An Empirical Simplification of the Temperature Penman-Monteith Model for the Tropics

A simple empirical equation (EPM) is presented to considerably shorten the computational steps required to estimate reference grass evapotranspiration (ETo) in the tropics using the FAO-56 Penman-Monteith equation (TPM) when the only available weather data are those of temperature. Generally EPM predicted TPM ETo with very high efficiencies, achieving statistical performance measures as high as 2 = 1.00 r , 1 = 0.98 E , 2 = 1.00 E , MAE=0.01 mm/day in tests on data from six locations in four countries in West Africa. EPM was of general form , = / /17000 b o EPM s a ET T R k R χ − , where o ET = reference grass ETo (MJ m 2 − d 1 − ), T = daily average air temperature ( o C), s R = estimated solar radiation (MJ m 2 − d 1 − ), a R = computed extraterrestrial radiation (MJ m 2 − d 1 − ); b , k , and χ , were parameters computed from local latitude and temperature data. The simplicity of EPM is expected to encourage wider usage of TPM ETo estimates which are more accurate than estimates obtained by using locally-uncalibrated versions of simpler ETo models where only temperature data are available.

); b , k , and χ , were parameters computed from local latitude and temperature data.The simplicity of EPM is expected to encourage wider usage of TPM ETo estimates which are more accurate than estimates obtained by using locally-uncalibrated versions of simpler ETo models where only temperature data are available.

Introduction
Reference evapotranspiration (ETo) estimates are very important in irrigation system design and operations.ETo estimates are used to estimate crop evapotranspiration (ETc) rates which are used to determine design peak irrigation system capacities and also irrigation water requirements and irrigation schedules (Keller & Bliesner, 1990).Because ETc is difficult and expensive to measure directly, it is usually conveniently estimated indirectly from mathematical models using climatic data inputs (Farahani et al., 2007).
Although the most widely accepted mathematical model for reference grass ETo estimation from meteorological data is the FAO-56 Penman-Monteith equation (PM) (Allen et al., 1998), it requires data, apart from temperature data, that are normally measured at few weather stations, even in the developed countries.For some locations the required climate data are available but of questionable quality, especially in developing countries (Droogers & Allen, 2002).However, as a minimum, many weather stations around the world, even in developing countries, collect temperature data of acceptable quality (Hargreaves & Allen, 2003).Therefore ETo models that require only temperature data, but are highly accurate, are very useful in data-poor areas of the world.
Where only temperature data are available, the FAO recommends using the temperature-only Penman-Monteith (TPM) version of PM, whereby all the unavailable data are estimated according to certain outlined procedures (Allen et al., 1998).The TPM has been reported to give reliable estimates in several locations around the world (Campbell Scientific, 1998;Jabloun & Sahli, 2008;Popova et al., 2006).Two other temperature ETo models are the Hargreaves & Samani (1985) (HG) and the Turc (1961) (TU) models which have been reported to give reliable ETo estimates (Lu et al., 2005;Yoder et al., 2005;Hargreaves & Allen, 2003), but only after proper local calibration (Gavilan et al., 2006).However because of lack of equipment for proper local calibration of TUand HG, the attractive simplicity of TU and HG, and discouraging complexity of TPM, uncalibrated HG and TU are often used in practice, especially in the developing world.
The goal of this study was to considerably simplify the computation of TPM ETo estimates with little loss of accuracy.
The main objective was to develop one single and simpler equation thatwould give essentially the same ETo estimates as TPM, but that would be as easy to use as simple temperature-based models such as TU and HG.

Materials and methods
The empirical Penman-Monteith equation (EPM) for the temperature Penman-Monteith equation (TPM) was developed using daily temperature data from six weather stations located in four countries in West Africa.The weather and other required data for the Accra site were obtained from the Water Research Institute (WRI) in Accra, while that for the other sites were obtained from TuTiempo (2008) and then processed for spreadsheet use.Where average wind speed data was not available the global average of 2 m/s reported in Droogers & Allen (2002) was assumed.
All the computations were executed in Microsoft Excel XP spreadsheet installed on Microsoft Windows XP operating system on an HP Pavilion dv6000 laptop computer with Intel (R) Core (TM)2 CPU T7200 2.00 GHz, 1.00 GB RAM.
EPM was developed by starting with a simple form of EPM, and then computing two sets of daily reference grass evapotranspiration (ETo) values (one by TPM and the other by EPM), and then tweaking the EPM parameter values using a numerical algorithm to make the EPM results as close as possible to those of TPM.Examination of the temporal plot of the errors informed the addition of other terms to EPM during the development process.Details of the forms of the TPM and EPM equations and the computational procedures are given in the sections that follow.

Computation of daily EPM and TPM ETo
The following parameters were computed for every day of the year from the daily temperature data for each weather station using the indicated equations: 1. Average temperature, T , using Eqn.( 14

ET
, using the developed EPM equation, Eqn. ( 19), with guessed initial values for the unknown parameters of the EPM equation.

Development of the form of EPM
After computing the daily values of the parameters in steps 1-14 above, an optimal form of EPM and its optimal parameter values were determined following these steps: 1.The modified coefficient of efficiency, 1 E , for E was computed and a new set of optimal parameter values was numerically determined using the solver in Microsoft Excel to maximize the new 1 E ; 6. Other simple forms of EPM were experimented with and the selected optimal form of EPM was that which yielded the highest 1 E values.

Calibration and tests of EPM
After the development of its form using the Accra data, EPM was calibrated for each of the other tropical sites using the following steps: 1.

ET
data and compared to their ideal values.

Temperature Penman-Monteith application (TPM)
TPM was used to estimate daily ETo from the maximum and minimum temperature data following the recommendations of Allen et al., (1998) and (Allen, 2002) where 0.77 ), was estimated using with Rs k set to 0.19 for locations near large water bodies, and 0.16 for other locations (Hargreaves & Allen, 2003).

Statistical performance measures
The main measure of the ability of EPM to predict TPM ETo values was the modofied coefficient of efficiency, 1 E , which is related to, but more discriminatory than the coefficient of efficiency, 2 E (Hall, 2001); In this paper they were defined generallly as c E , Bardsley (Bardsley & Purdie, 2007;Legates & McCabe, 1999) , , =1 , , =1 = 1 where c is a positive integer, equal to 1 for 1

General and specific EPM equations
The general form of the developed EPM equation was The optimal parameter values for the six sites during calibration and testing are shown in Table 1 3

.2 Minimizing EPM residuals
The final form of EPM consisted of two parts, the main part, EPMp, and the minor part, EPMc (see eqn. ).
The improvement in the , o TPM

ET
prediction efficiency of EPM by the introduction of EPMc was reflected in the reduced scatter and fluctuation in the temporal residual plots.For example when EPM=EPMp was calibrated with the average daily 1998-2006 Accra temperature data, 1 E was 0.95.But, when EPM=EPMp+EPMc was calibrated with that same temperature data 1 E increased to 0.99 with the EPM residuals plot almost flattened (Fig. 1b) compared to the EPM residuals plot of EPM=EPMp (Fig. 1a).
Although initially the residuals of EMP=EPMp were modeled independently and then added as a separate term to the EPMp to form a more effiicient EPM equation, no simple expression was found to closely model the EPM=EPMp residuals shown in Fig. 1(a).However, when the old EPMp and the resulting EPMc were then calibrated at the same time (as EPM=EPMp+EPMc), the scatter in the EPMp residuals was reduced and a more definite pattern emerged in the seasonal fluctutions which were then more easily modeled with a simple expression for EPMc.Thus calibrating EPM with both the EPMp and EPMc terms resulted in higher precision and 1 E values than calibrating EPMp and EPMc separately.

Precision and accuracy during calibration
During calibration the match between EPM and TPM was very good for all the sites as reflected in the performance parameters m , 2 r and 2 E being at their best possible values of 1.00.(Table 1).Even though EPM appeared to have performed equally well at all the sites, its performance at the Accra site was best using the 1 E measure which has a greater power of disrmination than 2 r and 2 E .However the difference betwen the best ( 1 = 0.99 E ) and worst ( 1 = 0.95 E ) performances at calibration was only 1 = 0.04 E Δ .The desirable value of c =0.00 mm d 1 − was also achieved at all the sites.Even though the desirable value of MAE=0 was not achieved at any of the sites the worst value was negligible for practical purposes, at 0.03 mm d 1 − .

Precision and accuracy under test
When the calibrated EPM, with the values of χ , k , and b obtained from calibration at each of the six weather stations, was used on new data to predict daily

ET
=6.0 mm d 1 − , and that 1 E for the year was as high as 0.96.Also, this involved only 8 out of 366 points (i.e., 2%) and thus no general conclusions can be drawn from this observation especially since it was not observed for the five other locations.

Similarities and differences with other models
EPM is similar, in form, to simple ETo models that use only temperature and solar radiation input data, such as the Hargreaves (Hargreaves et al., 1985), Turc (Turc, 1961) and Thornthwaite methods (Jacobs & Satti, 2001), but it is different from those simple models in that it is not another empirical temperature ETo model based on actual ETo data (Hargreaves & Allen, 2003;Hargreaves & Samani, 1985;Turc, 1961) but rather simply a shortcut method for estimating daily ETo using the Penman-Monteith equation when the only available data are maximum and minimum temperature data.EPM thus takes the place of the Penman-Moneith equation (eqn. 1) itself, and the nine equations-( 2), ( 3), ( 4), ( 5), ( 11), ( 12), ( 13), (15), and ( 17)-that are used to estimate the unavailable weather data.Therefore EPM is simply an empirical form of the Penman-Monteith equation for use when only temperature data are available and all the other data needed to use PM have to be estimated from temperature data.
solver function in Microsoft Excel was set up to maximize 1 E by numerically varying the values of the parameters of EPM at each stage of the development.The constraints were = 1 s , and = 0 c where s , and c are respectively, the slope and intercept of the straight line of the plot of , optimum values of the EPM parameters were the ones returned by the Microsoft Excel solver; 4. The residual, was plotted and modeled as an additional term to improve the efficiency of the current form of EPM; 5. A new value of 1 temperature at 2 m above ground ( o C); 2 = u wind speed at 2 m above ground surface (m s 1 pressure-temperature curve (kPa o C 1 − ), = γ psychometric constant (kPa o C 1 − ) Allen et al.(1998).When all the data required to use Eq. 1 is unavailable it can still be applied by using various formulas to estimate the missing data from temperature data; this method of applying PM from only temperature data is called the temperature PM method (TPM) in this study, for which ,

(
2005); Zhang et al. (2008); Stöckle et al. (2004).The other measures of the performance of EPM were coefficient of determination ( 2 r ), intercept of EPM versus TPM regression line ( c ), slope of the EPM versus TPM regression line ( m ), and annual mean absolute error (MAE).
(19)).The purpose of EPMp was to predict ETo by a very simple mathematical expression while that of EPMc was to minimize the prediction errors (i.e.,

Figure 1 .
Figure 1.Variations of the daily residuals of the calibrated EPM=EPMp (a) and and of EPM=EPMp+EPMc (b), showing how well EPMc models the modified residuals in the latter case, for the average 1998-2006 data for the WRI weather station.
The single equation, named the Empirical Penman-Monteith equation (EPM), should be applicable to data from other weather stations in West Africa apart from the original development site.EPM should use the same weather data as TPM, and should eliminate more than 10 of the intermediate computations required to apply TPM.
P , and G data were all estimated from maximum and minimum temperatures m Δ , γ ,