A SIMPLE PARAMETERIZATION OF BULK CANOPY RESISTANCE FROM CLIMATIC VARIABLES FOR ESTIMATING HOURLY EVAPOTRANSPIRATION

This paper examines a model for estimating canopy resistance ( r c ) and reference evapotranspiration ( ET o ) on an hourly basis. The experimental data refer to grass at two sites in Spain with semiarid and windy conditions in a typical Mediterranean climate. Measured hourly ET o values were obtained over grass during a four year period between 1997 and 2000 using a weighing lysimeter (Zaragoza, NE Spain) and an eddy covariance system (Córdoba, S Spain). The present model is based on the Penman-Monteith (PM) approach, but incorporates a variable canopy resistance r c as an empirical function of the square root of a climatic resistance r * that depends on climatic variables. Values for the variable r c were also computed according to two other approaches: with the r c variable as a straight line function of r * (Katerji and Perrier, 1983), and also as a mechanistic function of weather variables as proposed by Todorovic (1999). In the proposed model, the results showed that the ratio r c / r a (where r a is the aerodynamic resistance) presents a dependence on the square root of r * / r a , as the best approach with empirically derived global parameters. When estimating hourly ET o values, we compared the performance of the PM equation using those estimated variable r c


INTRODUCTION
Evapotranspiration is a component of the hydrological cycle whose accurate computation is needed for an appropriate management of water resources.A high degree of accuracy in the estimation of crop evapotranspiration may lead to important savings in water requirements in irrigated areas.The Food and Agriculture Organization (FAO) proposed a methodology for computing crop evapotranspiration (Doorenbos and Pruitt, 1977), based on the use of reference evapotranspiration (ET o ) and crop coefficients (K c ). Allen et al. (1998) redefined the concept of ET o and adopted the Penman-Monteith (PM) equation with constant canopy resistance (r c ) to estimate reference evapotranspiration.
The PM model has been applied in different regions of the world and has provided good results in comparison with other equations (Berengena et al., 2001;Hussein, 1999;Ventura et al., 1999;Allen et al., 1994a, b;Jensen et al., 1990;Allen et al., 1989).However, many of these studies suggest that in semi-arid and windy areas with a high evaporative demand, there is a tendency for the PM method to underestimate the higher values of measured ET o , and to overestimate the lower ET o values.The extent of this underestimation may range from 2% to 18% (Pereira et al., 1999;Todorovic, 1999;Ventura et al., 1999;Steduto et al., 1996).
The accuracy of the predictions of the PM model may be constrained by such simplifying assumptions as the 'big leaf' approximation with constant canopy resistance.Given the driving meteorological variables at a particular site, estimates made with the PM equation rely on the correct modeling of the effective values of two parameters: aerodynamic resistance r a and bulk (surface) canopy resistance r c .The FAO has adopted a standard PM equation that can be applied worldwide, using a constant value of 70 s m -1 for r c when calculating grass reference evapotranspiration (Allen et al., 1998;Allen et al., 1994a, b;Smith et al., 1991).However, this fixed value for r c may be the cause of the previously mentioned tendency for the Penman-Monteith equation to underestimate ET o (Rana et al., 1994).As r a can be calculated from meteorological conditions, in order to provide more accurate estimations of evapotranspiration using the PM equation, it may be necessary to parameterise canopy resistance as a primary factor in the evapotranspiration process (Monteith, 1965).
The net resistance to diffusion through the crop and soil surfaces is represented by the bulk surface resistance r c .This is not only a physiological parameter but also has an aerodynamic component (Alves et al., 1998).It is not easy to estimate r c for different climatic and crop water conditions as it is influenced by solar radiation, temperature, vapor pressure deficit and soil water content (Pereira et al., 1999;Alves et al., 1998;Huntingford, 1995;Kim and Verma, 1991;Noilham and Planton, 1989;Jarvis, 1976;Perrier, 1975).Even so, a simple method for modeling this resistance may yield a better estimation when the PM equation is applyed over both short and tall crops (Alves and Pereira, 2000;Pereira et al., 1999;Kelliher et al., 1995;Rana et al., 1997Rana et al., , 1994) ) and over other type of vegetation (Sommer et al., 2002;Monteith, 1995;Linacre, 1993).It could also be useful to incorporate the effects of the resistance due to vegetation into climatic and hydrological models (Crawford et al., 2000;Dickinson et al., 1991).The Jarvis's multiplicative model (1976) is one of the approaches to estimate bulk canopy resistance.This model relates r c , as computed by inverting the PM equation, and several climatic variables.However, this procedure only includes the physiological component of r c , and does not consider the aerodynamic component (Alves and Pereira, 2000).Todorovic (1999) has recently developed a new model to compute a variable r c that does not require any calibration.When it is applied to estimate hourly ET o with the PM equation, results exhibited a better adjustment to measured ET o than using a fixed r c value (Lecina et al., 2003;Todorovic, 1999).Katerji and Perrier (1983) proposed a linear model in which r c depends on climatic variables and aerodynamic resistance.After a previous calibration, this model has been tested and has yielded good results for practical purposes for a limited range of Bowen ratio values (Alves and Pereira, 2000;Rana and Katerji, 2000;Pereira et al., 1999;Rana et al., 1994).
However, the linear relationship represented by the Katerji and Perrier (1983) approach contains two parameters that are actually dependent on the Bowen ratio ($) value: a factor that it is not readily available.Over short periods of time it is expected to find only limited variations in the values of these two parameters (Alves and Pereira, 2000).In an attempt to avoid this dependence on $, we explored simple parametric models (based on the Katerji and Perrier approach) that relate canopy resistance to available meteorological data, in order to find the best approach involving global empirical parameters for any range of $ values.The next objective was to assess the behaviour of the PM method when a variable r c was estimated according to the previous approaches, and to evaluate whether the use of these variable rather than fixed r c values would improve the hourly ET o estimates obtained by the PM equation.Experimental measurements were carried out in two different regions: the Ebro and Guadalquivir valleys (Spain), which contain around 42 % of Spain's total irrigated land.
They correspond to semiarid conditions and exhibit high evaporative demand, particularly in summer.
Evapotranspiration estimates were compared with measured values using a weighing lysimeter (Ebro River valley) and an eddy covariance system (Guadalquivir River valley).

THEORETICAL CONSIDERATIONS
The Penman-Monteith combination model represents a basic general description of the evaporative process from a vegetative surface.It is expressed as (Allen et al., 1998): where 8E is the latent heat flux density (W m -2 ), R n and G are respectively the net radiation and soil heat flux (W m -2 ), ) is the saturation vapor pressure slope (Pa ºC -1 ), D a is the mean air density at constant pressure (kg m -3 ), c p is the specific heat of moist air (J kg -1 ºC -1 ), e * and e are respectively the saturation and actual vapor pressure of the air (Pa), ( is the psychrometric constant (Pa ºC -1 ), r a is the aerodynamic resistance (s m -1 ), and r c is the canopy or bulk surface resistance (s m -1 ).r a is the resistance to the turbulent transfer of vapor between the source and the reference level (Fig. 1).The source of vapor is at height d+z oh , which is considered the effective crop surface (Allen et al., 1998) where d (m) is the zero-plane displacement and z oh (m) is the roughness length for heat.r c is the bulk surface resistance of the entire vegetation canopy considered as a 'big leaf' (taken as the parallel sum of the stomatal resistances) or simply the canopy resistance.This single-layer or 'big leaf' model (Allen et al., 1998;Monteith and Unsworth, 1990) assumes that the latent heat lost by a crop is controlled by the bulk surface resistance, which represents the resistance of the whole canopy to the diffusion of water vapor from leaves to the atmosphere as a result of stomatal regulation.It is influenced by climatological and agronomical variables such as the structure of the canopy.Evaporation is initially governed by the difference between the saturated vapor pressure within the stomata and the vapor pressure outside them.Furthermore, the process depends on the opening of the stomata by means of the stomatal resistance of a single leaf r l .For a vegetated surface, r c is the combined resistance of all the leaves and of the soil surface, and the resistance to vapor transfer inside the canopy from these surfaces up to the 'big leaf' (Fig. 1).Thus, it can be considered that r c has an indirectly aerodynamic component (Alves et al., 1998), although it is commonly assumed that r c mainly represents a stomatal response.The resistance of the soil surface depends on the specific soil moisture conditions, while for an open water surface this resistance is zero (Berkowicz and Prahm, 1982).
The model proposed by Jarvis (1976) suggested that the environmental factors governing the stomatal resistance r l could be treated individually, by considering the stomatal conductance of a single leaf as a product of several functions.Each normalized function depends on one variable: solar radiation, temperature, vapor pressure deficit, leaf area index or soil moisture (Allen et al., 1996).This multiplicative model only includes the physiological component of r c but not the aerodynamic component, and each of the empirical functions contains one or more of the constants which must be determined previously (Kelliher et al., 1995;Monteith, 1995).Finally, it must be added that the model assumption that meteorological variables operate independently remains open to question.
The PM equation contains both a radiative and an aerodynamic term.Over a very large homogeneous, moist surface and under steady conditions, e tends to the saturation value e * and r c << r a .Consequently, the first term on the right or radiative term in Eq. ( 1) with r c .0, which is often described as the 'equilibrium' evaporation [ ] γ λ (Pereira et al., 1999;Monteith and Unsworth, 1990), may be considered as the lower limit of evaporation from moist surfaces.The aerodynamic term or second term on the right in Eq. ( 1), is therefore a measure of the departure from equilibrium in the atmosphere.The atmospheric boundary layer is hardly ever uniform and tends to maintain a humidity deficit, even over the oceans.Thus, equilibrium is rarely observed over a wet surface and the success of the PM model for estimating evapotranspiration may depend on the accurate modeling of average surface resistance r c .
Taking the equilibrium evaporation as a common factor, Eq. ( 1) can be written in the form Parameter r * is related to the isothermal resistance first introduced by Monteith (1965) and represents the surface resistance for equilibrium evaporation, because in Eq. (4) 8E = 8E e for r c = r * (Pereira et al., 1999;Rana et al., 1997;Huntingford, 1995).The value of r * mainly depends on climatic characteristics, although R n and G are also influenced by the characteristics of the vegetative surface.
The term r * can be called the climatic resistance for the surface.
Equation (4) depends on time, but in practice meteorological variables are measured continuously and averaged on a half-hourly or hourly basis, so the model can still be applied.The fraction on the right-hand side of Eq. ( 4) is a dimensionless quantity that can be considered as a cropclimatological coefficient.Perrier et al. (1980) and Katerji and Perrier (1983) showed that a link existed between r c /r a and r * /r a which depends on phenological state and soil water status.They proposed a linear relationship between these variables based on a dimensional analysis, while Rana et al. (1997Rana et al. ( , 1994) ) presented experimental results for reference crops such as grass and alfalfa in a Mediterranean climate.
The approach proposed by Katerji and Perrier (1983) is a linear link with the form . Parameters a and b can be empirically calibrated but, as pointed out by Alves and Pereira (2000), the model is constrained by the fact that they actually depend on the temporary value of the Bowen ratio $.Therefore, the empirical calibration should actually be made on $; the only factor that is not readily available.For practical purposes, this needs to be measured or estimated, though for well watered crops and over short periods of time $ is not expected to exhibit great variations.To avoid this dependence on $, and taking the Katerji and Perrier approach as a base, it is possible to explore any linear model that relates canopy resistance r c to r * .Our aim was to find the best approach for making estimations based on global empirical parameters, if they exist, that are valid for any range of $ values.These parameters will therefore be representative of the crop for any climatic conditions.With r c calibrated empirically and locally as a function of r * , only standard meteorological variables are required to estimate the bulk surface resistance.

Site description and data
This study was conducted at two locations that are representative of the central areas of the Ebro and Guadalquivir river valleys (which contain around 42% of Spain's total irrigated land) and lie in the provinces of Zaragoza and Córdoba (Figure 2).In the case of the Ebro site, the study was conducted at an experimental farm (225 m asl, 41º 43' N, 0º 49' W) located on the terraces of the river Gállego, in Zaragoza province, and about 8 km north of the confluence with the river Ebro.
Average annual precipitation is about 330 mm, and is mostly recorded in spring and autumn, though storm precipitation is also relatively frequent in summer.Average annual temperature is about 15 ºC.
The area is one of the windiest in Spain.Measurements were taken over a 1.2 ha plot with uniform grass cover (Festuca arundinacea Moench.).The plot was regularly irrigated and cut throughout the year so that its conditions resembled those of the reference standard as closely as possible A weighing lysimeter, 1.7 m in depth and covering a 6.3 m 2 effective surface area, was located at the centre of the plot.A load cell connected to a datalogger (CR500, Campbell Sci.Inc.) recorded lysimeter mass losses at 0.5 s intervals.These data were used to derive hourly evapotranspiration rates.
The combined resolutions of the load cell and the datalogger made it possible to detect mass losses of about 0.3 kg (0.04 mm water depth).Measurements were taken from March to October 1999 and from March to September 2000.Only days without incidences (irrigation, rainfall, lysimeter drainage and grass cutting), when measured grass height was between 0.10 and 0.15 m, were used for analyses.
An automatic weather station (CR10, Campbell Sci.Inc.) close to the lysimeter, recorded hourly averages for air temperature and relative humidity (HMP35D, Vaisala), and net radiation (Q-7, REBS) at a height of 1.5 m.Wind speed and direction were measured at a height of 2.0 m (switching anemometer A100R and wind vane W200P, Vector Instr.).The soil heat flux was measured using two plates (HFT1, REBS) buried at a depth of 8 cm and using an averaging soil temperature probe placed between 2 and 6 cm in depth (Allen et al., 1996).
In the Guadalquivir valley, the study was conducted at an experimental farm (70 m asl, 37º 51' N, 4º 51' W) located on the terraces of the Guadalquivir river, near Córdoba.Average annual precipitation is about 600 mm and is mainly recorded in winter, spring and autumn, with hardly any rainfall in summer.Average annual temperature is about 17 ºC.Advective conditions during summer are more frequent than in Zaragoza, but the area is significantly less windy than the middle section of the Ebro river valley.Measurements were taken over a 1.3 ha plot with uniform grass cover, regularly irrigated and cut all year round.The measurement periods were from July to October 1997 and from July to August 1998.Only days on which measured grass height was between 0.10 and 0.15 m were used for analyses.
An eddy covariance system (Campbell Sci.Inc.) was located at the centre of the plot to measure evapotranspiration.The sensors used included a krypton hygrometer (model KH20), a singleaxis sonic anemometer (model CA27), and two fine-wire thermocouples (models 127 and TCBR-3), which were attached to the two previously mentioned sensors.The hygrometer and the sonic anemometer were installed at a height of 0.32 m, in order to maintain a height:fetch ratio of more than 100.They were oriented towards the west, which was the predominant wind direction, and were placed 10 cm apart in direction north-south.Measurements of fluctuations in water vapor density, vertical wind speed and air temperature were recorded every 0.1 s and averaged every 10 minutes.
These readings were used to obtain hourly measured latent heat flux values as described by Villalobos (1997), which were then transformed to hourly ET o rates (Allen et al., 1998).An automatic weather station (CR10, Campbell Sci.Inc.), located close to the eddy covariance system, recorded standard meteorological variables in the same way as in the Ebro river valley.

Evapotranspiration estimation a. Constant canopy resistance: Penman-Monteith equation
The PM equation is given by Eq. ( 1).The important assumption behind this model is that, despite the variations within the canopy itself, the behaviour of all the stomata considered together is comparable with that of a single 'big leaf' from which heat and vapor escape.This 'big leaf' is located at a height d+z 0h , where d is the zero-plane displacement height and z 0h is the roughness length for heat (Fig. 1).In this paper, measured rather than estimated R n and G values were used.This was done in order to prevent potential uncertainties derived from the estimation of these two variables affecting the analysis involving the use of fixed and variable r c values.Units and computations (except R n and G) in Eq. (1) followed Allen et al. (1998), r c was considered constant and equal to 70 s m -1 and grass height was set to 0.12 m.Throughout the study period, Eq. ( 1) was applied to obtain hourly ET o estimates using hourly averages of meteorological variables.
For the following two models, the data set available for each location was divided into two groups: a calibration data set and a validation data set.Available days were organised by date and one of three days was selected for calibration, while the other two were selected for validation.

b. Todorovic method
This model is a new approach for modeling canopy resistance: it is based on the 'big leaf' approach but incorporates a variable r c .The model's input requires only standard meteorological data as in the PM combination approach.A full discussion of the theory and assumptions of the model can be found in Todorovic (1999).Here, we simply present the equations used to obtain the variable r c values.Todorovic (1999) considers a climatological resistance r i defined by: as used in the literature (Monteith and Unsworth, 1990;Monteith, 1965).This resistance, as seen by Then, Todorovic (1999) uses r i and r a , defined according to Allen et al. (1998), to set the following 2 nd degree equation a function of X = r c /r i , where the parameters are defined by Solving for Eq. ( 7), which has only one positive solution, the variable canopy resistance is obtained as r c = X r i .Eqs. ( 7) and (8) were applied to the validation data set to obtain hourly estimates of canopy resistance (r c,TD ) using the hourly averages of meteorological variables.A fixed value of r c =200 s m -1 was considered for night hours.These variable r c values were then used, assuming a grass height of 0.12 m, to obtain hourly ET o estimates by directly applying the PM equation (Eq.( 1)).
Preliminary testing of this approach has provided convincing results obtained on an hourly basis (Todorovic, 1999).

c. Variable canopy resistance: semiempirical approach
Based on the experimental relationship found by Perrier et al. (1980) between the bulk canopy, aerodynamic and climatological resistance r * (Eq.( 3)), several authors proposed a model in which r c /r a is a linear function of r * /r a .This model has been applied to wheat (Perrier et al., 1980), grass and alfalfa (Steduto et al., 2003;Rana et al., 1994;Katerji and Perrier, 1983), tomato (Katerji et al., 1988) and rice (Peterschmitt and Perrier, 1991).The coefficients of those linear relationships depend on the type of crop, its phenological state or soil water status (Rana et al., 1997).
In order to analyse this simple model, we propose exploring a functional relationship of the form for the reference crop, where a and b are global parameters that are to be determined empirically.
The most commonly used formulations for estimating aerodynamic resistance r a , consider that apparent sources of heat and vapor within the canopy are at a lower level than the apparent sink of momentum (Monteith and Unsworth, 1990).In the PM equation the big leaf is implicitly located at the height d + z oh , where z oh (m) is the roughness length for sensible heat transfer.When the leaves at the top of the canopy, between d + z oh and crop height h c , are the most important source of vapor flux, the use of the d + z oh level as the evaporative surface can lead to an overestimation of r a and to negative values for bulk surface resistance r c when it is computed by inverting the PM equation.To avoid this, r a can be computed between the top of the canopy h c and the reference height in the atmosphere as (Alves et al., 1998;Perrier, 1975) where z m and z h are the wind and air temperature measurement heights, respectively, h c is the crop height, k is the dimensionless von Karman constant, and u zm is the wind speed measured at height z m .
The term h c -d substitutes the roughness length for heat transfer used to compute r a according to Allen et al. (1998).
So, for both the calibration and the validation data set, the experimental or measured hourly values of bulk surface resistance r c (s m -1 ) were obtained by inverting the PM equation (top-down approach) as where D is the vapor pressure deficit (Pa) (D = e * -e in Eq. ( 1)), R n -G is the available energy for the entire surface (W m -2 ) and 8E is the latent heat flux (W m -2 ) measured by a lysimeter (Zaragoza) or by an eddy covariance system (Córdoba).
The hourly values measured for R n -G, D and 8E were used to compute hourly r c values in Eq.
(11), aerodynamic resistances in Eq. ( 10) and climatic resistances in Eq. ( 3).Using only the calibration data sets for Zaragoza and Córdoba, several simple relationships between r c /r a and r * /r a for any range of $ values were analyzed for the reference crop.The calibrated values of parameters a and b in Eq.
(9) allowed us to obtain, using only the validation data set, an estimated variable canopy resistance r c,est .This was then used to estimate hourly evapotranspiration values (8E est ) from Eq. (1) or Eq.(4).

Statistical methods
In the evaluation of the different relationships represented by Eq. ( 9), the ability to predict r c has been compared using the coefficient of determination R 2 as the measure of the goodness of fit, that is, of the performance of the model; R 2 is a measure of the total variance accounted for by the model.
Parameters a and b in Eq. ( 9) are then obtained for the best fit using the calibration data set.
A complication arises when the model that maximizes R 2 for r c may not maximize it when predicting evapotranspiration ET.Once the best parameterization of r c had been chosen and the optimum coefficient values in Eq. ( 9) had been found, comparisons between measured and estimated hourly ET values were carried out using the three models above.To test which of the resulting models was significantly better than the others, a simple linear regression between the predicted and the observed values was made, based on the criterion of trying to maximize R 2 .Nevertheless, in order to make a better quantitative comparison between the models, predicted results can be evaluated with reference to several statistics (Alexandris and Kerkides, 2003;Lee and Singh, 1998;Verhoef et al., 1997;Mayer and Butler, 1993;Willmot, 1981Willmot, , 1982)).
The first statistic is the bias or mean deviation error, The second is the root mean square error, ).In the above expressions, P i and O i represent the predicted (estimated) and observed (measured) values, n is the total number of data and O is the mean of observed data.RMSE provides a measure of the total difference between the predicted and observed values.The closer RMSE is to zero, the better the prediction.
The statistic we will use as the model selection criterion is the modeling efficiency (Lee and Singh, 1998) defined as , which is similar to the index of agreement (Willmot, 1982).Modeling efficiency is a dimensionless statistic that measures the fraction of the variance of the observed values which is explained by the model, so it provides a good measure of the model's performance.The best model is the one with a value of EF closest to 1 and with the lowest RMSE.

RESULTS AND DISCUSSION
The values of the statistics derived from air temperature and wind speed records for Zaragoza and Córdoba during the measurement periods are presented in Table 1.They are shown for descriptive purposes because, as a result of the different measurement periods, it was not possible to compare them directly.It should be noted that on 9% of the days at Zaragoza the average daily wind speed was greater than 4 m s -1 , whereas this did not occur at Córdoba.As previously mentioned, Zaragoza is located within one of the windiest areas in Spain.
The hourly values of r a obtained with Eq. (10) were lower than those obtained using the classical expression in the PM equation for the reference grass (Allen et al., 1998).This confirmed results obtained by other authors over both short and tall vegetation (Hall, 2002;Alves et al., 1998).
For these authors, aerodynamic resistance using the classical logarithmic profile equations for momentum exchange results in estimates that are systematically higher than values derived from direct measurements.The differences in the estimations of r a for dense or sparse canopies or even for grass, are, as reviewed by Verhoef (1995), the result of the problem of the parameterization of an excess resistance.Figure 3a shows the daily evolution of r a for a typical day at Zaragoza.As described elsewhere in the literature, bulk surface resistance is small and tends to remain relatively constant on average from 8h to 14h.After this, r c tends to increase gradually in the afternoon reflecting the daily course of the environmental variables that influence it (Fig. 3b).Negative values have been obtained for r c independently of the time of day, but these have mainly been registered early in the morning and at night, when the available energy (R n -G) tends to be small and with negative values and when the vapor pressure deficit (D) is also small.In well irrigated crops, factors external to the crops control the latent heat flux to a greater extent than purely physiological factors.
When a constant r c is considered, Table 2 shows the results of comparisons between the estimated and measured hourly values for evapotranspiration at the two locations, based on the validation data set for each measurement period.ET PM represents the estimated values obtained using the Penman-Monteith equation (Eq.( 1)) with a fixed value of r c = 70 s m -1 .The values for modeling efficiency (EF) suggest good agreement for the method at both locations, with a relatively small scatter of data but with a slope lower than one (Table 2).There is a tendency for the PM method to overestimate low values of measured ET o and to underestimate high ET o values (Fig. 4).This behaviour has already been reported at other Mediterranean locations (Steduto et al., 2003(Steduto et al., , 1996)).
The underestimation of high ET o values was smaller in Córdoba than in Zaragoza.This might have been due to the use of an eddy covariance system for measuring latent heat flux.This system, depending on the horizontal sensor separation and on measurement height, may underestimate evapotranspiration (Villalobos, 1997).Another reason for the differences in results between the two locations may be their different wind conditions.In Zaragoza, the agreement between lysimeter and ET PM values was found to decrease as wind speed increased.Under conditions of high evaporative demand (sunny days in summer), evapotranspiration rates are expected to further increase under windy conditions.In such situations, the PM method with constant r c is relatively robust, but seems to provide an insufficient representation of latent heat flux to the atmosphere.This method led to an underestimation of high ET o values in both zones under semiarid conditions (Steduto et al., 2003(Steduto et al., , 1996;;Ventura, 1999;Rana et al., 1994).
Since one of our aims was to analyse the performance of the PM equation with respect to different ways of considering canopy resistance, we have analysed the mechanistic approach represented by Eqs. ( 7) and ( 8) (Todorovic, 1999).The variable canopy resistance estimated with this method r c,TD was then applied to the PM equation (Eq.( 1)) and hourly estimates of evapotranspiration ET TD were obtained.The values of r c,TD in the Todorovic method only depend on weather variables and do not require calibration, they can therefore be directly applied to the validation data set for each measurement period at both sites.For the diurnal period, the values of r c,TD were, on average, 10% lower than the experimental values r c but, as shown in Fig. 5a, with a high degree of variability in the case of Zaragoza.
The drawback of the Todorovic method is that it can-not be generally applied for the nocturnal period when both vapor pressure deficit D and available energy R n -G are low or close to zero.In such cases both r i (Eq.( 5)) and r * have negative or non-defined values, so the solutions for X in Eq. ( 7) and r c are not defined.This model can only be applied when r i is positive, and therefore for cases when r i #0 (usually at night) a fixed value of r c,TD = 200 s m -1 must be considered (Allen et al., 1998).When the variable canopy resistance r c,TD was used in Eq. ( 1), estimated hourly values ET TD for evapotranspiration were obtained.

Results for the comparison between estimated ET TD and measured ET o values are presented in
Table 2 and Fig. 5b.The statistics showed that the relative RMSE(%) of estimates for ET TD was about 3% lower than that for ET PM , and that the slope of the linear regression with measured values for ET TD was closer to 1 than for ET PM .Furthermore, the relative mean deviation error (MBE(%)) indicated that ET TD tends to globally underestimate measured values less than ET PM does, as shown in Fig. 5b with respect to high evaporative demand.Even in Córdoba ET TD tends to globally overestimate measured ET o (negative MBE, Table 2).The values for the modeling efficiency taken toghether with the above results indicate that estimation of evapotranspiration using the Todorovic method (variable r c ) shows better performance than the PM method (fixed r c ).Thus, if there is a need to incorporate the effect of a variable r c , the Todorovic method may be useful at least under the semiarid and windy conditions such as those of the Ebro river valley.Results cited in a previous work (Lecina et al., 2003) showed that daily ET o estimates could be obtained with sufficient accuracy using the PM method.
The second way to estimate a variable r c is through the semi-empirical approach given by Eq. ( 9) which relates r c /r a to r * /r a .Results found by others authors (Steduto et al., 2003;Alves and Pereira, 2000;Rana et al., 1997) seem to show a linear relationship between these variables on an hourly basis.
These results were generally obtained under certain limitations corresponding to the diurnal period, or were obtained for a limited range of Bowen ratio $ values.Alves and Pereira (2000) found such results for $ values in the interval [-0.3, 0.3], corresponding to situations when crop evapotranspiration was maximum and when available energy R n -G had values corresponding to the middle of the day.
We carried out an analysis of the empirical relationship provided by Eq. ( 9) for our semiarid regions with the aim of amplifying this data interval and thereby covering the largest possible range of situations throughout the day, with the following results.Working with the complete set of calibration data for the two measurement zones, we found that a straight line did not represent the best relationship between r c /r a and r * /r a .This result was obtained independently of the interval of $ values considered.When latent and sensible heat fluxes are measured over grass on an hourly basis, 85% of the actual values for $ lie in the interval [-0.5, 0.5] throughout the day (Perez et al., 1999).As seen in Fig. 6a, when considering cases in which the $ values are between [-0.5, 0.5], the functional form of the dependence of r c /r a on r * /r a departs from linearity.It seems that r c /r a tends to some saturation value when r * /r a increases.As in this case we are including data corresponding to a longer diurnal period (07h to 19h) in our analisys, this result extends that reported by Alves and Pereira (2000).
The main drawback is that for the nocturnal period when vapor pressure deficit D and available energy R n -G are low, as at sunrise and sunset, the canopy resistance r c may register very high positive or very low negative values and the climatic resistance r * may register negative or nondefined values.For these situations and for comparison with the Todorovic model, when r * #0 we have considered a fixed value of 200 s m -1 for r c .Even so, about "5% is a typical error associated with the measurement of net radiation, while the resolution of the weighing lysimeter is 0.04 mm in the measurement of evapotranspiration.Therefore, the next step was to use in the analysis only the calibration data when R n -G $ 30 W m -2 and when measured ET o $ 0.04 mm h -1 , independently of $ values.In consequence this constraint effectively limited the analysis to the diurnal period going from 07h to 19h (Fig. 6b), but with $ values ranging from -0.8 to 2.0.
Table 3 shows the values of the empirical coefficients locally calibrated for each analysed relationship.Results show that the best-fit relationship corresponded to a dependence of r c /r a on the square root of r * /r a and not to a linear relationship, as the coefficient of determination R 2 indicates (Table 3).For the linear dependence (r c /r a ) est,2 , it can be observed that the values of the empirical coefficients for our semiarid zones (a 2 = 1.1, b 2 = 0.19), are similar to those obtained by Steduto et al. (2003) for grass in southern Italy (a 5 = 0.918, b 5 = 0.18).This finding also suggests that it may be possible to extrapolate the empirical coefficients of the proposed method (r c /r a ) est,1 (Table 3) to other climatic regions.
The statistics used for the comparison between estimated and experimental values of r c /r a , reported in Table 3 for the global set of calibration days at Zaragoza and Córdoba, indicate that for method (r c /r a ) est,1 relative RMSE is 13% lower and the modeling efficiency is higher than when using the estimating method (r c /r a ) est,2 .Although the Todorovic method does not need any previous calibration, the statistics show that r c,TD generally underestimates values of canopy resistance obtained by inverting the PM equation (MBE = 37.3%).This is mainly true for values of canopy resistance that exceed 50 s m -1 .Once empirically calibrated for our semiarid regions, the method (r c /r a ) est,1 performed better when estimating canopy resistance.
The variable r c estimated through the proposed methods (r c /r a ) est,1 and (r c /r a ) est,2 was used in Eq. ( 1) to estimate hourly values of evapotranspiration ET est,1 and ET est,2 , respectively.Table 4 shows the results of the comparison between the four estimating methods and the measured ET o values for the global set of validation days at Zaragoza and Córdoba.As seen, the relative RMSE of estimates for ET est,1 is about 2% lower than for ET PM , and the slope of the linear regression with measured values is closer to 1 for ET est,1 than for ET PM (Fig. 7).Overall, these results indicate that the proposed method ET est,1 using a variable r c tends to perform better than the PM method with constant r c .Although the improvement in the evapotranspiration estimation found in this work seems to be limited, the empirical approach involving r c /r a as a function of the square root of r * /r a , can be considered as an alternative to other approaches for modeling r c .It may also be useful to incorporate the effect of variable canopy resistance into climatic and hydrological models.

CONCLUSIONS
The results presented in this work show that hourly ET o estimates made with the Penman-Monteith equation and using a constant r c value of 70 s m -1 , tend to underestimate measured ET o during the summer period (when evaporative demand is high) and to overestimate it in winter (when ET o is low), under the semiarid conditions of both the Ebro and Guadalquivir river valleys.This may have been a consequence of considering canopy resistance as constant during the day.
The issue is that Eq. ( 11), which is used to calculate r c in a 'top down' approach, shows that r c depends on environmental variables.Therefore, an approach that models r c as a function of climatic conditions could be regarded as an alternative way of trying to overcome the limitations of the PM method.The Katerji and Perrier (1983) model proposed an r c variable as a function of climatic resistance r * .They showed experimentally that there is a linear link with two empirical coefficients between r c /r a and r * /r a , and that this depends on climatic conditions.
The results presented here for semiarid regions with a Mediterranean climate show that estimating r c as a function of the square root of r * on an hourly basis, constitutes a better empirical approach than the linear option.In the case of the linear relationship, the values of the empirical coefficients calibrated for our semiarid regions were similar to those obtained by Steduto et al. (2003) for grass in southern Italy.This result suggests that in the proposed approach (r c /r a ) est,1 , it may also be possible to extrapolate the locally calibrated empirical coefficients to other climatic regions.
When the different methods for estimating a variable r c are used in the PM equation, the results show that estimates of hourly evapotranspiration ET est,1 derived from the global validation data set tend to perform relatively better than ET PM with constant r c .The only apparent drawback is perhaps the fact that the minor improvement observed hardly justifies the effort to undertake the local calibration required.When the Todorovic mechanistic method for estimating r c is used and applied in the PM equation, the results obtained show that, in comparison with the other methods, ET TD also performs very well when it is used to estimate hourly evapotranspiration for our global data set.3.    (1) with: a constant r c = 70 s m -1 (ET PM ); a variable resistance r c,TD which were estimated by the   ) -hand side of the expression, the numerator can be expressed in the same way as the denominator, by defining a climatic resistance given

2,
which is considered as a better indicator of model performance than the correlation statistics.They can be expressed as percentages of the mean observed FIGURE CAPTIONS

FIGURE 3 .
FIGURE 3.a) Daily evolution of the aerodynamic (r a ) and canopy (r c ) resistances of grass for a specific day (March 5, 1999, Zaragoza).b) Experimental hourly averaged values for bulk surface resistance r c for the calibration data set for Zaragoza.These were obtained by inverting the PM equation using values of latent heat flux measured by lysimeter (Eq.(11)).Hourly averaged values for climatic resistance r*, available energy R n -G and vapor pressure deficit D are also shown.

FIGURE 4 .
FIGURE 4.-Hourly values of evapotranspiration estimated using the PM equation (ET PM ) with a fixed r c value of 70 s m -1 for the validation data sets of the measurement periods at each site (1:1 line is shown for comparison).Measured values (ET o ) were obtained with a weighing lysimeter (Zaragoza)and with an eddy covariance system (Córdoba).

FIGURE 5 .
FIGURE 5.a) Comparison between variable canopy resistances r c,TD estimated by the Todorovic method and the experimental values r c (diurnal period, Zaragoza).b) Hourly values of evapotranspiration ET TD estimated with the variable resistance r c,TD , compared with measured ET o values for the whole validation period at Zaragoza.

FIGURE 6 .-
FIGURE 6.-Variation of experimental values of r c /r a vs r * /r a on an hourly basis for the global calibration data set (Zaragoza and Córdoba) for: a) data corresponding to the diurnal period with R n -G $ 0 and $ values in the interval [-0.5, 0.5], b) data corresponding to the diurnal period with R n -G $ 30 (W m -2 ) independent of the $ values.The estimated values (r c /r a ) est,i respectively correspond to estimating models 1 and 2 in Table3.

FIGURE 7 .-
FIGURE 7.-Hourly values of estimated evapotranspiration ET PM (obtained from the PM equation with a fixed r c value of 70 s m -1 ) and ET est,1 (obtained from Eq. (1) with the variable canopy resistance obtained from (r c /r a ) est,1 ), compared with measured values ET o for the global validation data set (Zaragoza and Córdoba).

Table 1 .
TABLES Representative values of the meteorological conditions during the measurement period at Zaragoza and Córdoba.T x and T n are maximum and minimum air temperature, respectively, and u is average wind speed.

Table 2 .
Simple linear regression (ET est = c o + c 1 ET o ) and statistics of the comparison between estimated (ET est ) and measured (ET o ) hourly values of evapotranspiration (mm h -1 ) at the two locations for the respective validation data sets.Estimates were obtained using the Penman-Monteith equation with a fixed r c value of 70 s m -1 (ET PM ), and with the variable canopy resistance r c,TD (ET TD ).Measured

Table 3 .
Empirical coefficients locally calibrated for two functional forms of the dependence of r c /r a on r * /r a in Eq. (9): 1) for the experimentally found best-fit relationship As well as the Todorovic method r c,TD , models 4 and 5 proposed by other authors and applied to our data have also been included for comparison purposes.The last three columns show the values of the statistics for the comparison between estimated and experimental values of r c /r a .

Table 4 .
Simple linear regression (ET est = c o + c 1 ET o ) and statistics for the comparison between estimated (ET est ) and measured (ET o ) hourly values of evapotranspiration (mm h -1 ) for the global validation data set (Zaragoza and Córdoba).Estimates of evapotranspiration were obtained from Eq.
Todorovic method (ET TD ); and a variable resistance obtained from the estimating methods (r c /r a ) est,1 and (r c /r a ) est,2 (ET est,1 and ET est,2 , respectively).Measured ET o values were obtained using a weighing lysimeter (Zaragoza) and an eddy covariance system (Córdoba).