mid and end season
espec i ely a e abula ed in [
1
]. Speci ically,
Kc,mid =Kc,end =
1.05 o shallow lakes, i.e., o
hose wi h a dep h o up o
2m
. Fo deep lakes, i.e., wi h a dep h exceeding
5m
, he alues
Kc,mid =
0.65 and
Kc,end =
1.25 a e indica ed. The e o e, i should be bo ne in mind ha
(especially in he case o deep lakes) he esul o
EFAO
could lead o he unde es ima ion
o up o 35% o he o e es ima ion up o 25% du ing he season.
Since ou esea ch is no ocused only on he a ea o Lake Mos , bu also on lakes
ha a e only planned and does no exis a p esen , i is impossible o use limnological
and ba hyme ic da a, such as empe a u e p o ile o wa e dep h. This leads us o he
conside a ions o he a icle [
7
], which s a es ha in he case o missing limnological da a,
he lake coe icien
Kc=
1 can be selec ed and
EFAO
esul i sel could be conside ed o be
Wa e 2021,13, 1484 8 o 23
an open wa e e apo a ion es ima e. Hence, in all ou calib a ion and alida ion p ocesses,
he equa ion EFAO se es us as an e alon and all ou esul s a e compa ed agains i .
2.4. E apo a ion Es ima ion Me hods
Since he FAO Penman–Mon ei h equa ion
EFAO
(see
(1)
in Sec ion 2.3) is e y inpu -
in ensi e and complex in i s calcula ion p ocedu e, many o he me hods ha e been de i ed
o de e mine he a e o e apo a ion. Depending on he inpu s o he me hod, we di ide
hem in o empe a u e-, adia ion-, mass ans e -based, and combined me hods.
Tempe a u e-based me hod equa ions can be conside ed he simples ype o equa ions.
They p ima ily wo k wi h a single a iable, namely he a e age mean ai empe a u e
Ta
.
Qui e o en hese equa ions ha e a linea o m
E=p Ta+q
, bu hey also occu in he
o m
E=k Tm
a
, o he o m o exponen ial o mula
E=
10
p Ta+q
o
E=
e
p Ta+q
. Howe e ,
he g oup also includes ela ions in which he empe a u e occu s in combina ion wi h a
membe comp ising, o ins ance, ela i e humidi y
RH
o heo e ical leng h o he sola day
N
.
In his sec ion, we selec ed 7 simple (in compa ison o he complexi y o FAO Penman–
Mon ei h) e apo a ion models o he demons a ion o ou calib a ion app oach.
2.4.1. Reg ession De i ed Rela ions—Czech Republic
The ollowing h ee ela ions a e used in he Czech Republic. They a e de i ed by
eg ession be ween he obse ed e apo a ion and mean daily ai empe a u e, using s a is-
ical eg ession o ind bo h linea and exponen ial models. The model ela ions p esen ed in
his sec ion a e compa ed in he pape [34] wi h measu emen s on 20m2e apo a ion pan
placed in he me eo ological s a ion Hlasi o nea he ci y o Tábo 49
◦
C290
0
N, 14
◦
C450
0
E)
in he Sou h Bohemian Region. This s a ion is ope a ed by Výzkumný ús a odohospodᡠský
T. G. Masa yka (VUV, T. G. Masa yk Wa e Resea ch Ins i u e) and was buil in 1957 and has
a
20m2
e apo a ion ank, GGI-3000 pan and Class-A pan. The pan e apo a ion measu e-
men s he e a e ca ied ou om May o Oc obe , which is due o he empe a u es below
he eezing poin in he win e mon hs. The models a e gi en by
ES=100.0452 Ta−0.204, (10)
EBV =0.2157 Ta+0.1133, (11)
EVUV =0.2157 Ta+0.726 u2−1.2259, (12)
whe e
ES
is he equa ion acco ding o Še me [
35
],
EBV
acco ding o Be an and Vizina [
36
],
and
EVUV
acco ding o Adam Be an om VUV published in he o icial epo Model
p ˚ubˇehu me eo ologických eliˇcin p o oblas jeze a Mos do oku 2050 (The modelling o he
cou se o me eo ological quan i ies o Lake Mos a ea un il 2050). In all equa ions, he
e apo a ion a e is de e mined in
[mmday−1]
. Equa ion
(10)
has he o m o an exponen ial
unc ion and he e o e i s esul s can ne e be nega i e. Howe e , i mus be men ioned
ha he e a e limi a ions o Equa ions
(11)
and
(12)
: i he equa ion p oduces he nega i e
e apo a ion es ima ion, we se he alue equal o ze o. Fo ins ance,
EBV =
0 is se on days
wi h he mean empe a u e below −0.526 °C, since EBV would be nega i e in such cases.
To calib a e he models, we p esen a pa ame ic o mula ion o he
Equa ions (10)–(12) by
ES(θ) = 10θ1Ta+θ2, (13)
EBV(θ) = max{θ1Ta+θ2,0}, (14)
EVUV(θ) = max{θ1Ta+θ2u2+θ3,0}, (15)
whe e
θ
a e unknown pa ame e s, which will be calib a ed. We ex ended models by
p ojec ion o nonnega i e numbe s (using he ou e
max
unc ion) o en o ce he compu ed
nonnega i e e apo a ion.
Wa e 2021,13, 1484 9 o 23
2.4.2. Kha u a
The equa ion p esen ed by Kha u a in [
37
] is an example o a nonlinea empe a u e
o mula. I is mos ly w i en in he li e a u e in he o m
EK=0.34 p T1.3
a(16)
and esul s in he e apo a ion a e in
[mmday−1]
. In
(16)
, a iable
p
deno es he pe cen age
o o al day ime hou s o he daily pe iod ou o o al day ime hou s o he yea . This
o m is used, o ins ance, in [
38
,
39
]. The coe icien 0.34 was ound empi ically and i is
possible o e ine i wi h espec o he si e-speci ic condi ions. Fo example, in [
34
], he
o m
EK=
0.25
p T1.3
a
is gi en wi h he o mula being calib a ed o he condi ions o he
Hlasi o wea he s a ion in he Sou h Bohemian Region. In his s udy, he o m
(16)
wi h
coe icien 0.34 is used.
In his pape , we he calib a e model (16) in oducing he pa ame ic e sion
EK(θ) = (θ1pTθ2
ai Ta>0,
0 i Ta≤0, (17)
and calib a e he unknown pa ame e s θ∈R2.
2.4.3. Ha g ea es–Samani
Ano he me hod was in oduced by Ha g ea es in a icle [
40
] and u he modi ied
o he o m which can be ound in a icle [
41
]. Usually, he Ha g ea es–Samani equa ion is
gi en in i s basic o m
EHS =0.0023 RaT1
2
(Ta+17.8).
In his equa ion, a iable
T
deno es he di e ence be ween daily maximum and
minimum ai empe a u es [°C].
Al hough he o mula con ains a adia ion e m
Ra
, i is anked among he empe a u e-
based o mulae since he e m
Ra
he e is jus a heo e ical alue calcula ed acco ding o
Fo mula
(5)
. Using his compu a ion o
Ra
, he Ha g ea es–Samani equa ion akes any o
he ollowing equi alen o ms
EHS =0.0023 Ra
λ(Ta+17.8)√T =0.0023(0.408 Ra)(Ta+17.8)√T =0.00094 Ra(Ta+17.8)√T .(18)
The di e ence o hese o ms is only in he usage o di ision by he la en hea o apo -
iza ion o wa e
λ=2.45MJkg−1
, which is pe o med o ob ain he esul s in millime es
pe day.
In his pape , we conside he pa ame ic o m o Ha g ea es–Samani Equa ion (18)
EHS(θ) = maxθ1Ra(Ta+θ2)√T
λ,0(19)
wi h pa ame e s
θ∈R2
. These pa ame e s will be op imized du ing he calib a ion p ocess.
2.4.4. Schendel
In con as o he o mulae men ioned abo e, in which he ai empe a u e is su icien
o calcula e he e apo a ion a e, he ai ela i e humidi y
RH
measu emen is equi ed in
he ollowing Schendel equa ion. The o mula has a simple o m
ESch =16 Ta
RH. (20)
The equa ion can be ound in he o iginal Schendel pape [
42
]. I is used by many
au ho s, o example, see [19,20].
Wa e 2021,13, 1484 16 o 23
Table 3.
The compa ison o s a is ical measu es be o e and a e he calib a ion p ocess o he
Kha u a equa ion. The i s ow ep esen s he alues o a ious s a is ical measu es (see columns)
o he o iginal model. Remaining ows de e mine he objec i e s a is ical measu e, which espec o
he model was calib a ed. The columns ep esen he co esponding alues o a ious measu es.
Ob ained Value
NSE RMSE MAE PBIAS
Calib a ion
objec i e
O iginal 0.6675858 1.0387729 0.7795765 15.480837
NSE 0.8511901 0.6950192 0.5481185 −1.191246
RMSE 0.8511973 0.6950022 0.5482104 −1.156551
MAE 0.8509640 0.6955467 0.5478520 −2.451671
PBIAS 0.8149230 0.7750987 0.6029666 0.003074
Table 4.
The compa ison o s a is ical measu es be o e and a e he calib a ion p ocess o he
Ha g ea es–Samani equa ion. The i s ow ep esen s he alues o a ious s a is ical measu es (see
columns) o he o iginal model. Remaining ows de e mine he objec i e s a is ical measu e, which
espec o he model was calib a ed. The columns ep esen he co esponding alues o a ious
measu es.
Ob ained Value
NSE RMSE MAE PBIAS
Calib a ion
objec i e
O iginal 0.8935767 0.5877587 0.4377886 10.9323242
NSE 0.9300600 0.4764787 0.3440658 −1.2411942
RMSE 0.9300550 0.4764958 0.3442857 −1.0261792
MAE 0.9298460 0.4772072 0.3432423 −2.2165424
PBIAS 0.9297419 0.4775611 0.3461323 0.0023911
Table 5.
The compa ison o s a is ical measu es be o e and a e he calib a ion p ocess o he Tu c
equa ion. The i s ow ep esen s he alues o a ious s a is ical measu es (see columns) o he
o iginal model. Remaining ows de e mine he objec i e s a is ical measu e, which espec o he
model was calib a ed. The columns ep esen he co esponding alues o a ious measu es.
Ob ained Value
NSE RMSE MAE PBIAS
Calib a ion
objec i e
O iginal 0.6675858 1.03877290 0.7795765 15.4808373
NSE 0.8511901 0.6950192 0.5481185 −1.1912456
RMSE 0.8511973 0.6950022 0.5482104 −1.1565508
MAE 0.8509640 0.6955467 0.5478520 −2.4516705
PBIAS 0.8149230 0.7750987 0.6029666 0.0030740
As we men ioned abo e, he andom choice o calib a ion da a in he calib a ion
p ocess causes he andomness o he op imal pa ame e s. In Figu es 5–7, we p esen he
di e en op imal alues o calib a ed models wi h espec o a ious s a is ical measu es.
We can obse e ha using he c oss- alida ion app oach, he inal op imal pa ame e s a e
a andom a iable as well.
The calib a ion p ocess i s he model o he alues compu ed by he FAO Penman–
Mon ei h equa ion. We demons a e his capabili y on speci ic examples. Figu es 8–10
p esen he compa ison o mon hly e apo a ion compu ed by FAO Penman–Mon ei h
Equa ion (1)
, Ha g ea es–Samani Equa ion
(16)
, and calib a ed Ha g ea es–Samani Equa-
ion
(17)
in he o m o cumula i e sum and he sca e plo . The calib a ion has been
pe o med wi h espec o NSE.

Wa e 2021,13, 1484 17 o 23
Figu e 5.
The compa ison o he calib a ed Kha u a model pa ame e s ob ained by he c oss- alida ion calib a ion p ocess
on aining da a wi h espec o a ious s a is ical measu es.
Figu e 6.
The compa ison o he calib a ed Ha g ea es–Samani model pa ame e s ob ained by he c oss- alida ion
calib a ion p ocess on aining da a wi h espec o a ious s a is ical measu es.
Figu e 7. The compa ison o he calib a ed Tu c model pa ame e s ob ained by he c oss- alida ion calib a ion p ocess on
aining da a wi h espec o a ious s a is ical measu es.
Wa e 2021,13, 1484 18 o 23
Figu e 8.
The compa ison o he daily e apo a ion compu ed by FAO Penman–Mon ei h equa ion, Kha u a equa ion, and
calib a ed Kha u a equa ion wi h espec o NSE in he o m o cumula i e e apo a ion (le ) and he sca e plo ( igh ).
Figu e 9.
The compa ison o he daily e apo a ion compu ed by FAO Penman–Mon ei h equa ion, Ha g ea es–Samani
equa ion, and calib a ed Ha g ea es–Samani equa ion wi h espec o NSE in he o m o cumula i e e apo a ion (
le
) and
he sca e plo ( igh ).
Figu e 10.
The compa ison o he daily e apo a ion compu ed by FAO Penman–Mon ei h equa ion, Tu c equa ion, and
calib a ed Tu c equa ion wi h espec o NSE in he o m o cumula i e e apo a ion (le ) and he sca e plo ( igh ).
Wa e 2021,13, 1484 19 o 23
The inal esul is p esen ed in Figu e 11. He e, we demons a e he mon hly e apo a-
ion compu ed by Ha g ea es–Samani equa ion in he o m o a ime-line and his og am o
e apo a ion in mon hs. The pa ame e s o he calib a ed models can be ound in Table 2
and he imp o emen o he used s a is ical measu e on he whole da a se in Table 1.
Figu e 11.
The compa ison o he mon hly e apo a ion compu ed by FAO Penman–Mon ei h equa ion, Ha g ea es–Samani
equa ion, and calib a ed Ha g ea es–Samani equa ion. The calib a ion has been pe o med using a ious s ochas ic
measu es.
Wa e 2021,13, 1484 20 o 23
4. Discussion
In ou esul s p esen ed in he p e ious sec ion, we p ocessed he esul s o calib a ion
on andom da a om he c oss- alida ion p ocess (see Sec ion 2.6). Be o e he selec ion o
he mean model, we emo ed he ou lie s based on he qua ile h esholding. Fo ins ance,
in he case o he Kha u a equa ion calib a ed wi h espec o
NSE
, we emo ed 11 ou lie s
mo e han 1.5 in e qua ile anges (IQRs) below he i s qua ile o abo e he hi d qua ile.
Ou esul s show ha he calib a ed pa ame e s depend on he chosen s a is ical
measu e, see Table 2. Howe e , all o hem a e imp o ing he objec i e alue in compa ison
wi h he o iginal equa ions, see Table 1.
F om he ob ained esul s, we obse ed ha he calib a ion wi h espec o one se-
lec ed measu e imp o es no only his objec i e measu e bu also imp o es he emaining
measu es. See Tables 3–5, whe e we examined he Kha u a, Ha g ea es–Samani, and
Tu c model.
The measu es p esen ed in Sec ion 2.5 a e de ined as he sum o local di e ences. Ou
esul s o cumula i e e apo a ion p esen ed in Figu es 8–10 show he consequences o he
o mula ion o he objec i e unc ion in his o m— he cumula i e e apo a ion compu ed
by he op imal calib a ed model i s he cumula i e e apo a ion compu ed by he
FAO
equa ion. We obse ed his p ope y in he case o all measu es. Howe e , in he case o
daily e apo a ion (o mon hly e apo a ion), he local di e ence can be la ge, see
Figu e 11
.
E apo a ion in some mon hs has been unde es ima ed and in o he mon hs has been
o e es ima ed. In any case, his unde es ima ion and o e es ima ion a e always be e han
in he case o he o iginal equa ion.
The ob ained esul s ollow he equi alency o he calib a ion p ocess based on
NSE
maximiza ion and
RMSE
minimiza ion, i.e., Theo em 1. Please see Figu es 5–7, whe e we
demons a e he densi y o op imal pa ame e s o he calib a ed Kha u a, Ha g ea es–
Samani, and Tu c equa ion wi h espec o he andom da a spli in he c oss- alida ion
p ocess. The small di e ence be ween
NSE
and
RMSE
is caused by he e o o he i e a i e
algo i hm: he op imiza ion algo i hm has a s opping c i e ium based on he change o he
unc ion alue. Since he
NSE
and
RMSE
ha e di e en objec i e unc ions, he i e a i e
algo i hm s ops he op imiza ion p ema u ely (su icien ly app oxima ely) in di e en
op imize s. Especially in he case o Figu e 6, he di e ence is clea ly obse able. Howe e ,
in his case, we a e dealing wi h he Ha g ea es–Samani model
(19)
. We suppose ha
his di e ence is caused by he non-linea i y o he model (and he non-linea i y o used
s a is ical measu es). The di e ence be ween objec i e unc ions in he solu ions compu ed
by
NSE
and
RMSE
is app oxima ely 10
−2
(see Table 4), which is he alue used in he
s opping c i e ia o he i e a i e op imiza ion algo i hm. The si ua ion is simila o
RMSE
.
The esul s ob ained by ou analysis show ha he calib a ed Ha g ea es–Samani and
Tu c models seem o be he mos sui able simpli ica ion o he FAO Penman–Mon ei h
equa ion in he a ea o Lake Mos . Howe e , i is necessa y o men ion ha he inal choice
o he mos sui able calib a ed equa ion o e apo a ion modelling depends no only on he
inal alue o he s a is ical measu es bu also on he inpu da a equi emen s. The e o e,
we sugges using he Ha g ea es–Samani equa ion since his equa ion equi es only he
inpu o he ex a e es ial adia ion and he ai empe a u e, see Equa ion
(18)
. Figu e 11
p esen s he inal imp o ed e apo a ion es ima ion.
5. Conclusions
In his pape , we p esen ed he me hodology o he calib a ion o e apo a ion models
wi h he FAO Penman–Mon ei h equa ion and demons a ed i on selec ed simpli ied
models using he mos common s a is ical measu es. Addi ionally, we implemen ed a c oss-
alida ion p ocess o emo e he o e i ing o he calib a ed model. This app oach can be
easily applied o any model o in e es and any su icien ly easonable s a is ical measu e.
In he pape , we p esen ed a calib a ion wi h espec o heo e ical alues compu ed by
FAO Penman–Mon ei h equa ion; howe e , he me hodology can be used o calib a ion
wi h any heo e ical o measu ed e e ence alues o e apo a ion.
Wa e 2021,13, 1484 21 o 23
F om he p esen ed esul s, we sugges using he Ha g ea es–Samani equa ion o
model he e apo a ion on Lake Mos . This equa ion epo ed he su icien app oxima ion
o he FAO Penman–Mon ei h equa ion and addi ionally, i equi es only a ew inpu
pa ame e s, which can be easily (and cheaply) measu ed.
Du ing ou esea ch, we obse ed he global i ing p ope y o common s a is ical
measu es— he e apo a ion du ing cold days is unde es ima ed and he e apo a ion du ing
sunny days is o e es ima ed. To deal wi h his issue, we ocus ou u u e wo k on he
di ision o days in o g oups wi h di e en op imal models.
Au ho Con ibu ions:
Concep ualiza ion, V.D. and D.D.; me hodology, V.D., L.P.; so wa e, V.D.;
alida ion, V.D.; o mal analysis, L.P. and V.D.; in es iga ion, V.D. and D.D.; esou ces, D.D. and V.D.;
da a cu a ion, V.D. and D.D.; w i ing—o iginal d a p epa a ion, V.D. and D.D.; w i ing— e iew and
edi ing, L.P.; isualiza ion, V.D.; supe ision, D.D.; p ojec adminis a ion, D.D.; unding acquisi ion,
D.D. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This pape has been comple ed hanks o he inancial suppo p o ided o VSB-Technical
Uni e si y o Os a a by he Czech Minis y o Educa ion, You h and Spo s om he budge o
concep ual de elopmen o science, esea ch and inno a ions o he 2021 yea and he Depa men
o Ma hema ics a he Facul y o Ci il Enginee ing, VSB-Technical Uni e si y o Os a a.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The da a a e no publicly a ailable due o he equi emen s o con i-
den iali y and secu i y. Da a was ob ained om The Ins i u e o A mosphe ic Physics CAS and a e
a ailable om he au ho s wi h he pe mission o The Ins i u e o A mosphe ic Physics CAS.
Acknowledgmen s:
The au ho s would like o exp ess u he hanks o The Ins i u e o A mosphe ic
Physics CAS o he p o ided me eo ological da a om hei measu ing s a ion Kopis y.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
EFAO FAO Penman–Mon ei h Equa ion (1),
ESŠe me Equa ion (10),
EBV Be an-Vizina Equa ion (11),
EVUV he equa ion ecommended by T. G. Masa yk Wa e Resea ch Ins i u e (12),
EKKha u a Equa ion (16),
EHS Ha g ea es–Samani Equa ion (18),
ESch Schendel Equa ion (20),
EPT P ies ley–Taylo Equa ion (22),
PBIAS Pe cen age Bias (29),
MAE Mean Absolu e E o (28),
RMSE Roo Mean Squa e E o (27),
NSE Nash–Su cli e E iciency (26),
Ta he a e age ai empe a u e [°C],
Tmax,Tmin maximal and minimal ai empe a u e [°C],
T he di e ence be ween daily maximum and minimum ai empe a u es [°C],
Pa mosphe ic p essu e [kPa],
ppe cen age o o al day ime hou s o he pe iod used (daily o mon hly) ou o o al
day ime hou s o he yea .
RH ela i e humidi y [%],
Ra he ex a e es ial adia ion hMJm−2day−1i,
Rnne adia ion [kJ ·m−2·s−1],
Rs he sola adia ion hMJ m−2day−1i,

Wa e 2021,13, 1484 22 o 23
Rso he clea -sky adia ion hMJ m−2day−1i,
γ he psych ome ic cons an [m·s−1],
σS e an-Bol zmann cons an σ=4.903 ×10−9MJK−4m−2day−1
λla en hea o apo iza ion λ=2.45 MJ kg−1,
Cp he speci ic hea o ai , Cp=1013 Jkg−1°C−1,
ρa he ai densi y kg m−3,
∆ he slope o sa u a ion apo p essu e cu e [kPa ·[°C]−1],
u2wind speed a heigh 2m m s−1,
G he hea low in he soil [MJ m−2day−1],
esmean sa u a ion apo p essu e [kPa],
ea he cu en wa e apo p essu e [kPa],
es−ea apo p essu e de ici [kPa].
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