Optimal calibration of evaporation models against Penman-Monteith equation

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]. Re e ences 1. Allen, R.G.; Pe ei a, L.; Raes, D.; Smi h, M. C op E apo anspi a ion-Guidelines o Compu ing C op Wa e Requi emen s-FAO I iga ion and D ainage Pape 56; Uni ed Na ion-Food and Ag icul u e o ganisa ion: Rome, I aly, 1998; Volume 56. 2. Jensen, M.E.; Allen, R.G. E apo a ion, E apo anspi a ion, and I iga ion Wa e Requi emen s, 2nd ed.; Ame ican Socie y o Ci il Enginee s: Res on, VA, USA, 2016. [C ossRe ] 3. Gallego-El i a, B.; Baille, A.; Ma in-Go iz, B.; Maes e, J.; Ál a ez, V. E alua ion o e apo a ion es ima ion me hods o a co e ed ese oi in a semi-a id clima e (sou h-eas e n Spain). J. Hyd ol. 2012,458, 59–67. [C ossRe ] 4. C aig, I. Compa ison o p ecise wa e dep h measu emen s on ag icul u al s o ages wi h open wa e e apo a ion es ima es. Ag ic. Wa e Manag. 2006,85, 193–200. [C ossRe ] 5. Sca o, F.B.; Ma co Tina, G.; Gagliano, A. Compa ison o e apo a ion models o ee wa e basins su aces. In P oceedings o he 2019 4 h In e na ional Con e ence on Sma and Sus ainable Technologies (SpliTech), Spli , C oa ia, 18–21 June 2019; pp. 1–7. [C ossRe ] 6. Abd-Elhamid, H.F.; Ahmed, A.; Zeleˇnáko á, M.; V anayo á, Z.; Fa hy, I. Rese oi Managemen by Reducing E apo a ion Using Floa ing Pho o ol aic Sys em: A Case S udy o Lake Nasse , Egyp . 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