A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration

HYDROLOGICAL PROCESSESHydrol. Process. 20, 515–532 (2006)Published online 18 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.5919

A simple parameterization of bulk canopy resistance fromclimatic variables for estimating hourly

evapotranspiration

P. J. Perez,1* S. Lecina,2 F. Castellvi,1 A. Martınez-Cob2 and F. J. Villalobos3

1 Departamento de Medio Ambiente y Ciencias del Suelo, Universidad de Lleida, Av. Rovira Roure, 191, 25198 Lleida, Spain2 Laboratorio de Agronomıa y Medio Ambiente (DGA-CSIC), Departamento de Genetica y Produccion Vegetal (EEAD), Apdo. 202, 50080

Zaragoza, Spain3 Instituto de Agricultura Sostenible (CSIC) and Departamento Agronomıa, Universidad de Cordoba, Apdo. 4084, 14080 Cordoba, Spain

Abstract:

This paper examines a model for estimating canopy resistance rc and reference evapotranspiration ETo on an hourlybasis. The experimental data refer to grass at two sites in Spain with semiarid and windy conditions in a typicalMediterranean climate. Measured hourly ETo values were obtained over grass during a 4 year period between 1997 and2000 using a weighing lysimeter (Zaragoza, northeastern Spain) and an eddy covariance system (Cordoba, southernSpain). The present model is based on the Penman–Monteith (PM) approach, but incorporates a variable canopyresistance rc as an empirical function of the square root of a climatic resistance rŁ that depends on climatic variables.Values for the variable rc were also computed according to two other approaches: with the rc variable as a straight-linefunction of rŁ (Katerji and Perrier, 1983, Agronomie 3(6): 513–521) and as a mechanistic function of weather variablesas proposed by Todorovic (1999, Journal of Irrigation and Drainage Engineering, ASCE 125(5): 235–245).

In the proposed model, the results show that rc/ra (where ra is the aerodynamic resistance) presents a dependenceon the square root of rŁ/ra, as the best approach with empirically derived global parameters. When estimating hourlyETo values, we compared the performance of the PM equation using those estimated variable rc values with the PMequation as proposed by the Food and Agriculture Organization, with a constant rc D 70 s m�1. The results confirmedthe relative robustness of the PM method with constant rc, but also revealed a tendency to underestimate the measuredvalues when ETo is high. Under the semiarid conditions of the two experimental sites, slightly better estimates of ETo

were obtained when an estimated variable rc was used. Although the improvement was limited, the best estimateswere provided by the Todorovic and the proposed methods. The proposed approach for rc as a function of the squareroot of rŁ may be considered as an alternative for modelling rc, since the results suggest that the global coefficients ofthis locally calibrated relationship might be generalized to other climatic regions. It may also be useful to incorporatethe effects of variable canopy resistances into other climatic and hydrological models. Copyright 2005 John Wiley& Sons, Ltd.

KEY WORDS variable canopy resistance; evapotranspiration; Penman–Monteith equation; grass

INTRODUCTION

Evapotranspiration is a component of the hydrological cycle whose accurate computation is needed foran appropriate management of water resources. A high degree of accuracy in the estimation of cropevapotranspiration may lead to important savings in water requirements in irrigated areas. The Food andAgriculture Organization (FAO) proposed a methodology for computing crop evapotranspiration (Doorenbosand Pruitt, 1977), based on the use of reference evapotranspiration ETo and crop coefficients Kc. Allen et al.

* Correspondence to: P. J. Perez, Departamento de Medio Ambiente y Ciencias del Suelo, ETS Ingenieria Agraria, Av. Rovira Roure, 191,25198 Lleida, Spain. E-mail: [email protected]

Received 18 December 2003Copyright 2005 John Wiley & Sons, Ltd. Accepted 18 January 2005

516 P. J. PEREZ ET AL.

(1998) redefined the concept of ETo and adopted the Penman–Monteith (PM) equation with constant canopyresistance rc to estimate reference evapotranspiration.

The PM model has been applied in different regions of the world and has provided good results incomparison with other equations (Allen et al., 1989, 1994a,b; Jensen et al., 1990; Hussein, 1999; Venturaet al., 1999; Berengena et al., 2001). However, many of these studies suggest that, in semiarid and windyareas with a high evaporative demand, there is a tendency for the PM method to underestimate the highervalues of measured ETo and to overestimate the lower ETo values. The extent of this underestimation mayrange from 2 to 18% (Steduto et al., 1996; Pereira et al., 1999; Todorovic, 1999; Ventura et al., 1999).

The accuracy of the predictions of the PM model may be constrained by such simplifying assumptions asthe ‘big leaf’ approximation with constant canopy resistance. Given the driving meteorological variables at aparticular site, estimates made with the PM equation rely on the correct modelling of the effective values oftwo parameters, i.e. aerodynamic resistance ra and bulk (surface) canopy resistance rc. The FAO has adopteda standard PM equation that can be applied worldwide, using a constant value of 70 s m�1 for rc whencalculating grass reference evapotranspiration (Smith et al., 1991; Allen et al., 1994a,b, 1998). However, thisfixed value for rc may be the cause of the previously mentioned tendency for the PM equation to underestimateETo (Rana et al., 1994). As ra can be calculated from meteorological conditions, in order to provide moreaccurate estimations of evapotranspiration using the PM equation, it may be necessary to parameterize canopyresistance 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 resistancerc. This is not only a physiological parameter, it also has an aerodynamic component (Alves et al., 1998). It isnot easy to estimate rc for different climatic and crop water conditions, as it is influenced by solar radiation,temperature, vapour pressure deficit and soil water content (Perrier, 1975; Jarvis, 1976; Noilhan and Planton,1989; Kim and Verma, 1991; Huntingford, 1995; Alves et al., 1998; Pereira et al., 1999). Even so, a simplemethod for modelling this resistance may yield a better estimation when the PM equation is applied over bothshort and tall crops (Rana et al., 1994, 1997; Kelliher et al., 1995; Pereira et al., 1999; Alves and Pereira,2000) and over other types of vegetation (Linacre, 1993; Monteith, 1995; Sommer et al., 2002). It could alsobe useful to incorporate the effects of the resistance due to vegetation into climatic and hydrological models(Dickinson et al., 1991; Crawford et al., 2000).

The Jarvis (1976) multiplicative model is one of the approaches to estimate bulk canopy resistance. Thismodel relates rc, as computed by inverting the PM equation, and several climatic variables. However, thisprocedure only includes the physiological component of rc, and does not consider the aerodynamic component(Alves and Pereira, 2000). Todorovic (1999) has recently developed a new model to compute a variable rc

that does not require any calibration. When it is applied to estimate hourly ETo with the PM equation, theresults exhibit a better adjustment to measured ETo than using a fixed rc value (Todorovic, 1999; Lecinaet al., 2003). Katerji and Perrier (1983) proposed a linear model in which rc depends on climatic variablesand aerodynamic resistance. After a prior calibration, this model has been tested and has yielded good resultsfor practical purposes for a limited range of Bowen ratio values (Rana et al., 1994; Pereira et al., 1999; Alvesand Pereira, 2000; Rana and Katerji, 2000).

However, the linear relationship represented by the Katerji and Perrier (1983) approach contains twoparameters that are actually dependent on the Bowen ratio ˇ value, a factor that it is not readily available. Overshort periods of time it is expected that only limited variations in the values of these two parameters are to befound (Alves and Pereira, 2000). In an attempt to avoid this dependence on ˇ, we explored simple parametricmodels (based on the Katerji and Perrier approach) that relate canopy resistance to available meteorologicaldata, in order to find the best approach involving global empirical parameters for any range of ˇ values. Thenext objective was to assess the behaviour of the PM method when a variable rc was estimated according tothe previous approaches, and to evaluate whether the use of these variable, rather than fixed, rc values wouldimprove the hourly ETo estimates obtained by the PM equation. Experimental measurements were carried outin two different regions, i.e. the Ebro and Guadalquivir valleys (Spain), which contain around 42% of Spain’stotal irrigated land. They correspond to semiarid conditions and exhibit high evaporative demand, particularly

Copyright 2005 John Wiley & Sons, Ltd. Hydrol. Process. 20, 515–532 (2006)

PARAMETERIZATION OF BULK CANOPY RESISTANCE 517

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 PM combination model represents a basic general description of the evaporative process from a vegetativesurface. It is expressed as (Allen et al., 1998)

�E D �Rn � G� C �a cp�eŁ � e�/ra

C �

(1 C rc

ra

) �1�

where �E �W m�2� is the latent heat flux density, Rn �W m�2� and G �W m�2� are respectively the netradiation and soil heat flux, �Pa °C�1� is the saturation vapour pressure slope, �a �kg m�3� is the meanair density at constant pressure, cp �J kg�1 °C�1� is the specific heat of moist air, eŁ (Pa) and e (Pa) arerespectively the saturation and actual vapour pressure of the air, � �Pa °C�1� is the psychrometric constant,ra �s m�1� is the aerodynamic resistance, and rc �s m�1� is the canopy or bulk surface resistance. ra is theresistance to the turbulent transfer of vapour between the source and the reference level (Figure 1). Thesource of vapour is at height d C zo,h, which is considered the effective crop surface (Allen et al., 1998)where d (m) is the zero-plane displacement and zo,h (m) is the roughness length for heat. rc is the bulksurface resistance of the entire vegetation canopy considered as a ‘big leaf’ (taken as the parallel sum of thestomatal resistances), or simply the canopy resistance.

This single-layer or ‘big leaf’ model (Monteith and Unsworth, 1990; Allen et al., 1998) assumes that thelatent heat lost by a crop is controlled by the bulk surface resistance, which represents the resistance of thewhole canopy to the diffusion of water vapour 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. Evaporationis initially governed by the difference between the saturated vapour pressure within the stomata and thevapour pressure outside them. Furthermore, the process depends on the opening of the stomata by means ofthe stomatal resistance of a single leaf rl. For a vegetated surface, rc is the combined resistance of all theleaves and of the soil surface, and the resistance to vapour transfer inside the canopy from these surfaces upto the ‘big leaf’ (Figure 1). Thus, it can be considered that rc has an indirect aerodynamic component (Alveset al., 1998), although it is commonly assumed that rc mainly represents a stomatal response. The resistance

Figure 1. The concept of surface resistance in the PM or ‘big leaf’ method



of the soil surface depends on the specific soil moisture conditions, whereas for an open water surface thisresistance is zero (Berkowicz and Prahm, 1982).

The model proposed by Jarvis (1976) suggested that the environmental factors governing the stomatalresistance rl could be treated individually, by considering the stomatal conductance of a single leaf as aproduct of several functions. Each normalized function depends on one variable: solar radiation, temperature,vapour pressure deficit, leaf area index or soil moisture (Allen et al., 1996). This multiplicative model onlyincludes the physiological component of rc, not the aerodynamic component, and each of the empiricalfunctions contains one or more of the constants that must be determined in advance (Kelliher et al., 1995;Monteith, 1995). Finally, it must be added that the model assumption that meteorological variables operateindependently 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 eŁ and rc − ra. Consequently,the first term on the right, or radiative term, in Equation (1) with rc ³ 0, which is often described as the‘equilibrium’ evaporation �Ee D [/� C ��]�Rn � G� (Monteith and Unsworth, 1990; Pereira et al., 1999),may be considered as the lower limit of evaporation from moist surfaces. The second term, or aerodynamicterm, on the right in Equation (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 overthe oceans. Thus, equilibrium is rarely observed over a wet surface, and the success of the PM model forestimating evapotranspiration may depend on the accurate modelling of average surface resistance rc.

Taking the equilibrium evaporation as a common factor, Equation (1) can be written in the form

�E D�Rn � G�

[1 C �acp �eŁ � e�/ra

�Rn � G�

]

� C �� C �rc

ra

and after rearranging

�E D

C ��Rn � G�

1 C �acp�eŁ � e�

�Rn � G�ra

1 C �

C �

rc

ra

�2�

On the right-hand side of the expression, the numerator can be expressed in the same way as thedenominator, by defining a climatic resistance given as

rŁ D C �

�

�acp�eŁ � e�

�Rn � G��3�

so that, through substitution in Equation (2), we obtain

�E D

C ��Rn � G�

1 C �

C �

rŁ

ra

1 C �

C �

rc

ra

�4�

Parameter rŁ is related to the isothermal resistance first introduced by Monteith (1965) and represents thesurface resistance for equilibrium evaporation, because in Equation (4) �E D �Ee for rc D rŁ (Huntingford,1995; Rana et al., 1997; Pereira et al., 1999). The value of rŁ mainly depends on climatic characteristics,although Rn and G are also influenced by the characteristics of the vegetative surface. The term rŁ can becalled the climatic resistance for the surface.

Equation (4) depends on time, but in practice the meteorological variables are measured continuously andaveraged on a half-hourly or hourly basis, so the model can still be applied. The fraction on the right-hand



side of Equation (4) is a dimensionless quantity that can be considered as a crop–climatological coefficient.Perrier and co-workers (Perrier et al., 1980; Katerji and Perrier, 1983) showed that a link existed betweenrc/ra and rŁ/ra that depends on phenological state and soil water status. They proposed a linear relationshipbetween these variables based on a dimensional analysis, and Rana et al. (1994, 1997) presented experimentalresults 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 rc/ra D a C b�rŁ/ra�.Parameters a and b can be empirically calibrated, but, as pointed out by Alves and Pereira (2000), the model isconstrained by the fact that they actually depend on the temporary value of the Bowen ratio ˇ. Therefore, theempirical calibration should actually be made on ˇ, the only factor that is not readily available. For practicalpurposes, this needs to be measured or estimated, though for well-watered crops and over short periods oftime ˇ is not expected to exhibit great variations. To avoid this dependence on ˇ, and taking the Katerjiand Perrier approach as a base, it is possible to explore any linear model that relates canopy resistance rc torŁ. Our aim was to find the best approach for making estimations based on global empirical parameters, ifthey exist, that are valid for any range of ˇ values. These parameters will, therefore, be representative of thecrop for any climatic conditions. With rc calibrated empirically and locally as a function of rŁ, only standardmeteorological variables are required to estimate the bulk surface resistance.

MATERIALS AND METHODS

Site description and data

This study was conducted at two locations that are representative of the central areas of the Ebro andGuadalquivir river valleys (which contain around 42% of Spain’s total irrigated land) and lie in the provincesof Zaragoza and Cordoba (Figure 2). In the case of the Ebro site, the study was conducted at an experimentalfarm (225 m a.s.l., 41°430N, 0°490W) located on the terraces of the River Gallego, in Zaragoza province, andabout 8 km north of the confluence with the River Ebro. Average annual precipitation is about 330 mm, and

Figure 2. Locations of the study areas



is mostly recorded in spring and autumn, though storm precipitation is also relatively frequent in summer.Average annual temperature is about 15 °C. The area is one of the windiest in Spain. Measurements were takenover a 1Ð2 ha plot with uniform grass cover (Festuca arundinacea Moench.). The plot was regularly irrigatedand cut throughout the year, so that its conditions resembled those of the reference standard as closely aspossible.

A weighing lysimeter, 1Ð7 m in depth and covering a 6Ð3 m2 effective surface area, was located at thecentre of the plot. A load cell connected to a datalogger (CR500, Campbell Sci. Inc.) recorded lysimetermass losses at 0Ð5 s intervals. These data were used to derive hourly evapotranspiration rates. The combinedresolutions of the load cell and the datalogger made it possible to detect mass losses of about 0Ð3 kg (0Ð04 mmwater 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 grassheight was between 0Ð10 and 0Ð15 m, were used for analyses.

An automatic weather station (CR10, Campbell Sci. Inc.), close to the lysimeter, recorded hourly averagesfor air temperature and relative humidity (HMP35D, Vaisala) and net radiation (Q-7, REBS) at a height of1Ð5 m. Wind speed and direction were measured at a height of 2Ð0 m (switching anemometer A100R andwind vane W200P, Vector Instr.). The soil heat flux was measured using two plates (HFT1, REBS) buried ata depth of 8 cm and using an averaging soil temperature probe placed between 2 and 6 cm in depth (Allenet al., 1996).

In the Guadalquivir valley, the study was conducted at an experimental farm (70 m a.s.l., 37°510N, 4°510W)located on the terraces of the River Guadalquivir, near Cordoba. Average annual precipitation is about 600 mmand is mainly recorded in winter, spring and autumn, with hardly any rainfall in summer. Average annualtemperature is about 17 °C. Advective conditions during summer are more frequent than in Zaragoza, but thearea is significantly less windy than the middle section of the Ebro river valley. Measurements were takenover a 1Ð3 ha plot with uniform grass cover, regularly irrigated and cut all year round. The measurementperiods were from July to October 1997 and from July to August 1998. Only days on which measured grassheight was between 0Ð10 and 0Ð15 m were used for analyses.

An eddy covariance system (Campbell Sci. Inc.) was located at the centre of the plot to measureevapotranspiration. The sensors used included a krypton hygrometer (model KH20), a single-axis sonicanemometer (model CA27), and two fine-wire thermocouples (models 127 and TCBR-3), which were attachedto the two previously mentioned sensors. The hygrometer and the sonic anemometer were installed at a heightof 0Ð32 m, in order to maintain a height : fetch ratio of more than 100. They were oriented towards thewest, which was the predominant wind direction, and were placed 10 cm apart in a north–south direction.Measurements of fluctuations in water vapour density, vertical wind speed and air temperature were recordedevery 0Ð1 s and averaged every 10 min. These readings were used to obtain hourly measured latent heat fluxvalues as described by Villalobos (1997), which were then transformed to hourly ETo 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

Constant canopy resistance: PM equation. The PM equation is given by Equation (1). The importantassumption behind this model is that, despite the variations within the canopy itself, the behaviour of allthe stomata considered together is comparable to that of a single ‘big leaf’ from which heat and vapourescape. This ‘big leaf’ is located at a height d C zo,h, where d is the zero-plane displacement height and zo,h

is the roughness length for heat (Figure 1). In this paper, measured rather than estimated Rn and G valueswere used. This was done in order to prevent potential uncertainties derived from the estimation of thesetwo variables affecting the analysis involving the use of fixed and variable rc values. Units and computations(except Rn and G) in Equation (1) followed Allen et al. (1998), rc was considered constant and equal to70 s m�1, and grass height was set to 0Ð12 m. Throughout the study period, Equation (1) was applied toobtain hourly ETo estimates using hourly averages of meteorological variables.



For the following two models, the data set available for each location was divided into two groups: acalibration data set and a validation data set. Available days were organized by date; one of three days wasselected for calibration, and the other two were selected for validation.

Todorovic method. This model is a new approach for modelling canopy resistance. It is based on the ‘bigleaf’ approach, but incorporates a variable rc. The model’s input requires only standard meteorological data, asin the PM combination approach. A full discussion of the theory and assumptions of the model can be foundin Todorovic (1999). Here, we simply present the equations used to obtain the variable rc values. Todorovic(1999) considers a climatological resistance ri defined by

ri D �acp�eŁ � e�

��Rn � G��5�

as used in the literature (Monteith, 1965; Monteith and Unsworth, 1990). This resistance, as seen bycomparison with Equation (3), is

ri D [/� C ��]rŁ �6�

Then, Todorovic (1999) uses ri and ra, defined according to Allen et al. (1998), to set the followingsecond-degree equation

aX2 C bX C c D 0 �7�

a function of X D rc/ri, where the parameters are defined by

a D C ��ri/ra�

C ��ri/ra��e

Ł � e�

b D ��ri

ra

�

eŁ � e

C ��8�

c D �� C ��

eŁ � e

C �

Solving for Equation (7), which has only one positive solution, the variable canopy resistance is obtainedas rc D Xri. Equations (7) and (8) were applied to the validation data set to obtain hourly estimates of canopyresistance rc,TD using the hourly averages of meteorological variables. A fixed value of rc D 200 s m�1 wasconsidered for night hours. These variable rc values were then used, assuming a grass height of 0Ð12 m, toobtain hourly ETo estimates by directly applying the PM equation (Equation (1)). Preliminary testing of thisapproach has provided convincing results obtained on an hourly basis (Todorovic, 1999).

Variable canopy resistance: semi-empirical approach. Based on the experimental relationship found byPerrier et al. (1980) between the bulk canopy, aerodynamic and climatological resistance rŁ (Equation (3)),several authors proposed a model in which rc/ra is a linear function of rŁ/ra. This model has been appliedto wheat (Perrier et al., 1980), grass and alfalfa (Katerji and Perrier, 1983; Rana et al., 1994; Steduto et al.,2003), tomato (Katerji et al., 1988) and rice (Peterschmitt and Perrier, 1991). The coefficients of those linearrelationships depend on the type of crop, its phenological state and soil water status (Rana et al., 1997).

In order to analyse this simple model, we propose exploring a functional relationship of the form

rc

raD a C bf

(rŁ

ra

)�9�

for the reference crop, where a and b are global parameters that are to be determined empirically. WhenEquation (9) is substituted into Equation (4), the PM model contains only standard climatological variables.



The most commonly used formulations for estimating aerodynamic resistance ra consider that apparentsources of heat and vapour 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 C zo,h,where zo,h (m) is the roughness length for sensible heat transfer. When the leaves at the top of the canopy,between d C zo,h and crop height hc, are the most important source of vapour flux, the use of the d C zo,h

level as the evaporative surface can lead to an overestimation of ra and to negative values for bulk surfaceresistance rc when it is computed by inverting the PM equation. To avoid this, ra can be computed betweenthe top of the canopy hc and the reference height in the atmosphere as (Perrier, 1975; Alves et al., 1998)

ra Dln

(zm � d

zo,m

)ln

(zh � d

hc � d

)

k2uzm

�10�

where zm and zh are the wind and air temperature measurement heights respectively, hc is the crop height, kis the dimensionless von Karman constant, and uzm is the wind speed measured at height zm. The term hc � dsubstitutes the roughness length for heat transfer used to compute ra according to Allen et al. (1998).

So, for both the calibration and the validation data sets, the experimental or measured hourly values ofbulk surface resistance rc�s m�1� were obtained by inverting the PM equation (top-down approach) as

rc D ra

(

�

Rn � G

�E� C �

�

)C � cp

�

D

�E�11�

where D (Pa) is the vapour pressure deficit (D D eŁ � e in Equation (1)), Rn � G�W m�2� is the availableenergy for the entire surface and �E �W m�2� is the latent heat flux measured by the lysimeter (Zaragoza)or by the eddy covariance system (Cordoba).

The hourly values measured for Rn � G, D and �E were used to compute hourly rc values in Equation (11),aerodynamic resistances in Equation (10) and climatic resistances in Equation (3). Using only the calibrationdata sets for Zaragoza and Cordoba, several simple relationships between rc/ra and rŁ/ra for any rangeof ˇ values were analysed for the reference crop. The calibrated values of parameters a and b inEquation (9) allowed us to obtain an estimated variable canopy resistance rc,est using only the validationdata set. This was then used to estimate hourly evapotranspiration values �Eest from Equations (1) or (4).

Statistical methods

In the evaluation of the different relationships represented by Equation (9), the ability to predict rc hasbeen compared using the coefficient of determination R2 as the measure of the goodness of fit, i.e. of theperformance of the model; R2 is a measure of the total variance accounted for by the model. Parameters aand b in Equation (9) are then obtained for the best fit using the calibration data set.

A complication arises when the model that maximizes R2 for rc may not maximize it when predictingevapotranspiration ET. Once the best parameterization of rc had been chosen and the optimum coefficientvalues in Equation (9) had been found, comparisons between measured and estimated hourly ET values werecarried out using the three models above. To test which of the resulting models was significantly better thanthe others, a simple linear regression between the predicted and the observed values was made, based on thecriterion of trying to maximize R2. Nevertheless, in order to make a better quantitative comparison betweenthe models, predicted results can be evaluated with reference to several statistics (Willmott, 1981, 1982; Mayerand Butler, 1993; Verhoef et al., 1997; Lee and Singh, 1998; Alexandris and Kerkides, 2003).

The first statistic is the bias or mean deviation error MBE D [∑�Oi � Pi�

]/n. The second is the root-mean-

square error RMSE D √∑�Oi � Pi�2/n, which is considered as a better indicator of model performance

than the correlation statistics. They can be expressed as percentages of the mean observed value (relativeRMSE�%� D 100 ð RMSE/O). In the above expressions, Pi and Oi represent the predicted (estimated) and



observed (measured) values, n is the total number of data and O is the mean of observed data. The RMSEprovides a measure of the total difference between the predicted and observed values. The closer RMSE isto zero, the better the prediction.

The statistic we will use as the model selection criterion is the modelling efficiency (Lee and Singh, 1998),defined as EF D 1 � [∑

�Oi � Pi�2]

/[∑

�Oi � O�2]. This is similar to the index of agreement (Willmott,

1982). Modelling efficiency is a dimensionless statistic that measures the fraction of the variance of theobserved 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 unity and with the lowest RMSE.

RESULTS AND DISCUSSION

The values of the statistics derived from air temperature and wind speed records for Zaragoza and Cordobaduring the measurement periods are presented in Table I. They are shown for descriptive purposes because, asa result of the different measurement periods, it was not possible to compare them directly. It should be notedthat on 9% of the days at Zaragoza the average daily wind speed was greater than 4 m s�1, whereas this didnot occur at Cordoba. As mentioned previously, Zaragoza is located within one of the windiest areas in Spain.

The hourly values of ra obtained with Equation (10) were lower than those obtained using the classicalexpression in the PM equation for the reference grass (Allen et al., 1998). This confirmed results obtainedby other workers over both short and tall vegetation (Alves et al., 1998; Hall, 2002). For these workers,aerodynamic resistance using the classical logarithmic profile equations for momentum exchange results inestimates that are systematically higher than values derived from direct measurements. The differences in theestimations of ra for dense or sparse canopies, or even for grass, are, as reviewed by Verhoef (1995), theresult of the problem of the parameterization of an excess resistance. Figure 3a shows the daily evolution ofra for a typical day at Zaragoza.

As described elsewhere in the literature, bulk surface resistance is small and tends to remain relativelyconstant on average from 08 : 00 to 14 : 00 (Rana et al., 1994; Monteith, 1995; Alves et al., 1998; Pereiraet al., 1999; Todorovic, 1999; Alves and Pereira, 2000). After this, rc tends to increase gradually in theafternoon, reflecting the daily course of the environmental variables that influence it (Figure 3b). Negativevalues have been obtained for rc independently of the time of day, but these have mainly been registeredearly in the morning and at night, when the available energy (Rn � G) tends to be small and with negativevalues, and when the vapour pressure deficit D is also small. In well-irrigated crops, factors external to thecrops control the latent heat flux to a greater extent than purely physiological factors.

When a constant rc is considered, Table II shows the results of comparisons between the estimated andmeasured hourly values for evapotranspiration at the two locations, based on the validation data set for eachmeasurement period. ETPM represents the estimated values obtained using the PM equation (Equation (1))with a fixed value of rc D 70 s m�1. The values for modelling efficiency EF suggest good agreement for themethod at both locations, with a relatively small scatter of data but with a slope lower than one (Table II).

Table I. Representative values of the meteorological conditions during the measurement period at Zaragoza and Cordoba. Tx

and Tn are maximum and minimum air temperature respectively, and u is average wind speed

Variable Zaragoza Cordoba

Average Maximum Minimum Average Maximum Minimum

Tx (°C) 26Ð6 38Ð2 12Ð1 33Ð3 38Ð6 24Ð4Tn (°C) 11Ð3 20Ð1 �2Ð6 18Ð2 22Ð7 9Ð2u �m s�1� 2Ð1 7Ð6 0Ð4 1Ð7 3Ð6 0Ð7Days with u > 4.0 m s�1 (%) 9Ð0 0Ð0



Figure 3. (a) Daily evolution of the aerodynamic ra and canopy rc resistances of grass for a specific day (5 March 1999, Zaragoza).(b) Experimental hourly averaged values for bulk surface resistance rc for the calibration data set for Zaragoza. These were obtained byinverting the PM equation using values of latent heat flux measured by lysimeter (Equation (11)). Hourly averaged values for climatic

resistance rŁ, available energy Rn � G and vapour pressure deficit D are also shown

Table II. Simple linear regression (ETest D co C c1ETo) and statistics of the comparison between estimated (ETest) andmeasured (ETo) hourly values of evapotranspiration (mm h�1) at the two locations for the respective validation data sets.a

Estimates were obtained using the PM equation with a fixed rc value of 70 s m�1 (ETPM), and with the variable canopyresistance rc,TD (ETTD). Measured ETo values were obtained using a weighing lysimeter (Zaragoza) and an eddy covariance

system (Cordoba)

Method Location n c0 �mm h�1� c1 R2 RMSE (%) MBE (%) EF

ETPM Zaragoza 1776 0Ð013 0Ð87 0Ð97 23Ð8 5Ð50 0Ð961Cordoba 1392 0Ð022 0Ð89 0Ð98 18Ð7 0Ð74 0Ð976

ETTD Zaragoza 1776 0Ð004 0Ð97 0Ð97 20Ð6 0Ð20 0Ð971Cordoba 1392 0Ð014 0Ð94 0Ð98 16Ð0 �0Ð58 0Ð982

a n: number of hourly data; c0: intercept; c1: slope; R2: coefficient of determination; RMSE (%): relative root mean square error; MBE (%):relative mean deviation error; EF: modelling efficiency.

There is a tendency for the PM method to overestimate low values of measured ETo and to underestimate highETo values (Figure 4). This behaviour has already been reported at other Mediterranean locations (Stedutoet al., 1996, 2003).

The underestimation of high ETo values was smaller in Cordoba than in Zaragoza. This might have beendue to the use of an eddy covariance system for measuring latent heat flux. This system, depending on the



Figure 4. Hourly values of evapotranspiration estimated using the PM equation (ETPM) with a fixed rc value of 70 s m�1 for the validationdata sets of the measurement periods at each site (1 : 1 line is shown for comparison). Measured values ETo were obtained with a weighing

lysimeter (Zaragoza) and with an eddy covariance system (Cordoba)

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 windconditions. In Zaragoza, the agreement between lysimeter and ETPM values was found to decrease as windspeed increased. Under conditions of high evaporative demand (sunny days in summer), evapotranspirationrates are expected to increase further under windy conditions. In such situations, the PM method with constantrc 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 ETo values in both zones under semiarid conditions (Ranaet al., 1994; Steduto et al., 1996, 2003; Ventura et al., 1999).

Since one of our aims was to analyse the performance of the PM equation with respect to differ-ent ways of considering canopy resistance, we have analysed the mechanistic approach represented byEquations (7) and (8) (Todorovic, 1999). The variable canopy resistance rc,TD estimated with this methodwas then applied to the PM equation (Equation (1)) and hourly estimates of evapotranspiration ETTD wereobtained. The values of rc,TD in the Todorovic method only depend on weather variables and do not requirecalibration; they can, therefore, be applied directly to the validation data set for each measurement periodat both sites. For the diurnal period, the values of rc,TD were, on average, 10% lower than the experimentalvalues of rc, but, as shown in Figure 5a, with a high degree of variability in the case of Zaragoza.

The drawback of the Todorovic method is that it cannot be generally applied for the nocturnal period, whenboth vapour pressure deficit D and available energy Rn � G are low or close to zero. In such cases, both ri



Figure 5. (a) Comparison between variable canopy resistances rc,TD estimated by the Todorovic method and the experimental values rc(diurnal period, Zaragoza). (b) Hourly values of evapotranspiration ETTD estimated with the variable resistance rc,TD, compared with

measured ETo values for the whole validation period at Zaragoza

(Equation (5)) and rŁ have negative or non-defined values, so the solutions for X in Equation (7) and rc arenot defined. This model can only be applied when ri is positive; therefore, for cases when ri � 0 (usually atnight), a fixed value of rc,TD D 200 s m�1 must be considered (Allen et al., 1998). When the variable canopyresistance rc,TD was used in Equation (1), estimated hourly values ETTD for evapotranspiration were obtained.

Results for the comparison between estimated ETTD and measured ETo values are presented in Table IIand Figure 5b. The statistics show that the relative RMSE (%) of estimates for ETTD are about 3% lowerthan that for ETPM, and that the slope of the linear regression with measured values for ETTD was closerto unity than for ETPM. Furthermore, the relative mean deviation error MBE (%) indicated that ETTD tends,globally, to underestimate measured values less than ETPM does, as shown in Figure 5b with respect to highevaporative demand. Even in Cordoba ETTD tends, globally, to overestimate measured ETo (negative MBE,Table II). The values for the modelling efficiency taken together with the above results indicate that estimationof evapotranspiration using the Todorovic method (variable rc) shows better performance than the PM method(fixed rc). Thus, if there is a need to incorporate the effect of a variable rc, then the Todorovic method may beuseful, at least under the semiarid and windy conditions such as those of the Ebro river valley. Results cited



in a previous study (Lecina et al., 2003) showed that daily ETo estimates could be obtained with sufficientaccuracy using the PM method.

The second way to estimate a variable rc is through the semi-empirical approach given by Equation (9),which relates rc/ra to rŁ/ra. Results found by others workers (Rana et al., 1997; Alves and Pereira, 2000;Steduto et al., 2003) seem to show a linear relationship between these variables on an hourly basis. Theseresults were generally obtained under certain limitations corresponding to the diurnal period, or were obtainedfor a limited range of Bowen ratio ˇ values. Alves and Pereira (2000) found such results for ˇ values inthe interval [�0Ð3, 0Ð3], corresponding to situations when crop evapotranspiration was maximum and whenavailable energy Rn � G had values corresponding to the middle of the day.

We carried out an analysis of the empirical relationship provided by Equation (9) for our semiarid regionswith the aim of amplifying this data interval and thereby covering the largest possible range of situationsthroughout the day, with the following results. Working with the complete set of calibration data for the twomeasurement zones, we found that a straight line did not represent the best relationship between rc/ra andrŁ/ra. This result was obtained independently of the interval of ˇ values considered. When latent and sensibleheat fluxes are measured over grass on an hourly basis, 85% of the actual values for ˇ lie in the interval[�0Ð5, 0Ð5] throughout the day (Perez et al., 1999). As seen in Figure 6a, when considering cases in whichthe ˇ values are in the interval [�0Ð5, 0Ð5], the functional form of the dependence of rc/ra on rŁ/ra departs

Figure 6. Variation of experimental values of rc/ra versus rŁ/ra on an hourly basis for the global calibration data set (Zaragoza and Cordoba)for: (a) data corresponding to the diurnal period with Rn � G ½ 0 and ˇ values in the interval [�0Ð5, 0Ð5]; (b) data corresponding to thediurnal period with Rn � G ½ 30 W m�2 independent of the ˇ values. The estimated values �rc/ra�est,i respectively correspond to estimating

models 1 and 2 in Table III



from linearity. It seems that rc/ra tends to some saturation value when rŁ/ra increases. As in this case we areincluding data corresponding to a longer diurnal period (07 : 00 to 19 : 00) in our analysis, this result extendsthat reported by Alves and Pereira (2000).

The main drawback is that, for the nocturnal period, when vapour pressure deficit D and available energyRn � G are low, as at sunrise and sunset, the canopy resistance rc may register very high positive or very lownegative values and the climatic resistance rŁ may register negative or non-defined values. For these situations,and for comparison with the Todorovic model, when rŁ � 0 we have considered a fixed value of 200 s m�1

for rc. Even so, about š5% is a typical error associated with the measurement of net radiation, whereas theresolution of the weighing lysimeter is 0Ð04 mm in the measurement of evapotranspiration. Therefore, thenext step was to use in the analysis only the calibration data when Rn � G ½ 30 W m�2 and when measuredETo ½ 0Ð04 mm h�1, independently of ˇ values. As a consequence, this constraint effectively limited theanalysis to the diurnal period going from 07 : 00 to 19 : 00 (Figure 6b), but with ˇ values ranging from �0Ð8to 2Ð0.

Table III shows the values of the empirical coefficients calibrated locally for each relationship analysed.Results show that the best-fit relationship corresponded to a dependence of rc/ra on the square root of rŁ/ra

and not to a linear relationship, as the coefficient of determination R2 indicates (Table III). For the lineardependence �rc/ra�est,2, it can be observed that the values of the empirical coefficients for our semiaridzones (a2 D 1Ð1, b2 D 0Ð19), are similar to those obtained by Steduto et al. (2003) for grass in southernItaly (a5 D 0Ð918, b5 D 0Ð18). This finding also suggests that it may be possible to extrapolate the empiricalcoefficients of the proposed method �rc/ra�est,1 (Table III) to other climatic regions.

The statistics used for the comparison between estimated and experimental values of rc/ra (reported inTable III for the global set of calibration days at Zaragoza and Cordoba) indicate that the relative RMSE is13% lower and the modelling efficiency is higher for method �rc/ra�est,1 than when using the estimating method�rc/ra�est,2. Although the Todorovic method does not need any prior calibration, the statistics show that rc,TD

generally underestimates values of canopy resistance obtained by inverting the PM equation (MBE D 37Ð3%).This is mainly true for values of canopy resistance that exceed 50 s m�1. Once empirically calibrated for oursemiarid regions, the method �rc/ra�est,1 performed better when estimating canopy resistance.

The variable rc estimated through the proposed methods �rc/ra�est,1 and �rc/ra�est,2 was used inEquation (1) to estimate hourly values of evapotranspiration ETest,1 and ETest,2 respectively. Table IV showsthe results of the comparison between the four estimating methods and the measured ETo values for theglobal set of validation days at Zaragoza and Cordoba. As seen, the relative RMSE of estimates for ETest,1 isabout 2% lower than for ETPM, and the slope of the linear regression with measured values is closer to unityfor ETest,1 than for ETPM (Figure 7). Overall, these results indicate that the proposed method ETest,1 using avariable rc tends to perform better than the PM method with constant rc. Although the improvement in the

Table III. Empirical coefficientsa locally calibrated for two functional forms of the dependence of rc/ra on rŁ/ra inEquation (9): (1) for the experimentally found best-fit relationship �rc/ra�est,1 D a1 C b1

prŁ/ra; (2) for a linear dependence

�rc/ra�est,2 D a2 C b2 �rŁ/ra�. The Todorovic method rc,TD and models 4 and 5 proposed by other authors and applied toour 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 rc/ra

Model Function n ai bi R2 RMSE (%) MBE (%) EF

1. �rc/ra�est,1 Square root 738 �0Ð66 1Ð38 0Ð72 48Ð6 �0Ð31 0Ð7162. �rc/ra�est,2 Line 738 1Ð1 0Ð19 0Ð55 61Ð5 �2Ð1 0Ð5463. rc,TD 738 — — — 85Ð0 37Ð3 0Ð1334. Rana et al. (1997) Line — 0 0Ð16 — 86Ð2 59Ð7 0Ð1185. Steduto et al. (2003) Line — 0Ð918 0Ð18 — 62Ð2 9Ð4 0Ð536

a n: number of hourly data; ai: intercept; bi: slope; R2: coefficient of determination; RMSE (%): relative root mean square error; MBE (%):relative mean deviation error; EF: modelling efficiency.



Figure 7. Hourly values of estimated evapotranspiration ETPM (obtained from the PM equation with a fixed rc value of 70 s m�1) andETest,1 (obtained from Equation (1) with the variable canopy resistance obtained from �rc/ra�est,1�, compared with measured values ETo for

the global validation data set (Zaragoza and Cordoba)

Table IV. Simple linear regression (ETest D co C c1ETo) and statistics for the comparison between estimated (ETest) andmeasured (ETo) hourly values of evapotranspiration (mm h�1) for the global validation data set (Zaragoza and Cordoba).a

Estimates of evapotranspiration were obtained from Equation (1) with: a constant rc D 70 s m�1 (ETPM); a variable resistancerc,TD estimated by the Todorovic method (ETTD); a variable resistance obtained from the estimating methods �rc/ra�est,1 and�rc/ra�est,2 (ETest,1 and ETest,2 respectively). Measured ETo values were obtained using a weighing lysimeter (Zaragoza) and

an eddy covariance system (Cordoba)

Method n c0 �mm h�1� c1 R2 RMSE (%) MBE (%) EF

ETPM 3168 0Ð017 0Ð89 0Ð979 21Ð4 3Ð15 0Ð969ETTD 3168 0Ð009 0Ð96 0Ð978 18Ð4 �0Ð18 0Ð977ETest,1 3168 0Ð006 0Ð91 0Ð982 19Ð8 6Ð18 0Ð974ETest,2 3168 0Ð005 0Ð90 0Ð977 21Ð8 7Ð16 0Ð968

a n: number of hourly data; c0: intercept; c1: slope; R2: coefficient of determination; RMSE (%): relative root mean square error; MBE (%):relative mean deviation error; EF: modelling efficiency.



evapotranspiration estimation found in this study seems to be limited, the empirical approach involving rc/ra

as a function of the square root of rŁ/ra can be considered as an alternative to other approaches for modellingrc. It may also be useful to incorporate the effect of variable canopy resistance into climatic and hydrologicalmodels.

CONCLUSIONS

The results presented here show that hourly ETo estimates made with the PM equation and using a constant rc

value of 70 s m�1 tend to underestimate measured ETo during the summer period (when evaporative demandis high) and to overestimate it in winter (when ETo is low), under the semiarid conditions of both the Ebro andGuadalquivir river valleys. This may have been a consequence of considering canopy resistance as constantduring the day.

The issue is that Equation (11), which is used to calculate rc in a top-down approach, shows that rc dependson environmental variables. Therefore, an approach that models rc as a function of climatic conditions couldbe regarded as an alternative way of trying to overcome the limitations of the PM method. Katerji andPerrier (1983) proposed a model in which the rc variable is a function of climatic resistance rŁ. They showedexperimentally that there is a linear link with two empirical coefficients between rc/ra and rŁ/ra, and that thisdepends on climatic conditions.

The results presented here for semiarid regions with a Mediterranean climate show that estimating rc as afunction of the square root of rŁ on an hourly basis constitutes a better empirical approach than the linearoption. In the case of the linear relationship, the values of the empirical coefficients calibrated for our semiaridregions were similar to those obtained by Steduto et al. (2003) for grass in southern Italy. This result suggeststhat, in the proposed �rc/ra�est,1 approach, it may also be possible to extrapolate the locally calibrated empiricalcoefficients to other climatic regions.

When the different methods for estimating a variable rc are used in the PM equation, the results show thatestimates of hourly evapotranspiration ETest,1 derived from the global validation data set tend to performrelatively better than ETPM with constant rc. The only apparent drawback is perhaps the fact that theminor improvement observed hardly justifies the effort to undertake the local calibration required. Whenthe Todorovic mechanistic method for estimating rc is used and applied in the PM equation, the resultsobtained show that, in comparison with the other methods, ETTD also performs very well when it is used toestimate hourly evapotranspiration for our global data set.

ACKNOWLEDGEMENTS

This work was funded by the Spanish Ministry of Science and Technology under project REN2001-1630/CLIand by the Department of Universities and Research (Generalitat de Catalunya) under project 2001SGR-00306.The second author was also supported by a fellowship from the High Council of Research and Developmentof the Autonomous Government of Aragon. We would like to thank Jesus Gaudo, Miguel Izquierdo andEnrique Mayoral for their help with the fieldwork, and would also thank the reviewers for their commentsthat have helped to improve the final manuscript.

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A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration