Theor Appl Climatol DOI 10.1007/s00704-017-2213-7

ORIGINAL PAPER

Retrieving air humidity, global solar radiation, and reference evapotranspiration from daily temperatures: development and validation of new methods for Mexico. Part III: reference evapotranspiration P. Lobit 1

&

A. Gómez Tagle 2 & F. Bautista 3 & J. P. Lhomme 4

Received: 21 June 2016 / Accepted: 22 June 2017 # Springer-Verlag GmbH Austria 2017

Abstract We evaluated two methods to estimate evapotranspiration (ETo) from minimal weather records (daily maximum and minimum temperatures) in Mexico: a modified reduced set FAO-Penman-Monteith method (Allen et al. 1998, Rome, Italy) and the Hargreaves and Samani (Appl Eng Agric 1(2): 96–99, 1985) method. In the reduced set method, the FAOPenman-Monteith equation was applied with vapor pressure and radiation estimated from temperature data using two new models (see first and second articles in this series): mean temperature as the average of maximum and minimum temperature corrected for a constant bias and constant wind speed. The Hargreaves-Samani method combines two empirical relationships: one between diurnal temperature range ΔT and shortwave radiation Rs, and another one between average temperature and the ratio ETo/Rs: both relationships were evaluated and calibrated for Mexico. After performing a sensitivity analysis to evaluate the impact of different approximations on the estimation of Rs and ETo, several model combinations were tested to predict ETo from daily maximum and minimum * P. Lobit [email protected]

1

Instituto de Investigaciones Agropecuarias y Forestales, Universidad Michoacana de San Nicolás de Hidalgo, Km. 9.5 Carr. Morelia-Zinapécuaro, Unidad Posta Zootécnica, 58880 Tarímbaro, MICH, Mexico

2

Instituto de Investigaciones sobre Recursos Naturales, Universidad Michoacana de San Nicolás de Hidalgo, Avenida Juanito Itzicuaro SN, Nueva Esperanza, 58330 Morelia, MICH, Mexico

3

Centro de Investigaciones en Geografía Ambiental, UNAM, Antigua Carretera a Pátzcuaro No. 8701 Col. Ex-Hacienda de San José de La Huerta, 58190 Morelia, MICH, Mexico

4

UMR LISAH, 2 place Pierre Viala, 34060 Montpellier Cedex 1, France

temperature alone. The quality of fit of these models was evaluated on 786 weather stations covering most of the territory of Mexico. The best method was found to be a combination of the FAO-Penman-Monteith reduced set equation with the new radiation estimation and vapor pressure model. As an alternative, a recalibration of the Hargreaves-Samani equation is proposed.

1 Introduction Estimating crop irrigation requirements is indispensable for irrigation scheduling. Evapotranspiration (loss of water from soil and plants) is commonly estimated as the product of a crop coefficient (Kc), which depends on the phenological stage of the crop, by reference evapotranspiration (ETo), which varies with weather conditions. In the case of horticultural crops, where irrigation is applied at short time intervals, ETo should be calculated daily or weekly. The reference method to estimate ETo is that of PenmanMonteith, modified as in the FAO-56 document (Allen et al. 1998; Walter et al. 2001). It requires daily radiation, temperature, humidity, and wind speed as inputs. Although these data are routinely registered by automated weather stations (which usually also perform automatic ETo calculation), such stations are rather expensive and requires careful maintenance, which limits their use by small farmers, in particular in developing countries. Weather data from public networks, available in some cases, are often unreliable and/or difficult to access. For all these reasons, obtaining ETo from limited weather data acquired easily on the farm (in practice, maximum and minimum temperature) should be very useful for irrigation management. Several methods have been proposed for this purpose, for example, Blaney and Criddle (1950), Thornthwaite (1948), (Haude 1952), and Hargreaves and Samani (1985). In practice;

P. Lobit et al.

however, very few people use them, probably because they are perceived as unreliable, limited in their applicability, or lacking a theoretical basis. Also, little work has been done to evaluate and calibrate these methods in different environments, which further limits their use. The FAO-Penman-Monteith equation Allen et al. (1998), which is based on the physical calculations of the energy fluxes at the canopy, is considered the reference. When humidity and wind speed are not available, Allen et al. (1998) recommends to estimate humidity from minimum temperature and use a constant wind speed. This reduced set equation, however, still requires solar radiation as input. When measurements are not available, it has to be combined with an empirical radiation model, such as the one we proposed in the first and second paper of this series. To our knowledge, there has been no study to evaluate the impact of these approximations on the prediction of ETo. Among all empirical methods, that by Hargreaves and Samani (1985) is probably the one with the soundest physical basis. It results from the combination of two relations: one between diurnal temperature range and shortwave radiation and another one between shortwave radiation, average temperature, and ETo (Hargreaves 2003). Other methods, like those of Blaney-Criddle or Thornthwaite, in which the only solar input is day length, may be accurate at a monthly timescale, but are unlikely to predict the fluctuations of ETo on shorter timescales, when it fluctuates with changes of cloud cover and radiation. Haude’s (1952) model, which only takes into account humidity, may also miss short term variations of solar radiation. Therefore, the Hargreaves-Samani method remains the most promising among empirical methods for daily ETo estimation. In practice, this equation, initially calibrated for continental USA only, has been tested for some other locations around the globe, with variable results (Bautista et al. 2009; Oliveira and Yoder 2000; Yoder et al. 2005; Stefano and Ferro 1997; Borges and Mendiondo 2007). However, such testing has often been limited to an evaluation of the quality of fit of the model and/or the determination of empirical correction coefficients. In the case of Mexico, these studies have been performed only at regional scales and their results cannot be extended to the whole country (Bautista et al. 2009). The purpose of this work was (1) to identify the sources of error in temperature-based ETo models based on the modified reduced set FAO-Penman-Monteith equation coupled with a previously developed radiation model, (2) to evaluate and recalibrate the Hargreaves-Samani equation for Mexico, and (3) to compare it to the modified reduced set FAO-PenmanMonteith method.

2 Material and methods

Inifap (Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias (Inifap: http://clima.inifap.gob.mx) and 27 stations from the APEAM (avocado producers’ association of the state of Michoacán: http://www. apeamclima.org). The Inifap network, designed to provide farmers with weather data, covers most of the agricultural areas of Mexico: the regions without coverage are mostly mountains or deserts. The geographical distribution of the stations is shown in Fig. 5. A detailed description of available data, quality checks, and corrections is given in the second article of this series. The reference FAO-Penman-Monteith equation This equation, standardized by FAO-56 (Allen et al. 1998) and later by the ASWERI (Walter et al. 2001), is ETo ¼

  .   0:408 Δ ðRn−GÞ þ γ 900 ðTmean þ 273:16Þ u2 ðes−eaÞ ðΔ þ γ ð1 þ 0:34 u2ÞÞ

The main input variables are as follows: – – – – –

ea is the vapor pressure u2 is the wind speed, measured at 2 m altitude Tmean is the daily mean average temperature Rn is the net radiation at the crop surface G is the ground heat flux (usually neglected for daily measurements).

The details of the calculation of the internal variables Δ, γ, es, Rnl, G, Rso, and Ra can be found in Allen et al. (1998). The reduced set Penman-Monteith method This method calculates an approximation of the reference evapotranspiration: [RS]ETo ≈ ETo. [RS]ETo uses the following approximations to obtain the variables used in the FAO-PenmanMonteith equation (excepted Rs) from Tmax and Tmin: T mean≈ðT max þ T min Þ=2−ϵ

–

ð2Þ

daily mean temperature (normally defined as the average of all hourly temperatures) is approximated by the average between maximum and minimum temperature, decreased by ϵ. ϵ accounts for the fact that, since the diurnal evolution of temperature is not a symmetrical curve, mean temperature (taken as the average of hourly measurements) is slightly smaller than the average of maximum and minimum temperature. For Mexico, we found ϵ = 0.42°C (data not shown). 

Weather data This study used data from 759 automated weather stations belonging to the national network of the

ð1Þ

ea≈0:6108 exp

17:27 T min T min þ 237:3

 ð3Þ

Retrieving reference evapotranspiration from daily temperatures

–

–

actual vapor pressure is approximated by saturated vapor pressure at minimum temperature. Alternatively, we proposed an improved estimation, described in the first article of this series.

–

 u2≈u2

½HS ETo ¼ k 2 ðTmean þ bÞ ½HS Rs ðwith k 2 ¼ a=KT Þ

ð4Þ

wind speed is approximated by a constant value (in all following calculations, we take the average mean speed measured in Mexico: u2 = 1.2 m/s).

The reduced set FAO-Penman-Monteith equation can be used only when Rs data are available. Otherwise, Rs has to be estimated. In the second article of this series, we proposed a method that provided a good estimate from minimum and maximum temperature data for most of Mexico. In this work, we evaluated the effect of introducing, one by one, the following approximations on the estimation of ETo: (1) mean temperature using Eq. 2, (2) vapor pressure using either Eq. 3 or our new method, (3) wind speed constant at its average value for the whole of Mexico (1.2 m s−1), and (4) radiation using either the recalibrated Hargreaves-Samani relation (see below) or our new method. Finally all approximations were applied simultaneously to evaluate the quality of prediction of the FAO-Penman-Monteith reduced set equation coupled to our humidity and radiation model. In all cases, the quality of fit was evaluated as described below.

The Hargreaves-Samani method This equation calculates another approximation of reference evapotranspiration [HS]ETo ≈ ETo. It can be written in the form of the following equation ½HS ET o ¼ ða=2:45Þ ðTmean þ bÞ Ra

pffiffiffiffiffiffiffiffiffiffi ΔT

ð5Þ

where a and b are empirical parameters and ΔT = Tmax − Tmin is the daily temperature range. The values proposed by Hargreaves and Samani are a = 0.0023 and b = 17.8. The factor 2.45 is used as a conversion between energy (MJ m−2 day−1) and evaporation (mm day−1). This equation actually results from the combination of two equations: –

The first one estimates global radiation from diurnal temperature range:

½HS Rs ¼ KT Ra

pffiffiffiffiffiffiffiffi ΔT

The second one states that ratio between [HS]ETo and [HS]Rs is a linear function of daily mean temperature

ð7Þ

where KT, a, and b are parameters. Neither Eq. 6 nor Eq. 7 have been evaluated for Mexico. In this work, they were verified and calibrated as follows. In Eq. 6, Hargreaves and Samani use a single parameter, KT, to account for the relation between temperature range and radiation. For Mexico, this parameter was estimated by fitting Eq. 6 to measured radiation data using pooled daily records of all available weather stations. Parameter optimization was done using the nls procedure in the R software (R Development Core Team 2012). The quality of prediction of Rs was then evaluated. A study by Knapp et al. (1980) showed that, in the USA, KT varied from place to place, and they proposed to estimate it more accurately as KT = 0.00185 ΔT2 − 0.0433 ΔT + 0.4023. Following a similar approach, we calculated, for each station in our database, the value of KT required for Eq. 6 to match measured radiation  pffiffiffiffiffiffiffiffi as: KT ¼ Rs= Ra ΔT . We then looked for relationships between KT and various climate statistics: annual average ΔT, humidity (vapor pressure, relative humidity and vapor pressure deficit), latitude, altitude, and distance to the coast. Equation 7 states that the ratio ETo/Rs can be approximated by a linear function of Tmean: in other words, the fraction of incoming radiation dissipated as latent heat L (where L = 2.45 ETo) depends mostly on mean temperature. The original equation was ETo/Rs = (a + b Tmean ), with parameters a = 0.2403/2.45 = 9.808 10−2 and b = 0.0135/2.45 = 5.51 10−3. As for [HS]Rs, the validity of this approximation has not been established for Mexico. In order to evaluate and recalibrate it, we estimated a and b by fitting ETo predicted using measured Rs values to reference ETo. Parameter estimation was done again using the Bnls^ procedure. Then, indices of quality of fit were calculated. Measurement of the quality of fit of the models The quality of fit between predicted and measured variables was estimated for each station using the coefficient of determination (R2) and the following three indices: root mean square error (RMSE), bias (BIAS), and concordance index (CI), as described in the first article of this series.

3 Results ð6Þ

where Ra is extraterrestrial radiation (equations to calculate it are given in Allen et al. (1998)) and KT a parameter

Sensitivity analysis of the reduced set FAO-PenmanMonteith equation Figure 1 presents the quality of fit between the reduced set FAO-Penman-Monteith ETo and

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FAO-Penman-Monteith ETo with each of the following variables estimated in turn: Tmean (mean temperature), ea (vapor pressure), u2 (wind speed), and Rs (shortwave radiation). Estimating Tmean from the average of maximum and minimum temperature according to Eq. 2 produced a small but not negligible error: with ϵ set to 0, there was a positive bias in ETo (BIAS = 0.07 mm day−1), with the corresponding impact on the prediction error (RMSE = 0.19 mm day−1) (Fig. 1a). With ϵ = 0.42°C (average difference between mean temperature and average of Tmax and Tmin), the bias on ETo dropped to 0 mm day − 1 but the estimation error remained (RMSE = 0.17 mm day−1) (data not shown). Estimating ea

had a moderate impact on the estimation of ETo (Fig. 1b). Using the dew-point method (Allen et al. 1998) caused a small averageerror(RMSE=0.35mm day−1,BIAS=−0.05mmday−1), but this error was concentrated on the stations and/or days with higher ETo. For ETo ≥5 mm day−1, the negative bias could be as high as 1 mm day−1. When using the new model of humidity prediction (data not shown), the average error was reduced to RMSE = 0.31 mm day−1, the bias was zero and its geographical variation was reduced. The main source of estimation error was from considering wind speed constant (Fig. 1c). The average RMSE was high (RMSE = 0.51 mm day−1), and this approximation induced a considerable negative bias at high ETo (up to 1 mm day−1 for ETo ≥6 mm day−1). Finally, estimating Rs using

Fig. 1 Sensitivity analysis of the reduced set Penman-Monteith equation when different inputs are estimated as recommended by Allen et al. (1998) (all other inputs being known). a Daily average temperature estimated as Tmean = (Tmax + Tmin)/2. b Vapor pressure estimated as

saturated vapor pressure at the minimum temperature. c Wind speed considered constant (u2 = 2 m s−1). d Rs estimated by the new radiation model

Retrieving reference evapotranspiration from daily temperatures

the new model caused rather small errors (RMSE = 0.39 mm day−1, BIAS = 0.02 mm day−1). In particular, almost no bias was found for the smallest or highest values of ETo (Fig. 1d). Possible correlations between errors were looked for

RMSE ¼

(data not shown): no correlation appeared, that is, errors on mean temperature, humidity, wind, and radiation estimation were independent. In this case, the expected error when applying all approximations simultaneously would be

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   RMSETmean 2 þ RMSEea 2 þ RMSEU 2 2 þ RMSERs 2 ¼ 0:74 mm day−1

which was close to the value found for the integrated model (0.72 mm day−1) (Fig. 4d).

Estimating Rs (original Hargreaves-Samani equation and new model) When fitting the Hargreaves-Samani radiation

Fig. 2 a Relationship between temperature range and transmittance observed (points), predicted by the Hargreaves model with its original parameter (KT = 0.162) (dotted line), and with KT fitted for Mexico (KT = 0.1557, solid line). b Rs predicted using the Hargreaves-Samani

model with KT fitted for Mexico. c Transmittance measured (gray points) and predicted by the new radiation model (black points). d Rs predicted by the new radiation model

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model to the data from all Mexico, we found KT = 0.155, a value approximately 5% smaller than defined for continental USA (KT = 0.162). However, as observed in Fig. 2a, this model did not reproduce the shape of the relation between ΔTand transmittance: transmittance was overestimated at low (ΔT < 10°C) and high (ΔT > 20°C) ΔT values and underestimated around average (ΔT ≈ 15°C) values. As a result, Rs the quality of fit was low, with RMSE = 3.4 MJ m−2 day−1 and CI = 0.88 (Fig. 2b). An attempt to find a correlation between KT and annual average climate statistics, following Knapp et al. (1980), showed that the only variable correlated to which KT was  : KT ¼ 0:194398− the annual average temperature range ΔT 2 −16  0:002476 ΔT (R = 0.51, p ≤ 2.2×10 ). When trying to correlate KT with other annual average climate variables in addi , a relation with annual average vapor pressure was tion to ΔT  −0:0088303 ea  f o u n d : KT ¼ 0:2369933−0:0042762 ΔT 2 −16 (R = 0.68, p = 2.2×10 ). However, the quality of fit of Rs was not improved when using any of these relations (data not shown): for practical purposes, a unique value of KT can be used for the whole country. The new radiation model, in contrast with Hargreaves method, reproduced the shape of the transmittance curve well (Fig. 2c) and produced unbiased estimates of Rs (Fig. 2d). The quality of fit for the whole of Mexico was acceptable with RMSE = 3.06 MJ m−2 day−1, BIAS = 0.05 MJ m−2 day−1 and CI = 0.91.

and reference ETo was rather high (R2 = 0.87). The parameters of this linear correlation were as follows: ETo = 0.896 [HS]ETo— 0.171. After correction, the indices of quality of fit were RMSE = 0.81 mm day−1, BIAS = −0.02 mm day−1, and CI = 0.88 (data not shown). The recalibrated HargreavesSamani model ([HS′]ETo, with its coefficients fitted for Mexico), resulted in the same quality of fit: RMSE = 0.81 mm day−1 and CI = 0.88 (Fig. 4b). Although the overall bias (−0.02 mm day−1) was negligible, this model overestimated low ETo and underestimated high ETo.

Relation between Rs and ETo The Hargreaves-Samani equation states that the ratio between ETo and Rs is approximately a linear function of mean temperature. Defining the latent heat flux as L = 2.45 ETo, the ratio L/Rs can be interpreted as the fraction of Rs used for evapotranspiration (latent heat flux) instead of sensible heat flux. Figure 3a shows that such a linear relation was also found in our data. However, it was rather loose (R2 = 0.23) and its parameters were different from those established in the USA: we found L/Rs ≈ 0.2442 + 0.0105 Tmean, against L/Rs ≈ 0.2403 + 0.0135 Tmean according to Hargreaves and Samani. That is, at 20 °C (the average mean temperature for Mexico), 45% of shortwave radiation was used for potential evapotranspiration, against 51% according to the original parameterization. Using this recalibrated relation to predict ETo from known Rs (Fig. 3b) led to the following quality of fit: RMSE = 0.6 mm day −1, BIAS = 0 mm day−1 and CI = 0.95. Evaluation of the Hargreaves-Samani method Figure 4 compares the predictions of ETo obtained using different model combinations to predict first Rs, then ETo in function of Rs. When using the Hargreaves-Samani model ([HS]ETo) Bas is^ to predict ETo (Fig. 4a), the quality of fit was low (RMSE = 1.18 mm day −1 , CI = 0.82), with a bias of 0.84 mm day−1. This method overestimated ETo by more than 20% on average. However, the correlation between estimated

Fig. 3 a Fraction of Rs used for evapotranspiration (L/Rs = 2.45 ETo/Rs) against mean temperature. Hargreaves original equation (dotted line): ETo/Rs = 0.0981 + 0.00551 Tmean; best fit to Mexico data: ETo/ Rs = 0.0937 + 0.00392 Tmean (solid line). b ETo estimated from Rs using the recalibrated Hargreaves equation. The solid line represents the 1:1 relation

Retrieving reference evapotranspiration from daily temperatures

Fig. 4 a Estimation of ETo by the Hargreaves-Samani formula with parameters proposed by Hargreaves. b Same method, after calibration of KT, a and b for Mexico. c ETo calculated using the new radiation

model and recalibrated Hargreaves relation between Rs and ETo. d ETo calculated using the reduced set FAO-Penman-Monteith method with the new radiation model

The best prediction of ETo (Fig. 4d) was obtained with the new radiation estimation model combined with the FAO-PenmanMonteith reduced set equations ([RS(Tmean, ea, u2, Rs)]ETo): RMSE = 0.72 mm day−1 , BIAS = 0.02 mm day −1, and CI = 0.92. However, almost the same quality of prediction was obtained with the new radiation estimation model combined with the recalibrated Hargreaves-Samani relation between ETo/Rs and Tmean ([HS′(Rs)]ETo): RMSE = 0.75 and BIAS = 0.02 mm day−1 and CI = 0.91 (Fig. 4c).

set equation with the new vapor pressure and radiation models), evaluated through the root of the mean squared error of prediction and bias, respectively. Prediction of ETo was excellent in the center of the country and on the west coast (RMSE <0.5 mm day−1) and good in the south-east and the east coast (RMSE <0.75 mm day−1). Most of the stations with high errors (RMSE >1 mm day−1) were located inland in the north of the country, in the dry areas of the central plateau. There were also areas with high prediction errors on the northwestern coast (state of Tamaulipas) and in the Tehuantepec isthmus. In contrast with the errors caused by estimations of humidity and/or wind, many of the places where the radiation model presented high errors (in the mountain ranges in the

Geographical variations in the quality of fit Figures 5 and 6 show a map of the quality of prediction of ETo by [RS(Tmean, ea′, u2, Rs′)]ETo (modified FAO-Penman-Monteith reduced

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Fig. 5 Map of the root mean square error of the improved reduced set FAO-Penman-Monteith ETo model. Colors correspond to a scale of rainbow colors from 0.5 (dark blue) to 1 (red)

east of the country, in states of Veracruz, Hidalgo and Nuevo León, see first article of this series) still maintained a good estimation of ETo. Seasonal variations in the quality of fit Figure 7 compares ETo predicted using the new reduced set model to FAO-PenmanMonteith ETo for a whole year at four different stations. The best prediction was obtained at the BLas Estacas^ (Guanajuato) station, with almost no seasonal bias (excepted for a small overestimation in December and January). Most of the stations in the central-western part of the country also presented low overall prediction error (Fig. 5), consistent with the absence of seasonal bias (data not shown). Another station with low prediction error was BLa Perseverancia^ (Sonora), with the exception of a slight overestimation of ETo between October and December. At the other two stations BIch-ek (Campeche) and BEl colorín^ (Aguascalientes), ETo prediction was good most of the year, but inaccurate between March and May (i.e., during the dry season). During this period, ETo was underestimated and its shortterm fluctuations were not well predicted. In contrast, during the rest of the year, sudden drops in ETo (due to either cold or cloudy days) were well predicted.

4 Discussion The Penman-Monteith equation accounts for two components that control evapotranspiration: energy supply, which depends mostly on global solar radiation (and to a lower extent downwelling long-wave radiation, which in turns depends on temperature and cloud cover), and ventilation, which depends on wind speed and vapor pressure deficit. The sensitivity analysis shown in Fig. 1 confirmed that accurate estimation of solar radiation was critical to predict ETo (vapor pressure and average temperature were less important). At several stations, mostly in the north of the country, ETo prediction was less accurate during the dry season (April-May). At these times, the cloud cover was usually low, so that radiation, close to clear-sky radiation, could be estimated accurately. Therefore, strong fluctuations of ETo during this season were mostly attributable to the ventilation term, with fluctuations of wind speed in conditions of high vapor pressure deficit. Wind during the dry season is often due to thermal instability, with bursts during the early afternoon. This was probably the situation in places like Aguascalientes or Campeche, where ETo was not well predicted. The coast of Sonora may have more regular

Retrieving reference evapotranspiration from daily temperatures

Fig. 6 Map of the bias of the improved reduced set FAO-Penman-Monteith ETo model. Colors correspond to a scale of rainbow colors from −0.5 (red) to 0.5 (blue). The red squares tagged from A to D mark the geographical positions of the stations shown in Fig. 7

winds, while the plains of Guanajuato present little wind and lower vapor pressure deficit. Concerning radiation, the model presented in the second article of this series provided estimates of Rs that were accurate enough to predict ETo. In contrast, the Hargreaves-Samani method produced skewed and biased estimate. Using the parameters calibrated for continental USA, Rs was overestimated by about 5%. This was probably a consequence of a thicker, more humid and less transmissive tropical troposphere. The different shapes of the relation between transmittance and ΔTwere probably due to the type of clouds found in the tropics, with cumulus and cumulonimbus which strongly reduce transmittance under unstable atmospheric conditions. Concerning ventilation, the relationship between ETo and Rs was predicted with a similar accuracy either by the recalibrated Hargreaves-Samani relation (with ETo/Rs modeled as a linear function of Tmean) or by the reduced set FAO-PenmanMonteith method (which took into account vapor pressure). This suggests that vapor pressure was not a critical variable to predict ETo, while wind speed probably was. The average wind speed in Mexico was found to be around 1.20 m s−1 and using this value in the reduced set FAO-Penman-Monteith model produced no bias for the country as a whole but there were strong regional

and seasonal biases, consistent with corresponding fluctuations of wind speed. Stronger winds along the coasts are probably one of the reasons why our model underestimated ETo in the northeast coast (north of Veracruz and Tamaulipas) and near the Tehuantepec isthmus. Unfortunately, we found no way to estimate wind speed from temperature measurements alone. It is also noteworthy that at the same temperature, the fraction of Rs used for evapo-transpiration was approximately 10% smaller than in the USA: this was probably explained by the increased importance of the convective component of the surface energy balance (and consequently lower importance of the latent heat term) under unstable atmospheric conditions. Another source of errors in the north-west and on the west coast was probably the influence of synoptic weather events: in contrast with most of the country where cloud formation is mostly controlled by atmospheric instability, these regions are often affected by cold fronts (which affect both temperature range and cloud types): this phenomenon probably accounted for the lower accuracy of the radiation model, with the corresponding impact on predicted ETo. For practical purposes, the new method based on an improved reduced set FAO-Penman-Monteith method provided rather accurate and unbiased predictions of ETo for most of the country. Surprisingly, at places where radiation could not be estimated

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Fig. 7 Comparison between reference ETo (points) and ETo estimated using the FAO-Penman-Monteith reduced set model with the new radiation estimation model (solid line) at four locations during 2011. a

Campeche. b Sonora. c Guanajuato. d Aguascalientes. The geographical coordinates and altitude of each station are indicated on the corresponding graph

accurately (mostly mountain areas), ETo still was well predicted. This is probably due to the fact that errors in the estimation of Rs occurred during mostly humid and cloudy days, when ETo was already low due to low vapor pressure deficit. Most of the places with inaccurate prediction of ETo were found aligned east of the BSierra Madre^ cordillera that borders the pacific coast of Mexico. These places present very low humidity due to the foehn effect and maybe thermal winds due to the proximity of the mountains: as a result, ETo is underestimated in most of these places. A few other places, in particular in the Tehuantepec Isthmus and in the north east, also present errors, probably to local wind conditions. Apart from these exceptions, ETo estimation was accurate for most of the rest of the country.

5 Conclusion The best method to predict ETo from measurements of maximum and minimum temperature in Mexico was a modified reduced set FAO-Penman-Monteith equation with solar radiation (Rs) and vapor pressure (ea) estimated with our corresponding improved models and using a constant value (average for the whole country) for wind speed. Once recalibrated, the Hargreaves-Samani relation between ETo/Rs and Tmean also appeared applicable in most of Mexico. In contrast, all models based on the estimation of Rs through the HargreavesSamani method were inaccurate: not only was RMSE about 10% higher than in the modified reduced set FAO-Penman-

Retrieving reference evapotranspiration from daily temperatures

Monteith, but these models were also overestimated ETo for low values and underestimated it for high values. However, for practical purposes (for example, irrigation management when computing tools are not available), the recalibrated Hargreaves equation provided a good enough alternative. With its estimated parameters, this equation became ETo ¼ 0:1555 Ra

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tmax þ Tmin ðTmax−Tmin Þ 9:967 10−2 þ 4:280 10−3 2

where Ra is extraterrestrial radiation.

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