Arab J Sci Eng (2012) 37:521–534 DOI 10.1007/s13369-012-0194-5

R E S E A R C H A RT I C L E - C I V I L E N G I N E E R I N G

Mostafa A. Benzaghta · Thamer A. Mohammed · Abdul Halim Ghazali · Mohd Amin Mohd Soom

Validation of Selected Models for Evaporation Estimation from Reservoirs Located in Arid and Semi-Arid Regions

Received: 18 February 2010 / Accepted: 7 October 2010 / Published online: 20 March 2012 © King Fahd University of Petroleum and Minerals 2012

Abstract This paper investigates evaporation at Algardabiya Reservoir, Sirte, Libya. Evaporation from Algardabiya Reservoir is the main problem faced which contributes in water loss. Libya is considered as arid region with limited water resources. Modeling of evaporation from the reservoir is crucial for water management at the area. Three evaporation models, Linacre, DeBruin and Harbeck, were selected to predict reservoir evaporation, and meteorological data were input to the models from a weather station at the reservoir site. Daily evaporation also was measured, and statistical tests were conducted to check model accuracy. These tests showed that the models produced reasonable accuracy, with biases in model prediction of 6.5, 8.5 and 9.4% for the DeBruin, Linacre and Harbeck models, respectively. Based on the model testing, the Linacre model can be used to predict evaporation from the Reservoir when available meteorological data are limited to air temperature only. However, we recommend application of the DeBruin and Harbeck models when more meteorological data are available. Keywords Modeling · Evaporation · Lake · Reservoir · Algardabiya · Arid region

M. A. Benzaghta (B) · T. A. Mohammed · A. H. Ghazali Civil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia E-mail: [email protected] M. A. Benzaghta Sirte University, Sirte, Libya M. A. M. Soom Biological and Agricultural Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

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1 Introduction Evaporation is a fundamental element of the hydrologic cycle, because it impacts river-basin agricultural yield, reservoir capacity, and consumptive use by crops. Evaporation in arid and semiarid regions is much higher than other elements of the hydrologic cycle, such as precipitation, runoff, and groundwater flow. It exacerbates water shortages, and is therefore considered the most critical aspect of the hydrologic cycle. Data of evaporation depth from open surfaces will help hydrologists assess the impact of evaporation on water resources. However, in the published literature, many works have highlighted the difficulty of accurate evaporation estimation [1]. Evaporation losses should be considered in the design of various water resource and irrigation systems. In areas with little rainfall, evaporation losses can be a significant part of the water budget for a lake or reservoir, and may greatly contribute to lowering water surface elevation [2]. Therefore, accurate estimation of evaporation from water bodies is of primary importance for monitoring and allocating water resources. The main factors affecting evaporation are wind speed, water vapor deficit, air and water temperatures, atmospheric pressure and solar radiation [3,4]. In hydrologic practice, there are two approaches for estimating evaporation from water surfaces, direct and indirect. In the direct method, evaporation is directly measured using an evaporation pan. The Class A pan is one of the most common instruments for evaporation measurement [3], and has been used in research worldwide [5–9]. Indirect methods include estimation of evaporation using empirical models that are based on meteorological data [10–16]. Many researchers have applied indirect methods. For example, Kohler et al. [17] used empirical methods, Shuttleworth [18] used water budget methods, Anderson [19] used energy budget methods, Harbeck [20] used mass-transfer methods and Penman [21] used combination methods. These methods varied greatly in their ability to define the magnitude and variability of evaporation. Most researchers developed models for estimating free water evaporation [14,21–23]. DeBruin [24] applied a simplified model, by combining the Priestley–Taylor and Penman equations. He indicated that the model made good predictions for periods of 10 days or more. Singh and Xu [1] evaluated and compared 13 evaporation models (based on mass transfer), running them with climatological data. Singh and Xu [25] examined the sensitivity of mass transfer-based evaporation equations to errors in daily and monthly input data. Also, Xu and Singh [26] evaluated and generalized temperature-based methods for evaporation calculations. Mosner and Aulenbach [27] compared four empirical methods of evaporation estimation, including the Papadakis, Priestley-Taylor, DeBruin-Keijman, and Penman equations for Lake Seminole in southwest Georgia and northwest Florida, from April 2000 to September 2001. It was found that the average monthly bias in evaporation estimation derived from empirical equations was 16%. Finch [28] estimated evaporation from water temperature over a period from 1956 to 1962, demonstrating that mean annual evaporation can be estimated with considerable confidence using the model. He found that the model was less successful in estimating monthly evaporation, with a root mean square error of 49%, twice the estimated average error in measured values. Rosenberry et al. [16] reported that methods requiring measurement of both solar radiation and air temperature were not substantially better than methods requiring only air temperature. This finding was based on the application of 14 different models using data from Mirror Lake, USA. Model outputs were compared with the Bowen-Ratio-Energy Budget model (BREB model). Although there are some published works on evaporation from lakes and reservoirs, it appears that Algardabiya Reservoir has not been studied before. Thus, the major goals of the present research are to highlight the seriousness of evaporative water loss from the reservoir and to check the accuracy of several selected models, which are based on mass transfer and energy balance. This will aid evaporation estimation for Algardabiya Reservoir, by identifying models with reasonable accuracy.

2 Study Area and Observation Data 2.1 Study Area Libya is considered to have limited renewable water resources, because most parts of the country are semi-arid or arid, with average annual rainfall less than 100 mm and average annual evaporation of 2,500 mm, much greater than the rainfall [29]. This is a case study for Algardabiya Reservoir (Sirte, Libya, 31◦ 09 30.71"N; 16◦ 40 58.02"E, 50 m.a.s.l), which is part of the Libyan Man-Made River Project, shown in Fig. 1.

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Fig. 1 Layout of Pipes and Algardabiya Reservoir for Manmade River Projects Table 1 Descriptive statistics of the meteorological data Variables

Units

Minimum

Maximum

Mean

SD

CV (SD/mean)

CR with EVP

MaxTemp ˚C 10 44 25.70 6.88 0.27 0.91 MinTemp ˚C 4 32 14.97 5.57 0.37 0.89 AvTemp ˚C 8.5 36 20.18 5.80 0.29 0.98 WS Km/h 5 50 14.63 7.78 0.53 0.16 RH % 30 95 60.82 12.28 0.20 0.77 EVP mm/d 3.7 15.2 7.66 2.29 0.30 1 MaxTemp maximum air temperature, MinTemp minimum air temperature, AvTemp average air temperature, Ws wind speed, RH relative humidity, EVP pan evaporation, CV variation coefficient, CR correlation

The reservoir is an earth embankment type, located 10 km southeast of the city of Sirte, adjacent to the coastal highway. The reservoir has a crest diameter of 887.66 m and an operating depth of 12.5 m, giving it a maximum volume of 6.9 million cubic meters. Water seepage from the reservoir is controlled by a geomembrane, which covers the entire inner slope and floor of the reservoir. A 400 mm diameter unplasticized polyvinyl chloride (UPVC) slotted drain runs completely around the inner toe of the reservoir, and drains at an outlet chamber adjacent to the reservoir spillway. The reservoir has an apical diameter of 887.66 m, bottom diameter of 794.080 m, and surface area of 593,860 m2 [30].

2.2 Observation Data The meteorological data used to estimate evaporation from Algardibyia Reservoir was acquired from the meteorological observatory of Great Man-Made River Authority (GMRA), Sirte, Libya. The meteorological data include maximum and minimum air temperature, relative humidity, wind speed, and class A evaporation pan. The average monthly rainfall was 270 mm during the study period. The pan evaporation is multiplied by a factor of 0.69 to get the actual evaporation from the reservoir. This factor is obtained by measuring the evaporation from both class A pan and reservoir before operation (no inflow and outflow) [31]. After reservoir operation, evaporation data are obtained using the class A pan only. The class A pan is widely used as a robust instrument for measuring evaporation, but the physical conditions for the class A pan and the reservoir are different, and this renders the evaporation depth from them different, though they are in the same area and subjected to the same weather conditions. This is why the readings from the class A pan are multiplied by 0.69 to estimate actual evaporation depth from the reservoir. Three years of daily evaporation records from 2004 to 2006 were used to study the reservoir evaporation. Table 1 shows the various meteorological data and their descriptive statistics. Details of the equipment used for measuring the meteorological data are shown in Table 2.

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Table 2 Equipment used for measuring the meteorological data Name of the apparatus

Measurement

Units

Time interval

Accuracy

Hygrothermograph (MT 1500, SIAP, Bologna) Hygrothermograph (MT 1500, SIAP, Bologna) Anemometer (Munro-1056-44) Rain gauge (UM 8100, SIAP, Bologna) Class A pan model 254-210

Air temperature Relative humidity Wind speed Rainfall Evaporation

˚C % Km/h mm mm

Hourly Hourly Hourly Hourly Daily

±1◦ C ±3% ±1.8 km/h 0.4 mm 0.02 mm

3 Models There are many models to estimate evaporation from an open water body, also known as lake evaporation; here it is reservoir evaporation. Some of these models are used in the water budget, energy budget, eddy correlation, mass transfer, Penman and pan methods, and the combination equation [32,33]. The main disadvantage of most of these methods is that they require measurement of several meteorological variables, such as air temperature, wind speed, humidity, and solar radiation, to estimate reservoir evaporation. Inaccurate measurements of solar radiation and humidity will produce an error in the predicted evaporation. This error is estimated to be four times the error in the predicted evaporation from inaccurate measurement of wind speed and temperature [34,35]. For instance, a 30% error in the wind value creates on error of only 5% in estimated evaporation depth [35]. So, accurate wind speed records may not be essential. The pan method is the only existing method that does not require site-specific measurements, and is therefore commonly used to estimate evaporation from lakes and reservoirs [12,36–39]. Many models are not applicable to the current case study, because of data unavailability. The available data for Algardibyia Reservoir are mentioned above; for example, solar radiation is not measured. There are many methods that can estimate solar radiation, but validation of these estimates is essential. So it is preferable to use the available meteorological data for estimation of evaporation from the reservoir. Thus, only three relevant empirical models can be applied to the case study, which are Linacre, DeBruin and Harbeck models. The predicted evaporation from each of these models was compared with the measured evaporation from the reservoir. The selected models are based on standard approaches such as mass transfer and energy balance, which make them appropriate for arid and semi-arid regions. For example, Anyadike [40] applied the Linacre model in the West African region. 3.1 Pan Method Evaporation is regularly measured at a weather station 200 m from Algardibyia Reservoir using a class A evaporation pan, which is one of the weather station elements from 2004 to 2006. Each day, an observer fills the evaporation pan with water to a predefined level. The change in water level from evaporation and rainfall is observed on the following day. The evaporation rate is calculated by adjusting the change in water level with rainfall depth recorded by the rain gauge. For determining reservoir evaporation, the evaporation pan (E p ) method is used, which is a common substitute in regions where input climate variables are not available. Lake evaporation estimates are obtained by multiplying the pan data by an appropriate coefficient K p , as follows. E = KP EP,

(1)

where E is the amount of evaporation from the reservoir in unit depth, K P is a pan coefficient, and E P is the amount of evaporation from the class A pan, in unit depth. For Algardibyia Reservoir, K P is computed based on three years of daily evaporation data, and it is 0.69 [31]. 3.2 Linacre Model To overcome the difficulty of using the Penman formula, Linacre [22] introduced a simplified Penman formula requiring only temperature, dew point, elevation and latitude. The Penman formula for estimating the rate of evaporation from an open water surface is written as follows. L E = (Rn + ρ c S/ ra )/(1 + γ /),

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(2)

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where LE is evaporation rate (J m−2 ), Rn is net radiation (J m−2 ), ρ is the density of air (1.3 kg m−3 ), c is the specific heat of air (0.24 cal g−1 ˚C−1 ), S is the average saturation-deficit of the air (Pa), ra is the diffusion resistance between water and air (s m−1 ),  is the slope of the curve of saturation vapor pressure with respect to temperature (Pa ˚C−1 ) andγ is the psychrometric constant (0.67 mbar ˚C−1 ) The parameters S/ and (1 + γ /) replaced by equivalent expressions involving temperature are as follows. S = (T − Td )  1+

γ = 2 (1 − 0.0125 T ), 

(3)

(4)

where T is the mean temperature and Td the dew point temperature. γ Both S and 1 +  are included in the Penman formula, which is based on the physics of the evaporation process. To reduce the difficulty with this formula, Linacre [22] developed an approximation to the above variables. The approximation is based solely on temperature measurement. So, Eqs. 3 and 4 can be described as semi-empirical equations. Linacre [22] reported that from 432 sets of monthly mean actual values of (Ta − Td ), were collected from 37 places in Australia and New Zealand in order to be used for model assessment. By applying Linacre model for arid region it was found that the mean error was 1 mm/day in estimated evaporation for data sets. Fortunately, the resulting error in estimating the evaporation rate in arid conditions is made less important by the large rate there, i.e., the error is a relatively small fraction. The data were taken from the hottest and coldest months for places in Africa, Australia and South America. The net radiation term Rn in Eq. 2 can be replaced by a temperature function, using relationships derived by Linacre [11], as follows. Rn = Rs (0.75 − α),

(5)

where Rs is global-radiation (cal cm−2 s−1 ) and α the albedo, which is 0.05 for an open water surface [22]. There is an empirical relationship between Rs and temperature (T ) for places at latitude (L a ) and elevation (h) in meters [41], which can be written as follows. Rs = Tm /60 (100 − L a ),

(6)

where Tm is the sea-level equivalent of the measured mean temperature (T ), expressed as Tm = T + 0.006 h

(7)

By combination of Eqs. 2, 3, 4 and 5, one obtains the following expression for the rate of evaporation in mm/day from a lake [22]: E=

700 (Ta + 0.006 h)/(100 − L a ) + 15 (Ta − Td ) , 80 − Ta

(8)

where Ta , is the mean air temperature (˚C), h is the elevation (meters) above mean sea level, L a is latitude (˚); Td is the mean dew point temperature (˚C). Linacre presented an equation for estimating (Ta − Td ) that can be written as (Ta − Td ) = 0.0023 h + 0.37 Ta + 0.53 R + 0.35 Rann − 10.9,

(9)

where R is the monthly mean daily temperature (˚C) and Rann is mean temperature of the hottest and coldest month (˚C). Monthly mean values of the term (Ta − Td ) can be obtained from Eq. 9, provided that precipitation is at least 5 mm month−l and (T- Td) at least 4˚C [22]. An absence of rain appears to weaken the correlation of (Ta − Td ) with other climatic data. The multiple correlation coefficients of actual values with those calculated from the best available regression proved to be only 0.73 in the case of 63 sets for which the monthly precipitation was less than 5 mm, and the standard error of (Ta − Td ) estimation in this case was 5.2˚C. On the other hand, the coefficient was 0.822 and standard error only 2.3˚C for 294 sets in months when rainfall exceeded 4 mm. In the latter case, the best regression equation depends on R, T , Rann , h and precipitation, in order of decreasing importance [22].

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3.3 Mass Transfer Model Mass transfer (aerodynamic)-based methods use the concept of eddy motion transfer of water vapor, from an evaporating surface to the atmosphere. All such methods are based on Dalton’s law. The mass transfer methods give satisfactory results in many cases [42–47], normally use easily measured variables, and have simple model form. Mass transfer is one of the oldest methods [21,48] and is still attractive for estimating evaporation from free water surfaces, because of its simplicity and reasonable accuracy. The mass transfer method is based on the Dalton equation for free water surface, and can be written as E = C L (es − ea ),

(10)

where E is free water surface evaporation, es is the saturation vapor pressure at the temperature of the water surface (mbar ˚C−1 ), ea is the vapor pressure in the air (mbar ˚C−1 ), and C L is a coefficient that is dependent on barometric pressure, wind velocity and other variables. Practically all mass transfer equations have one common factor, that evaporation is directly proportional to the product of vapor pressure differences and wind velocity u at 2 m above the water surface (miles/h). Therefore, Eq. 10 can be expressed as E = f (u) (es − ea )

(11)

Yamamuto [49] and Marciano and Harbeck [50] were among the earliest investigators of the evaporation concept in terms of atmospheric turbulence. Harbeck [20] presented an empirical mass transfer equation for evaporation from Lake Hefner [51], as follows. E = N u 8 (es − ea ),

(12)

where E is evaporation in inches/day, N is the mass-transfer coefficient, u 8 is wind speed at 8 m above the water surface in miles/h, es is saturation vapor pressure at temperature Ts (millibars), and ea is the vapor pressure of the air (millibars). He stated that the mass-transfer coefficient N generally represents a combination of many variables, such as the size of the lake, roughness of the surface, atmospheric stability, barometric pressure, and density and kinematic viscosity of the air. He suggested a substitution for mass transfer coefficient to calculate evaporation losses from lakes, as follows. E = 0.00338 A−0.05 u 2 (es − ea ),

(13)

where A is the lake area (acres) and u 2 is wind speed at 2 m above the water surface (miles/h). The values of es and ea were calculated by substituting air temperature Ta (˚C) and relative humidity RH (%), respectively, in the following equations [39].   ea = 33.8639 (0.00738 Ta + 0.8072)8 − 0.000019 |1.8 Ta + 48| + 0.001316 (14) es = ea/R H

(15)

3.4 DeBruin Model In 1978, DeBruin [24] presented a model to estimate evaporation when only three variables are known— air temperature, wind speed and relative humidity. He derived this model by combining the Priestley and Taylor [52] and Penman [21] equations. He solved the Priestley and Taylor equation for the difference in net radiation and heat flux, and substituted the result into the Penman equation as     D γ E= (16) (Rn − G) + F(u) (es − ea ) D+γ D+γ The Priestly and Taylor formula can be written as   D E=B (Rn − G) D+γ

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(17)

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Rearranging Eq. 17 yields   B (Rn − G) = E

D D+γ

 (18)

Combining Eqs. 16 and 18 produces B E= B−1



γ D+γ

 F(u) (es − ea ),

(19)

where E is evaporation in W m−2 , B is the Priestley and Taylor coefficient (B = 1.26), γ is the psychrometric constant, D the slope of the saturation vapor pressure-temperature curve; es and ea ,the vapor pressures in millibars, can be computed by Eqs. 12 and 13, where F(u) is a wind function that can be written as in Sweers [53]: F(u) = 2.9 + 2.1 u,

(20)

where u is the wind speed measured at 2 m height (ms−1 ) and F(u) is in W m−2 mb−1 . The variation of wind speed with height, or wind profile in the friction layer, is usually expressed by one of two general relationships, namely, the logarithmic velocity or power law profiles. In hydrology, the relationships are mostly used to estimate the wind speed in the surface boundary layer, i.e., the thin layer of air between the ground surface and anemometer level. The latter is usually about 10 m, but often lower at special test sites or experimental stations. No standard specific anemometer height has been generally recommended in the literature. However, adjustments can be made for differences in height. Linsley et al. [3] recommended wind speed adjustments for the friction layer and over the sea of about 40 and 70%, respectively. The common requirement is to use wind speed above a snow or water surface for computations of snowmelt and evaporation, respectively [3]. According to the operating authority for Algardabiya Reservoir, the anemometer for wind speed recording is fixed at a height of 10 m above the ground. Here, the wind speed at 2 m height was calculated using the recommended concept above, and then used to estimate evaporation depth. The wind speeds measured at 10 m height were converted with the one-seventh power law to give the wind speeds at 2 m [54]. 4 Models Testing Performance of each model was evaluated by its statistical performance. To ensure rigorous model comparison, an extended analysis was done using different statistical indices for estimated values. Fox [55] and Willmott [56] pointed out that commonly used correlation measures, such as (R) and (R 2 ) in general testing of statistical significance, are often inaccurate or misleading when comparing model-predicted and observed variables. The two most widely used statistical indicators in estimation models are the root mean square error (RMSE) and mean bias error (MBE) [56–58]. According to Fox [55], the mean absolute error (MAE) is less sensitive to extreme values than the RMSE, so it is not used in model testing. The root mean square error and mean bias error are defined as n 1/ 2 1

RMSE = (E i, pr ed − E i,obs )2 (21) n i=1

MBE =

n 1

(E i, pr ed − E i,obs ), n

(22)

i=1

where Ei,obse is observed evaporation (mm/day), Ei, pr ed is predicted evaporation (mm/day), and n is the number of data pairs. The RMSE is said to provide information on model short-term performance, by allowing a term-by-term comparison of the actual difference between observed and predicted values [59]. The smaller the value is, the better the model performance. A drawback of this test is that a few large errors in the sum can produce a significant increase in RMSE. In addition, the test does not differentiate between under- and overestimation.

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Fig. 2 Monthly evaporation at the Algardabiya Reservoir over period 2004–2006

The MBE is said to provide information on model long-term performance. A positive value gives the average amount of overestimation in the estimated values, and vice versa [59]. The smaller the absolute value is, the better the model performance. It is obvious that the RMSE and MBE statistical indicators, if not used in combination with one another, may not be adequate indicators of model performance. However, RMSE and MBE generally provide a reasonable procedure for model comparison. They do not indicate objectively whether model estimates are statistically significant, however. Thus, we use an additional statistical indicator, the t-statistic. This indicator allows models to be compared, and can also indicate whether their estimates are statistically significant at a particular confidence level. The t-statistic is defined through the root mean square error and mean bias error, as in [58]:  t=

(n − 1) M B E 2 RM SE2 − M B E2

1/ 2 (23)

The smaller the t value, the better the model performance. To verify statistical significance of model estimates, one simply has to obtain a critical t value from standard statistical tables, i.e., tα/2 at level of significance (α) and degrees of freedom (n − 1). For model estimates to be statistically significant at the 1 − α confidence level, the calculated t value must be less than the critical t value. The level of significance can vary between 0 and 1, but is usually 0.05 or 0.01 [58]. In the present study, the level of significance was chosen to be α = 0.05, so that the corresponding critical t value from the statistical tables is t = 2.576, for n − 1 degrees of freedom.

5 Results The Algardabiya Reservoir (Fig. 1) is characterized by a Mediterranean semi-arid climate, with warm and dry summers and mild winters. Typical annual rainfall at the reservoir site is about 170 mm. Most rainfall occurs during winter. Generally, the observed data over 3 years show the same yearly trend in reservoir evaporation (Fig. 2). The highest annual evaporation from the reservoir was 11.2 mm day−1 , which occurred in August and September 2006. The lowest annual value was 4.8 mm day−1 , in January 2004. The maximum evaporation in 2006 was a result of the highest temperature (44˚C), which was recorded in August. The last column of Table 1 represents the correlation (CR) between model input variables, such as air temperature (Ta), wind speed (WS), relative humidity (RH) and evaporation (EVP) from the reservoir. The correlations show that the sequence of significant weather variables is air temperature, relative humidity, and wind speed. In the absence of solar radiation data for the reservoir, temperature is taken as a measure of reservoir evaporation, although the latter is largely driven by solar radiation and air temperature is considered a secondary variable governing evaporation. The performance of the Linacre, Harbeck, and DeBruin models were evaluated using RMSE, MBE and R 2 tests. Results are presented in Table 3. From Table 3, it is obvious that the DeBruin model performed best, since RMSE values are smaller. The Linacre model follows with respect to RMSE values, with the Harbeck model last. Interestingly, with respect to (lower) MBE values, the DeBruin model performed best, followed by the Linacre and Harbeck models.

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Table 3 Statistic analysis for the model Model

RMSE (mmd−1 )

Linacre Harbeck deBruin

0.73 0.91 0.66

MBE (mmd−1 ) −0.65 0.79 0.53

R2

t

0.98 0.96 0.97

66 58 45

Fig. 3 Scatter plot for the Linacre model prediction

Table 3 also shows the correlation coefficients R 2 for the three models. For example, R 2 between measured reservoir evaporation and predicted evaporation using the Linacre, Harbeck, and DeBruin models is 0.98, 0.96 and 0.97, respectively. These results are in agreement with Warnaka and Pochop [10], who found R 2 for the Linacre and DeBruin models at 0.89 and 0.86, respectively. The t test result in Table 3 shows level of confidence for the above models (α = 0.05). There is no significant difference between them. The Linacre model required only temperature data to run, and its output agreed with the measured reservoir evaporation and predicted evaporation from the other models (Harbeck and DeBruin) that require more data. This finding can help overcome the data shortage and make model application relatively easy. In his West African study, Anyadike [40] reported that the Linacre model is superior to the Thornthwaite and Penman models in ease of use and accuracy. This supported our selection of Linacre and the other models for accuracy testing. The evaporation estimates of each model are given in Figs. 3, 4, and 5, in the form of scatter plots versus observed evaporation. These confirm that the Linacre model prediction is reasonable. Figures 6, 7, and 8 show monthly bias bars for the models during the study period (2004–2006). The absolute biases from the models ranged from 0.3 to 1.5 mm. This shows that model biases are not large, and confirms that the model predictions are reasonable. Figure 9 shows monthly observed and predicted evaporation for Algardabiya Reservoir, for the period 2004–2006. The Harbeck and DeBruin models overestimated evaporation, as confirmed by their positive MBE values. The Linacre model underestimated evaporation, as confirmed by its negative MBE values (Table 3). Figure 9 presents model behavior, including time lag and model accuracy. The total evaporation from reservoir for the study period (2004–2006) was found to be 8,407.3 mm, while the estimated evaporation from the Linacre model was 7,692.7 mm. This shows that this model underestimated reservoir evaporation by 8.5%, while the other models overestimated it, by 9.4% (Harbeck) and 6.5% (DeBruin).

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Fig. 4 Scatter plot for the Harbeck model prediction

Fig. 5 Scatter plot for the DeBurin model prediction

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Fig. 6 Observed and predicted monthly evaporation for Algardabiya Reservoir (2004–2006), using the Linacre model. Vertical bars represent 95% confidence of observed evaporation

Fig. 7 Observed and predicted monthly evaporation for Algardabiya Reservoir (2004–2006), using the Harbeck model. Vertical bars represent 95% confidence of observed evaporation

Fig. 8 Observed and predicted monthly evaporation for Algardabiya Reservoir (2004–2006), using the DeBruin model. Vertical bars represent 95% confidence of observed evaporation

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Fig. 9 Observed and predicted monthly evaporation for Algardabiya Reservoir (2004–2006)

6 Conclusion Evaporation rates from Algardabiya Reservoir (Sirte, Libya) were measured directly by class A pan, and estimated using three selected models. Various statistical measures, such as the root mean square error (RMSE), mean bias error (MBE) and coefficient of determination (R 2 ), were used to evaluate model performance. The models were the Linacre, DeBruin and Harbeck. Statistical tests showed that measured and predicted evaporation rates were in agreement. Based on those tests, it is recommended that the Linacre model be used whenever air temperature is the only available data. This model underestimated evaporation from Algardabiya Reservoir by 8.5%. The other models (DeBruin and Harbeck) can be applied when other meteorological data are available. These models overestimated the reservoir evaporation by 9.4% (Harbeck) and 6.5% (DeBruin). Acknowledgments The lead author of this study gratefully acknowledges the financial assistance extended by the Libyan Ministry of Higher Education for pursuing his doctoral research at University Putra Malaysia. The assistance of the Great Man-Made River Authority (GMRA), Sirte, Libya, in providing the meteorological data is gratefully acknowledged. Also, the author is thankful to the anonymous reviewers for their valuable suggestions, which greatly improved the quality of this paper.

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