The Ethiopian highlands is the region in Ethiopia above 1500 m and covers a total area of 537,000 km² (43% of Ethiopia) [29] [30] . It constitutes more than half of all the highland areas of Africa [31] . Most of the area of the highlands is cultivated land and land degradation is a major threat. One of the main rivers is the Blue Nile with a watershed of 180,000 km². It has a monsoonal climate with rainfall varying between 800 and 3000 mm per year and evapotranspiration between 1400 and 1681 per year [32] . The major rain phase for the highlands of Ethiopia is between June and September.

One of the difficulties in the Ethiopian highlands is that pan evaporation data is measured only at the Bahir Dar Station for long period. It is located 2.2 km from Lake with latitude of 11˚35'59" and longitude of 37˚21'36". The elevation is 1805 m (Figure S1 in the Supplementary Material). The station has more than 30 years of data but Pan Evaporation started only in 2005 (Table S1 in the Supplementary Material list the collected data). Daily maximum and minimum temperature, relative humidity, sunshine hour, wind speed at 2 meter, pan and Piche evaporation data were obtained for the 11 years from 2005 to 2015 from Bahir Dar Meteorology Directorate.

FAO-Penman Monteith, Priestley & Taylor, Hargraves, Thonthwaite, Blaney-Criddle, Enku, Pan-Allen, Pan-Pereira, Piche-Stanhill and Piche-Adam & Ahmed methods were selected. The first four are used locally by experts and by researchers for hydrological modeling and in computations of irrigation water requirement. Piche-Stanhill and Piche-Adam & Ahmed methods were included because of availability Piche readings in several weather stations in the Ethiopian highlands. Pan-Allen and Pan-Pereira methods were used for validation of the rest of the empirical models. Since the Penman has not been validated for the Ethiopian highlands, the Class A pan data using coefficients proposed by Pereira et al. [5] and Allen et al. [6] are used as the direct measure of reference evaporation against which the calculated values are compared.

1) FAO-Penman Monteith Method (FAO-PEN)

FAO-Penman Monteith method computes reference evapotranspiration (ET₀) from meteorological data. It has been recommended as the standard method for computation of the reference evapotranspiration. The method considers all parameters that govern energy exchange and corresponding latent heat flux and requires air temperature, relative humidity, sunshine intensity/hour, wind speed data and elevation as input parameters.

The modified Penman-Monteith equation can be written as [6] :

E T 0 = 0.408 Δ ( R n − G ) + γ 900 T + 273 u 2 ( e s − e a ) Δ + γ ( 1 + 0.34 u 2 ) (1)

where ET₀ reference evapotranspiration [mm day⁻¹], R_n net radiation at the crop surface [MJ m⁻² day⁻¹], G soil heat flux density [MJ m⁻² day⁻¹], T mean daily air temperature at 2 m height [˚C], u₂ wind speed at 2 m height [m s⁻¹], e_s saturation vapor pressure [kPa], e_a actual vapor pressure [kPa], e_s − e_a saturation vapor pressure deficit [kPa], Δ slope vapor pressure curve [kPa ˚C⁻¹], γ psychrometric constant [kPa ˚C⁻¹]. Further details how these parameters are calculated are given in the Supplementary Material Section.

2) Priestley & Taylor Method (PT)

The Priestley & Taylor form of the equation is [15] [33] [34] :

E T 0 = α Δ Δ + γ ( R n − G ) λ (2)

where;

α = 1 + γ Δ + γ ∗ r s r a (3)

where; r_s is surface resistance [s m⁻¹] which is 70 s m⁻¹ for the grass reference surface, r_a is aerodynamic resistance which helps to determine the transfer of heat and water vapor from the evaporating surface into the air above the canopy. The aerodynamic resistance r_a [s m⁻¹] for the grass reference surface becomes:

r a = 208 u 2 (4)

where u₂ is the wind speed [m s⁻¹] at 2 m.

3) Hargraves Method (HAR)

This method requires daily maximum and minimum air temperature, and extraterrestrial solar radiation, which is computed from the latitude of the study site. The Hargreaves and Samani [35] equation is defined as follows:

E T 0 = 0.0023 ( T max − T min ) 0.5 ( T m e a n + 17.8 ) R a (5)

where T_min daily minimum temperature [˚C], T_max saturation vapor pressure at daily maximum temperature and R_a extraterrestrial radiation [MJ m⁻² day⁻¹].

4) Thornthwaite Method (TH)

This formula is based mainly on temperature with an adjustment factor for the number of daylight hours. The potential evapotranspiration, calculated on a monthly basis, is [36] :

P E ′ = 16 ( 10 T m ¯ I ) a (6)

where the subscript m indicates the months 1, 2, 3, …, 12, T_m is the monthly mean temperature (˚C), I is the heat index for the year, given by:

I = ∑ i m = ∑ ( T m ¯ 5 ) 1.5 (7)

where, i_m is monthly values of heat index and a is empirical coefficient defined as

a = 6.7 × 10 − 7 I 3 − 7.7 × 10 − 5 × I 2 + 1.8 × 10 − 2 + 0.49 (8)

Therefore,

E T 0 = P E ′ ( d 12 ) ( N m 30 ) (9)

where, N_m is the monthly adjustment factor related to hours of daylight, ET₀ reference evaporation [mm/day], d duration of daylight [hr].

5) Blaney-Criddle Method (BC)

The Blaney-Criddle equation has been used to calculate the reference crop evapotranspiration ET₀ when pan evaporation is not available. It can be expressed as [37]

E T 0 = p ( 0.46 T m e a n + 8 ) (10)

where ET₀ is reference crop evapotranspiration (mm/day) averaged over the month T_mean = mean daily temperature (˚C), and p is mean daily percentage of annual daytime hours and varies between 0.26 and 0.29 for Bahir Dar.

6) Enku’s Simple Temperature Method (ENKU)

The new simple empirical temperature method, which was named by “Enku’s simple temperature method”, was developed [19] and tested in Ethiopia with Penman Montieth reference Evaporation and the Piche Evaporimeter. The equation is:

E T 0 = ( T max ) n k (11)

where ET₀ is the reference evapotranspiration (mm day⁻¹); n = 2.5 for the Lake Tana area; k is the coefficient, which is calibrated for local conditions ranging from about 600 for lower mean annual maximum temperature areas to 1300 for higher mean annual maximum temperature areas. The coefficient, k, was found by Enku and Melesse [19] as k = 48 ∗ T m m − 330 for combined wet and dry conditions or k = 73 ∗ T m m − 1015 for the dry phase, and k = 38 ∗ T m m − 63 for the rain phase, where T_mm (C) is the long term daily mean maximum temperature for the seasons under consideration.

7) Pan Methods

The evaporation rate from Class A Pans filled with water is easily obtained. In the absence of rain, the amount of water evaporated during a period (mm/day) corresponds with the decrease in water depth. Pans provide a measurement of the integrated effect of radiation, wind, temperature and humidity on the evaporation from an open water surface [6] .

There are two types of Pan Models to estimate reference evaporation.

a) Pereira Model (PAN-P)

According to Pereira et al. [5] , the reference evaporation, ET₀ from pan data calculated with:

E T 0 = E p a n ⋅ K 1 (12)

where; ET₀ reference evapotranspiration [mm/day], K₁ pan coefficients [-], E_Pan Class A pan evaporation [mm/day].

K 1 = 0.85 ( Δ + γ ) [ Δ + γ ( 1 + 0.33 u 2 ) ] (13)

b) Allen Model (PAN-A)

As Allen et al. [6] , ET₀ from pan data calculated with:

E T 0 = E p a n ⋅ K 2 (14)

The pan coefficient K₂ for Class A pan with green fetch can be found as [6] :

K 2 = 0.108 − 0.0286 u 2 + 0.0422 ln ( F ) + 0.1434 ln ( R H )     − 0.000631 [ ln ( F ) ] 2 ln ( R H ) (15)

where F is fetch or distance of the reference grass (m) and RH is the relative humidity in percent.

8) Piche Methods

The Piche evaporimeter is a type of atmometer used to measure the rate of evaporation from a wet disc of absorbent paper. It is used mainly in hot, dry climates where water loss through evaporation must be observed regularly. Since the results are dependent on wind speed past the disc, as well as the wet bulb saturation deficit, it is almost essential to expose the evaporimeter inside a meteorological screen [14] [38] [39] .

There are two types of Piche models used in this study to estimate evapotranspiration.

a) Stanhill Method (PI-S)

Stanhill [13] suggested that it may be possible to estimate the second term in Penman’s equation from available sheltered Piche evaporation data (E_pi) as follows.

E T 0 = Δ R n Δ + γ + a E p i + b (16)

where a slope and b is intercept of a linear relation with the aerodynamic term of Penman equation:

a E p i + b = γ Δ + γ E a (17)

where;

E a = f ( u ) ( e s − e a ) (18)

where f(u) is an aerodynamic wind function and (e_s − e_a) is the difference between saturated vapour pressure and actual vapour pressure in hPa, evaluated at mean air temperature and at 2 m above the ground or water surface.

The equivalent to the wind function of Penman [40] is

f ( u ) = 0.263 ( a u + b u a ) (19)

where u is the wind speed in m s⁻¹ at 2 m elevation, and a, and b, are empirical coefficients. Penman [40] , Penman [41] suggested using values equivalent to 1 and 0.537 for a, and b, respectively for a short grass cover when u is measured in m s⁻¹.

b) Adam & Ahmed Method (PI-ADAH)

Adam and Ahmed [14] showed that the ratio of Penman estimated and Piche evaporation have relation with relative humidity exponentially. That is:

E T 0 E P i = a e b R H (20)

where a and b are constants that can be find from the exponential relation and the RH humidity in percent

The collected meteorological data for Bahir Dar (2005-2015) was of good quality with less than 5% of missing data. A simple arithmetic mean was used to determine the missing data. Reference evaporation was computed at daily and monthly time step using the 10 methods listed above after data preparation.

In addition, we compared the calculated reference evaporation with the ten methods using three techniques: visual inspection, descriptive statistics and statistical methods to test methods of efficiency and their standard error. The visual inspection of plotted ET₀ prediction methods reveals whether the calculated reference evaporation is in agreement with either other methods or an outlier. Descriptive statistics like maximum, absolute minimum, median value, total range of values, standard deviation, and coefficient of variation were used to compare the ET₀ prediction methods.

In this study, every single model was correlated with the pan ET₀s to assess the model performances and standard error. Pearson’s correlation and root mean square error (RMSE) equations were used for this purpose.

The formula for the Pearson product moment correlation coefficient, r, is:

r = ∑ ( x − x ¯ ) ( y − y ¯ ) ∑ ( x − x ¯ ) 2 ∑ ( y − y ¯ ) 2 (21)

The equation for the standard error of the predicted y is:

1 n − 2 [ ∑ ( y − y ¯ ) 2 − [ ∑ ( x − x ¯ ) ( y − y ¯ ) ] 2 ∑ ( x − x ¯ ) 2 ] (22)

where x ¯ and y ¯ are the sample means.

A sensitivity analysis was carried out to evaluate the percentage response of calculated ET₀ to selected weather variables. Changes of model outputs and their variability induced by change in weather variables were evaluated. The change in variables was analyzed for values 25% above and 25% below the mean value.

Mann Kendall trend test was used to indicate whether there are trends in the ET₀ computed using the ten methods. Non-parametric Mann Kendall Trend Test is useful to examine the temporal variation trend. It is based on the significance of differences, not directly on the random values. Therefore the trend that’s been determined is less affected by outliers [42] . The nonparametric Mann-Kendall trend test has been applied in many studies to identify whether monotonic trends exist in hydro-meteorological data such as temperature, rainfall and stream flow [43] . The Mann Kendall Trend Test, S is calculated by using the equation below [44] [45] .

S = ∑ i = 1 N − 1 ∑ j = i + 1 N sgn ( x i − x j ) , sgn ( x i − x j ) = { + 1 → x i − x j > 0 0 → x i − x j = 0 − 1 → x i − x j < 0 (23)

where x_j and x_i are the sequential data value and j greater than i, N is the length of the data set.

As indicated in Mann [44] and Kendall and Stuart [46] , when N ≥ 8, the distribution of S approaches the Gaussian form with mean E(S) = 0 and variance Var(S) given by:

V a r ( S ) = N ( N − 1 ) ( 2 N + 5 ) − ∑ m = 1 N t i ( m − 1 ) ( 2 m + 5 ) m 18 (24)

where: t_i is the number of ties of length m.

The statistic S is then standardized (Z), and its significance can be estimated from the normal cumulative distribution function.

Z = { S − 1 V ( s ) → S > 0 0 → S = 0 S + 1 V ( s ) → S < 0 (25)

The positive Z value indicates an increasing trend while a negative Z value indicates a decreasing trend. When testing two sided trends at a selected level of significance a, the null hypothesis (H₀) of no trend is rejected if the absolute value of Z is greater than Z_a/2 where a represents the chosen significance level (5% with Z_0.025 = 1.96).

The data in Figure 2 are further summarized in Figure 4 where the monthly averaged reference evaporation, ET₀, of each of the 10 methods for the eleven years

(2005-2015) are depicted. As in Figure 2 the Hargraves (HR), the Priestley & Taylor (PT) and the Piche-Stanhill method (PI-S) over predicts the reference evaporation greatly and will not be discussed in any detail. These three methods should not be applied in the Ethiopian highlands to calculate evaporation.

While all methods have the smallest reference evaporation during the rain phase (Figure 4), the Blaney-Criddle (blue solid line) is the exception. At the end of dry phase, the ET₀ calculated with the Blaney-Criddle method is in the same order as other indirect measurement that predicts realistically ET₀ such as the FAO-Penman and the Enku methods, but severely over predicts during the rain phase. It has also the smallest variability compared to all other methods (Figure 4). The reason is that the reference evaporation is calculated as linear function of the average monthly temperature (Equation (10)) which in the countries near the equator varies little during the year and thus the ET₀ values are within a narrow band as well (Figure 4, Figure 5). By replacing the constant values for the whole year in the Blaney-Criddle equation (Equation (10)), by monthly varying constants and calibrating these, the model will fit much better [48] . The modified Blaney-Criddle Equation becomes in this way similar to the Enku method (Equation (11)). Interestingly, in Figure 4 has shown that during the rain phase starting in May (when the first rains fell) through September (when the rains ended), the reference evaporation, ET₀, calculated with the three indirect methods (the FAO-Penman Monteith, the Thornthwaite and the Enku methods) compared well with the direct measurement of both pan evaporation methods and the Piche evaporimeter using coefficients proposed by Adam and Ahmed [14] .

During the rain phase when precipitation exceeds evaporation, the condition of a well-watered surface on which the reference evaporation is based is similar to that in the Ethiopian highlands. So, during the wet phase the direct and indirect

measurements should agree and they do for the FAO-Penman Monteith, the Thornthwaite and the Enku methods with the two pan methods and one of the Piche methods (Figure 4). During the dry phase, the reference evaporation for these seven methods started to deviate. Initially after the rains stopped and the surface of soil was still wet, only the Thorntwaite method predicted higher ET₀ values while the remaining five agreed well. In January to April when the landscape is very dry and rivers are without water, the direct measurements of the ET₀ are much greater than the calculated values using the meteorological data (Figure 4). The reason for the deviation can be explained with the complementary relationship of actual evaporation and apparent (or pan) evaporation originally introduced by Bouchet [49] and refined by Brutsaert and Stricker [50] . The theory is as follows: When the soil is dry and the evaporation is less than the reference evaporation, the energy that under wet conditions is used for evaporation is converted to the sensible heat and causes an increase in the evaporative demand of the atmosphere. The reference evaporation rates derived from the pan during the dry season are therefore greater than it would be during the wet season under otherwise similar atmospheric conditions. The FAO-Penman method predicts reference evaporation rates of a well-watered irrigation field independent of the condition of the landscape. Hence, the discrepancy between the direct and indirect methods under extreme dry conditions in the period is from January through April (Figure 4).

The question whether the direct or indirect measurement of the reference evaporation is more appropriate during the dry phase is a mood point because the evaporation for most of the landscape, where the soil is dry, is limited by the soil and not by the atmosphere. Only for irrigated fields and lakes, the reference evaporation determines the rate of evaporation during the dry phase. As pointed out by Bouchet [49] the upward wind site of an irrigated field, the evaporation rate is similar to the pan measurement while in the remaining of the area the evaporation rate is equal to reference evaporation calculated with the Penman Monteith.

In addition to plotting the averaged reference evaporation, ET₀, in Figure 5, it is also of interesting to investigate the variation in the reference evaporation for each of the 10 methods. Therefore, the maximum, minimum, mean, lower and upper quartile of the long term daily reference evaporation for the period are shown in Figure 5 from 2005 to the end of 2015. Figure 5 indicates that the Blaney-Criddle method has the least variation because it is only dependent of the average temperature that (despite what the Ethiopians claim) varies little throughout the year. The FAO-Penman Monteith has only a slightly larger variation in ET₀ than the Blaney-Criddle method because it calculates likewise the ET₀ based on meteorological variables that vary little throughout the year except for the cloud cover which is the reason of the additional spread in ET₀ values. The Enku method that depends on temperature has a larger variation than the Blaney-Criddle, because besides temperature it is dependent on a few other fitted functions so that it can simulate the lower ET₀ during the month with clouds. The two pan measurements have a large variation, because they are based on measured data. The measured ET₀ values have a large spread because of the sensible heat during the dry phase that increases the pan evaporation but is not included in any of the indirect methods.

In the Pearson correlation statistics, we looked the performance of eight reference evaporation methods with the pan evaporation methods (Table 1). As expected form the discussion above, the FAO-Penman Monteith and Enku simple maximum temperature methods were highly correlated with correlation coefficient of respectively 0.91 and 0.93 (monthly), and 0.64 and 0.69 (daily). Thornthwaite method holds the third place of good ET₀ monthly estimator with correlation of 0.89 for both pan ET₀ methods on a monthly basis.

Rácz et al. [15] stated that methods having smallest RMSE have the lowest systematic error in predicting ET₀. Table 2 confirms our findings above that the Penman, Enku, Piche and Pan are most similar because they have the smallest RMSE (Table 2). It is surprising that the Blaney-Criddle model has such a small RMSE with the pan on daily time step. It is quirk of how the daily ET₀ values are calculated.

In the sensitivity analysis, we looked the relative effect on reference evaporation of a 25% change in temperature, sunshine hours, relative humidity and wind speed (Figure 6, Figure S2). As expected, an increase in temperature increased reference evaporation (Figure 6). For the indirect methods (Enku, Thornthwaite and Hargreaves methods), the reference evaporation increased the same percentage or more as the temperature. The direct methods were the least sensitive to the change in temperature. Changing sunshine hours, relative humidity and wind speed did not change or increased the reference evaporation with the exception that an increase in relatively humidity for the FAO-Penman Monteith and an increase in winds Speed for both pan methods decreased the reference evaporation (Figure S2).

The trend analysis results of the reference potential evaporation obtained by applying the Mann-Kendall method is shown in Table 3. To our surprise despite our short 11-year record, there were strong trends for the majority of the 10

methods. Only the direct measurement of the ET₀ with the two pan methods did not change over the eleven year period (i.e., accept the null hypothesis that there is not a trend in the ET₀ values, Table 3, Figure 7(b)). Methods that calculated the reference evaporation from meteorological data (e.g., FAO-Penman Monteith (FAO-PEN), Hargraves (HAR), Thornthwaite (TH), Blaney-Criddle (PT), Enku (ENKU)) increased with time (Table 3, Figure 7(a) and Figure S3). The Priestly and Taylor methods that was ill suited for the Ethiopian highlands was an exception. Thus, since these methods were directly based on measured meteorological data (two only on temperature) it clearly indicates that the climate is changing in the Ethiopian Highlands. In other words, since three of the indirect methods were primarily based on temperature, it is getting warmer fast in Bahir Dar! Finally, the Piche methods that not directly depend on the measured standard indicated that the reference evaporation was decreasing.

This paradox of increasing reference evaporation calculated by indirect methods and either not changing (both pan measurements) or decreasing (Piche measurement) by direct measurement has been noted before in a slightly different context by Brutsaert and Parlange [51] as one of the first. They noted that pan evaporation rates were generally decreasing and related that to increased terrestrial evaporation because more runoff occurring with climate change. In terms of the complementary relationship introduced earlier, they explain that more of the incoming energy is used for evaporation and consequently less is converted to sensible heat. This in turn reduces pan evaporation. For the Bahir Dar meteorological station, the explanation might be slightly different. Temperatures in Bahir Dar are increasing either due to climate change or because of rapid urbanization. The dry season has been always so dry that all evaporative energy was converted to sensible heat and increasing temperatures did not affect pan evaporation. The Piche evaporimeter measurements were likely not affected either by the higher temperatures but the relatively humidity decreased due to the higher temperature decreasing the ET₀ value as can be seen from Equation (20).