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provided by Lirias Theor Appl Climatol (2010) 100:439–465 DOI 10.1007/s00704-009-0168-z

ORIGINAL PAPER

Evaporation estimates from Nasser Lake, Egypt, based on three floating station data and Bowen ratio energy budget

Mohamed Elsawwaf & Patrick Willems & Andrea Pagano & Jean Berlamont

Received: 15 January 2009 /Accepted: 26 June 2009 /Published online: 26 August 2009 # Springer-Verlag 2009

Abstract Nasser Lake is located in a hyper-arid region in conducted based on quasi-Monte Carlo sequences (Latin the south of Egypt. Evaporation is by far the most Hypercube sampling). The standard deviation of the total important factor in explaining the water losses from the uncertainty on the BREB evaporation rate was found to be lake. To obtain better management scenarios for Nasser 0.62 mm day−1. The parameter controlling the change in Lake, an accurate estimation of the lake evaporation losses stored energy, followed by the BR parameter, was found to be thus is essential. This paper presents an update of previous the most sensitive parameters. Several of the six conventional evaporation estimates, making use of local meteorological methods showed substantial bias when compared with the and hydrological data collected from instrumented plat- BREB method. These were modified to eliminate the bias. forms (floating weather stations) at three locations on the When compared to the BREB-based values, the Penman lake: at Raft, Allaqi, and Abusembel (respectively at 2, 75, method showed most favorably for the daily time scale, while and 280 km upstream of the Aswan High Dam). Results for the monthly scale, the Priestley–Taylor and the deBruin– from six conventional evaporation quantification methods Keijman methods showed best agreement. Differences in were compared with the values obtained by the Bowen ratio mean evaporation estimates of these methods (against the energy budget method (BREB). The results of the BREB BREB method) were found to be in the range 0.14 and method showed that there is no significant difference 0.36 mm day−1. All estimates were based calculations at the between the evaporation rates at Allaqi and Abusembel. daily time scale covering a 10-year period (1995–2004). At Raft, higher evaporation rates were obtained, which were assumed to be overestimated due to the high uncertainty of the Bowen ratio (BR) parameter. The average 1 Introduction BR value at Allaqi and Abusembel was used to eliminate this overestimates evaporation. Variance-based sensitivity Nasser Lake is located in the lower Nile River Basin at the and uncertainty analyses on the BREB results were border between Egypt and Sudan at 182 m above mean sea level (amsl; Fig. 1). The lake was created from the late 1960s to the 1970s together with the construction of the Aswan High Dam (AHD) upstream of the old Aswan dam, M. Elsawwaf (*) : P. Willems : J. Berlamont about 5 km south of the city of Aswan. The AHD created a Hydraulics Laboratory, Department of Civil Engineering, multi-purpose storage reservoir to provide adequate water Katholieke Universiteit Leuven, supply in summer, hydropower, flood protection, and Kasteelpark Arenberg 40, improved river navigation. The lake has an area of about 3001 Heverlee, Belgium 2 e-mail: [email protected] 6,540 km and a length of about 500 km, 350 km of which lies in Egypt and 150 km in Sudan. The lake has an average A. Pagano width of about 10 km, a maximum width of about 60 km, Econometrics and Applied Statistics Unit, an average depth of 25 m, and a maximum depth of about EC Joint Research Centre, 9 3 Via Fermi 2749, 90. The total capacity of the reservoir is 162.3×10 m at its 21027 Ispra (VA), Italy highest water level (Sadek et al. 1997; Omar and El-Bakry 440 M. Elsawwaf et al.

clear, resulting in massive storage of solar energy in a Aswan High Dam considerably greater volume. The bulk temperature of the Raft station lake (the temperature below about half a meter) will fluctuate to a small degree over the course of a day and the evaporation rate will change correspondingly. Methods that use only net radiation to estimate evaporation rates must fail because net radiation is not providing the energy required for evaporation. For example, these models will Allaqi station predict a small or zero evaporation rate at night since the net radiation is small. Over a water body at night, the evaporation rate is in fact nearly as large as it is during the Western Desert day. Even on a daily or weekly basis, conventional methods based on net radiation will not work well. For example, between June and September, the lake temperature will rise Eastern Desert significantly as an extremely large amount of energy is stored in the water. Methods that use the net radiation will Abusembel station Egypt assume that all of the available energy is going into sensible or latent energy, yet a significant portion is used to warm the Arqeen station lake. On a daily basis, the overestimation of evaporation is Egypt-Sudan border not large but consistently, which might lead to large errors when the evaporation rates are aggregated over longer time periods. Similarly in the fall, when the lake cools, these N Aswan High Dam Sudan methods will underestimate the evaporation rates. Working FS Accounting for the above considerations, observational FS under setting up studies of lake evaporation have used a variety of different methods to estimate evaporation rates. These include the Fig. 1 Location map of the Nasser Lake area showing the floating Bowen ratio energy budget (BREB) method and eddy- stations (FS) sites and the major physical features of Nasser Lake correlation techniques, the water-budget method, methods of the so-called Dalton group such as the bulk aerodynamic method, the methods in the so-called combination group 1981). For many years, the Egyptian Ministry of Water such as the Penman, Priestley–Taylor, and deBruin–Keijman − Resources and Irrigation adopted the figure of 7.54 mm day 1 methods, and the methods in the temperature group such as as the annual mean evaporation rate with a maximum value the Papadakis method among others (Lenters et al. 2005; − in June of 10.8 mm day 1 and a minimum in December of Winter 1981;Winteretal.1995). Another method that is − 3.95 mm day 1 (Whittington and Guariso 1983). used in numerous studies is the mass transfer method (e.g., Evaporation from open-water bodies is quite different Yu and Knapp 1985; Ikebuchi et al. 1988; Laird and from these above-land surfaces (Eichinger et al. 2003). The Kristovich 2002) because of its ease of application and sun’s energy penetrates the water to depths of as much as suitability for modeling (e.g., Hostetler and Bartlein 1990; 30 m in clear water, somewhat less in turbid water, and is Blodgett et al. 1997; Lenters et al. 2005). The water-budget stored throughout the water column (Eichinger et al. 2003). method (e.g., Myrup et al. 1979) can potentially provide a The water column is mixed by surface motion and becomes most reliable estimate of evaporation, as long as each water- the source of energy that drives evaporation (Eichinger et budget component is accurately measured or modeled, which al. 2003). Because of the large heat storage capacity of is often a difficult task, especially for the groundwater losses − water (1.006×106 Jm3) and the fact that water is (Lenters et al. 2005). In general, however, the BREB and approximately 1,000 times more dense than air, the eddy-correlation techniques are considered to be the most temperature of deep, clear, water bodies does not change accurate methods, albeit at the cost of additional, high- significantly throughout the day when compared to the quality instrumentation data (Lenters et al. 2005; Rosenberry atmosphere (Eichinger et al. 2003). The amount of et al. 2007; Winter 1981). As reported by Sene et al. (1991) available energy at the surface is nearly constant throughout and Stannard and Rosenberry (1991), except for a recent the day and night, leading to a nearly constant evaporation study by Blanken et al. (2000), most applications of the rate. The water in Nasser Lake lies between the two eddy-correlation technique have been limited to applications extremes of a clear water body and a land surface (desert). with short-term data series. The energy-budget method is being The lake is deep over much of its area and is relatively the preferred technique for evaporation estimation based on Evaporation estimates from Nasser Lake, Egypt 441 accurate, long-term monitoring data (Winter et al. 2003; evaporation studies for Nasser Lake applied conventional Lenters et al. 2005; among others). methods, except Omar and El-Bakry (1981) and Sadek et Despite this preference, few BREB-based lake evaporation al. (1997), who also applied the BREB method, but with studies based on long-term data series (at least 5 years) very limited data. All previous studies relied on ground are found. Lenters et al. (2005) presented a comprehensive, station data, except the Omar and El-Bakry (1981) study. 10-year BREB-based analysis of seasonal, intraseasonal, and The evaporation estimates by Sadek et al. (1997) have a interannual variations in lake evaporation for the Sparkling rather narrow range, from 5.70 to 7.05 mm day−1 (3.25 Lake in Northern Wisconsin (USA). Other BREB-based billion m3 difference at its highest water level), or only studies have been conducted by Robertson and Barry (1985) slightly above 4% of the natural annual Nile flow at the for the Perch Lake in Ontario (Canada), also based on AHD (84 billion m3). Sadek et al. (1997) excluded the 10-year data series, and by Sturrock et al. (1992)forthe results of the bulk aerodynamic method due to its severe Williams Lake in Minnesota (USA), based on a 5-year limitations. After excluding these results, they obtained an analysis. Several other studies used data of shorter duration. average annual evaporation estimate of 6.0±0.3 mm day−1, Myrup et al. (1979) provided a review of some of the earlier or 20% less than the 7.5 mm day−1 adopted by the irrigation studies of lake energy budgets, as well as their own 3-year authorities in Egypt and Sudan. Omar and El-Bakry (1981) analysis of the energy and water budgets of Lake Tahoe in estimated the mean annual evaporation from Nasser Lake to California-Nevada (USA). Lee and Swancar (1997)computed be 7.35 mm day−1 with the maximum rate in June the evaporation for the Lucerne Lake, a seepage lake in (10.9 mm day−1) and the minimum rate in January Florida, using the BREB method to describe the influence of (3.8 mm day−1). During periods when the lake level is the evaporation losses to the hydrologic budget of the lake. highest, the yearly evaporation increased to about 17.5 More recently, Winter et al. (2003) conducted a comprehen- billion m3. Another study by Morton (1983b) estimated the sive 6-year BREB-based analysis of evaporation rates for mean annual Nasser Lake evaporation to be 5.7 or Mirror Lake in New Hampshire (USA). 5.9 mm day−1. The latter result includes the values of change Nemours short-term BREB studies have compared the in heat storage and net inflow published by Omar and El- BREB methods with other evaporation methods over lakes Bakry (1981), which led to higher evaporation rates than the and reservoirs. For example, Rosenberry et al. (2004) earlier estimates by Morton (1979)of4.66mmday−1 compared evapotranspiration rates determined with several (increased to 5.23 mm day−1 after some adjustment). empirical methods for a wetland in semiarid North Dakota. The present study of Nasser Lake evaporation quantifi- Rosenberry et al. (2007) compared the results from 14 cation was designed to be similar to those conducted for alternate evaporation methods during six open-water seasons Mirror Lake in New Hampshire (Rosenberry et al. 2007), with values from the BREB method for a small mountain Williams Lake in Minnesota (Winter et al. 1995), and lake (Mirror Lake) in the northeastern USA. Sacks et al. Sparkling and Lucerne Lakes in Florida (Lee and Swancar (1994) present a short term, but interesting comparison of 1997). The BREB method was taken as the standard one, BREB rates for two subtropical lakes with different depths in but its results were compared with those of six other Florida (USA). Winter et al. (1995) compared a nearly conventional methods. Three of these conventional meth- identical suite of evaporation methods, applied to a medium- ods quantify both the energy and advective terms and are sized lake in Minnesota. Rasmussen et al. (1995)usedseven therefore called “combination methods”. One method empirical methods to estimate evaporation from nine lakes in requires measurements of one or more of the following Minnesota. Dalton et al. (2004) compared evaporation rates variables: air temperature, humidity, water-surface temper- determined with several methods for Lake Seminole, a large ature, and wind speed. Also, the water-budget method, reservoir that borders Georgia and Florida. which expresses the change in the Nasser Lake water The studies that used BREB as the standard (e.g., storage over a certain period of time as the difference Rosenberry et al. 2007; Rosenberry et al. 2004; Winter et between the in- and outflow volumes over the same period, al. 1995; Dalton et al. 2004; Harbeck et al. 1958; Anderson has been tested, as well as an additional simple method that 1954) found that traditional methods, which emphasize an only requires measurement of air temperature. Sensitivity assessment of energy fluxes, provide the best estimates of and uncertainty analysis, moreover, is conducted using a water loss to the atmosphere. Rankings of alternate methods variance-based method in order to analyze the accuracy of for estimating evaporation varied among these studies. In the BREB results and its sensitivity to errors in the input some cases, the robustness of a particular empirical method data. The sensitivity analysis also provides information of was dependent on the ambient climate. the most sensitive parameters, which may need extra Previous estimates of average annual evaporation from caution. The research aimed to investigate the variability of Nasser Lake range from 1.7 to 2.9 m or from 4.66 to the output from the different alternative methods as compared 7.94 mm day−1 (Sadek et al. 1997). Most of the previous to the BREB values in order to determine the relative 442 M. Elsawwaf et al. significance of different evaporation drivers in a wide, desert to reconstruct the time series on the basis of the chosen year lake like Nasser Lake. The suitability of the use of less robust and on the variance of the entire data set. In other words, and less expensive evaporation methods (that require fewer we repeat periodically the information of our reference year data) is discussed and, where appropriate, the estimates from adding some random disturbances whose variance depends the conventional methods adjusted to generate better evapo- on the variance of the initial data set. ration estimates for the Nasser Lake case. Coming to the second aspect, the outliers exist for Ta during 2000, for RH2 during 2003, and for U2 during 2003 at Raft station and for Ta during 1996 and for RH2 during 2 Data sources and quality 1995 at Allaqi station. Abusembel station does not have outliers for all meteorological measurements during the Aside from the lake surveys, all measurements are made at available period. In this case, we considered it a good idea 1-h intervals and averaged to daily values. The hourly to smooth out what can be an inconsistency, by using measurements are collected from three floating stations, moving average filtering. In particular, the moving average operated by the AHD Authority. The three stations have filter applied to the Ta series reduces the effects of possible worked with full capacity since 1995 for the stations at Raft outliers (see section on sensitivity).

(2 km upstream of the AHD) and Allaqi (75 km upstream The result of this analysis is shown in Table 1. Ta values of the AHD) and from 2000 for the station at Abusembel at Raft station are subjected to high uncertainty compared (280 km upstream of the AHD). The AHD Authority has with values at Allaqi and Abusembel stations (Table 1). three more floating stations (Fig. 1), at the Amada Temple This can be explained by the longest time span covered at

(185 km upstream of the AHD), at Toshka (240 km Raft station. Also, the Ta mean value at Raft is higher than upstream of the AHD), and at Arqeen (331 km upstream the Tamean at Allaqi and Abusembel, while the mean value of the AHD), but these stations were not used due to its of the water temperature at this station is lower.The mean limited meteorological data available. Measured variables value of U2 at Abusembel station is increased by 23% and (see Table 1) include net radiation, lake level and temper- 53% from the mean values at Raft and Allaqi stations, ature profiles, air temperature, relative humidity, wind respectively (Table 1). The evaporation methods that speed, and atmospheric. contain the U2 variable consequently may have unduly Here, we present results from 10 years of data collection higher evaporation estimates at Abusembel station (in (1995–2004). Dealing with data quality, we need to address comparison with the other two stations). The maximum two different aspects: On one side, we need to analyze the error of RH2 at Allaqi and Abusembel stations are the same, amount of missing data in our time series; on the other side, but higher than for Raft station (Table 1). There is a slight we want to detect the presence of outliers. Moreover, with change in the mean and uncertainty values of the Pa respect to missing data, it is also important to evaluate the parameter at all floating stations (Table 1). Despite these length of consecutive periods when data are not available. uncertainties in RH2 and Pa parameters, the resulting The three stations are quite different with respect to the first impact on the energy-budget-derived evaporation rates is aspect. Raft station data are almost complete. They span the rather small (see Section 5). period February 1995–September 2004. Abusembel station Next to these climatologic data, also, hydrological data presents an almost complete set of data going from September were collected in order to calculate the different water 2000 to September 2004. Quite different is the case of the balance terms for the lake. Data on the inflow in the lake Allaqi station where the amount of missing data is really were based on the measured flow at Dongola station, significant. We remind that, in principle at the Allaqi station, 780 km upstream of the AHD. The inflow discharge values we have coverage from February 1995 to September 2004, are obtained from the water level-discharge rating curve but, as already observed, there are a lot of missing periods. where the discharge is measured using the standard velocity Therefore, at Allaqi, we may proceed in recovering missing area method, and the water level is measured by means of a data at the daily time step, putting aside the issue of filling standard staff gauge from the Nile Water Sector, Ministry of longer periods (months, quarters, or even years). Water Resources and Irrigation in Egypt (Sadek et al. To recover the missing data, even in an extreme case as 1997). As the river section at this station essentially the Allaqi station, we may employ several methods, one of consists of a rock surface, the rating curve remains them being interpolation after the analysis of seasonal constant, a condition that is verified by periodic measure- components. Since the data coming from Raft station are ments. In addition, natural inflow data at Aswan were almost complete and they show a strong seasonal patterns collected. This is a calculated figure for the inflow that (as one should expect),we may infer that the same would arrive in Aswan if there was no water use in Sudan. would happen for Allaqi. So, we may consider one specific The outflow from the lake during the study period was year for which we have a good coverage and then try through the AHD gates and the Toshka spillway. Toshka vprto siae rmNse ae Egypt Lake, Nasser from estimates Evaporation Table 1 List of primary variables along with their mean value and standard deviation and percent data filled with methods: interpolation, analysis of seasonal components, and error bound estimates at the three floating stations

Variable Mean Standard deviation Filled (%) Error bounds

Raft Allaqi Abusembel Raft Allaqi Abusembel Raft Allaqi Abusembel Raft Allaqi Abusembel

Ta 26.7 25.5 25.4 6.7 5.6 6.2 4.9 2.1 5.7 [0.5 1.5] [0.5 1.3] [0.5 1.1] T0 22.0 24.9 24.3 4.0 4.8 3.9 5.6 2.1 5.7 [0.5 1.2] [0.5 1.0] [0.5 1.0] T-2 21.9 23.8 24.1 3.9 4.2 3.9 10.2 4.2 14.4 [0.5 1.2] [0.5 1.0] [0.5 1.0] T-5 21.5 23.7 23.5 3.8 4.2 3.7 12.5 4.4 14.3 [0.5 1.0] [0.5 1.0] [0.5 1.0] T-10 20.8 22.4 22.9 3.4 4.0 3.6 12.9 4.6 14.1 [0.5 1.0] [0.5 1.0] [0.5 1.0] T-15 20.4 22.2 22.4 3.0 3.6 3.4 12.8 4.5 14.1 [0.5 1.0] [0.5 1.0] [0.5 1.0] T-20 19.9 21.1 21.8 2.6 2.8 3.2 12.8 4.5 14.1 [0.5 1.0] [0.5 1.0] [0.5 1.0] T-30 19.0 20.2 – 1.8 2.2 – 12.8 4.5 – [0.5 1.0] [0.5 1.0] – T-40 18.3 18.8 – 1.3 1.3 – 12.8 4.5 – [0.5 1.0] [0.5 1.0] – T-50 18.1 19.1 – 1.0 1.3 – 13.3 5.1 – [0.5 1.0] [0.5 1.0] – Qn 164.0 134.0 44.6 38.0 –– – [5.8 23.2] [5.8 20.0] Qv 18.0 18.0 18.0 52.8 52.8 52.8 –– – [−5.8 23.2] [−5.8 23.2] [−5.8 23.2]

Qx 2.5 −12.4 −5.5 293.9 397.9 329.1 –– – [−23.2 23.2] [−22.2 22.2] [−20.2 20.2] BR −0.25 −0.05 −0.03 0.2 0.3 0.10 –– – [−0.9 0.9] [−0.4 0.4] [−0.2 0.2]

RH2 36.35 49.9 43.6 10.30 10.0 9.7 –– – [1.0 3.6] [1.0 5.0] [1.0 5.0] Pa 99190 99450 98964 454 459 414 –– – [50 150] [50 150] [50 150] U2 3.5 2.8 4.3 1.2 1.0 1.3 –– – [0.2 0.7] [0.2 0.5] [0.2 0.7]

Ta air temperature at 2 m above the lake surface (°C), T0 lake surface temperature (°C), T-2 lake water temperature at 2-m depth (°C), T-5 lake water temperature at 5-m depth (°C), T-10 lake water temperature at 10-m depth (°C), T-15 lake water temperature at 15-m depth (°C), T-20 lake water temperature at 20-m depth (°C), T-30 lake water temperature at 30-m depth (°C), T-40 lake water −2 −2 temperature at 40-m depth (°C), T-50 lake water temperature at 50-m depth (°C), Qn net radiation (W m ), Qv net energy advected by streamflow, ground water, and precipitation (W m ), Qx −2 change in stored energy (W m ), BR Bowen ratio, dimensionless, RH2 relative humidity at 2 m above the lake surface (%), Pa atmospheric pressure at the lake surface (Pa), U2 wind speed at 2 m − above lake surface (m s 1 ) 443 444 M. Elsawwaf et al. spillway is located 270 km upstream of the AHD. This temperature (Pa) spillway spills the water at 178 m above the mean sea level ea Vapor pressure at 2 m above the lake (Pa) to four natural depressions (Toshka depressions) located to the east side of Nasser Lake. The spillway was, however, The BR value of every month of the year was obtained only used during the water years from 1988–1999 to 2000– using the hourly data for the stations Raft, Allaqi, and 2001. Most of the outflow thus goes through the AHD gates; Abusembel. The monthly values of BR for different stations these outflow discharges were obtained from the water level and the values obtained by Omar and El-Bakry (1981)are measurements and the hydraulic characteristics of the AHD given in Table 2. From this table, it is clear that the BR gates. The gates’ hydraulic characteristics are under regular values at Raft station are more negative than the values at calibration. The lake storage and surface areas were Allaqi and Abusembel stations and the values used by Omar determined from the reservoir level measurements together and El-Bakry (1981). It is explained by the higher negative with the reservoir level–volume relationship obtained in values of the term (T −T )inEq.2 at Raft station (Fig. 2). Elsawwaf et al. (2005). Both the in- and outflow data were 0 a Use of the BR in developing Eq. 2 has been recognized provided through the annual reports of the Ministries of as a source of uncertainty in the BREB method (e.g., Lee Water Resources and Irrigation in Egypt and Sudan. and Swancar 1997; Anderson 1954). When the evaporation rate is high, the BR variable acts as a small correction factor. However, when the evaporation rate is low, the BR 3 Material and methods factor can cause unrealistic evaporation estimates. This occurs when the temperature gradient (T −T ) in Eq. 2 is 3.1 BREB method 0 a negative or when the vapor pressure gradient (e0−ea) approaches zero. Plots of the daily vapor pressure and The BREB method equation (Lee and Swancar 1997; temperature gradients in Fig. 2d–i show that these con- Lenters et al. 2005; Rosenberry et al. 2007) for calculating ditions occurred during the entire year, except for the winter open-water evaporation can be stated as: months (from November to March). þ ¼ Qn Qv Qx : 7 ð Þ The effect of small or inverted temperature and vapor EEB ½ðÞþþ ðÞ 8 64 10 1 rw L 1 BR cT0 Tb pressure gradients on the BR calculations can be seen in the plot of daily BR values (Fig. 2a–c). The validity of this where ratio is questionable if it is less than −1.0 or greater than 1.

EEB Volume of evaporating water by the BREB method When this occurs, normally acceptable errors in daily (mm day−1) measurements of temperature and vapor pressure result in −3 w Density of evaporating water (assume 998 kg m ) unrealistic evaporation estimates. Only Raft station L Latent heat of vaporization (J kg−1) c Specific heat capacity of water (4,186 Jkg−1°C−1) Tb Reference base temperature (°C) Table 2 The mean BR values (computed from the daily values) at all floating stations and the values used by Omar and El-Bakry (1981) The multiplier 8.64×107 that appears in Eq. 1 is to convert output to mm day−1. Month Bowen ratio (BR) Raft Allaqi Abusembel Omar and 3.1.1 Bowen ratio station station station El-Bakry (1981)

− The Bowen ratio (BR) variable can be seen as the proportion January 0.014 0.071 0.105 0.175 − − of the energy utilized for evaporation. As neither the energy February 0.023 0.005 0.063 0.006 conducted and convected from the lake to the atmosphere as March −0.358 −0.079 −0.065 −0.041 sensible heat nor the energy used for evaporation can be April −0.327 −0.034 −0.089 −0.047 measured directly, the BR has been widely calculated using: May −0.379 −0.138 −0.105 −0.064 June −0.348 −0.189 −0.095 −0.115 c P T T ¼ pa a 0 a ð Þ July −0.220 −0.100 −0.088 −0.095 BR : 2 0 622L e0 ea August −0.359 −0.064 −0.116 −0.080 where September −0.351 −0.125 −0.084 −0.068 October −0.222 −0.111 −0.022 −0.055 c The specific heat of air at constant pressure pa November −0.244 0.013 0.053 0.034 (1,011 Jkg−1°C−1) December −0.132 0.092 0.123 0.175 e0 Saturation vapor pressure at the water-surface Evaporation estimates from Nasser Lake, Egypt 445 contains many BR values outside that range. These which we correct using regressions against data available values are due to zero and negative values of the vapor (Table 1). pressure gradient (e0−ea;Fig.2d) and uncertainties in the measurements of Ta and the T0 (Fig. 2 g). The BREB 3.1.3 Net energy advected (Qv) evaporation estimates consequently will be less accurate for this station. Qv is considered one of the most important inputs for the energy-budget equation in the case of Nasser Lake. It is

3.1.2 Net radiation (Qn) based on the surface and ground water in- and outflows to and from the lake. Qv may be a difficult variable to measure Qn is measured at four floating stations: Raft (during accurately in a lake or reservoir with large surface water in- October–December 2004 and February 2005– October and outflow as is the case for Nasser Lake. The error in this 2006), Toshka (during August–October 2006), Abusembel measurement therefore can be a limiting factor to the (during July 2005–April 2006 and October–November successful application of the BREB method. Inflow from 2006), and Arqeen (during November–December 2002 the upper Nile catchments is assumed to be the only source and August–September 2004) stations. Occasionally miss- of advected heat to Nasser Lake. Advected heat from ing or bad data are filled using empirical equations ground water is assumed to be negligible for this study. The described by the FAO-56 procedure (Allen et al. 1998). long-term average rainfall at Aswan is 1.2 mm year−1 (e.g., This method utilizes measurements of air temperature, Griffiths 1972; Shahin 1985). This depth corresponds to humidity, and cloudiness to estimate downwelling long- some 5 million m3 year−1 or less than 0.01% of the inflow wave radiation. Here, we use temperature and humidity to the reservoir. The Qv portion explained by rainfall over measurements from the floating stations and estimate Nasser Lake thus also can be neglected. Consequently, Qv cloudiness from the shortwave radiation data (relative to a is calculated from the daily in- and outflow volumes and theoretical clear sky). The net shortwave radiation resulting temperatures through the following formula: from the balance between incoming and reflected solar radiation is adjusted using an Albedo reflection coefficient cr q ðÞT T cr q ðÞT T Q ¼ w i i b w o o b ð3Þ of 0.08 as obtained from Maidment (1993). We find that n A this method provides Q estimates, which are in very good n where agreement with the observations (R2=0.82, 0.89, and 0.92 3 for Raft, Allaqi, and Abusembel stations, respectively). qi Inflow to the lake (m ) 3 There is, however, a slight bias in the empirical estimates, qo Outflow from the lake (m )

a b 3 c 3 3 2 2 2 1 1 1 0 0 0 –1 –1 –1 –2 –2 –2 –3 BR, (dimensionless) –3 –3 Jan–96 Jan–98 Jan–00 Jan–02 Jan–04 Jan–95 Jan–97 Jan–99 Jan–01 Jan–03 Jan–05 Jan–01 Jan–02 Jan–03 Jan–04 Jan–05 d e f 5000 5000 5000 2500 2500 2500 , (Pa)

a 0 0 0 e – 0 e –2500 –2500 –2500

–5000 –5000 –5000 Jan–96 Jan–98 Jan–00 Jan–02 Jan–04 Jan–95 Jan–97 Jan–99 Jan–01 Jan–03 Jan–05 Jan–01 Jan–02 Jan–03 Jan–04 Jan–05 g 20 h 20 i 20 10 10

C) 10 ° , ( a 0 0 0 T – 0

T –10 –10 –10

–20 –20 –20 Jan–96 Jan–98 Jan–00 Jan–02 Jan–04 Jan–95 Jan–97 Jan–99 Jan–01 Jan–03 Jan–05 Jan–01 Jan–02 Jan–03 Jan–04 Jan–05

Fig. 2 a–c Daily average BR value for the period 1995–2004 at Raft the same periods. g–i Daily average temperature difference between and Allaqi stations and for the period 2000–2004 at Abusembel the lake surface and 2 m above the lake surface at Raft, Allaqi, and station. d–f Vapor pressure difference between the lake surface and Abusembel stations for the same periods 2 m above the lake surface at Raft, Allaqi, and Abusembel stations for 446 M. Elsawwaf et al.

Ti Daily average temperature of water inflow (°C) bottom of the AHD at a level of 121.30 m amsl. Figure 3 To Daily average temperature of water outflow (°C) represents the computed monthly average from the daily A Daily surface lake area (m2) calculations of the net advected heat for the period 1995– 2004 and the values of Omar and El-Bakry (1981). Equation 3 needs to be multiplied by 11.6×10−6 to −2 obtain a unit of W m for Qv. 3.1.4 Change in stored energy (Qx) In Eq. 3, the inflow is based on the measured flow at Dongola station, while the outflow from the lake was based The change in stored energy (Qx) is an essential component on the flow data at the AHD gates and the Toshka spillway of the energy budget because the large specific heat (see Section 2). capacity of water allows even a small lake to store and The temperature of the inflowing water is obtained from exchange large amounts of heat energy. Daily stored heat the daily measured lake water temperature column at was computed from the thermocouples based hourly water Arqeen station (331 km upstream of the AHD) for the temperature measurements at depths 0, 2, 5, 10, 15, 20, 30, – period April September 2003. Daily water temperature 40, and 50 m. Since the temperature varies with depth in measurements were also available at Arqeen and Abusem- the lake, the stored heat was numerically integrated using bel stations. Power relations between the water temper- increments of volume for each of the nine layers in which atures at Arqeen and Abusembel and between Arqeen and the temperature was measured and assumed constant (as Allaqi stations were constructed to estimate the missing also done in, e.g., Saur and Anderson 1955): periods of the temperature of water inflow at Arqeen. These P P r n ðÞ Δ r m ðÞ Δ power relations are given by: ¼ c w i¼1 T2i Tb V2i c w i¼1 T1i Tb V1i Qx n A ¼ ð Þ TiArqeen mT0Abusembel=Allaqi 4 ð5Þ where where m and n The model parameters 1 and Refers to conditions at the beginning and the end 2 of the period (day) TiArqeen Daily water temperature of water inflow to Nasser Lake at Arqeen station (°C) n and Refers to the water layer number m T0Abusembel=Allaqi Daily water temperature at Abusembel or Allaqi stations (°C) T1i Temperature of layer (water body) i at the beginning of the day (°C)

The parameters, which gave the best fittings, are given in T2i Temperature of layer (water body) i at the end of Table 3. The error standard deviations were 1.48°C and 0.53°C the day (°C) – – for Arqeen Allaqi and Arqeen Abusembel models, respec- ∆V1i Volume of water in the layer i at the beginning of 2 tively. The mean square error (MSE) was 2.21 and 0.284[°C] the day (m3) – – for Arqeen Allaqi and Arqeen Abusembel models, respec- ∆V2i Volume of water in the layer i at the end of the day 3 tively. The error standard deviation in Qv associated with the (m ) missing data for the water temperature at Arqeen and the use of these power functions is 2.6 Wm−2. Multiplication of Eq. 5 by 11.6×10−6 is required to −2 The outflow temperature at Toshka spillway was taken convert the unit of Qx to W m . from the surface water temperature at Abusembel station. The water volume of each layer was determined from the The water temperature of the outflow release from the AHD daily lake level data, which were obtained from the AHD gates was estimated using the measured water temperature Authority using the rating curve developed by Elsawwaf et at 50-m depth at Raft station. This deepest observation al. (2005). The monthly Qx values are shown in Fig. 4 and depth was taken because the water is flowing out at the compared with the values by Omar and El-Bakry (1981). It

Table 3 The power model regression parameters m and n Model Model regression Regression residuals of daily temperature values at parameters Arqeen station based on Allaqi or Abusembel stations using mn Mean Mean squared Residual standard 2 Eq. 4 residual, [°C] residual, [°C] deviation [°C]

Arqeen–Allaqi 2.073 0.7554 1.00 0.003 1.11 Arqeen–Abusembel 0.5014 1.234 1.00 0.0003 1.03 Evaporation estimates from Nasser Lake, Egypt 447

Fig. 3 Net energy advection −2 (Qv)inWm calculated using Eq. 3 with the values of Omar and El-Bakry (1981)

is noteworthy that the net increase or decrease in Qx equals to calculate potential evapotranspiration because the evap- 22.97, 53.01, and −7.66 Wm−2 over the entire period at orating surface of Nasser Lake is open water, they are Raft, Allaqi, and Abusembel stations, respectively. These assumed here to represent evaporation. The methods are values should be close to zero when using the average data. grouped in Table 4 according to the type of method. The net increase in the heat content by Omar and El-Bakry Combination methods include an available energy term and (1981) was 122.96 Wm−2, which is higher and biased an aerodynamic term. These methods are the most data because they used fewer water temperature measurements intensive and require measurements for some or all of the and a shorter time period (1970–1971). terms Qn, Qx, Ta, U2, and ea. The Dalton-type method, one of which is compared here, requires measurements of U2, 3.2 Traditional estimates of evaporative fluxes T0, Ta, and ea. The mass transfer Dalton-type method (conventional methods) requires an empirical coefficient (N) that is site dependent. The last method listed in Table 4 requires measurements

Several most commonly used evaporation methods (e.g., only of Ta to calculate the saturated vapor pressures at Stewart and Rouse 1976; deBruin and Keijman 1979; maximum and minimum air temperatures. Brutsaert 1982; Harbeck et al. 1958; McGuinness and Bordne 1972) were selected for comparison with the BREB 3.3 Water-budget method values. The methods also were selected to represent a range of method complexity with regard to data requirement This method expresses the change in the reservoir content (Table 4). Although many of these methods were developed over a certain period of time as the difference between the

Fig. 4 Change in heat content −2 (Qx)inWm , at all floating stations, calculated using Eq. 5 with the values used by Omar and El-Bakry (1981) 448 M. Elsawwaf et al.

Table 4 Traditional methods for evaporation estimation, the results of which are compared to the results from the BREB method, in mm day−1

Method Reference Equation formula Developed for hi – ¼ Δ Qn Qx : 6 Priestley Taylor Stewart and Rouse (1976) EPT a Δþg Lr 86 4 10 10 days or greater

Δ ðÞQnQx 6 deBruin–Keijman deBruin and Keijman (1979) E ¼ 86:4 10 Daily dB K 0:85Δþ0:63g Lr ¼ Δ Qn Qx : 6 Penman Brutsaert (1982) EPen Δþg Lr 86 4 10 Greater than 10 days þ g ½: ðÞ: þ : ðÞ −2 gþΔ 0 26 0 5 0 54U2 es ea 10 6 Mass transfer Harbeck et al. (1958) EHar ¼ NU 2ðÞe0 ea 86:4 10 Depends on calibration of N ¼ : ½ −2 ðÞ −2 10 Papadakis McGuinness and Bordne (1972) EPap 0 5625 es max 10 es min 10 2 d Monthly

The multipliers 86.4×106 and 10 that appear in the several equations are to convert output to mm day−1 α=1.26=Priestley–Taylor empirically derived constant, dimensionless, Δ slope of the saturated vapor pressure–temperature curve at mean air −1 −1 temperature (Pa°C ), γ psychometric “constant” (depends on temperature and atmospheric pressure; Pa°C ), U2 wind speed at 2 m above lake water surface (ms−1 ), N coefficient of efficiency of vertical transport of water vapor by eddies of the wind (used 1.458×10−11 Pa−1 for Nasser −1 Lake as calculated by Omar and El-Bakry (1981); Pa ), es saturated vapor pressure at temperature of the air (Pa), d number of days in the month, esmax and esmin saturated vapor pressures at daily maximum and minimum air temperatures (Pa) in- and outflow volumes over the same period. The consistent among years at all stations and occurred during hydrological budget of the lake can be conceptually August for 1997, 1999, and 2001–2003, during October for calculated by means of an ordinary differential equation 1995, June for 1996, September for 1998, and July for 2000 (see Maidment 1993) such that at Raft station. The maximum monthly rates at Allaqi station occurred (when only the years with full series are dVðhÞ ¼ qi þ P O EðhÞSðhÞBðhÞQTðhÞð6Þ considered) during October for 1996, June for 2000, and dt August for 2001. At Abusembel, the maximum occurred where for all years during the month of August. The Allaqi and − Abusembel averages (total average during the study period) V Volume of water (storage) of the lake (106×m3day 1) 6 3 −1 differ from the value at Raft station with 1.44 and P Direct precipitation on the lake (10 ×m day ) −1 − 1.32 mm day , respectively. This value is higher when O Release rate from the lake (106×m3day 1) comparing the difference between Allaqi and Abusembel E Evaporated volumetric rate of water from the lake −1 − (0.12 mm day ). surface (106×m3day 1) − As discussed in Section 3.1.1, the BR values at Raft S Regional throughflow rate (106×m3day 1) − station are subject to large errors due to the higher B Change in bank storage of the lake (106×m3day 1) − uncertainty of the T of the lake. The number of time Q Flow rate to Toshka’s depression (106×m3day 1) a T moments where the BR value is outside the acceptable h Water level in the lake (m) range from −1 to +1 is significantly higher for Raft station than for the other two stations. When the average of the BR values at Allaqi and Abusembel stations were used in the 4 BREB results and discussions BREB estimations for Raft station, the higher evaporation estimates at Raft station were eliminated. After this 4.1 BREB rates modification, the new monthly evaporation rates at Raft station ranged from 2.5 to 11.2 mm day−1 and averaged − Comparisons of monthly evaporation rates were made for 5.90 mm day 1 (Fig. 5b), with the latter being − 109 months during 1995–2004 at Raft station, 58 months 1.32 mm day 1 higher than the value before the modifica- − during 1995–2004 at Allaqi station, and 40 months during tion. A difference of 1.32 mm day 1 in the average monthly 2000–2004 at Abusembel station. The values of BR at Raft evaporation rate corresponds to approximately 3.15 billion station gave BREB monthly evaporation rates ranging from m3 at the highest water level. This is a bit less than 3.8% 3.4 to 13.3 mm day−1 and averaged 7.22 mm day−1 during from the average annual Nile natural inflow at Aswan. the 10-year study period (Fig. 5a). The BREB values fall in Next to the sensitivity of the BREB results to the BR −1 the range from 2.9 to 9.1 mm day and averaged value, also, their sensitivity to the Qv values was inves- −1 5.78 mm day at Allaqi station and from 1.8 to tigated. To do so, random errors were applied to the Qv − 9.8 mm day−1 and averaged 5.90 mm day−1 at Abusembel estimates based on a standard deviation of ±2.6 Wm 2. station (Fig. 5c–d). Maximum monthly rates were not This error only included the power function based gap Evaporation estimates from Nasser Lake, Egypt 449

Fig. 5 Daily evaporation estimates from Nasser Lake: a 109 months averaged per month, as determined by the BREB method, 1995–2004 at Raft station using BR values from Raft station, b 109 months at (data from HADA, Egypt). Error bars indicate the estimated Raft station using BR values from Allaqi and Abusembel stations, maximum uncertainty (≅10%) of the BREB estimates as indicated in c 58 months at Allaqi station, and d 41 months at Abusembel station, Section 5 filling. A difference of 0.09 mm day−1 (or 1.5%) in the Abusembel station. Maximum monthly BREB/inflow ratios mean evaporation estimate was found. This means that the consistently occurred during the May–July season when the BR parameter could be considered as one of the most inflow to Nasser Lake is low (Fig. 6a–c). The maximum sensitive and thus important parameters among all param- ratio was 60.5% during May 2004 at Raft station, 59.9% eters in the BREB method. The accuracy in obtaining the during May 2003 at Allaqi station, and 66.98% during June measurements (e.g., air temperature) underlying the BR 2003 at Abusembel station (Fig. 6a–c). The minimum ratio values clearly determines the success of the BREB method. occurred during the flood season (from August to October) since Nasser Lake receives the highest inflows in that 4.2 BREB/inflow ratios period. The minimum ratio was 4.6% during September 1996 at Raft station, 5.7% during August 2003 at Allaqi Although the Nasser Lake evaporation changes only station, and 5.2% during August 2001 at Abusembel slightly over time, its variation presents two maxima and station. The mean BREB/inflow ratios for the entire period two minima in the year. The annual variability of the lake were very close for the three floating stations (from 26.9% level depends greatly on the monthly ratio of the lake at Raft station to 27.6% at Abusembel station). The mean of evaporation over the lake inflow, particularly during the the differences in BREB/inflow ratios has a value of 0.62% months from August to October when water storage is the when the results at Raft are compared with the results at highest. Figure 6 shows the ratios between the BREB Allaqi, 0.50% for Raft versus Abusembel, and 0.60% for estimates and the lake inflow rates for the different months, Allaqi versus Abusembel. The standard deviations of the seasons, and years at the location of the three floating differences in BREB/inflow ratios equal for these compar- stations. The analysis of the monthly BREB/inflow ratios isons 6.7%, 8.6%, and 5.9%, respectively. The maximum shows that the evaporation rate changes very considerably difference in BREB/inflow ratio is 16.41% (Raft–Allaqi) from month to month. The coefficient of variation ranges and 23.5% (Raft–Abusembel) during June 2003. This is from a maximum value of 56.3% during November to a due to the lower estimated BREB value (5.3 mm day−1) minimum value of 14% during April at Raft station, from during June 2003 compared with the estimated values at 32% during October to 3% during August at Allaqi station, Allaqi (6.4 mm day−1) and Abusembel (7.9 mm day−1) and from 41.5% during November to 13% during June at stations. 450 M. Elsawwaf et al.

Fig. 6 BREB/inflow ratios for a 109 months at Raft station, Abusembel station, and g 8 years at Raft station. Error bars indicate the b 58 months at Allaqi station, c 41 months at Abusembel station, d 35 estimated maximum uncertainty (≅10%) of the BREB estimates as seasons at Raft station, e 11 seasons at Allaqi station, f 11 seasons at indicated in Section 5 Evaporation estimates from Nasser Lake, Egypt 451

Figure 6d–f shows the mean seasonal BREB/inflow and after this event, the necessity of changing the AHD ratios at the three floating stations. Similar trends are operation policy (to decrease the huge amount of the water observed with maximum BREB/inflow ratios during the loss by evaporation) became clear. February–April and May–July seasons and minimum ratios during the flood and November–January seasons. The coefficient of variation ranges from a maximum value of 5 BREB sensitivity analysis 35% during the May–July season to a minimum of 12% during the February–AprilseasonatRaftstation.The In this section, global sensitivity analysis has been applied coefficient of variation at Allaqi and Abusembel could not to quantify the relative importance of the input variables be defined due to the limited available seasonal data. The (Qx, Qn, Qv, BR, and T0)oftheBREBmethodin values of the maximum and minimum ratios during the entire determining the evaporation output. More specifically, a season varied from time to time. At Raft station, the values of global sensitivity analysis tries to quantify the BREB the maximum and minimum ratios ranged from 12.4% uncertainty due to the uncertainty in the input variables, during 2002 to 6.3% during 1998 for the flood season, from both singly and in combination with one another. Variance- 29.1% during 2002 to 13.9% during 1998 for the November– based sensitivity analysis (SA) techniques are intended to January season, from 41.4% during 2003 to 29.8% during estimate how much output variability is explained by each 2002 for the February–April season, and from 58% during of the input variables. Variance-based methods for SA were 2004 to 17.9 during 1996 for the May–July season. At Allaqi used first by chemists in the early 1970s (Saltelli et al. 2008 station, the ratios have narrowed down to 8% and 30% and Cukier et al. 1973). Cukier and colleagues not only during the flood and February–April seasons, respectively. proposed conditional variances for a SA based on first- During the May–July season, the ratio values ranged from a order effects but were already aware of the need to treat maximum of 32.3% during 2003 to a minimum of 19.7% higher-order terms and of the underlying variance decom- during 1996. Medium range values were found during the position theorem assumptions (Saltelli et al. 2008 and November–January season, from a maximum of 27% during Cukier et al. 1978). 2001 to a minimum of 18.9% during 1999. The maximum Variance-based methods are described in detail in many and the minimum value of the ratios ranged from 39.2% studies (e.g., Cukier et al. 1973, 1978; Hora and Iman 1990; during the May–July season of 2002 to 8.2% during the Saltelli et al. 1999, 2004) and can be defined by referring to flood season of 2003 at Abusembel station. a decomposition of the model (BREB model) Y=f (X) itself On an annual basis, the BREB/inflow ratio values into terms increasing dimensionality (high-dimensional ranged from 11.5% during 1998 to 21.1% during 2002 model representation (HDMR), e.g., Ratto et al. 2007; (Fig. 6g) with an average value of 15.9%. The later values Rabitz et al. 1999; Rabitz and Alis 2000; and Sobol’ are based on the measured inflow to Nasser Lake at 1990a), i.e. Dongola station. If the natural inflow data at Aswan would P P P fXðÞ¼1; X2; :::; Xk f0 þ fi þ fij þþf12:::k be considered, these values decrease from a maximum of i i j>i 15.9% to a minimum of 10.9%, with average value of 12.9%. ð7Þ The designers of the AHD reservoir estimated the average 3 annual evaporation at Aswan as 10% (or 9 billion m ) from where each term in the input factors X (Qx, Qn, Qv, BR, and 3 the mean annual of the natural inflow (84 billion m )at T0) is a function only of the factors in its index, i.e., fi= 3 Aswan. This is lower than the value of 15.4 billion m f (Xi), fij=f (Xi,Xj) and so on. The various terms are defined as −1 ¼ ðÞ= ¼ = ; ðÞ= (based on 7.5 mday ) adopted by the irrigation authorities f0=E(Y), fi EY X i f0 and fij EYXi Xj EY X i in Egypt and Sudan. In this study, the mean annual EY=Xj f0. evaporation for the entire period using the BREB method It is obvious that VfðÞi =V ¼ VEYðÞðÞ=Xi =V;inother is calculated to be 12.1 billion m3 or 34% higher than the 9 words, the main effect in Eq. 7 equals the variance of the billion m3 AHD design value. This difference can be first-order term of the HDMR decomposition scheme of explained not only by the more advanced calculation the total unconditional variance V(Y), equivalent to method and floating stations data but also due to the higher HDMR, can be derived (e.g., Sobol’ 1990a, 1993;Ratto flows recorded during the study period. During in the so- et al. 2007) called drought period of 1978–1988, the water level of X X X Nasser Lake indeed dropped from 177.47 m amsl during VðYÞ¼ Vi þ Vij þþV12:::k ð8Þ November 1978 to its minimum of 150.72 m amsl during i i j>i July 1988 (Elsawwaf 2002). The total water loss during this 3 3 ¼ ðÞðÞ= ¼ ; period was 78 billion m (annual average 8.8 billion m ), where Vi VXi EXi Y X i ,Vij VXi; Xj EXij Y Xi Xj 3 ¼ ; ; whereas the total water deficit was 4.6 billion m . During Vi Vj, Vijl VXi;Xj;Xl EXijl Y Xi Xj Xl Vij Vil V jl 452 M. Elsawwaf et al.

Vi Vj V l and so on. Normalizing by V from Eq. 8,we input variables follow different distributions. One can observe obtain the closed identity for sensitivity indices: that Qn and T0 follow a uniform distribution, Qx and BR X X X follow normal distribution and Qv follows beta distribution. Si þ Sij þþS12:::k ¼ 1 ð9Þ The upper and lower bounds for the five input variables are i i j>i reported in Table 1. The SA was done for two different BR bounds. The first is using the upper and lower bounds of the where the Si are the first-order sensitivity indices of the input original values at Raft station and the other is using the upper parameters and Sij the second-order sensitivity indices and lower values at Allaqi and Abusembel stations (Table 1). (accounting for the possible interaction between two parame- These two cases were considered to investigate the sensitivity ters), and so on. of the uncertainty results to the BR bounds.

The sensitivity indices are nicely scaled in the range [0, 1], In both cases, the Qx is the most influential input and for the main effects, the summation of Si is less than 1, parameter among the other input parameters, while the T0 where the equality holds for purely additive models. For parameter is non-influential as its total sensitivity index, independent inputs, the total sensitivity index STi can also be which is negligible (Fig. 7a). The second influential defined as the sum of all effects containing the factor Xi, i.e. parameter for both cases is Qv (case 1) and BR (case 2). X X In both cases, the sum of the first-order effects on the total þ þ þþ ¼ ð Þ Si Sij Sijl S12:::k STi 10 variance is nearly 0.987 (Fig. 7a), and the sum of the j>i l>j>i second-order effects is 0.017 (Fig. 7b), while the sum of the total indices is 1.004. As the first-order sensitivity indices In this study, the sensitivity indices of the BREB method sums are very close to 1, there are no significant have been computed by applying the smoothing spline interactions among the input parameters in the BREB ANOVA recursive (SS-ANOVA-R) method, which has model (Fig. 7b). With the BREB model outputs calculated been developed by Ratto et al. 2007 (http://eemc.jrc.ec. for the 512 input errors sampled, the uncertainty analysis of europa.eu/softwareSS-ANOVA-R.htm). In most quasi- the BREB method is performed. The total variance of the Monte Carlo computations, low-discrepancy sequences are BREB error using the BR values from Raft station (case 1) used because they provide the best convergence properties is 0.96 (mm day−1)2, which corresponds to an error (e.g., Sobol’ 1976; Sobol’ et al. 1992; Bratley et al. 1993), standard deviation of 0.98 mm day−1. This value is very which allow fast computations. For more details, the reader close to the estimated error (variance of 0.83 (mm day−1)2 is referred to Ratto et al. 2007. In this study, 512 quasi- and standard deviation of 0.91 mm day−1) calculated from Monte Carlo sequences (Latin Hypercube sample “LHs”) the monthly running mean of BREB values with BR values have been generated for each of the five inputs variables of from Raft station and with the mean BR values from Allaqi the BREB method to estimate the sensitivity indices and the and Abusembel stations. When the lower and upper bounds total uncertainty for the BREB model. The LHs coupled of the BR values at Allaqi and Abusembel stations (case 2) with an optimization algorithm (maximin LHs), as in Ratto are used, the variance of the BREB error decreases to a et al. (2007), were used to reduce the computational load. The value of 0.20 (mm day−1)2, which corresponds to an error

Fig. 7 a First- and b second- order sensitivity indices obtained with the SS-ABOVA-R method (512 LHs) and based on the variance decomposition theory applied to the BREB input parameters for two differ- ent cases: case 1, bounds of BR obtained as the original values at Raft station; case 2, bounds of BR obtained as the mean values at Allaqi and Abusembel stations Evaporation estimates from Nasser Lake, Egypt 453 standard deviation of 0.62 mm day−1. The difference bias during August, September, and October (Figs. 8e, 9e, between the two estimations is considered small (0.74 and 10e). billion m3 at the highest water level of Nasser Lake). Evaporation values from the mass transfer method varied subsequently in comparison with the BREB values at different stations. The bias values were negative (average of 6 Comparisons of BREB method with conventional 2.95 mm day−1) during most of the period at Abusembel methods station, as would be expected since the mean values of wind speed at 2 m from the lake surface is much higher than the Most of the alternate approaches for determining evapo- mean values at Raft and Allaqi stations (Table 1). At the other ration compared well with the mean value of BREB. At stations, the bias was lower: −0.14 and 0.15 mm day−1 at Raft Raft station, four of the six alternate methods (Priestley– and Allaqi stations, respectively. The Papadakis method that

Taylor, deBruin–Keijman, mass transfer, and water budget) requires measurements of the maximum and minimum Ta provided mean evaporation values that were for all the provided similar large negative bias at Raft and Abusembel months within 1 mm day−1 (2.05 billion m3 at maximum stations (Figs. 8d and 10d). The average bias values water level) of the BREB values (Fig. 8a, c, and f). At the are −3.26 mm day−1 at Raft station and −2.54 mm day−1 at Allaqi station, all methods provided evaporation values Abusembel station. Due to their simplicity, values from within 1 mm day−1 (Fig. 9a-f). The Priestley–Taylor and the Papadakis method indicated higher negative bias during the deBruin–Keijman methods were within 1 mm day−1 of the entire period of analysis. The higher negative bias is due to the BREB evaporation values at Abusembel station. Two of the higher value of the coefficient 0.5625 that appeared in Table 4, three combination methods (Priestley–Taylor and deBruin– which was calibrated by McGuinness and Bordne (1972), but Keijman) compared very well with BREB values, having also due to the higher air temperature at these stations. small bias for different months except the months of the Values from the water-budget method (Figs. 8f, 9f, and flood season (August, September, and October), which 10f) showed a large negative bias in some months during have large deviations (seasonal bias). This seasonal bias is 1998, 2001, and 2003 and a large positive bias in few due to the impact of the absence of the Qv term in the months during 1996, 1998, 1999, 2001, and 2003 at combination methods. The Penman method had a negative different stations, but otherwise provided good comparison bias during most of the months, except the positive seasonal with the BREB values, particularly during 1995, 1997,

Fig. 8 Difference in calculated evaporation at Raft station, between the five methods presented in Table 4: a Priestley–Taylor, b mass transfer method, c deBruin–Keijman method, d Papadakis method, e Penman method, f water-budget method versus the BREB method 454 M. Elsawwaf et al.

Fig. 9 Difference in calculated evaporation at Allaqi station, between the five methods presented in Table 4: a Priestley–Taylor, b mass transfer method, c deBruin–Keijman method, d Papadakis method, e Penman method, f water-budget method versus the BREB method

2000, and 2002. The maximum higher differences (Table 6) operation actions that were taken during these years. For relative to the BREB values were 2.66 billion m3 during example, actions such as the artificial closure of the Toshka October 1998 at Raft station, 1.8 billion m3 during October spillway by an earth dam and the limitation of the 2001 at Allaqi station, and 1.42 billion m3 during July 2003 downstream release took place. Moreover, when the small at Abusembel station. These higher positive and negative earth dam was cleared afterward, the spillway efficiency biases during 1998 to 2003 were due to several irregular bounced back to 100%. Therefore, it is very difficult to

Fig. 10 Difference in calculated evaporation at Abusembel station, between the five methods presented in Table 4: a Priestley–Taylor, b mass transfer method, c deBruin–Keijman method, d Papadakis method, e Penman method, f water-budget method versus the BREB method Evaporation estimates from Nasser Lake, Egypt 455

Fig. 11 Mean ± standard deviation of the differences between the b Allaqi station, and c Abusembel station. The two gray lines display monthly evaporation estimates between the alternate modified the standard deviation value associated with the BREB estimates as evaporation methods and the BREB method at a Raft station, determined from Section 5 calculate accurately the evaporation losses using the water- to unity, but the regression relation explained only 32% and budget method during the period between 1998 and 2004. 79% of the variance at Raft and Abusembel stations, Standard deviations of differences between the BREB and respectively. The method which had zero correlation with water-budget methods at all three floating stations are very the BREB results was the water-budget method. high (Fig. 11): standard deviations of 7.2, 6.2, and The alternate evaporation methods also were ranked based 6.6 billion m3 at Raft, Allaqi, and Abusembel, respectively. on the percentage of monthly periods during which values The results from the alternate evaporation methods were from alternate methods differ less than 5, 10% and 20% of related to the BREB values using least-squares linear BREB values. At each station, the methods were ordered in regression with BREB estimates. A transformation model in Table 5 based on the 20% criterion. The rank of each method is the form lnðZÞ¼lnðaÞþb lnðuÞþlnðeÞ was constructed to different from one station to another. At Raft station, the compare the results of the alternate evaporation methods Priestley–Taylor method has the first rank with 77% of the (dependent variable Z) with the results of the BREB method monthly values differing less than 20% of the BREB values, (independent variable u). In that model, ln(a) is the intercept whereas the Penman method has the last rank. The deBruin– parameter, b the slope parameter, and ln(e) a random term, Keijman method has the best rank at Allaqi and Abusembel which expresses the model residual with the BREB results. stations having percentages of 76% and 80%, respectively, In fact, most of the alternate evaporation methods deviated whereas the water budget and mass transfer methods have the significantly from the BREB results (Table 5). All methods last rank at Allaqi and Abusembel stations, respectively. The ranked very badly based on the coefficient of determination of deBruin–Keijman method has the best rank at Raft and the regression (R2). The Penman and mass transfer methods Abusembel stations if the 5% criterion is used. The percentages ranked best based on the mean squared value of the regression are for this period and criterion 46% and 39% at Raft and residuals at the different stations. Using the regression slope b Abusembel stations, respectively. The two best methods, based as the ranking criterion (analyzing the proximity to the on the 20% and 10% criteria, are the combination methods, regression slope of 1), only the mass transfer method ranked which require the greatest number of measured input variables. best, at Raft and Abusembel stations. In several cases, the The Penman method has a bad rank because the wind function degree of correlation with the BREB values did not coincide term in the Penman equation is uncertain and has higher with the regression slope being near unity. For example, the fluctuations. However, the Papadakis method ranked fourth slope coefficient for the mass transfer method was very close and fifth and requires measurements of Ta only. The later 456 M. Elsawwaf et al.

Table 5 R2, slope parameter (b), intercept parameter ln(a)of Method Regression versus BREB % of results that did not differ more than the regression of the monthly 2 evaporation estimates by the R b ln(a) MSE 5% of BREB 10% of BREB 20% of BREB alternate methods versus the BREB values, the MSE of Raft station the regression, and the percent- Priestley–Taylor 0.15 0.21 1.36 0.14 38 64 77 age of monthly estimates that deBruin–Keijman 0.15 0.22 1.3 0.14 39 62 76 did not differ more than 5%, − 10%, and 20% from the BREB Mass transfer 0.32 1.09 0.76 0.08 16 34 58 values Water budget 0 −0.04 1.76 0.35 6 12 25 Papadakis 0.17 0.59 0.95 0.10 3 10 21 Penman 0.23 0.19 1.82 0.04 3 6 13 Allaqi station deBruin–Keijman 0.15 0.14 1.42 0.10 20 54 76 Priestley–Taylor 0.04 0.11 1.46 0.12 27 53 71 Mass transfer 0.09 0.33 1.16 0.15 6 14 39 Penman 0.07 0.13 1.69 0.07 2 20 38 Papadakis 0.21 1.30 −1.23 0.15 10 21 38 Water budget 0.00 −0.26 2.20 0.48 7 16 28 Abusembel station deBruin–Keijman 0.26 0.36 1.06 0.15 46 64 80 Priestley–Taylor 0.26 0.38 1.05 0.16 41 62 79 Water budget 0 −0.09 1.76 0.26 13 20 30 Papadakis –––– 13 20 28 Penman 0.4 0.26 1.7 0.03 5 13 18 Mass transfer 0.79 0.99 0.03 0.01 0 0 3 method provided values that did not differ more than 10% of was determined using the available 10-year meteorological BREB values for nearly 10% to 20% of the monthly measurements. The coefficient “N” reflects the efficiency of estimates. vertical transport of water vapor by turbulent eddies of the wind (Herting et al. 2004). N was calculated using the following formula (e.g., Herting et al. 2004): 7 Method adjustments 0:622r 0:16 N ¼ a hi ð11Þ P r 2 The alternate evaporation methods were adjusted following a w ln zm zd the same procedure of Rosenberry et al. (2007) to obtain z0 close relation with the BREB values. The results from the alternate methods compare much more fruitfully with where −3 BREB values if a simple offset is provided. ρa Density of air [1.220 kg m ] The three combination evaporation methods typically do Zm Height at which wind speed and air vapor pressure are not include the energy term Qv (Table 4). As mentioned measured; Zm=2 for this Nasser Lake case [m] previously, this term is usually significantly high. However, Zd Zero-plane displacement [m]; Zd=0 over typical water Nasser Lake, situated in an arid climate region with covered surfaces −4 Nubian sandstones mixed with igneous rocks, occasionally Zo Roughness height of the surface [m]; Zo=2.30×10 m receives a huge amount of inflow every year, which then is over typical water surfaces warmed in the lake and lost to surface outflow and to groundwater, giving Qv a greater significance. The combi- nation methods thereof were modified to include the Qv − term to the terms (Qn Qx) in Table 4, to be in the form of Values of Zd and Zo are obtained from Brutsaert (1982), as (Qn+Qv−Qx). The Papadakis multiplier (0.5625) was cali- suggested by Dingman (2002). They increase as wind brated to be consistent with the BREB values at Nasser Lake. speed increases due to the effects of waves. The new The new multiplier has an average value of 0.458. In coefficient N for Nasser Lake was calibrated using Eq. 11 − − addition, the mass transfer coefficient (N=1.458×10 11 Pa 1) to be 1.488×10−11 Pa−1. vprto siae rmNse ae Egypt Lake, Nasser from estimates Evaporation Table 6 Highest differences of the monthly and annual evaporation values obtained by the traditional methods versus the release rate from Nasser Lake, at Raft, Allaqi, and Abusembel stations

Method Monthly comparison from BREB Yearly comparison from BREB

Highest differences, Highest differences, Release rate from Highest difference Highest differences, Highest differences, Release rate from Highest difference − − mm day 1 billion m3 Nasser Lake (O), over O, % mm day 1 billion m3 Nasser Lake (O), over O, % billion m3 billion m3

Raft station Priestley–Taylor −0.74 (Dec. 96) 0.13 2.12 (Dec. 96) 6.13 0.20 (1995) 0.39 55.70 (1995) 0.70 deBruin–Keijman 2.00 (May 03) 0.32 6.48 (May 03) 4.94 0.52 (2003) 1.00 56.23 (2003) 1.78 Penman −1.70 (Nov. 03) 0.29 3.53 (Nov. 03) 8.22 −0.70 (1996) 1.35 55.15 (1996) 2.45 Mass transfer −7.00 (Nov. 03) 1.20 3.53 (Nov. 03) 33.99 −0.71 (1998) 1.50 65.52 (1998) 2.29 Papadakis −4.75 (June 98) 0.68 7.50 (June 98) 9.07 −1.07 (2002) 2.22 61.93 (2002) 3.58 Water budget −13.50 (Oct. 98) 2.66 6.94 (Oct. 98) 38.33 −2.78 (1998) 5.8 65.52 (1998) 8.85 Allaqi station Priestley–Taylor 1.22 (June 02) 0.20 7.94 (June 00) 2.52 No year with completely available data deBruin–Keijman 1.90 (June 02) 0.32 7.94 (June 02) 4.03 Penman 1.90 (June 02) 0.32 7.94 (June 02) 4.03 Mass transfer −6.90 (Nov. 95) 1.15 2.14 (Nov. 95) 53.74 Papadakis −5.78 (Dec. 96) 1.05 2.12 (Dec. 96) 49.53 Water budget −9.53 (Oct. 01) 1.80 6.09 (Oct. 01) 29.56 Abusembel station Priestley–Taylor 1.10 (Oct. 02) 0.19 3.78 (Oct. 02) 5.03 Only one meteorological year (2001) available deBruin–Keijman −1.60 (Mar. 04) 0.27 4.34 (Mar. 04) 6.22 Penman 2.30 (May 04) 0.36 5.73 (Mar. 04) 6.28 Mass transfer −5.20 (Apr. 02) 0.94 5.30 (Apr. 02) 17.74 Papadakis −4.25 (Apr. 02) 0.77 5.30 (Apr. 02) 14.53 Water budget −9.72 (July 03) 1.42 6.98 (July 03) 20.34 457 458 M. Elsawwaf et al.

8 Comparisons of BREB method with adjusted methods The method that requires measurements of only Ta, the Papadakis method, requires a 36% reduction in the 0.5625 Nasser Lake has a large evaporation volume every water multiplier to eliminate the difference with the BREB results year. The importance of the evaporation is for this lake at Raft station, 8% reduction to eliminate this difference at greater than for any other large reservoir in the world. For Allaqi station, and 18% reduction at Abusembel. The the same reason, the significance of errors is perhaps average reduction in the 0.5625 multiplier for Nasser Lake greater for this lake than anywhere else. Table 6 shows the is estimated to be 19%, which gives a new multiplier of maximum differences between the BREB results and the 0.458. In spite of the mean monthly values of the modified results of the other conventional methods in billion m3 and Papadakis and BREB values being nearly identical and the as a percentage of the release rate from the reservoir. These differences between these mean values being lower than the give an indication of the importance of the evaporation standard deviation of the BREB errors, still, the monthly volume and its uncertainty in the Nasser Lake water differences between the two methods have large biases balance, also in relation to the water demand (by irrigation, (Table 6). The standard deviation of the differences municipal and industrial uses, navigation, etc.) for Egypt. between the BREB and modified Papadakis methods range When the three combination evaporation methods are from three to six times the results obtained from the three modified to include Qv, the bias and the standard deviation combination methods (Fig. 11). of the differences (Fig. 11) become clearly smaller. After Comparison of the modified mass transfer method with the modification, the modified Priestley–Taylor method BREB values at Abusembel station showed much higher showed best agreement with the BREB results at the three differences than the estimates based on the coefficient floating stations. The maximum monthly differences in developed by Omar and El-Bakry (1981). The differences are billion m3 are 0.13 during December 1996, 0.20 during negative (average bias of 6.56 billion m3), as would be June 2002, and 0.19 during October 2002 at Raft, Allaqi, expected since the calibrated N is higher than the N value and Abusembel, respectively (Table 6). The corresponding taken from the study of Omar and El-Bakry (1981). At the relative differences (relative against the BREB results) vary other stations, the bias is much lower compared with the bias from2.54% at Allaqi station to 6.13% at Raft station. The at Abusembel: −0.53 and 0.10 billion m3 at Raft and Allaqi mean differences relative to the BREB values of the entire stations, respectively. The mass transfer method may be period were 0.33, 0.02, and 0.82 billion m3 at Raft, Allaqi, unduly sensitive to wind speed when applied to Nasser Lake. and Abusembel stations, respectively. During periods when the wind speed is substantially larger or The modified Penman method indicates maximum smaller than normal, the mass transfer method provides monthly differences with the BREB results in billion m3 evaporation values that are substantially different from the of 0.32 (4.03% relative to the release rate from Nasser BREB values. Lake) during June 2002 and 0.36 (6.28%) during May 2004 The modified conventional evaporation methods moreover at Allaqi and Abusembel stations, respectively, whereas it were compared with the BREB results based on the annual indicates maximum differences of 0.29 billion m3 (8.22%) estimates at Raft station (Table 6). At this time scale, the during November 2003 at Raft station (Table 6). The mean results from the Priestley–Taylor method compare much evaporation difference relative to the BREB value is 0.96, favorably followed by the deBruin–Keijman method. The 0.29, and 0.72 billion m3 at Raft, Allaqi, and Abusembel mass transfer method showed much smaller differences stations, respectively. The negative biases at Raft and (Table 6) compared with the differences from the monthly Abusembel stations are due to the higher wind values at comparison. The surprisingly good performance of this these stations. The modified deBruin–Keijman method evaporation method, which makes use of measurement of 3 gives a maximum monthly difference of 0.32 billion m at e0, ea,andU2, indicates that wind speed is better correlated both Raft and Allaqi stations and 0.27 billion m3 at with the evaporation process at Nasser Lake for the longer Abusembel station. It gives close mean differences relative time periods than for the short periods. Of the remaining two to the BREB values at Allaqi and Abusembel stations, 0.33 methods, the large Papadakis and water-budget departures and 0.47 billion m3, respectively, while it shows a mean from the BREB values (Tables 5, 6,and7) indicate that these difference of 0.94 billion m3 at Raft station. The standard methods are not well suited for use at Nasser Lake. This deviations of the differences between the results of the conclusion is based on the low regression slope value of 0.23. Priestley–Taylor, deBruin–Keijman and Penman methods, Re-ranking of the evaporation methods following the modi- and the BREB results were small at each of the three fied methods leads to no change in the ranking of the three floating stations (Fig. 11). All three modified combination best combination methods (Table 7). The mass transfer evaporation methods provide mean values that differ from method ranked very badly based on the R2 values at all the BREB results less than the standard deviation on the stations except at Abusembel station where the method has a BREB error (0.92 billion m3). higher R2 value of 0.78. The combination group also ranked Evaporation estimates from Nasser Lake, Egypt 459

Table 7 R2, slope parameter (b), intercept parameter ln(a)of Method Regression versus BREB % of results that did not differ more than the regression of the monthly 2 evaporation estimates by the R b ln(a) MSE 5% of BREB 10% of BREB 20% of BREB alternate methods versus the BREB values, the MSE of Raft station the regression, and the percent- Priestley–Taylor 0.97 1.03 −0.08 0.003 64 93 100 age of that did not differ more deBruin–Keijman 0.94 0.95 0.07 0.007 47 73 96 than 5%, 10%, and 20% from − the BREB values Penman 0.89 1.05 0.27 0.008 31 56 79 Mass transfer 0.33 1.19 −1.04 0.080 17 35 57 Papadakis 0.23 0.73 0.95 0.100 6 13 40 Allaqi station Penman 0.89 1.15 −0.55 0.008 40 72 94 deBruin–Keijman 0.91 1.35 −0.99 0.009 32 74 92 Priestley–Taylor 0.92 1.64 −1.81 0.008 41 71 92 Papadakis 0.23 0.98 −0.22 0.140 13 18 40 Mass transfer 0.1 1.02 0.40 0.15 10 16 37 Abusembel station Priestley–Taylor 0.98 1 −0.01 0.003 56 85 100 deBruin–Keijman 0.95 0.87 0.29 0.008 54 79 97 Penman 0.78 1.96 −3.00 0.030 10 38 71 Papadakis ––– – 20 21 44 Mass transfer 0.78 1.04 −0.12 0.014 0 0 3 the best based on the MSE and R2 values at all stations, using to the BREB values that used the mean BR values from proximity to a regression slope of 1 as the ranking criterion. Allaqi and Abusembel stations. The water budget, Penman, The modified evaporation methods were ranked based on the mass transfer, Papadakis, Priestley–Taylor, and deBruin– percentage of monthly periods during which the evaporation Keijman methods indicate biases relative to the BREB estimates were within 5%, 10%, and 20% of the BREB values obtained using the BR value from Raft station of values. Table 7 showed the order of the modified methods at 0.71, 1.60, 1.84, 2.5, 2.78, and 3.32 billion m3, respec- each station based on the 20% criterion. The rank of each tively, and standard deviations of 7.87, 2.36, 4.69, 4.2, method is different from one station to another. The Priestley– 2.27, and 2.66 billion m3, respectively. These standard Taylor method has the first rank with 100% at Raft and deviations are considered high compared with the values Abusembel stations and the third rank with 92% at Allaqi illustrated in Fig. 11a. The differences in BREB estimates at station. The deBruin–Keijman method has the second rank Raft station (after applying the two BR values) correspond with 96%, 92%, and 97% at Raft, Allaqi, and Abusembel to the following percentages of the mean annual inflow to stations, respectively. The Penman method has the first rank Nasser Lake (estimated to be 55.5 billion m3): 0.6%, 1.3%, with 94% at Allaqi station and the third rank at Raft and 1.5%, 2.33%, 2.98%, and 3.40% for the Papadakis, water Abusembel stations with 79% and 71%, respectively. budget, mass transfer, Penman, Priestley–Taylor, and deBruin–Keijman methods, respectively.

9 Errors made by BR values from Raft station 10 Uncertainty analysis for the modified methods The Raft evaporation rate, associated with the selected BR The total uncertainty for the modified evaporation methods parameter value, was estimated by calculating the daily was quantified simply based on the variance-based method evaporation with the BR value obtained from the Raft data (as reported in Section 5). Equatio 8 can be rewritten as: and the one obtained from Allaqi and Abusembel stations. The difference in BREB values over the 10-year period was VEðÞ¼BREB Ei VBREB þ Vi þ VBREB;i ð12Þ analyzed using the BR value from Raft station, the BREB where values compared badly with the alternate modified conven- tional methods and better with the water-budget method. V(EBREB−Ei) First-order error variance for the difference This conclusion is consistent with the results of the between the BREB method and the comparison made before for the modified methods relative modified method i 460 M. Elsawwaf et al.

VBREB First-order error variance for the BREB temperature only are cost effective, but provide error values method (VBREB=0.2, as illustrated in that are much higher than the errors of the combination Section 5) modified methods. This means that these methods are not

Vi First-order error variance for the modified suitable to be used for evaporation estimation at Nasser Lake. method i

VBREB,i Second-order error variance between the BREB method and the modified method i 11 Results comparison with other dryland lakes

In Eq. 12, the second-order error variances account for Nasser Lake lies on hyper-arid regions or so-called the correlations that exist between the different uncertainty drylands (based on the dryland definition given in sources or inputs in the evaporation equations. The values Williams 2005). The area indeed has a very low annual of the second-order error variances in Table 8 show that rainfall total of 1.2 mm (e.g., Griffiths 1972;Shahin1985) interactions exist between the modified evaporation methods and a high evaporation total. There are many large lakes in considered. The higher interaction values exist in the the world (Caspian Sea, Aral, Chad, Balkhash, Eyre, and evaporation methods that rely on few input variables like Torrens) as well as reservoirs (Razza Dyke in Iraq, Glen the Papadakis and mass transfer methods. Canyon, and Hoover in USA), which lie in drylands. Following the modifications, all the methods of the Table 9 presents average daily and total annual evapo- combination group provide differences (percentage of ration estimates, based on literature, for some lakes and the error standard deviation over the mean BREB value) with reservoirs. Given that they have climate conditions similar the BREB mean value that are within 20% of the mean to the ones of Nasser Lake, the evaporation estimates are evaporation values (Table 8). The Priestley–Taylor, deBruin– also close. Lake Mead in the USA, for instance, is similar Keijman, and Penman methods have for each of the three to Nasser Lake not only for its climate conditions but also floating stations the smallest differences. The Papadakis and for its hydrologic basin properties and reservoir operations. mass transfer methods have the highest error standard When the BREB estimates are compared with these of deviation. Surprisingly, at Abusembel station, the mass transfer Nasser Lake, the total annual average evaporation differ- − and Papadakis methods have a value of standard deviation that ence is small: −122 or −0.32 mm day 1. The variance and considered small comparing with their values at Raft and the standard deviation of the overall monthly differences Allaqi stations (Table 8;Fig.11c). Given their simplicity, the between the BREB estimates from Nasser Lake and Lake − methods based on wind speed, water, and air temperature or on Mead also is small: 0.36 versus 0.6 mm day 1. The BREB

Table 8 The mean evaporation, first-order error variances, Methods Mean First- Second- Error standard Error standard second-order error variances, evaporation order order deviation deviation over −1 −1 standard deviation of error, and (mmday ) variance variance (mmday ) mean BREB value (%) percentages of error standard deviation over the mean BREB Raft station value for the modified evapora- Priestley–Taylor 5.74 0.93 1.17 0.96 16.3 tion methods at Raft, Allaqi, and deBruin–Keijman 5.54 0.91 1.16 0.96 16.3 Abusembel stations obtained with the variance-based method Penman 6.37 1.29 1.38 1.14 19.3 Papadakis 5.87 4.35 2.53 2.08 35.3 Mass transfer 6.16 4.63 2.62 2.15 36.4 Allaqi station Priestley–Taylor 5.77 1.37 1.42 1.17 20.2 deBruin–Keijman 5.62 1.32 1.16 1.15 19.9 Penman 5.64 1.22 1.34 1.11 19.2 Papadakis 5.84 5.62 2.88 2.37 41.0 Mass transfer 5.73 6.70 3.14 2.59 44.8 Abusembel station Priestley–Taylor 5.60 0.94 1.18 0.97 16.4 deBruin–Keijman 5.67 1.22 1.34 1.11 18.8 Penman 6.25 1.63 1.55 1.28 21.7 Mass transfer 9.09 2.10 1.75 1.44 24.4 Papadakis 5.80 2.40 1.90 1.56 26.4 vprto siae rmNse ae Egypt Lake, Nasser from estimates Evaporation Table 9 Published daily evaporation estimates from some large man-made reservoirs and endorheic fresh and saline lakes located in arid and semiarid climates distributed over the world

Reservoir or Location Surface Volume Evaporation Estimation method Lake type Climate Reference lake name area (km2) (billion m3) Annual averaged Daily averaged − − (mmyear 1 ) (mmday 1 )

Mead Arizona and 658.42 33.77 2,276 6.22 (data BREB Man-made reservoir Arid Westenburg et al. (2006) Nevada, USA 1997–1999) Mead Arizona and 658.42 33.77 1,951 5.34 (data Unknown Man-made reservoir Arid Westenburg et al. (2006) Nevada, USA 1953–1995) Chad Sahara Desert, Africa 22,000.00 72.00 2,300 6.30 Unknown Permanent freshwater Arid Thieme et al. (2005) Eyre South Australia 9,690.00 30.10 2,090 5.72 Satellite-derived Permanent saline Arid Prata (1990) Torrens South Australia 5,932.00 7.52 2,000 5.48 American class Permanent saline Arid Schmid (1988, 1990) A pan Aral Central Asia 66,500.00 1,023.00 970 2.66 Water balance Permanent saline Arid Szesztay (1974), Shnitnikov (1973), and Williams (2005) Caspian Central Asia 395,000.00 76,040.00 980 2.68 Water balance Permanent saline Arid Shahgedanova (2003) Balkhash Central Asia 18,300.00 112.00 1,047 2.87 Water balance Permanent saline Arid Shahgedanova (2003) Titicaca South America 8,500.00 1,147.5 1,350 3.70 Bulk transfer Partially permanent Semiarid Taylor and Aquize (1984), freshwater Delclaux et al. (2007) Titicaca South America 8,500.00 1,147.5 1,900 5.20 BREB Partially permanent Semiarid Richerson et al. (1977), freshwater Delclaux et al. (2007) Ihotry Southwestern Max 116.00 Max 159.00 1,725 (Mean) 4.73 (Mean) Pan-evaporation Permanent saline Semiarid Vallet-Coulomb et al. (2007) Madagascar Ihotry Southwestern Max 116.00 Max 159.00 1,900 (Mean) 5.20 (Mean) Complementary Permanent saline Semiarid Vallet-Coulomb et al. (2007) Madagascar relationship lake evaporation Manton Northern Territory, 4.40 0.02 1,971 5.40 Penman Man-made reservoir Semiarid tropical Vardavas and Australia Fountoulakis (1996) Manton Northern Territory, 4.40 0.02 2,139 5.86 Priestley–Taylor Man-made reservoir Semiarid tropical Vardavas and Australia Fountoulakis (1996) 461 462 M. Elsawwaf et al. difference is within the uncertainty range of the BREB BREB method and six traditional methods. The BR values estimates for Nasser Lake. at Raft station are subject to large errors due to the higher

Another example is Lake Chad, located in the Sahara Desert uncertainty of Ta and T0. These errors gave higher BR of Africa. It is the largest endorheic freshwater lake over the negative values and many values outside the range of [−1 world and also has a climate close to the one of Nasser Lake. 1]. Applying these values, the BREB evaporation rates The mean annual evaporation estimate (2,300 mm year−1)is ranged from 3.4 to 13.3 mm day−1 and averaged also for this lake close (to the corresponding value at Nasser 7.22 mm day−1. These values were modified using a Lake; Table 9). Also, the endorheic saline lakes located in correction to the BR parameter value based on the average South Australia (Eyre and Torrens) have very close evapo- of the values at Allaqi and Abusembel stations. After this ration estimates (Table 9): 2,088 and 2,000 mm year−1, correction, new evaporation rates were obtained ranging respectively. These estimates again compare surprisingly from 2.5 to 11.2 mm day−1 and averaged 5.90 mm day−1. well with the mean annual BREB estimate (2,170 mm) for They ranged from 2.9 to 9.1 mm day−1 and averaged Nasser Lake. The evaporation methods used in the case of 5.78 mm day−1 at Allaqi station and 1.8 to 9.8 mm day−1 the Lakes Eyre and Torrens were satellite-derived. Other and averaged 5.9 mm day−1 at Abusembel station. lakes, such as the ones located in Central Asia (Aral, BREB/inflow ratios were calculated for all months, Caspian, and Balkhash), are out of the expected range. seasons, and years to provide an indication of the volume These lakes are located in arid and semiarid climates, but of water contained in evaporation in comparison to the their published evaporation rates are much lower in inflow to Nasser Lake. When the natural flow at Aswan is comparison with the rates obtained for the lakes Nasser, used, the BREB/inflow ratio has a value of 14.5%. This Mead, Chad, Eyre, and Torrens (Table 9). The reason for this value is much higher than the value of 10% adopted by the is that the evaporation estimates for these lakes are not AHD designers. The coefficient of variation of the BREB/ sufficiently precise. Lougheed (2006) mentioned that the inflow ratio exhibits, however, a wide range: from a available evaporation data for the Caspian Sea are of very maximum of 56.3% during November at Raft station to low accuracy and are only useful for making general 3% during August at Allaqi. The mean BREB/inflow ratios estimates. The estimate for the twentieth century is for the entire period are very close: 26.9%, 27.1%, and 970 mm year−1 (e.g., Lougheed 2006;Rodionov1994). 27.6% at Raft, Allaqi, and Abusembel, respectively. The Another lake for which evaporation estimates are available seasonal BREB/inflow ratios at Raft station have lower in the literature is Lake Titicaca located in a high-altitude values for the coefficient of variation compared with the semiarid region in the Northern Andean Altiplano. It is a very monthly values. They range from a maximum of 35% largelake(1,147.5billionm3) with partially permanent during the May–July season to a minimum of 12% during freshwater. For this lake, the results again compare favorably. February–April season. The mean values of the seasonal The difference in total annual evaporation with Nasser Lake is BREB/inflow ratio are 10%, 25.7%, 24.4%, and 32.8% for rather high (270 mm) but logical since Lake Titicaca is the flood seasons, November–January, February–April, and located in a semiarid region where the evaporation is lower May–July, respectively, at Abusembel station. At Allaqi than in arid regions. Lake Ihotry, as another example, is station, these mean ratios are 8%, 22.8%, 31.1%, and located in a semiarid region in Southwestern Madagascar and 27.4% for the previous seasonal sequences. Based on the has a capacity volume of 159.00 billion m3, which is close to measured inflow to Nasser Lake at Dongola station, the the Nasser Lake capacity. Also, in this case, evaporation annual BREB/inflow ratio values are ranged from 11.5% values compare reasonably well with the BREB values for during 1998 to 21.1% during 2002 with an average annual Nasser Lake (Table 9). The differences in mean annual value of 15.9%. When the natural annual flow at Aswan is evaporation rates are within 0.74 mm day−1 (CRLE model) taken, this ratio decreases from a maximum of 15.9% to a and1.22mmday−1 (pan-evaporation method) of the Nasser minimum of 10.9% during the same time, with average value Lake BREB values. A final example is the Manton reservoir, of 12.9%. The mean annual evaporation for the entire period is located in a semiarid region in Northern Territory, Australia. 12.1 billion m3 or 34% higher than the value (9 billion m3) Considering its small volume and surface area, evaporation adopted by the designer of AHD. This value increases to values compare surprisingly well with the BREB values 15.4 billion m3 based on the 7.5 mday−1 adopted by the (Table 9). irrigation authorities in Egypt and the Sudan. SA has been applied to quantify the relative importance of the different input variables of the BREB method in 12 Summary and conclusions determining the value of the evaporation output. We have focused on the computation of sensitivity indices that are The present paper has provided a comprehensive 10-year based on decomposing the total variance of the BREB analysis of seasonal variations in lake evaporation using the evaporation error in a quantitative fashion. Sensitivity Evaporation estimates from Nasser Lake, Egypt 463 indices for the BREB input variables are computed based Quantification of the total uncertainty for the modified on the variance-based method and based on Latin Hyper- evaporation methods shows that the combination methods cube sampling. The SA is done for two different BR have the higher accuracy evaporation estimates. The error bounds. The first is using the upper and lower bounds of variances of these methods ranged from 0.93 (mm day−1)2 the original values at Raft station and the other is using the for Priestley–Taylor method at Raft station to 1.63 upper and lower values at Allaqi and Abusembel stations. (mm day−1)2 for the Penman method at Abusembel station. The SA for the two cases shows that the most influential The mass transfer and Papadakis methods provide much input error is the error in the Qx followed by the BR input higher variances at all the floating stations compared with parameter. The other inputs are less influential as their the variance values for the combination methods. sensitivity indices are negligible. The sum of the first- and This study has been the first one studying evaporation second-order variance terms are 0.987 and 1.004, for the for all available stations along Nasser Lake. In anticipation two cases. Given that these values are very close to 1, it can be of future studies, we have to continue to collect the assumed that the interactions among the input parameters in meteorology of Nasser Lake beyond the 10-year period of the BREB model are negligible. The total uncertainty of the the current study. Also, further analysis at additional BREB method has an error standard deviation of stations could be done. We anticipate that these extensions 0.91 mm day−1 for case one and 0.62 mm day−1 for case two. would help to reduce some of the uncertainties present in The values of the six traditional methods were compared the current study. with the BREB values. Evaporation methods that include available energy and aerodynamic terms (combination meth- Acknowledgments The financial support from the Applied Training ods) provide the best comparisons with the BREB evapo- Project (ATP) of Nile Basin Initiative (NBI) to accomplish this research ration. The three combination methods provided values that and the contributions from the ATP team are gratefully acknowledged. were within 20% of BREB values during more than 88% of We thank the AHD Authority for the use of their data and guidance and the Water Resources Research Institute (WRRI) of the National Water the energy-budget periods considered. The advected energy Research Center (NWRC) in Egypt for their support and cooperation. Qv associated with surface water fluxes improved the evaporation estimates when compared with the BREB values. With the inclusion of the Qv term, values from the References Priestley–Taylor method were within 20% of the BREB values during 45 of the 49 months considered at Allaqi Allen R, Pereira L, Raes D, Smith M (1998) Crop evapotranspiration— station and during the entire period considered at Raft and guidelines for computing crop water requirements-FAO Irrigation Abusembel stations. The values from the Penman method and drainage paper 56. FAO-Food and Agriculture Organization of were within 20% of the BREB values during 79%, 94%, and the United Nations, Rome, M-56 ISBN 92-5-104219-5, http://www. fao.org/docrep/x0490e/x0490E00.htm. 71% of the months at Raft, Allaqi, and Abusembel stations, Anderson ER (1954) Energy-budget studies, in water-loss investiga- respectively. Values from deBruin–Keijman method were tions: Lake Hefner studies. 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