Review of ET o calculation methods and software

Climate change and Bio-energy Unit (NRCB) Review of ET o calculation methods and software Technical report Delobel François February 2009 1

Table of Content List of figure... 4 1. Introduction... 8 2. Programmes description... 10 2.1 ETo_Calc... 10 2.1.1 Overall description... 10 2.1.2 ET o calculation method... 10 2.1.2.1 Daily calculation... 10 2.1.2.2 Dekadal calculation... 11 2.1.3 Input limits and accuracy... 12 2.1.4 Output limits and accuracy... 13 2.2 AgroMetShell... 13 2.2.1 Overall description... 13 2.2.2 ET o calculation method... 14 2.2.2.1 Daily calculation... 14 2.2.2.2 Dekadal calculation... 16 2.2.3 Programme-related issues... 16 2.2.3.1 Value limits and constraints... 16 2.2.3.2 Calculation of the solar radiation (Rs)... 18 2.2.3.3 Wind speed profile relationship... 22 2.2.3.4 Penman-Monteith... 24 2.3 CROPWAT... 25 2.3.1 Overall description... 25 2.3.2 ET o calculation method... 25 2.3.2.1 Daily calculation... 25 2.3.2.2 Dekadal calculation... 26 2.3.3 Input limits and accuracy... 26 2.3.4 Output limits and accuracy... 27 2.4 Some other programmes... 27 2.4.1 DailyET (Cranfield)... 27 2.4.2 AB@ITC ETo Calculator... 28 3. Daily calculation... 29 3.1 Calculation of intermediate parameters... 29 3.1.1 Actual vapour pressure... 29 Tangier (1990):... 30 Laayoune (1990)... 32 Paraburdoo (1982)... 33 Flinders Airport (1996)... 35 Kairi Research Station (1990)... 36 3.1.2 Daily mean saturation vapour pressure... 37 3.1.3 Slope of saturation vapour pressure curve... 41 3.2 ET o calculation equation: Original Penman-Monteith vs FAO Penman-Monteith... 44 3.3 Data availability and accuracy... 45 3.3.1 Sensitivity to humidity data availability... 45 3.3.2 Sensitivity to radiation data availability... 48 3.3.3 ET o calculation formulas: FAO Penman-Monteith vs Hargreaves... 54 4. Dekadal calculation... 57 4.1 Methodology... 57 2

4.2 Results and analysis... 57 5. Conclusion... 61 Annex 1: Standard ETo calculation method equations (FAO paper No. 56)... 62 Annex 2: Additional intermediate variables... 67 Annex 3: Additional Relationships... 69 Annex 4: Climatic diagrams... 70 Bibliography... 74 3

List of figures Figure 1 Comparison between Tmin and Tdew estimated by Linacre. Tmin values are -5 (gray line), 5 C ( red line), 15 C (blue line) and 25 (green line).... 15 Figure 2 AgroMetShell: errors in daylight durations for the day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line).... 19 Figure 3 AgroMetShell: errors in daylight durations for the latitudes: 0 (blue line), 23 N (light red line), 65 N (light green line), 23 S (dark red line) and 65 S (dark green line).... 19 Figure 4 AgroMetShell: errors on tan δ according to the latitude for four day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line).... 20 Figure 5 AgroMetShell: ratio between theoretical values of tan(δ) and those deducted from the programme s results.... 21 Figure 6 AgroMetShell: variation of the errors in extraterrestrial radiation (MJ m -2 day -1 ) according to the latitude for the day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line).... 21 Figure 7 AgroMetShell: errors in solar radiation for the latitudes: 0 (blue line), 23 N (light red line), 65 N (light green line), 23 S (dark red line) and 65 S (dark green line)... 22 Figure 8 AgroMetShell: converted wind speed corresponding to a 5 m s -1 wind observed at different height of measurement for AgroMetShell (blue line) and manual computation (red line)... 23 Figure 9 Comparison of the wind speeds at 2 meters obtained with AgroMetShell (blue line) and manually (red line) from fictive wind speeds measured at 10 meters... 23 Figure 10 AgroMetShell: Error on daily Eto for the station of Laayoune in 1990... 24 Figure 11 Tangier: comparison between actual vapour pressures derived from RH min and RH max (abscises) and the results of five tested methods: equation 19 (upper left), Smith 1992 (upper right), equation 18 (centre left), equation 48 (centre right) and finally Linacre s estimated Tdew (lowest). The units are kpa.... 31 Figure 12 Laayoune: comparison between actual vapour pressures derived from RH min and RH max (abscises) and the results of five tested methods: equation 19 (upper left), Smith 1992 (upper right), equation 18 (centre left), equation 48 (cntre right) and finally Linacre s estimated Tdew (lowest). The units are kpa.... 33 Figure 13 Paraburdoo: comparison between actual vapour pressures derived from RH min and RH max (abscises) and the results of five tested methods: equation 19 (upper left), Smith 1992 (upper right), equation 48 (lower left) and finally Linacre s estimated Tdew (lower right). The units are kpa... 34 Figure 14 Flinders airport station: comparison between actual vapour pressures derived from RH min and RH max (abscises) and the results of five tested methods: equation 19 (upper left), Smith 1992 (upper right), equation 48 (lower left) and finally Linacre s estimated Tdew (lower right). The units are kpa.... 35 Figure 15 Kairi research station: comparison between actual vapour pressures derived from RH min and RH max (abscises) and the results of five tested methods: equation 19 (upper left), Smith 1992 (upper right), equation 48 (lower left) and finally Linacre s estimated Tdew (lower right). The units are kpa.... 36 Figure 16 Tangier: differences between FAO paper No. 56 (red line) and CROPWAT (blue line) saturation vapour pressure calculation methods (using Tetens formula) with FAO paper No. 56 method using Goff-Gratch formula... 38 Figure 17 Laayoune: differences between FAO paper No. 56 (red line) and CROPWAT (blue line) saturation vapour pressure calculation methods (using Tetens formula) with FAO paper No. 56 method using Goff-Gratch formula... 39 4

Figure 18 Paraburdoo: differences between FAO paper No. 56 (red line) and CROPWAT (blue line) saturation vapour pressure calculation methods (using Tetens formula) with FAO paper No. 56 method using Goff-Gratch formula... 39 Figure 19 Flinders Airport: differences between FAO paper No. 56 (red line) and CROPWAT (blue line) saturation vapour pressure calculation methods (using Tetens formula) with FAO paper No. 56 method using Goff-Gratch formula... 40 Figure 20 Kairi Research Center: differences between FAO paper No. 56 (red line) and CROPWAT (blue line) saturation vapour pressure calculation methods (using Tetens formula) with FAO paper No. 56 method using Goff-Gratch formula... 40 Figure 21 Tangier: Slopes of the saturation vapour pressure curve calculated by means of equation 13 of the FAO paper No. 56 (blue line) and CROPWAT's method (red line).. 42 Figure 22 Laayoune: Slopes of the saturation vapour pressure curve calculated by means of equation 13 of the FAO paper No. 56 (blue line) and CROPWAT's method (red line).. 42 Figure 23 Paraburdoo: Slopes of the saturation vapour pressure curve calculated by means of equation 13 of the FAO paper No. 56 (blue line) and CROPWAT's method (red line).. 43 Figure 24 Flinders Airport: Slopes of the saturation vapour pressure curve calculated by means of equation 13 of the FAO paper No. 56 (blue line) and CROPWAT's method (red line)... 43 Figure 25 Kairi research station: Slopes of the saturation vapour pressure curve calculated by means of equation 13 of the FAO paper No. 56 (blue line) and CROPWAT's method (red line)... 44 Figure 26 Tangier: Errors in ET o resulting from the calculation of the actual vapour pressure by means of Smith s method (blue line) or using Linacre s estimation of dew point temperature (red line)... 46 Figure 27 Laayoune: Errors in ET o resulting from the calculation of the actual vapour pressure by means of Smith s method (blue line) or using Linacre s estimation of dew point temperature (red line)... 46 Figure 28 Paraburdoo: Errors in ET o resulting from the calculation of the actual vapour pressure by means of Smith s method (blue line) or using Linacre s estimation of dew point temperature (red line)... 47 Figure 29 Flinders Airport: Errors in ET o resulting from the calculation of the actual vapour pressure by means of Smith s method (blue line) or using Linacre s estimation of dew point temperature (red line)... 47 Figure 30 Kairi Research Station: Errors in ET o resulting from the calculation of the actual vapour pressure by means of Smith s method (blue line) or using Linacre s estimation of dew point temperature (red line).... 48 Figure 31 Tangier: Correlation between solar radiations (MJ m -2 day -1 ) computed with Angstrom and Hargreaves equations... 49 Figure 32 Tangier: Errors in reference evapotranspiration calculations caused by the use of Hargreaves solar radiation formula... 49 Figure 33 Laayoune: Correlation between solar radiations (MJ m -2 day -1 ) computed with Angstrom and Hargreaves equations... 50 Figure 34 Laayoune: Errors in reference evapotranspiration calculations caused by the use of Hargreaves solar radiation formula... 50 Figure 35 Paraburdoo: Correlation between solar radiations (MJ m -2 day -1 ) computed with Angstrom and Hargreaves equations... 51 Figure 36 Paraburdoo: Errors in reference evapotranspiration calculations caused by the use of Hargreaves solar radiation formula.... 51 Figure 37 Flinders Island Airport: Correlation between solar radiations (MJ m -2 day -1 ) computed with Angstrom and Hargreaves equations... 52 5

Figure 38 Flinders Airport: Errors in reference evapotranspiration calculations caused by the use of Hargreaves solar radiation formula... 52 Figure 39 Kairi Research Station: Correlation between solar radiations (MJ m -2 day -1 ) computed with Angstrom and Hargreaves equations... 53 Figure 40 Kairi Research Station: Errors in reference evapotranspiration calculations caused by the use of Hargreaves solar radiation formula.... 53 Figure 37 Tangier: correlation between reference evapotranspirations (mm day -1 ) calculated with Hargreaves formula and FAO modified Penman-Monteith formula... 54 Figure 38 Laayoune: correlation between reference evapotranspirations (mm day -1 ) calculated with Hargreaves formula and FAO modified Penman-Monteith formula... 55 Figure 39 Paraburdoo: correlation between reference evapotranspirations (mm day -1 ) calculated with Hargreaves formula and FAO modified Penman-Monteith formula... 55 Figure 40 Flinders Airport: correlation between reference evapotranspirations (mm day -1 ) calculated with Hargreaves formula and FAO modified Penman-Monteith formula... 56 Figure 41 Kairi Research Station: correlation between reference evapotranspirations (mm day - 1 ) calculated with Hargreaves formula and FAO modified Penman-Monteith formula. 56 Figure 42 Tangier: Error on dekadal calculations caused by the following methods: ETo_Calc using Rn and ea (blue line), ETo_Calc using n and ea (red line), ETo using n and RHmean (gray line) and CROPWAT (green line)... 58 Figure 43 Laayoune: Error on dekadal calculations caused by the following methods: ETo_Calc using Rn and ea (blue line), ETo_Calc using n and ea (red line), ETo using n and RHmean (gray line) and CROPWAT (green line)... 58 Figure 44 Paraburdoo: Error on dekadal calculations caused by the following methods: ETo_Calc using Rn and ea (blue line), ETo_Calc using n and ea (red line), ETo using n and RHmean (gray line) and CROPWAT (green line)... 59 Figure 45 Flinders Airport: Error on dekadal calculations caused by the following methods: ETo_Calc using Rn and ea (blue line), ETo_Calc using n and ea (red line), ETo using n and RHmean (gray line) and CROPWAT (green line)... 59 Figure 46 Kairi Research Station: Error on dekadal calculations caused by the following methods: ETo_Calc using Rn and ea (blue line), ETo_Calc using n and ea (red line), ETo using n and RHmean (gray line) and CROPWAT (green line)... 60 Figure 47 Sensitivity of the latent heat of vaporisation to the daily mean temperature... 67 Figure 48 Sensitivity of the psychrometric constant to the latent heat of vaporisation for under three different atmospheric pressure: 101.3kPa (green line), 91.3kPa (or around 880m of altitude for a standard atmosphere) (blue line) and 81.3kPa (or around 1850m of altitude for a standard atmosphere) (red line).... 67 Figure 49 Tangier: Maximum (blue line), minimum (green line) and average (red line) daily temperature... 70 Figure 50 Tangier: Daily potential ET (green line) and rainfall (red line) temperature.... 70 Figure 51 Laayoune: Maximum (blue line), minimum (green line) and average (red line) daily temperature... 71 Figure 52 Laayoune: Daily potential ET (green line) and rainfall (red line) temperature.... 71 Figure 53 Paraburdoo: Maximum (blue line), minimum (green line) and average (red line) daily temperature... 72 Figure 54 Paraburdoo: Daily potential ET (green line) and rainfall (red line) temperature.... 72 Figure 55 Flinders Airport: Maximum (blue line), minimum (green line) and average (red line) daily temperature.... 72 Figure 56 Flinders Airport: Daily potential ET (green line) and rainfall (red line) temperature.... 73 6

Figure 57 Kairi Research Station: Maximum (blue line), minimum (green line) and average (red line) daily temperature... 73 Figure 58 Kairi Research Station: Daily potential ET (green line) and rainfall (red line) temperature... 73 7

1. Introduction The concept of evapotranspiration (ET) plays a central role in crop yield modelling. Combining the water losses both from the soil by evaporation and from the vegetation cover through transpiration, evapotranspiration is indeed decisive in biomass production. The Food and Agriculture of the United Nations (FAO) has been playing a central role in the standardisation of ET calculation. The FAO Irrigation and Drainage Papers numbers 24 Crop Water Requirements and 56 Guidelines for Computing Crop Water Requirements (Allen et al., 1998) are commonly used as reference publications and the FAO Penman Monteith method is usually considered as the standard evapotranspiration calculation method. However several variants appeared in ET o computations, which generate different results from different programmes. In order to avoid this kind of conflict in the future, it has been decided to scrutinize the possible sources of differences and to standardize ET o calculation methods again. By definition, evapotranspiration depends on four types of factors: meteorological conditions, crop characteristics, environment and management practices. Since each of these factors involves a wide range of variables, evapotranspiration is differentiated into two other concepts in order to make computations easier: the potential evapotranspiration (PET), which represents the evaporating power of the atmosphere, and the crop factors, which encompass the other three factors. One of the most common methods to compute ET is the Penman-Monteith equation: ( es ea ) Δ ( Rn G) + ρ ac p ra λ ET = rs Δ + γ (1 + ) ra Where: ET: reference evapotranspiration (MJ m -2 day -1 ) Δ: slope of the saturation vapour pressure temperature relationship (kpa C -1 ) R n : net radiation at the crop surface (MJ m -2 day -1 ) G: soil heat flux density (MJ m -2 day -1 ) ρ a : mean air density at constant pressure (kg m -3 ) Cp: air specific heat at constant pressure (MJ kg -1 C -1 ) e s : saturated vapour pressure (kpa) e a : actual vapour pressure (kpa) γ: psychrometric constant (kpa C -1 ) r s : surface crop resistance (s m -1 ) r a : aerodynamic resistance (s m -1 ) In order to make potential evapotranspiration calculations comparable in different locations and at different seasons, the FAO Expert Consultation on Revision of FAO methodologies for Crop Water Requirements defined in 1990 a reference grass cover with the following characteristics: well-watered, 0.12 m of height, 70 s m -1 of surface resistance and 0.23 of albedo. The water vapour released by such a surface, called reference crop evapotranspiration 8

(ET o ), is then exclusively depending on weather conditions. ET o is computed according the FAO Penman-Monteith equation, formulated like this: 900 0.408Δ( Rn G) + γ u2 ( es ea ) T + 273 ETo = Δ + γ (1+ 0.34u2) Where: ET o : reference evapotranspiration (mm day -1 or kg H 2 O m -2 day -1 ) T: mean air temperature ( C) u 2 : mean wind speed measured at 2 meter height (m s -1 ). The methodology used for this research is relatively simple. Firstly several commonly used programmes, namely ETo_Calc, AgroMetShell and CROPWAT, are reviewed one after the other, as well as the methods that they apply. Secondly tests are carried out with a view to revealing possible programming issues. Finally the methods are tested with data from a range of different climatic conditions and their results are compared. This paper gathers all the outcomes of the analysis. In addition to this report, two other final products came out of this research: a documented source code for ETo calculation based on ETo_Calc and a MS Excel programme. The report is laid out according to the following structure. In a first step, daily ET o calculations are considered. The three main programmes are analysed and the methods are compared. Then, dekadal computations are studied, according to the same approach. Finally, a brief discussion is proposed with a presentation of the documented source code and the Excel spread sheets. Throughout the report, the equation numbering refers to that of the FAO Irrigation and Drainage paper No 56. All equations are listed in Annex 1. 9

2. Programmes description 2.1 ETo_Calc 2.1.1 Overall description ETo_Calc is a public domain programme developed by Dirk Raes and Giovanni Munoz to compute ET o for daily, dekadal and monthly time steps. The programming language is Pascal. The version used is the third one, dated June 2008. 2.1.2 ET o calculation method 2.1.2.1 Daily calculation ETo_Calc basically offers the possibility to use any of the different options to compute the parameters as presented in the FAO paper No. 56 (Allen et al., 1998), though it imposes a hierarchy between these options. The method strictly follows the FAO Penman-Monteith method, as explained in the manual. In the scope of this research, it is important to point out some of the procedures (FAO, 2009a and FAO, 2009b) As far as the calculation of the ET o is concerned, the equation that the software uses is the FAO Penman-Monteith one (equation 6) The minimum required data are T min and T max. However, it is highly recommended to add humidity, wind speed and sunshine data in order to significantly improve the accuracy. The atmospheric and astronomic parameters are calculated exactly like in the FAO paper No. 56, as well as the slope of the saturation vapour pressure curve and the mean saturation vapour pressure. When possible, the actual vapour pressure is uppermost computed from the dew point temperature (equation 14). If it is not available, the programme uses the temperature of the psychrometer s thermometers (T wet and T dry ) (equation 15). If they are not available either, the calculation is carried out from the relative humidity and temperature data in the following order: - from RH min, RH max, T max and T min, using equation 17, - from RH max and T min, using equation 18, - from RH mean and T mean, using the following expression (Smith, 1992): e = e a o ( T ) mean RH 100 This latter formula does not appear in the FAO paper No. 56, which proposes the equation 19 instead (e a derived from RH mean, T min and T max ). If none of these data are available, the user is mean 10

them proposed to compute e a assuming that T dew = T min for humid conditions, T dew = T min 2 for arid conditions. About the radiation parameters, the user can choose to take into account the elevation in the calculation of R s0 or not. It is also possible to use calibrated Angstrom coefficients to compute that parameter. By default, the programme applies the elevation correction and attributes a s and b s their standard value, which are respectively 0.25 and 0.50. If not directly entered by the user, R n can be computed from R s through the normal procedure (equations 38, 39 and 40). If R s is not available either, it can be calculated in different ways. It is derived from the sunshine duration (n), from the sunshine fraction (n/n) if n is not entered. If there are no data about the direct sunshine, the Hargreaves radiation formula (equation 50) is employed. The soil heat flux G is always assumed to be insignificant. It is thus worth 0. Finally, if the wind speed data are not at hand, the programme proposes to make an approximation of it from the average strength of the wind in the studied area (table 1). Table 1 General classes of monthly wind speed data Wind strength Wind speed (m s -1 ) Light 0.5 light to moderate 2 moderate to strong 4 strong 5.5 This classification is based on the one in FAO paper 56, Chapter 3, table 4. That approximation is said to be acceptable for monthly periods since the fluctuations around months averages are relatively small. However, ETo_Calc allows its use for daily and dekadal calculation as well. 2.1.2.2 Dekadal calculation Regarding the input variables, the ETo calculator from the Land and Water Development Division of the FAO basically offers the same options as for daily computations. Air humidity data can be entered under the form of mean relative humidity (%), minimum and maximum relative humidity (%), mean dew point temperature (%) or mean actual vapour pressure (%). Concerning the radiation data, mean duration of sunshine (hours), sunshine fraction (%), solar radiation (MJ m -2 day -1 ) or net radiation (MJ m -2 day -1 ) can be indistinctly employed. Besides that, minimum and maximum temperatures, wind speed, station latitude and altitude are obviously required. If mean hours of sunshine, sunshine fraction or solar radiation are used, ETo_Calc computes the astronomic parameters for the fifth day of the dekad. This is consistent with the FAO paper No. 56. Even if this is not stated explicitly, that is the method used in the example 17. 11

2.1.3 Input limits and accuracy Inputs are needed at two different stages. Firstly, the latitude and the altitude of the station have to be entered in the station characteristic panel. Here are their characteristics (table 2): Table 2 ETo_Calc input variables: station characteristics Station Characteristics Variable Unit Range Accuracy Default value φ -69 69 0.01 Required z m -500 5000 1 Required All the inputs related to the weather are then to be inserted in the Meteorological data and ET o panel, according to the ticked items in the Input data description menu. The details about these inputs are listed hereafter (table 3): Table 3 ETo_Calc input variables: weather data Meteorological data and ET o Variable Unit Range Accuracy Default value J day 0 365 or 366 1 Required T max C T min 45 0.1 Required T min C -15 T max 0.1 Required T mean C -15 45 0.1 (T max +T min )/2 RH mean % 15 100 0.1 - RH max % RH min 100 0.1 - RH min % 15 RH max 0.1 - T dew C -15 45 0.1 T min (-2) e a kpa 0.2860 9.5825 01 - T dry C -15 45 0.1 - T wet C -15 45 0.1 - U m s -1 0 8 0.01 - h m 0.5 6 0.1 2.0 n hours 0 N 0.01 - n/n - 0 1 0.01 - R s MJ m -2 day -1 0 R s0 0.01 - R n MJ m -2 day -1 0 R s0 0.01 - Note: the units indicated in this table are the default units. The software allows the conversion of almost all variables into other units. The default value of T dew is T min of which 2 C can be subtracted under arid climatic conditions. However, this default value is ignored if the mean actual vapour pressure e a is provided. The limits of ea correspond to the saturation vapour pressures for temperatures of -10 C and +45 C, which is sufficiently wide to cover most of situations. 12

The limitation of the wind speed range to 8 m s -1 (= 28.8 kmph) is insufficient for computations in windy areas, especially for daily operations. 2.1.4 Output limits and accuracy The sole output is the final result of the ET o calculation (table 4). Table 4. ETo_Calc output: ET o Variable Unit Accuracy ET o mm day-1 0.1 The accuracy on ET o is relatively low as, in most cases, there are only two significant digits. However, it may be sufficient for practical use. Some intermediate values of parameters can be displayed as well. To do so, the results have to be exported with the specification intermediate values in the report options. The following values can then be obtained (table 5): Table 5 ETo_Calc output: intermediate variables Variable Unit Accuracy e a kpa 0.01 e s kpa 0.01 U 2m m s -1 0.01 Ra MJ m -2 day -1 0.01 n hours 0.01 N hours 0.01 Rs MJ m -2 day -1 0.01 Rnl MJ m -2 day -1 0.01 Rn MJ m -2 day -1 0.01 The main advantage of the programme is that it considers all the possible situations about the data availability. However, there is a danger for the non experts to use simplified options which confers a low accuracy to the results. The user should know about the limitations of the different options. 2.2 AgroMetShell 2.2.1 Overall description AgroMetShell (FAO, 2007) is a programme aimed at crop modelling and yields forecasting. Coded in Delphi, Visual Basic and C++, there are three modules able to compute reference evapotranspirations. The first one, accessible in the menu Database/Calculate/Formula, calculates ET o for the stations in the database on the basis of their meteorological data. The other two can be found under the same path (Tools/Potential ET) and compute ET o from meteorological data either from a file to be imported or manually entered. There exists also 13

the same programme under the form of a MS Excel Spreadsheet (Donatelli, 2003a and 2003b). This version is freely available on www.sipeaa.it/asp/asp2/et_csdll.asp. 2.2.2 ET o calculation method 2.2.2.1 Daily calculation The programme offers the choice between two calculation methods: Priestley-Taylor and Penman-Monteith. In this paper, only the latter will be considered. The minimum data required to run the computation are the minimum and maximum temperatures (T min and T max ), the solar radiation (R s ) and the wind speed (U 2 ). Rs can be calculated from the actual duration of sunshine (n) or the sunshine fraction (n/n), the number of the day (J), the latitude of the station (φ) and Angstrom coefficients (a s and b s )). The atmospheric parameters are calculated as indicated in the FAO paper No. 56, except for the latent heat of vaporisation which is not considered as a constant (2.45 MJ kg -1 ) but instead as a function of the mean temperature (T mean ): λ = 2.501 0. 002361 T mean This difference may cause small differences in the value of the latent heat of vaporisation (see Annex 2). As a result, since it depends on the latent heat of vaporisation, the psychrometric constant γ may also show some differences in comparison with that computed in the FAO Penman- Monteith method. However the effect on ET o calculation is negligible.. In absence of humidity data, AgroMetShell estimates the temperature of the dew point according to (Linacre, 1992), whereas the standard method estimates the dew point temperature by the minimum temperature (subtracted of 2 or 3 C under arid and semi-arid climates). T dew = 0.52T + 0.6T 9T min max 2 max 2 Unlike the approximation by T min, the estimation of Linacre is sensitive to the amplitude of the range of temperature (see Figure 1). 14

30 25 20 15 Tdew ( C) 10 5 0-10 0 10 20 30 40 50 60 70-5 -10 Tmax ( C) Figure 1 Comparison between Tmin and Tdew estimated by Linacre. Tmin values are -5 (gray line), 5 C ( red line), 15 C (blue line) and 25 (green line). The numbers do not pretend to be realistic, however as we can see, Linacre s T dew estimation is very close to T min for T min values around 15 C and in general when the temperature range is small. In case of very cold or very warm conditions, the difference between the two values can be of an order of 5 or more. This estimated dew point temperature is then used to calculate the actual vapour pressure by the mean of the equation number 14 in FAO paper 56. From this point, every other variable and the eventually reference evapotranspiration can be calculated. The clear-sky solar radiation is programmed slightly differently from the recommendations of FAO paper No. 56. Indeed, instead of being merely calculated with equations 36 or 37, AgroMetShell assigns it the maximum value among R s and 0.75*R a. According to the Help file, the programme uses the original Penman-Monteith formula instead of the FAO modified Penman-Monteith equation. λ ET = Δ ( R n G) + ρ c a p r Δ + γ (1 + r ( e s a ) s e r a a ) The use of that formula implies that the rounding in the second term of the sum in the numerator is not done. Indeed, in the FAO paper 56 we can see in the box number 6 the following expression: c p r λ a ρ a 900 γ u T + 273 2 15

This value of 900 being a rounded aggregation of all the constants involved. However, the real value is: 86400 ε 0.622 = 86400 891.3 1.01 208R 1.01 208 0.287 = The original Penman-Monteith method thus comes down to use this value of 891.3 instead of 900, even though it does not explicitly achieve this step that way. The results can be refined by entering more meteorological variables. The use of a measured dew point temperatures eliminates the uncertainty related to the approximation by Linacre 1992. AgroMetShell can also use minimum and maximum relative humidity data in order to calculate the actual vapour pressure. This option is actually used in priority by the model. The elevation can also be entered in order refine atmospheric pressure and psychrometric constant values. Clear-sky radiation does not take it into account though (eq. 36). Finally the programme also allows the use of wind speed measured at heights other than 2 meters. The wind speed at 2 meters is then calculated from the wind speed profile relationship (eq.47). 2.2.2.2 Dekadal calculation AgroMetShell apparently uses exactly the same method as ETo_Calc, i.e. that astronomic parameters for the fifth day of the dekad. The inputs can be slightly different though. Averaged maximum and minimum relative humidity are required, as well as one variable among solar radiation, sunshine duration and sunshine fraction. 2.2.3 Programme-related issues 2.2.3.1 Value limits and constraints Inputs If we only consider the calculation of the Potential Evapotranspiration (it is also possible to compute actual evapotranspiration), eleven different inputs can be introduced. Four of them are strictly required for any calculations: minimal and maximal temperatures, solar radiation (called global radiation here) and the wind speed. The next table (table 6) inventories all these inputs and their characteristics. Table 6 AgroMetShell input variables Variable Unit Range Accuracy Default value J day 0 365 or 366 1 0 T max C T min 0.1 Required T min C T max 0.1 Required R s MJ m -2 day -1 R s0 5 Required U m s -1 0.1 41532 1 Required RH max % RH min 100 0.01-16

RH min % 0 - RH max 0.01 - h m 0.1 < 1 2 φ -66 66 0.01 0 z m...... 1 0 T dew C...... 0.1 (Linacre 1992) If no day number is entered, the default value is 0, which is not realistic; the range should be 1 to 365 or 366. Users should not forget to enter the day number. There is no lower limit for R s. The elevation and the dew point temperature can be any value, either positive or negative. It is also important to note that the wind cannot be 0. In the manual version of the module, some of the inputs of the previous table can in fact be deducted from other parameters: the day number from the date, the solar radiation from the sunshine, and the relative humidity from the vapour pressure. That makes few extra inputs (table 7). Table 7 AgroMetShell extra input variables Variable Unit Range Accuracy Default value Date day - 1 - n hours 0 N 0.01 - n/n -...... 1 - a s -...... 1 0.25 b s -...... 1 0.50 ea mean kpa C -1 e 0 (T max ) 0.01 - ea min kpa C -1 e 0 (T max ) 0.01 - ea max kpa C -1 e 0 (T max ) 0.01 - The sunshine fraction basically accepts whatever number, positive and negative, but the range of realistic values is comprised between 0 and 1 (n = N). It is also important to note that Rs is rounded to the first decimal when computed from the sunshine duration or the sunshine fraction. Outputs AgroMetShell provides the user with the following outputs (table 8). Table 8 AgroMetShell output variables Variable Unit Range Accuracy ET o mm day-1-0.01 r a s m-1 0.01 2077 0.01 R nl MJ m-2 day-1-0.01 λ MJ kg-1-0.01 R n MJ m-2 day-1-0.01 R a MJ m-2 day-1 0.05 41.72 0.01 N hours 1.75 22.25 0.01 e s kpa 0.01 0.01 Δ - 0.01 0.01 Vol Heat Cap. MJ m -3 C -1 001 001 17

VPD kpa 0.01 0.01 Max VPD kpa 0.01 0.01 fh 0.34 0.01 e a kpa 0.01 0.01 The volume heat capacity is equal to the psychrometric constant γ times the latent heat of vaporization λ. VPD stands for Vapour Pressure Deficit and corresponds to (e s e a ). Max VPD corresponds to the difference between e 0 (T min ) and e 0 (T max ). The air humidity correction factor f h is defined as: f = 0.34 0. 14 h e a This coefficient is one of the terms in the formula determining the net longwave radiation (R nl ). 2.2.3.2 Calculation of the solar radiation (R s ) Unlike the Excel version, the versions of the module integrated in AgroMetShell allows the computation of the solar radiation with the Angstrom formula. However, this routine is producing some small errors, which do not come from the Angstrom formula itself, but rather from the values of daylight duration and extraterrestrial radiation, themselves both depending on the latitude and the day number. These errors, of an order of few tenths, could have two origins: either some of the variables are rounded, which can lead in some cases to significant errors through an accumulation effect, or the error comes from a difference in the equations formulations. These issues are dealt with in the next paragraphs. Daylight duration (N) Figure 2 shows the difference between the theoretical values of the day duration calculated in an excel spreadsheet, and the values computed by the programme, for three Julian days: 81, 172, 264 and 355, i. e. the four dates of solstices and equinoxes. As we can see, the errors are larger at high latitudes and they vary according to the season of the year. This demonstrates that the errors are not only due to a rounding effect, as they would be stepwise. The errors reach up to plus or minus 0.40 hours, or 24 minutes, during the fall equinox. 18

0.50 0.40 0.30 ) Errors on N (hours 0.20 0.10-80 -60-40 -20 0 20 40 60-0.10-0.20 80-0.30-0.40-0.50 Latitude ( ) Figure 2 AgroMetShell: errors in daylight durations for the day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line). The same operation has been conducted with fixed values of latitude and varying day numbers. Figure 3 confirms the observations made above, i.e. that the errors are much larger at high latitudes, with a peak at 0.46 hours (about 28 minutes) on the 300 th day at +/-65 of latitude. There is basically no error at the equator. It must also be stressed out that for four days in the year the error is nul at any latitude: the 24 th, the 103 rd, the 172 nd and the 355 th. As could be expected from the figure above, the errors are symmetric with regard to the North and South latitude but not with the seasons. 0.60 0.40 Errors on N (hours) 0.20 0 50 100 150 200 250 300 350 400-0.20-0.40-0.60 Day number Figure 3 AgroMetShell: errors in daylight durations for the latitudes: 0 (blue line), 23 N (light red line), 65 N (light green line), 23 S (dark red line) and 65 S (dark green line). 19

One of the reasons that could explicate the errors is a possible daylight duration correction in the programme, in order to take into account the effect of atmospheric refraction. However, this is apparently not the case: the programme does not exclusively overestimate the daylight duration; there is no symmetry between equinoxes etc. The fact that, for programming reasons, AgroMetShell uses the equations 26 and 27 instead of equation 25 to compute the sunset hour angle is not responsible for the errors either. Both ways are indeed completely equivalent... However, if the sunset hour angle equation is inversed, the error on the argument of the arc cosine function becomes a multiple of the tangent of the latitude for fixed days. The argument as computed by AgroMetShell can be deducted this way: N AMS π 24 ( arg) = Cos N AMS being the daylight duration computed with AgroMetShell. If the argument deducted from AgroMetShell results is then divided by the tangent of the latitude, the remaining errors happen to be a constant depending on the day of the year (Figure 4). Indeed, the solar declination (δ) uniquely depends on the day number. The straightforward conclusion of this would be that AgroMetShell under- or over-estimates the solar declinations during some periods in the year. 0.03 0.02 0.01 Errors on tan(δ) -80-60 -40-20 0 20 40 60-0.01-0.02 80-0.03-0.04-0.05 Latitude ( ) Figure 4 AgroMetShell: errors on tan δ according to the latitude for four day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line). In figure 5, the ratio between theoretical values of tan(δ) and those deducted from AgroMetShell results is given. The misestimating is very small during most of the year but can be relatively large around the two equinoxes, when the solar declination is close to zero. 20

1 8.00 6.00 4.00 Ratio 2.00 0 50 100 150 200 250 300 350 400-2.00-4.00-6.00 Day number Figure 5 AgroMetShell: ratio between theoretical values of tan(δ) and those deducted from the programme s results. Further investigations were carried out over the solar declination but they did not bring better clues about the source of the errors. Indeed, the errors on δ or on the day number vary very similarly to the ones previously analysed. Therefore there does not seem to be a straight forward explanation for the errors in daylight duration calculation... Extraterrestrial radiation (R a ) 1.50 1.00 Errors on Ra (MJm-2day-1) 0.50-80 -60-40 -20 0 20 40 60 80-0.50-1.00-1.50 Latitude ( ) Figure 6 AgroMetShell: variation of the errors in extraterrestrial radiation (MJ m -2 day -1 ) according to the latitude for the day numbers: 81 (gray line), 172 (red line), 264 (green line) and 355 (blue line). 21

Since extraterrestrial radiations are computed on the basis of sunset hour angles and solar declinations, which are affected by unexplained errors, errors on this variable cannot be explicated either. However, it is still interesting to observe how these errors vary according to the latitude and the day number. Similarly to errors on the daylight duration, the further from the equator the station lies, the larger the error becomes (Figure 6 and 7). It reaches more than 1.5 MJ m -2 day -1 during the fall at high South latitudes. The highest absolute error is reached on day 296 with 1.5 MJ m -2 day -1 for an expected value of 29,16 MJ m -2 day -1 (6%), while the highest relative error happens 100 days earlier, with a value of 9,1%. In comparison with daylight duration errors, the autumn is also the season during which absolute errors are the largest. However there is not any symmetry between South and North latitudes, and the days that the curves cross the x- axis do not correspond to each other anymore. 1.50 1.00 Errors on Ra (MJ m-2 day-1) 0.50 0 50 100 150 200 250 300 350 400-0.50-1.00-1.50-2.00 Day number Figure 7 AgroMetShell: errors in solar radiation for the latitudes: 0 (blue line), 23 N (light red line), 65 N (light green line), 23 S (dark red line) and 65 S (dark green line) The errors may have four origins: the formulation of the extraterrestrial radiation equation, the solar declination, the sunset hour angle and the inverse Earth-Sun distance. 2.2.3.3 Wind speed profile relationship Another source of error is related to the wind speed computation. AgroMetShell provides the possibility to enter in the programme wind speed data measured at heights different from 2 meters along with the actual height of measurement and calculates the corresponding wind speed at two meters; supposedly by means of the wind speed profile relationship of the FAO paper No. 56 (eq. 47). However, this step seemed to produce some errors. The next figure shows the difference between the wind speeds at 2 meters obtained with AgroMetShell and with equation 47 of the FAO paper No. 56 for a fixed value of wind (5 m.s - 1 ) and for a wide range of heights of measurement (m). The range of heights includes unrealistic heights on purpose: it gives a better picture of the shape of the curve. As Figure 8 22

shows, the programme uses another logarithmic relationship, with an intersection for the height h = 2m. The wind profile relationship actually employed by AgroMetShell is: U 2 = U h 6.969 ln( h) + 6.216 10 9 8 7 Wind speed (m s-1) 6 5 4 3 2 1 0 0 20 40 60 80 100 120 Height of measurement (m) Figure 8 AgroMetShell: converted wind speed corresponding to a 5 m s -1 wind observed at different height of measurement for AgroMetShell (blue line) and manual computation (red line) It is very common in weather stations to measure wind speeds at 10 meters above the ground. The impact of the difference in wind profile relationship can be observed in the following figure (Figure 9). 14 12 Wind speed at 2m (m s-1) 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 Wind speed at 10m (m s-1) Figure 9 Comparison of the wind speeds at 2 meters obtained with AgroMetShell (blue line) and manually (red line) from fictive wind speeds measured at 10 meters. 23

The difference between the two methods in the conversion of wind speeds measured at 10 meters reaches about 0.7 m.s -1 for a 5 m.s -1 wind and 1.4 m.s -1 for a 10 m.s -1 wind. Even though for a global wind speed average (2 m.s -1 ) the difference is only of 0.28 m.s -1, in case of strong wind these differences may have a sensible effect on the computation of the reference evapotranspiration. 2.2.3.4 Penman-Monteith Finally, it seems that independently from the problems described above another small error is due to the calculation of the reference evapotranspiration itself. Indeed, even if AgroMetShell calculated values of net radiation (R n ) are taken into account, some errors still remain in the final values of ET o. An example is hereunder provided with weather data from the station of Tangier, Morocco, in 1990 (some extra details about the climate in this station are given in section 3.1 and in Annex 4). Reference evapotranspirations were computed from minimum and maximum temperatures, minimum and maximum relative humidity and wind speed. The use of relative humidity allows to avoid problems with actual vapour pressure calculations. The wind speed is used as if it were measured at 2 meters to avoid the wind profile relationship problem, and the net radiation computed by AgroMetShell is forced into the control calculation. 0.80 0.70 0.60 Error on ETo (mm day-1) 0.50 0.40 0.30 0.20 0.10 0 50 100 150 200 250 300 350 400 Day number Figure 10 AgroMetShell: Error on daily Eto for the station of Tangier, 1990 Despite these measures, there are still errors in the results. As we can see in Figure 10, the ET o is still overestimated by the programme, with, in this case, some peaks beyond 0.7 mm in the summer. On average, the error represents 5% throughout the year. Forcing the net radiation indeed produces very small changes in the control calculations (about 0.1 or 0.2 mm day -1 at maximum). Reference evapotranspiration does not seem to be sensitive to so small variations, at least under this kind of climatic conditions. Since the other parameters (latent heat of vaporisation, saturation and actual vapour pressure, slope of the saturation vapour 24

pressure curve and psychrometric constant) displayed by the programme are consistent with the control experiment, the source should be the formulation of the equation of Penman- Monteith. In a nutshell, the analysis reported in this paper show several problems with the ET o calculation tools of AgroMetShell, even though they could not be entirely explained. Firstly there are errors in the computation of the daylight duration and the extraterrestrial radiation, which both depend on the day number and the latitude. Secondly, the programme does not apply the wind profile relationship as in the FAO irrigation and drainage paper 56. Finally the final step of ETo calculation (original Penman-Monteith equation) seems to be a source of error as well. From this point of view, the opinion of the developers might be very enlightening! 2.3 CROPWAT 2.3.1 Overall description CROPWAT is a programme developed by the Water Resources Development and Management Service of UN-FAO aimed at estimating the crop water and irrigation requirements as well as proposing irrigation schedules. Calculation procedures are in theory based on the Irrigation and drainage paper 56 of the FAO. CROPWAT is written in Visual Delphi 4.0 and is freely available on http://www.fao.org/nr/water/infores_databases_cropwat.html. The programme has a set of different modules, of which only the on called Climate/ETo is relevant to this research. The module is indeed the one used for reference evapotranspiration computation, on daily, dekadal or monthly basis. 2.3.2 ET o calculation method 2.3.2.1 Daily calculation CROPWAT Version 8.0 (Smith, 1992) calculates ET o from five strictly required climatic variables: minimum and maximum temperatures, mean relative humidity, wind and sunshine hours, in addition to the station characteristics (latitude and altitude). As outputs, the programme displays the solar radiation and the ET o in mm per day, as well as their monthly averages. Other intermediate values are not displayed. According to the Help and About files, the programme is closely following the FAO paper No. 56 procedures. In fact, three important steps are achieved in a different way: the computations of the actual vapour pressure (e a ), the saturation vapour pressure (e s ) and slope of the vapour pressure curve (Δ). Firstly, the actual vapour pressure is computed the same way as in ETo_Calc, i. e. according to Smith 1992: e = e a o ( T ) mean RH 100 mean 25

Secondly, the daily saturation vapour pressure is estimated as the saturation vapour pressure corresponding to the mean temperature: o e = e T s ( ) Finally, the slope of the vapour pressure curve is calculated as the average of the slopes computed for T min and T max : = 2049 e ( T ) mean + e ( T min max Δ 2 2 ( T + ) ( + ) min 237.3 Tmax 237.3 Knowing the non-linearity of the relationships, we can reasonably expect differences in outcomes. In addition, also note that like in AgroMetShell the equations 26 and 27 (arctan relationship) are used to compute the sunset hour angle, instead of the equation 25. That does not have any incidence on sunset hour angle nor ET o calculations though. ) 2.3.2.2 Dekadal calculation CROPWAT also applies its daily calculation method to dekadal calculations, i.e. deriving the actual vapour pressure from the average daily mean temperature and the average daily mean relative humidity (Smith 1992), the saturation vapour pressure from the average daily mean relative humidity, and the slope of the saturation vapour pressure curve from the average of this parameter calculated with the mean maximum and minimum temperatures. The required inputs are the minimum and maximum temperatures, the mean relative humidity, the mean sunshine duration and the mean wind speed (km day -1 ). A particularity in the routines is that CROPWAT computes the average solar radiation over dekads from day lengths and extraterrestrial radiations calculated for every day. 2.3.3 Input limits and accuracy All the inputs must be entered in the main window. They are on the one hand the characteristics of the station (table 9), and on the other hand the weather data (table 10). Table 9 CROPWAT input variables: station characteristics Station Characteristics Variable Unit Range Accuracy Default value φ -90 90 0.01 Required z m -200 8000 0.1 Required Other characteristics about the station are to be entered, like the longitude, the name etc. but they do not play any part in ET o calculations. The weather variables are the following: 26

Table 10 CROPWAT input variables: weather data Meteorological data and ET o Variable Unit Range Accuracy Default value T max C T min 60 0.1 Required T min C -80 T max 0.1 Required RH mean % 0 99 1 Required U km day -1 0 800 1 Required n hours 0 24 0.1 Required The value of relative humidity is limited to two digits, but that is not a problem because practically it is very unlikely to have an average of 100% of relative humidity over a full day. The wind speed unit is kilometre per day, which is not the most common. Users should not forget to convert the data at the risk of biasing severely the results. About the ranges, they are in general wider than in ET o _Calc, which gives more freedom and increases the variety of locations and climate conditions that can be considered for computations. 2.3.4 Output limits and accuracy CROPWAT displays two outputs: the solar radiation and the reference evapotranspiration (table 11). Table 11 CROPWAT outputs Variable Unit Accuracy R s MJ m -2 day -1 0.1 ET o mm day -1 0.01 The main advantages of CROPWAT are its straightforward use and its user-friendliness. The drawbacks are its rigidity regarding the inputs requirement, the fact that it does not display more intermediate values and eventually the methods used to compute the saturation vapour pressure and the slope of the vapour pressure curve, which brings divergent results with other programmes. This is the subject of the next section. 2.4 Some other programmes Internet abounds with various programmes meant for the computation of ET o, developed by universities, research institutes or individuals. Here is a selection of freely downloadable ones. 2.4.1 DailyET (Cranfield) DailyET is a programme quite similar to AgroMetShell developed by Tim Hess, from the Cranfield University at Silsoe, UK. The programme is free and available on (http://www.cranfield.ac.uk/sas/naturalresources/research/projects/dailyet.jsp) but the source code is not available. 27