ORIGINAL PAPER

Estimation of open water evaporation using land-basedmeteorological data

Fawen Li1 & Yong Zhao2

Received: 15 March 2017 /Accepted: 28 September 2017# Springer-Verlag GmbH Austria 2017

Abstract Water surface evaporation is an important processin the hydrologic and energy cycles. Accurate simulation ofwater evaporation is important for the evaluation of waterresources. In this paper, using meteorological data from theAixinzhuang reservoir, the main factors affecting water sur-face evaporation were determined by the principal componentanalysis method. To illustrate the influence of these factors onwater surface evaporation, the paper first adopted the Daltonmodel to simulate water surface evaporation. The resultsshowed that the simulation precision was poor for the peakvalue zone. To improve the model simulation’s precision, amodified Dalton model considering relative humidity wasproposed. The results show that the 10-day average relativeerror is 17.2%, assessed as qualified; the monthly averagerelative error is 12.5%, assessed as qualified; and the yearlyaverage relative error is 3.4%, assessed as excellent. To vali-date its applicability, the meteorological data of Kuanchengstation in the Luan River basin were selected to test the mod-ified model. The results show that the 10-day average relativeerror is 15.4%, assessed as qualified; the monthly averagerelative error is 13.3%, assessed as qualified; and the yearlyaverage relative error is 6.0%, assessed as good. These results

showed that the modified model had good applicability andversatility. The research results can provide technical supportfor the calculation of water surface evaporation in northernChina or similar regions.

1 Introduction

Water surface evaporation is an important process in the hy-drologic and energy cycles (Fu et al. 2004; Guzha et al. 2015;Hu et al. 2017a, b). Estimates of the amount and rate of evap-oration from open water surfaces are required in water re-source management projects such as the design of storagereservoirs, computation of basin water balance, estimation ofmunicipal and industrial water supply and management ofwetlands (Brutsaert 1982; Marsh and Bigras 1988; Finch2001). Therefore, accurate simulation of water evaporationis important for evaluating the reliability of the distributionand quantity of water resources.

Several methods are currently used to estimate evaporationfor open-water reservoirs based on meteorological data. Theyare generally categorised into temperature and radiation (Xuand Singh 2000, 2001), mass transfer (aerodynamic) (Singhand Xu 1997), pan coefficient (Fu et al. 2004) and energybudget and combinat ion methods (Gianniou andAntonopoulos 2007; Rosenberry et al. 2007). The accuracyand reliability of different methods for estimating open waterevaporation were discussed in detail by Craig and Hancock(2004). Different methods for estimating water evaporationhave been and are still used. For example, Dalton-basedmethods are widely used in engineering practice because oftheir ease of application and suitability in modelling exercises(e.g. Hostetler and Bartlein 1990; Blodgett et al. 1997;Armstrong et al. 2008; Korzukhin et al. 2011; Jodat et al.2012). The Dalton model has a simple structure and few

* Fawen Lilifawen@tju.edu.cn

Yong Zhaozhaoyong@iwhr.com

1 State Key Laboratory of Hydraulic Engineering Simulation andSafety, Tianjin University, Tianjin 300072, People’s Republic ofChina

2 State Key Laboratory of Simulation and Regulation of Water Cyclein River Basin, China Institute of Water Resource and Hydro-powerResearch, Beijing 100038, People’s Republic of China

Theor Appl Climatolhttps://doi.org/10.1007/s00704-017-2281-8

intermediate links, thereby reducing the uncertainties of themodel. Further, parameter optimization in the model requireslimited data. Dalton stated that water evaporation is propor-tional to the difference between vapour pressures at the sur-face of the water and in the ambient air and that the velocity ofthe wind affects this proportionality (Amin et al. 2012). Thegeneral form of Dalton is as follows:

E ¼ f uð Þ ew−eað Þ ð1Þ

where ew is the saturated vapour pressure at the water surfacetemperature, hPa; ea is the partial vapour pressure in the air at aspecified location height, hPa; u is wind speed, m/s; andf(u) isa wind speed function. The form of f(u) certainly depends onwhether u and ea are measured over the land upwind of thewater body or over the water body itself. Other factors thatinfluence f(u) include the size of the water body, the measure-ment height, near surface buoyancy effects, the formation ofwaves and the roughness of the terrain surrounding the waterbody. Consequently, wind functions derived for specific waterbodies may not be accurate when applied to other locations(Mcjannet et al. 2012). Additionally, the simulation of watersurface evaporation at different stations by the Dalton modelhas different system errors with distinct signs in each season(Li et al. 2013). The primary reason for this is that the relativehumidity of each season varies greatly, causing the sign of thefitting error to alternate with the seasons. However, the Daltonwater surface model does not consider the effect of relativehumidity on evaporation. Therefore, it is necessary to improvethe Dalton model and make it a more adaptable water surfaceevaporation model.

This paper selected the Ziya River basin located in thesouthern portion of the Haihe River basin and the LuanRiver basin located in the northeast portion of the HaiheRiver basin as its study areas. At present, there is littleresearch on water surface evaporation in northern China. Mu(2014) adopted the Dalton model to simulate water surfaceevaporation in the Ziya River basin, and the precision of themodel was relatively low. Dong et al. (2015) analysed watersurface evaporation using the Dalton and Penman equations inthe southern Horqin sandy area of Inner Mongolia, China, andfound that the stability of the Dalton model was worse thanthat of Penman. All the results of the above studies showedthat the Dalton model’s precision was low, and the applicabil-ity of the model remained unverified. Moreover, the influenceof other meteorological factors on the model, especially rela-tive humidity, was not considered in the above studies. In thispaper, according to the meteorological data for theAixinzhuang reservoir, the main factors affecting water sur-face evaporation were determined by the principal componentanalysis (PCA) method. To illustrate the influence of thesefactors on water surface evaporation, the paper first adoptedthe Dalton model to simulate water surface evaporation. The

results showed that the simulation precision of the Daltonmodel was poor for the peak value zone. Therefore, to im-prove the simulation precision of the model, a modifiedDalton model considering these factors was proposed thatshowed high accuracy. Moreover, the modified model wasapplied to the Kuancheng reservoir located in the LuanRiver basin, where it also showed high accuracy, demonstrat-ing that the modified model had good applicability and versa-tility. The research results can provide technical support forthe calculation of water surface evaporation in northern Chinaor similar regions.

2 Study sites and meteorological data

2.1 Study sites

The paper selected the Aixinzhuang and Kuancheng reser-voirs as its cases. The two reservoirs are located in the ZiyaRiver basin and Luanhe River basin, respectively. Figure 1shows the location of the study areas.

The Ziya River basin lies in the central southern portion ofthe Haihe River basin. As the second largest basin of theHaihe River, it stretches across Shanxi, Hebei and Tianjin.The eastern, western, southern and northern boundaries areBohai Bay, Taihang Mountain, Zhangwei River and DaqingRiver, respectively. The Ziya River basin has a drainage areaof 48,165 km2. Located in a temperate continental monsoonclimate zone, it has long summers and winters, while springsand autumns are short. The annual average temperature rangesfrom 11.8 to 12.9 °C. The annual average precipitation is540 mm. The annual precipitation is distributed unevenly;approximately 80% of the annual precipitation occurs inJuly, August and September.

The Luanhe River basin is located in the north-eastern partof the Haihe River basin. Originating from the northern foot ofBayanguer Mountain in Zhangjiakou, Hebei Province, theLuanhe River travels through Hebei Province, InnerMongolia autonomous region and Liaoning Province and fi-nally flows into Bohai Bay. The Luanhe River basin has adrainage area of 44,750 km2. The basin is in a temperatesemiarid continental monsoon climate, with an annual averagetemperature of − 0.3 to 11 °C, an annual average precipitationof 400–700 mm and an annual average potential evapotrans-piration of 950–1150 mm. The spatial and temporal distribu-tions of the precipitation are uneven, and approximately 70–80% of the annual precipitation occurs from June toSeptember.

2.2 Meteorological data

The data, including the daily maximum/minimum tempera-tures, daily wind speed, daily relative humidity, water surface

F. Li, Y. Zhao

temperature, water vapour pressure and vapour pressure defi-cit, were provided by the Chinese Meteorological DataSharing Service System (http://cdc.cma.gov.cn/home.do).Data of water evaporation were provided by the Hydrologyand Water Resource Survey Bureau of Hebei Province. Thedata were available from 2007 to 2012 for the Ziya Riverbasin and Luanhe River basin.

3 Methods

3.1 Determination of impact factors

Water surface evaporation is very sensitive to weather condi-tions; the empirical formula of water surface evaporation has astrong local characteristic. For open water and specific waterquality, the impact factors of water surface evaporation can besummarised into two categories, namely, thermodynamic fac-tors and dynamic factors. At present, the primary meteorolog-ical impact factors, including wind speed, water vapour pres-sure, water vapour pressure difference, relative humidity, wa-ter temperature and temperature difference, are considered in

most studies of water surface evaporation (Xu and Singh2000, 2001). However, there are interactions and informationoverlaps between various factors; some components of themeteorological factors need to be transformed into indepen-dent indicators before the impact factors are determined. PCA,as a statistical method to simplify factors, takes into accountthe relationship between various factors, using a dimensionreduction method to determine several comprehensive factors.These factors are unrelated to one another to represent thenumerous original variables and reflect the information oforiginal factors as much as possible (Li et al. 2007; Rao2014; Abou Zakhem 2016; Ahmadalipour et al. 2017; Huet al. 2017a, b).

The paper used the PCA method to analyse the sensitivefactors of water surface evaporation and determine themain factors affecting water surface evaporation. The pa-per selected six primary meteorological factors, includingwind speed, water vapour pressure difference, water tem-perature, relative humidity, water vapour pressure and tem-perature difference, as the impact factors, which were alsofrequently considered in some empirical formulas of watersurface evaporation (Shakir et al. 2008; Zhang et al.2010;

Fig. 1 Location of the study sites

Estimation of open water evaporation using land-based meteorological data

Zheng and Wang 2014; Li et al.2016). On this basis, factoranalysis was carried out by running factor analysis (ana-lyse > dimension reduction > factor) and checking the boxfor BKMO (Kaiser-Meyer-Olkin) and Bartlett’s test ofsphericity^ in SPSS20.0 statistical software. KMO andBartlett’s test of sphericity are measures of sampling ade-quacy that are recommended to check the case to variableratio for the analysis being conducted. While the KMOranges from 0 to 1, the accepted index worldwide is over0.6. Additionally, the Bartlett’s test of sphericity relates tothe significance of the study and thereby shows the validityand suitability of the responses collected to the problembeing addressed through the study. For factor analysis tobe recommended as suitable, the Bartlett’s test of spheric-ity must be less than 0.05. The results of the test are shownin Tables 1 and 2.

Table 1 shows the correlation between various factors. Itcan be seen from Table 1 that the correlation coefficients ofwater vapour pressure, water vapour pressure difference andtemperature difference are relatively high, which means thatthere is significant correlation among them. In addition, watertemperature and water vapour pressure are also significantlycorrelated, while the correlation between wind speed, relativehumidity and other factors is not obvious.

The KMO statistic in Table 2 is an indicator used to com-pare the simple correlation coefficient and partial correlationcoefficient among variables. The closer the value of KMO isto 1, the more suitable the variables are to carry out factoranalysis. The Bartlett sphericity test assumes the correlationcoefficient matrix to be a unit matrix. If the statistical value isrelatively large and its corresponding probability value is lessthan the significance level specified by the user, the null hy-pothesis is rejected and the variables are suitable for factoranalysis. As seen from Table 2, the value of KMO is 0.600,which means that the variables selected are suitable for factoranalysis. If the Sig value is 0, smaller than the significancelevel of 0.05, so the null hypothesis is rejected, indicating thatthe variables are correlated and suitable for factor analysis.

To perform a factor analysis, we first need to select anBextraction method^ and a Brotation method.^ The

Bextraction^ button is selected to specify the extraction meth-od. Second, the box for a Bscree plot^ is checked. This willgive a scree diagram. Third, the section labelled Bextract^ isviewed. The default setting is for SPSS to use the Kaiserstopping criterion (i.e. all factors with eigenvalues greater than1) to decide howmany factors to extract. Lastly, the button forBrotation^ is clicked. The outputs are shown in Tables 3, 4 and5 and Fig. 2.

The eigenvalue indicates whether a variable has a strongeffect on that principal component, and the criterion forextracting the number of principal components is an eigenval-ue larger than 1. Table 4 shows that the first two cumulativecontributions reach 72.113%, so it is considered that the firsttwo components are sufficient to represent the basic informa-tion of all factors. It can be seen from the scree plot that theline slope of the first two principal components is steep at firstbut gradually becomes gentler. Moreover, the eigenvalues ofthe first principal component and the second principal com-ponent are larger than 1, while the eigenvalues of the rest ofthe components are smaller than 1. Therefore, the first twoprincipal components are the main factors to extract.

The component matrix describes the linear relationshipbetween the initial variables and the principal components,so the main initial variables affecting the principal compo-nents can be determined according to the coefficient ofeach variable. It can be seen from Table 5 that the factorscan be divided into two categories. One is the first princi-pal component, whose variance contribution is 51.914%,and the loadings of water vapour pressure difference, watertemperature, water vapour pressure and temperature differ-ence are relatively large. The first principal componentreflects the energy supply of the evaporation environmentand plays a leading role in the evaporation process. Fromthe perspective of physical mechanisms, the temperature

Table 1 The correlation matrix

Windspeed

Vapour pressuredifference

Watertemperature

Relativehumidity

Vapourpressure

Temperaturedifference

Wind speed 1.000 0.111 − 0.130 − 0.127 − 0.064 − 0.064

Vapour pressuredifference

0.111 1.000 0.597 − 0.285 0.829 − 0.749

Water temperature − 0.130 0.597 1.000 − 0.095 0.830 − 0.358

Relative humidity − 0.127 − 0.285 − 0.095 1.000 − 0.087 0.266

Vapour pressure − 0.064 0.829 0.830 − 0.087 1.000 − 0.700

Temperature difference − 0.064 − 0.749 − 0.358 0.266 − 0.700 1.000

Table 2 KMO andBartlett’s test KMO 0.600

χ2 of Bartlett’s test 554.418

df 15

Sig. 0.000

F. Li, Y. Zhao

difference determines the saturated water vapour contentand water vapour transmission speed in air, and the watertemperature determines the active degree of water. Thehigher the temperature difference and water temperatureare, the larger the evaporation rate. Since water vapourpressure difference is a function of water temperature, tem-perature difference and relative humidity, the influence ofwater temperature can be reflected by the water vapourpressure difference. Therefore, the first principal compo-nent can be characterised by only one factor—water va-pour pressure difference. The other is the second principalcomponent, whose variance contribution is 20.199%, andthe loadings of wind speed and relative humidity are rela-tively large. The second principal component reflects theaerodynamics around the evaporation surface, which playsan important role in the process of evaporation as well.Wind speed reflects the intensity of turbulent diffusion,and the larger the wind speed is, the faster the moleculesdiffuse and the larger the evaporation rate. Relative humid-ity reflects not only the saturation degree of air but also thedifferences between dry and wet climates of different sea-sons and regions. Therefore, the second principal compo-nent can be characterised by wind speed and relativehumidity.

Through PCA method analysis, the main impact factors ofwater surface evaporation inAixinzhuang are water vapour pres-sure difference, wind speed and relative humidity. The originalDalton model was a two-factor model including water vapour

pressure difference Δe and wind speed u (see formula 1).Obviously, the water surface evaporation of this area is affectedby relative humidity, so formula (1) is transformed into an im-proved water surface evaporation model with three factors:

E ¼ f uð Þ ew−eað Þg rð Þ ð2Þwhere g(r) represents the function of relative humidity.

3.2 Determination of model

Through the above analysis, we can see that relative humidityis an important impact factor in water surface evaporation. Toanalyse its influence on water surface evaporation, the paperfirst adopted the original Dalton model to simulate the watersurface evaporation and then compared the simulation preci-sion with the modified model considering relative humidity,thus proving the significance of relative humidity for watersurface evaporation in the study region.

In the Dalton formula, the wind speed function mainlyadopts two forms:

f uð Þ ¼ Aþ Buα ð3Þf uð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAþ Bu2

pð4Þ

where A, B and α are parameters of the wind speed function.

Table 3 Common factor variance

Initial Extraction

Zscore (wind speed) 1.000 0.601

Zscore (water vapour pressure difference) 1.000 0.866

Zscore (water temperature) 1.000 0.732

Zscore (relative humidity) 1.000 0.487

Zscore (water vapour pressure) 1.000 0.945

Zscore (temperature difference) 1.000 0.695

Table 4 Total variance explained

Component Initial eigenvalues Extraction Rotation

Total % of variance Cumulative % Total % of variance Cumulative % Total % of variance Cumulative %

1 3.115 51.914 51.914 3.115 51.914 51.914 3.065 51.089 51.089

2 1.212 20.199 72.113 1.212 20.199 72.113 1.261 21.023 72.113

3 0.845 14.077 86.190

4 0.589 9.808 95.998

5 0.185 3.082 99.080

6 0.055 0.920 100.000

Table 5 Component matrix

Component

1 2

Zscore (wind speed) 0.014 − 0.775

Zscore (water vapour pressure difference) 0.922 − 0.126

Zscore (water temperature) 0.783 0.345

Zscore (relative humidity) − 0.302 0.629

Zscore (water vapour pressure) 0.949 0.210

Zscore (temperature difference) − 0.812 0.189

Estimation of open water evaporation using land-based meteorological data

For f(u) =A +Buα, α is usually set as a constant smaller than1, so d2f(u)/du2 < 0 and f(u)~u is a convex curve, which is betterfitted with points for higher wind speed but more poorly fitted

with points for lower wind speed. For f uð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAþ Bu2

p,

d2f(u)/du2 > 0 and f(u)~u is a concave curve, which is better fittedwith points for lower wind speed but more poorly fitted withpoints for higher wind speed.

Based on the above two approaches to wind speed formu-las, Min (2005) proposed that the wind speed should be set asa piecewise function:

f uð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAþ Bu2

pu < u1ð Þ

Aþ Bu u1≤u≤u2ð ÞAþ Bu

uþnmu u > u2ð Þ

8<: ð5Þ

where u1 is the wind speed equal to 1 m/s, u is called low windspeed when u is smaller than 1 m/s; u2 is the wind speed equalto 4 m/s, u is called high wind speed when uis larger than 4 m/s; and m and n are parameters of the wind speed function.

The 10-day average wind speed data of the Aixinzhuangstation in the Ziya River basin during 2007–2010 were statisti-cally analysed. As shown in Fig. 3, approximately 96% of thewind speed is below 1m/s, which belongs to the lowwind speed

zone. Therefore, the function type in accordance with low windspeed was adopted (formula (4)). Finally, the Dalton formulawas obtained by substituting formula (4) into formula (1):

E ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAþ Bu2

pew−eað Þ ð6Þ

To determine the parameter values of A and B in the wind

speed function, E ¼ ΔeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAþ Bu2

pwas transformed into

EΔe

� �2 ¼ Aþ Bu2; taking measured 10-day average water sur-face evaporation, water vapour pressure difference (Δe) andwind speed during 2007–2010 as the basic data, the relation-

ship diagram of EΔe

� �2∼u2 was drawn, which is shown inFig. 4.

The values of A (= 19.718) and B (= 0.5796) can be iden-tified from Fig. 4, and the Dalton formula expression ofAixinzhuang was obtained:

E ¼ 0:1Δeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ 0:5796u2

pð7Þ

The water surface evaporation during 2007–2010 was cal-culated by formula (7), and the simulated values and measuredvalues were compared, as shown in Figs. 5 and 6.

Fig. 2 The scree plot

Fig. 3 Ten-day average windspeed during 2007–2010

F. Li, Y. Zhao

It can be seen from Fig. 6 that the correlation coef-ficient between the simulated values and observedvalues reaches 0.9214, indicating that the overall fittingis good. The differences between the simulated valuesand observed values are mainly located in the peakvalue zone of evaporation; moreover, the simulated peakvalue is larger than the observed peak value (Fig. 5).The differences between the simulated and observedvalues in the peak value zone are shown in Table 6.The relative errors are greater than 20%, which showsthat the errors are greater in the peak value zone.

It can be seen from Fig. 5 that the water vapourpressure difference at the peak value of evaporation islarger than the others. The 4-year average water vapourpressure difference is 6.2 hPa, while the peak value ofwater vapour pressure difference is generally larger than10 hPa, with a maximum of 23.2 hPa; this is signifi-cantly higher than the water vapour pressure of the oth-er periods in the year. The simulated value is obviouslygreater than the observed value when the water vapourpressure difference is larger than 10 hPa. The reason ismainly because the differences in relative humidity in

these periods are large, leading to larger errors in modelfitting. However, the relative humidity was not consid-ered in formula (1), which resulted in a larger error inthe peak value zone. Through the above principal com-ponent analysis, we found that relative humidity is animportant impact factor in the study region. Therefore,the paper proposed a modified Dalton model and used itto modify the simulated value of evaporation when thewater vapour pressure difference was larger than10 hPa.

The function of the relative humidity is usuallyexpressed as follows (Min 2003):

g rð Þ ¼ aþ b 1−r2� �0:5 ð8Þ

where a and b are parameters of the model; r is relativehumidity, %.

Substituting it into formula (2):

E ¼ Δeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ 0:5796u2

paþ b 1−r2

� �0:5h ið9Þ

Fig. 5 Process diagram forsimulated and observed watersurface evaporation

Fig. 6 Relationship diagram for simulated and observed water surfaceevaporation

Fig. 4 The relationship diagram for EΔe

� �2∼u2

Estimation of open water evaporation using land-based meteorological data

To determine the parameters of the relative humidity func-tion, formula (9) is transformed into the following:

E

Δeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ 0:5796u2

p ¼ aþ b 1−r2� �0:5 ð10Þ

A linear relationship between EΔe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ0:5796u2

p and (1

− r2)0.5 is constructed, and the linear relationship dia-gram is drawn to determine the parameters (a = 0.09,b = − 0.01).

Therefore, the modified Dalton water surface evaporationmodel is as follows:

E ¼ 0:1Δeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ 0:5796u2

pΔe < 10 hPað Þ

Δeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi19:718þ 0:5796u2

p0:09−0:01 1−r2

� �0:5h iΔe≥10 hPað Þ

(

ð11Þ

Formula (11) was used to simulate and calculate thewater surface evaporation, and the results are shown inFig. 7.

The modified Dalton model shows an obvious im-provement at the peak value. It is seen in Table 6 thatthe relative errors are less than 20% in the peak valuezone, and the precision of the model is improved. Thisresult indicates that taking relative humidity r as a fac-tor of the water surface evaporation model is an impor-tant strategy for improving the simulation accuracy ofwater surface evaporation.

To validate the precision of the modified model, weset 2011–2012 as the validation period. The simulationresults for water surface evaporation in Aixinzhuangduring 2011–2012 were achieved by applying formula(11), as shown in Fig. 8. It can be seen in Fig. 8 thatthe correlation coefficient between the simulated valuesand observed values during 2011 and 2012 reaches0.9267 and 0.9212, respectively. In 2011, the observedpeak value is 58.1 mm, and the simulated peak value is60.9 mm, while the relative error is 4.8%. In 2012, theobserved peak value is 68.0 mm, and the simulatedpeak value is 73.5 mm, while the relative error is8.1%. The overall fit is good during the validationperiod.

3.3 Model fitting error test criterion

In this paper, the average relative error δ and qualified rate ηwere adopted as the error evaluation indicators.

δ ¼ 1

nΔEE

�������� ð12Þ

where n is the number of samples; ΔE is the difference be-tween the simulated value and observed value, mm; and E isthe observed value, mm.

η ¼ n0

nð13Þ

where n′ is the number of samples whose absolute value ofrelative error is within 20%.

Fig. 7 Comparison diagram formodified simulated evaporationand observed evaporation

Table 6 Differences between simulated and observed values in thepeak value zone

Year Original model Modified model

2007 Observed (mm) 69.0 69.0

Simulated (mm) 89.0 72.3

Relative error (%) 29.0 4.8

2008 Observed (mm) 63.0 63.0

Simulated (mm) 90.7 73.6

Relative error (%) 43.9 16.8

2009 Observed (mm) 76.3 76.3

Simulated (mm) 103.6 84.2

Relative error (%) 35.7 10.3

2010 Observed (mm) 53.7 53.7

Simulated (mm) 71.1 55.1

Relative error (%) 24.9 2.6

F. Li, Y. Zhao

The evaluation results were divided into four grades,with corresponding grades of excellent, good, qualifiedand unqualified, as shown in Table 7. The longer the

calculation period is, the higher the standard for thesame grade.

4 Results and discussion

4.1 Model fitting error test

Based on the results of calculation during 2007–2012 inAixinzhuang, error evaluation results of the water sur-face evaporation model were obtained by applying for-mulas (12) and (13), as shown in Table 8. Table 8shows that the evaluation results of the model fittingerror test are different for various calculation periods.With a 10-day calculation period, all average relativeerrors are below 20%, and all qualified rates η are larg-er than 60%, so all simulated results are assessed asqualified during both calibration and validation periods.With a 1-month calculation period, all simulated resultsexcept for 2010 are assessed as qualified during thecalibration period, and the result of 2010 is assessedas good, while all results are assessed as qualified

Fig. 8 Comparison diagram for simulated and observed evaporationwith calculation periods of 10 days. a Process diagram for simulatedand observed value in 2011. b Relationship diagram for simulated and

observed value in 2011. c Process diagram for simulated and observedvalue in 2012. d Relationship diagram for simulated and observed valuein 2012

Table 7 Error evaluation standards of water surface evaporationformula

Calculation period Grade δ (%) η (%)

10 days Excellent ≤ 10 ≥ 80

Good 10–15 70–80

Qualified 15–20 60–70

Unqualified > 20 < 60

A month Excellent ≤ 7.5 ≥ 90

Good 7.5–10 80–90

Qualified 10–15 70–80

Unqualified > 15 < 70

A year Excellent ≤ 5 100

Good 5–7.5 90–100

Qualified 7.5–10 80–90

Unqualified > 10 < 80

Estimation of open water evaporation using land-based meteorological data

during the validation period. With a 1-year calculationperiod, all simulated results except for 2009 areassessed as excellent during the calibration period, andthe result of 2009 is assessed as good, while the resultsof 2011 and 2012 are assessed as good and excellentduring the validation period, respectively.

4.2 Model applicability test

In this paper, the water surface evaporation model wasconstructed based on the measured data of theAixinzhuang station in the Ziya River basin to validateits applicability, and the meteorological data ofKuancheng station in the Luan River basin were select-ed to test the modified model.

It is seen from Fig. 9 and Table 9 that the evaluationresults of the model fitting error test are different forvarious calculation periods. With a 10-day calculationperiod, all relative errors and qualified rates of 2007–2012 meet the requirements, and moreover, for 2007,2010, and 2012, the simulated values reach the goodstandard. With a 1-month calculation period, all relativeerrors and qualified rates of 2007–2012 meet the

requirements. With a 1-year calculation period, all rela-tive errors and qualified rates of 2007–2012 meet therequirements, and for 2007, 2009, and 2011, the evalu-ation grade achieves the excellent standard. From thepoint of view of total average, the 10-day average rela-tive error and qualified rate are 15.4 and 74.7%, respec-tively, and assessed as qualified; the monthly averagerelative error and qualified rate are 13.3 and 80.6%,respectively, and assessed as qualified; and the yearlyaverage relative error and qualified rate are 6.0 and100%, respectively, and assessed as good.

By validating the model, it can be seen that the sim-ulated water surface evaporation results of Kuanchengstation have high precision, indicating that the watersurface evaporation model proposed in this paper hascertain applicability in northern China or similarregions.

5 Conclusions

(1) Water surface evaporation is very sensitive to weatherconditions, so the Dalton model may have different

Table 8 Error evaluation resultsof water surface evaporationformula

Calculation period Year δ (%) η (%) Grade± 20%

10 days Calibration period 2007 18.2 69.4 Qualified

2008 17.0 63.9 Qualified

2009 15.3 77.8 Qualified

2010 17.8 72.2 Qualified

Average 17.1 70.8 Qualified

Validation period 2011 16.5 61.1 Qualified

2012 18.5 63.9 Qualified

Average 17.5 62.5 Qualified

1 month Calibration period 2007 14.6 75.0 Qualified

2008 13.4 83.3 Qualified

2009 12.4 83.3 Qualified

2010 9.7 91.7 Good

Average 12.5 83.3 Qualified

Validation period 2011 13.2 75.0 Qualified

2012 11.8 83.3 Qualified

Average 12.5 79.2 Qualified

1 year Calibration period 2007 0.9 100 Excellent

2008 2.4 100 Excellent

2009 5.8 100 Good

2010 1.0 100 Excellent

Average 2.5 100 Excellent

Validation period 2011 6.0 100 Good

2012 4.5 100 Excellent

Average 5.3 100 Good

F. Li, Y. Zhao

performances in different regions. Accordingly, whenconstructing a water surface evaporation model, weshould first consider the main factors affecting watersurface evaporation for the special study region. On thisbasis, we propose a modified Dalton model that is suit-able for the special study region.

(2) The PCA method was used to analyse the sensitive fac-tors of water surface evaporation and determine the mainfactors affecting water surface evaporation. ThroughPCA method analysis, the main impact factors of watersurface evaporation in Aixinzhuang were shown to bewater vapour pressure difference, wind speed and rela-tive humidity.

(3) The paper first adopted the original Dalton modelto simulate the water surface evaporation. Theoverall fit of the model is good, while the simula-tion precision is poor for the peak value zone. Thewater vapour pressure difference for the peak valuezone is generally larger than 10 hPa, and the

difference of relative humidity in this zone is large,leading to a larger error in model fitting. Therefore,the modified Dalton formula that considered rela-tive humidity was adopted when the water vapourpressure difference was larger than 10 hPa.

(4) Taking 2007–2010 and 2011–2012 as the calibra-tion period and the validation period, respectively,the modified Dalton model had a higher fitting ac-curacy. The results show that the 10-day averagerelative error is 17.2%, assessed as qualified; themonthly average relative error is 12.5%, assessedas qualified; and the yearly average relative erroris 3.4%, assessed as excellent.

(5) Applying the modified model to the Luan Riverbasin, the meteorological data of Kuancheng stationwere selected to test the modified model. The re-sults show that the 10-day average relative error is15.4%, assessed as qualified; the monthly averagerelative error is 13.3%, assessed as qualified; and

Fig. 9 Comparison diagram for simulated and observed 10-day water surface evaporation at Kuancheng station in the Luan River basin

Estimation of open water evaporation using land-based meteorological data

the yearly average relative error is 6.0%, assessedas good. The simulation of water surface evapora-tion at Kuancheng station has high precision, indi-cating that the modified Dalton model proposed inthis paper has applicability in northern China.

Funding information The authors would like to acknowledge the finan-cial support for this work provided by the National Key R & D programof China (Grant no. 2016YFC0401407) and the National Natural ScienceFoundation of China (Grant no. 51579169).

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Table 9 Evaluation results of water surface evaporation at Kuanchengstation

Calculation period Year δ (%) η (%) Grade± 20%

10 days 2007 12.3 83.3 Good

2008 18.5 62.3 Qualified

2009 16.2 69.4 Qualified

2010 14.9 77.8 Good

2011 15.9 77.8 Qualified

2012 14.7 77.8 Good

Average 15.4 74.7 Qualified

1 month 2007 10.5 91.7 Qualified

2008 14.6 75 Qualified

2009 14.9 75 Qualified

2010 13.6 75 Qualified

2011 14.1 83.3 Qualified

2012 12.2 83.3 Qualified

Average 13.3 80.6 Qualified

1 year 2007 3.5 100 Excellent

2008 9.8 100 Qualified

2009 1.3 100 Excellent

2010 7.8 100 Qualified

2011 3.8 100 Excellent

2012 9.8 100 Qualified

Average 6.0 100 Good

F. Li, Y. Zhao

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Estimation of open water evaporation using land-based meteorological data

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