do energy-based pet models require more input data than temperature-based models? - an evaluation at...

DO ENERGY-BASED PET MODELS REQUIRE MORE INPUT DATA THAN TEMPERATURE-

BASED MODELS? — AN EVALUATION AT FOUR HUMID FLUXNET SITES1

Josephine A. Archibald and M. Todd Walter2

ABSTRACT: It is well established that wet environment potential evapotranspiration (PET) can be reliably esti-mated using the energy budget at the canopy or land surface. However, in most cases the necessary radiationmeasurements are not available and, thus, empirical temperature-based PET models are still widely used, espe-cially in watershed models. Here we question the presumption that empirical PET models require fewer inputdata than more physically based models. Specifically, we test whether the energy-budget-based Priestley-Taylor(P-T) model can reliably predict daily PET using primarily air temperature to estimate the radiation fluxes andassociated parameters. This method of calculating PET requires only daily minimum and maximum tempera-ture, day of the year, and latitude. We compared PET estimates using directly measured radiation fluxes toPET calculated from temperature-based radiation estimates at four humid AmeriFlux sites. We found goodagreement between P-T PET calculated from measured radiation fluxes and P-T PET determined via air tempera-ture. In addition, in three of the four sites, the temperature-based radiation approximations had a stronger correla-tion with measured evapotranspiration (ET) during periods of maximal ET than fully empirical Hargreaves,Hamon and Oudin methods. Of the three fully empirical models, the Hargreaves performed the best. Overall, theresults suggest that daily PET estimates can be made using a physically based approach even when radiationmeasurements are unavailable.

(KEY TERMS: potential evapotranspiration; empirical model; physically based model; Priestley-Taylor equation;temperature; radiation.)

Archibald, Josephine A. and M. Todd Walter, 2013. Do Energy-Based PET Models Require More Input Datathan Temperature-Based Models? — An Evaluation at Four Humid FluxNet Sites. Journal of the AmericanWater Resources Association (JAWRA) 50(2): 497-508. DOI: 10.1111/jawr.12137

INTRODUCTION

Potential evapotranspiration (PET) is the amountof evaporation that would occur if there were anabundant supply of water in the landscape. Earlyinterpretations of this concept allowed modelers toestimate the upper limit of evapotranspiration (ET)using meteorological data (Thornthwaite, 1948). Fur-ther refinements of this idea have created different

interpretations of the precise meaning of PET andthe particular situation that a PET equation is meantto model. For example, reference crop ET is usuallythe PET from a uniform, well-watered short grass oralfalfa. Also, the concept of the equilibrium PET rep-resents the PET from an environment in which thevapor-pressure gradient between the canopy and theair is very small such that the energy budget isthe primary control on the ET. However, despitethese differences, hydrologic watershed modelers

1Paper No. JAWRA-13-0002-P of the Journal of the American Water Resources Association (JAWRA). Received January 3, 2013; acceptedAugust 27, 2013. © 2013 American Water Resources Association. Discussions are open until six months from print publication.

2Respectively, Graduate Student and Associate Professor, Department of Biological and Environmental Engineering, Cornell University,Riley Robb Hall, Ithaca, New York 14853 (E-Mail/Archibald: [email protected]).

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JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

Vol. 50, No. 2 AMERICAN WATER RESOURCES ASSOCIATION April 2014

commonly use these terms synonymously as an indexof the “drying power” of the meteorological conditions(e.g., Dingman, 2002, p. 310).

The PET concept is widely adopted by watershedmodelers as a precursor to estimate actual ET. Spe-cifically, the models calculate PET and then scale itbased on modeled plant development (e.g., leaf areaindex, rooting depth) and soil moisture status to esti-mate actual ET. Many PET equations have been pro-posed ranging from fully empirical equations,generally based on air temperature, to physicallybased equations that relate PET to the surfaceenergy budget. Although empirical or temperature-based PET equations are generally not recommendedfor estimating average PET over durations less thana week to a month (e.g., Shuttleworth, 1993), hydro-logic modelers routinely use them in daily time-stepwatershed models. The justification for using thetemperature-based PET models is that they requiremuch less data than energy-budget-based models,especially radiation fluxes and other data that areoften not readily available. Although we acknowledgethat radiation fluxes are not measured ubiquitously,there has been considerable effort in developingmethods to estimate these types of data (e.g., seeAllen et al., 1998 or ASCE-EWRI, 2005), so it is notclear that there is a good justification for not usingenergy-budget-based PET equations. Campbell andNorman (1998) state that “the data requirements aresubstantial, but one usually obtains better resultsusing [a physically based] equation and estimatingmissing data, than by using a simpler equation thatdoes not use such a mechanistic approach.” Weacknowledge that the idea of a hybrid between empiri-cal temperature-based and a fully physically basedmodel is not new (Allen et al., 1998; Walter et al.,2005), but here we will try to introduce as littleempiricism as possible while restricting our meteoro-logical data requirements to daily maximum andminimum temperature.

The three most widely used energy-budget-basedPET equations are the Penman (1948) equation, thePenman-Monteith (P-M) (Monteith, 1965) equation,and the Priestley-Taylor (P-T) (Priestley and Taylor,1972) equation. Both the P-M and P-T equations werederived from the Penman equation. The P-M equationis the accepted standard method, used by the Food andAgriculture Organization (FAO) of the United Nationsand the American Society of Civil Engineers (ASCE).Because the P-M equation incorporates the plant con-trols on ET via the inclusion of a resistance term thatis a function of various environmental factors thatmight stimulate the opening or closing of stomata, itcan be used to calculate actual ET. However, the P-Mequation is often used to calculate reference crop ET orPET by using parameters for ideal plant growth. Both

the Penman and P-M equations have advection termsthat require wind speed information. In this study, wechose to use the P-T equation which omits the advec-tion or vapor-deficit term that is in the Penman and P-M equations, thus, alleviating the need for wind speeddata. Omitting the advection term implies that thesurface and plant canopy are in equilibrium, thus, theP-T equation is generally presumed to represent equi-librium PET, or PET from wet environments whereET is predominantly energy limited. Interestingly,variations of P-T equation have sometimes been foundto estimate PET equally well or more accurately thanthe P-M equation (Stannard, 1993; Utset et al., 2004;Sumner and Jacobs, 2005; Rosenberry et al., 2007). Inaddition, Martinez and Thepadia (2010) found that,when using ASCE methods to estimate missing radia-tion and wind data, the P-M equation did not performas well as more empirical methods in coastal areas ofFlorida.

In this study, we apply the P-T equations on adaily time step and we can assume that the dailyground heat flux is negligible compared to the dailynet radiation, RN (Allen et al., 1998). Using thisapproximation, the P-T equation is simply a functionof temperature and RN.

PET ¼ aqk

� �D

Dþ cRN ð1Þ

where RN is daily net radiation (kJ/m2/day), a is thePriestley-Taylor constant (unitless), q is the densityof water (1,000 kg/m3), k is the latent heat of vapori-zation (2,500 kJ/kg), c is the psychrometric constant(0.066 kPa/°C), and D is the slope of the saturationvapor pressure-temperature curve (kPa/°C), whichcan be estimated from air temperature, T (°C)(Tetens, 1930):

D ¼ 2508:3

ðT þ 237:3Þ2 exp17:3T

T þ 237:3

� �ð2Þ

The typical value for a is 1.26 for conditions withrelative humidity greater than 60% (Shuttleworth,1993).

As part of this study we wanted to include someempirical temperature-based models for comparisonpurposes. There are lots of such models but we choseto use Hargreaves (Hargreaves and Samani, 1985;Equation A1), Hamon (1963; Equation A2) and Oudin(Oudin et al., 2005; Equation A3) because they arecommonly used in watershed modeling and the formsof the equations were quite unique from one another.In addition, each equation was developed for uniqueapplications: Hargreaves was developed for a reference

JAWRA JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION498

ARCHIBALD AND WALTER

crop (Hargreaves and Samani, 1985), early applica-tions of Hamon were for small, mixed land coverwatersheds (Hamon, 1963), and Oudin was specifi-cally developed for watershed modeling (Oudin et al.,2005). In practice these equations along with theenergy-budget-based equations are often used inter-changeably to approximate PET over a wide varietyof ecosystems or over whole watersheds. For example,the Soil and Water Assessment Tool (SWAT) hydro-logic model has the option to use P-M, P-T, orHargreaves methods, with each of these havingprogressively less data requirements (Neitsch et al.,2011). In addition, the empirical Hamon PET modelwhich uses only daily mean temperature has beenused in many watershed modeling applications suchas the Soil Moisture Routing Model (Frankenbergeret al., 1999) and the U.S. Geological Survey’s(USGS’s) Precipitation Runoff Modeling Systems(USGS, 2013).

One impetus for this study is to explore options forresearchers who are interested in understandinghydrology in the context of climate change, and toexpand on ideas proposed in an earlier comment(Archibald and Walter, 2013). Bae et al. (2011) notedthat a wide variety of PET models, including temper-ature-based models such as Hamon and Hargreaves,and radiation-based models, P-T and P-M, are usedto estimate PET (and subsequently actual ET) underfuture climate scenarios — unsurprisingly theseresearchers found large differences in PET predic-tions based on the formulation used. Burns et al.(2007) used the temperature-based Hamon model toestimate PET trends in the mainly forested region ofthe Catskill Mountains in New York State over thepast half century. Some of these types of empiricaltemperature-based PET equations are overly sensi-tive to increasing temperatures (Kingston et al.,2009; Lofgren et al., 2011; Milly and Dunne, 2011;Shaw and Riha, 2011).

We have limited the scope of this study to humidenvironments to avoid complications associated withthe complimentary relationship between actual ETand PET. In humid environments, PET, actual ET,and equilibrium PET converge and we would expectP-T PET to closely match actual ET (Szilagyi, 2008).In other words, we expect these humid sites to beenergy limited, causing advection to be less importantin driving ET here.

Estimating Net Radiation, RN

Net radiation is the difference between net long-wave radiation and net incoming solar radiation, i.e.,RN = S + LA � LT, where RN is net radiation, S isnet incoming solar radiation, and LA and LT are

atmospheric and terrestrial longwave radiation,respectively.

Solar radiation is estimated from geographic lati-tude, day of the year, surface albedo, and atmo-spheric transmissivity (Equations 3-5).

S ¼ ð1� aÞTr SP ð3Þ

where S is solar radiation (kJ/m2/day), a is albedo(unitless), Tr is transmissivity (Equation 5, unitless),and SP is potential solar radiation at the edge of theatmosphere (Equation 4, kJ/m2/day). A general valuefor the albedo of vegetation is 0.23 (e.g., Shuttle-worth, 1993), and was used here. We calculatedpotential solar radiation at the edge of the atmo-sphere using Equation (4):

SP ¼ CS arccosð�tanðdecÞ tanðlatÞÞ sinðlatÞ� sinðdecÞ þ cosðlatÞ cosðdecÞ� sin ðarccosðtanðdecÞ tanðlatÞÞÞ=p

ð4Þ

where dec is solar declination (radians ¼0:4102 sin p Jday�80

180

� �), Jday is the day of the year

(1-366, unitless), lat is latitude (radians), and CS is asolar constant (117,500 kJ/m2/day).

Atmospheric transmissivity (unitless) is calculatedfollowing Bristow and Campbell (1984):

Tr ¼ Af1� expf�½0:036 expð�0:154DTÞ�DTBggð5Þ

where DT is the average temperature range in the15 days before and after the current day (°C), ΔT isthe daily temperature range (Tmax-Tmin) (°C), A is themaximum clear sky transmissivity (unitless) — herewe used 0.75, and B is an energy partitioning coeffi-cient (unitless) — here we use 2.4 following Bristowand Campbell (1984).

In this study, we estimated transmissivity usingthe Bristow-Campbell equation (5), which was devel-oped with the understanding that diurnal tempera-ture range is driven by solar radiation, i.e., days witha large difference between maximum and minimumtemperature are expected to have greater atmo-spheric transmissivities than days with a smallerrange (Bristow and Campbell, 1984). Note, while thisis not, strictly speaking, a physically based equation,it is physically logical and, because it uses the dailytemperature range and not the temperature itself, weanticipate that it will not have the same problemsthat empirical PET models have in making futurepredictions under increasing air temperatures (e.g.,Shaw and Riha, 2011). Other studies have attempted


DO ENERGY-BASED PET MODELS REQUIRE MORE INPUT DATA THAN TEMPERATURE-BASED MODELS? — AN EVALUATION AT FOUR HUMID FLUXNET SITES

to further refine the transmissivity equation (e.g.,Thornton and Running, 1999), however, to avoidadditional parameterization, we are confining thisanalysis to the original equation suggested by Bri-stow and Campbell (1984).

Longwave radiation is calculated with the Stefan-Boltzmann equation (6). Terrestrial emissivity of 0.97is a mid-range average emissivity of a vegetatedsurface (Jin and Liang, 2006).

L ¼ erðTÞ4 ð6Þ

where e is emissivity (unitless, terrestrial = 0.97,atmospheric emissivity is calculated using Equa-tion 7), T is temperature (°K), and r is the Stefan-Boltzmann constant (4.89 9 10�6 kJ/m2/K4/day).

Brutsaert (1975) developed a physically basedequation for clear sky emissivity that is dependent onaverage temperature and vapor pressure near thesurface (first term in Equation 7). In this study, weestimated daily vapor pressure as the saturatedvapor pressure at the minimum daily temperature,which we expect to be most accurate under moist-airconditions such that dew forms at night (e.g., Walteret al., 2005) (Equation 8). Using this approximationfor vapor pressure, we were able to determine theclear sky emissivity with only temperature data.However, because emissivity is also related to cloudi-ness, we modified Brutsaert’s equation according toMonteith and Unsworth (1990) (Equation 7), basedon the assumption that cloudiness is linearly relatedto transmissivity (Equation 9).

e ¼ 1:24� VP

T

� �1=7 !

ð1� 0:84 CÞ þ 0:84 C ð7Þ

where VP is vapor pressure estimated as the satu-rated vapor pressure at Tmin (millibars), T is temper-ature (°K), and C is cloudiness (unitless) — linearlyrelated to transmissivity (see below). Note: the first

term on the left-hand-side of the equation is theBrutsaert (1975) clear sky emissivity.

Vapor pressure is estimated using Tetens’ (1930)equation for saturated vapor pressure at the dailyminimum temperature (Allen et al., 1998; Walteret al., 2005; Crago et al., 2010):

VP ¼ 0:6108� exp17:27Tmin

237:3þ Tmin

� �ð8Þ

where Tmin is daily minimum temperature (°C).Cloudiness was related linearly to transmissivity:

C ¼0; Tr[ 0:751; Tr\0:15

1� Tr�0:150:6 ; Tr� 0:15;Tr� 0:75

8<: ð9Þ

METHODS

We tested our hypothesis that PET can be reason-ably well approximated by P-T PET even if RN is esti-mated primarily from daily temperature data usingdata from four AmeriFlux network sites (Oak RidgeNational Laboratory Distributed Active Archive Cen-ter; accessed 2012. http://fluxnet.ornl.gov) (Table 1).These sites were chosen because of the availability ofradiation and water vapor flux records. In addition,they are either designated as fully humid sites or irri-gated during a well-defined growing season (Mead).This allows us to compare the vapor flux with PET inthe growing season, as P-T PET is expected to matchET only when water is not limiting over a large area(e.g., Brutsaert and Stricker, 1979). When water islimiting, advection will play a dominant role in ET,and we would expect P-T PET to underestimate PETand ET from an irrigated surface, and overestimateET from a dry surface (e.g., Shuttleworth and Calder,1979). We accessed hourly or 30-min temperature,radiation, ET (vapor flux), and precipitation data

TABLE 1. AmeriFlux Sites Used in This Study. Only days that were missing less than 5%of data for evapotranspiration, precipitation, RN, and T were used.

Site Location Description Years (# days used) Citation

Morgan Monroe State Forest Indiana, USA. Latitude: 38.32 Broadleaf forest 2001-2004, 2007-2009(2,100 days)

Schmid et al. (2000)

Mead-irrigated continuousmaize

Nebraska, USA. Latitude: 41.17 Irrigated agriculture,conservationplow system

2001-2005 (1,648 days) Suyker and Verma(2009)

Bartlett Experimental Forest New Hampshire, USA. Latitude: 44.06 Mixed forest 2004-2009 (292 days) Jenkins et al. (2007)UCI 1964 wet (referred toas “Boreal Forest”)

Manitoba, Canada. Latitude: 55.92 Boreal forest,poorly drained

2004-2005 (434 days) Goulden et al. (2011)



from the four wet Ameriflux sites. From these data,we generated a daily dataset — omitting days thatwere missing more than 5% of these four parameters.

We compared the net radiation calculated using themethodology described above to measured RN at eachsite. The FAO and ASCE suggest a method for esti-mating RN using temperature data, so we also com-pared measured RN values at these sites to thismethod, which we call the ASCE method here for sim-plicity (ASCE-EWRI, 2005; Allen et al., 1998) (Fig-ure 1) (Equations A4-A6). Because the approachproposed here has a more physically based atmo-spheric emissivity component, and a tighter relation-ship with measured RN based on slope and R2, wecontinued our analysis of PET using only theapproach presented here.

We calculated daily P-T PET using both mea-sured radiation and using the methods we proposedfor estimating RN. We also calculated PET usingthe empirical Hargreaves (Hargreaves and Samani,1985), Hamon (1963), and Oudin (Oudin et al.,2005) Equations (A1, A2, and A3, respectively) andcompared them to the P-T RN estimated PET using

all days of the year with complete records. We alsocompared the temperature-based PET estimates tomeasured ET during the periods of the year wherewe expect the highest ET rates: in the three humidforested sites this was June 1-August 31 and at theMead-irrigated site this was during irrigation —between 60 and 100 days after first planting (Suy-ker and Verma, 2009). The four sites chosen for thisanalysis have humid summers or are irrigated, sowater limitations should not be a large control onET rates at these sites, and consequently we expectmodeled PET to be strongly related to measuredET.

In addition, we were interested in the sensitivity ofthese temperature-based methodologies to increasesin temperature. For this analysis, we used the datafrom the Mead-irrigated site, i.e., the most completedataset. We approximated the effect of warmer cli-mates by systematically increasing the observed dailyaverage (and max and min) temperature. Using thisadjusted input, we calculated the average PET usingthe five approaches: P-T with measured radiation,P-T with estimated RN, and the three empirical,

050

100

150

200

250

300

Mead

RMSE R2

ASCE RN modelProposed RN model

36 0.82 30 0.85

Morgan Monroe

RMSE R2


32 0.76 32 0.78

050

100

150

200

250

300

0 50 100 150 200 250 300

Bartlett

RMSE R2


34 0.68 37 0.69

0 50 100 150 200 250 300

Boreal

RMSE R2


41 0.74 38 0.77


ASCE regressionProposed RN regression

Measured RN Wm−2

Mod

eled

RN

W

m− 2

FIGURE 1. Comparison of Measured RN to Temperature-Based Estimates of RN. Gray triangles are the American Society of Civil Engineers(ASCE)/Food and Agriculture Organization method, and the black open circles are estimated using the methodology outlined here. Linearregressions of modeled RN to measured RN are shown as gray dashed lines for ASCE method and solid black lines for the proposed method.Slopes for the ASCE method range from 0.66 to 0.8; slopes of the proposed method range from 0.87 to 1.03. R2 is the coefficient of determina-tion; RMSE is the root mean square error (W/m2).



temperature-based equations. This is an oversimplifi-cation of the changes we can expect under climatechange — we are not considering changes in diurnaltemperature range, or the differential increases intemperature by season. Thus, here we only show thesensitivity of these methods to a uniform increase intemperature.

The equations and R computer code used in thisanalysis can be found in the EcoHydrology packagein R (R Core Team 2012, R: A language and environ-ment for statistical computing, accessed 2012, http://www.R-project.org) (Fuka, D.R., Walter, M.T., Archibald,J.A., Steenhuis, T.S., and Easton, Z.M., EcoHydRology:A community modeling foundation for Eco-Hydrology.accessed 2012, http://CRAN.R-project.org/package=EcoHydRology).

RESULTS

We found good agreement (RMSE <1 mm) betweenP-T PET calculated using model-estimated and mea-

sured RN at each site (Figure 2). Linear R2 betweenradiation-dependent and temperature approximatedP-T PET varied from 0.76 at Bartlett Forest to 0.89at the Mead-irrigated site. When we removed dayswith nonzero precipitation values, the relationshipsimproved (Figure 2). By comparison, the empiricalHamon and Oudin PET estimates correlated less-well to the P-T PET using the measured RN

(RMSE ≥ ~1 mm, R2 between 0.59 and 0.84) (Fig-ure 3). The Hargreaves model did the best of thethree empirical models when compared to P-T PET,with RMSE between 0.93 and 0.97 mm, and R2

between 0.76 and 0.88 at the four sites. Overall, themethod of estimating net radiation for use in the P-Tequation had the best fit to the P-T method usingmeasured radiation values, with slopes closest to 1and lowest intercepts for this methodology (Table 2).

When comparing PET estimates to measured ETduring the growing season, the P-T PET using mea-sured RN consistently had the highest degree ofcorrelation, as expected (Figure 4). Our approachof using P-T PET with approximated RN had a stron-ger correlation with measured ET than the Oudinand Hamon empirical PET models (Figure 4 — note,

02

46

8 Mead Irrigated

RMSE R2

All daysRain−free days 0.69 0.92

0.84 0.89

Morgan Monroe Forest

RMSE R2


0.87 0.85

02

46

8

0 2 4 6 8

Bartlett Forest

RMSE R2


0.97 0.76

0 2 4 6 8

Boreal Forest

RMSE R2


0.79 0.83

Days with no rainDays with rain

P−T PET from Measured RN (mm/day)

P−

T P

ET

from

Mod

eled

RN

(m

m/d

ay)

FIGURE 2. Comparison of Priestley-Taylor (P-T) Potential Evapotranspiration (PET) Calculated Using Radiation BudgetApproximations vs. P-T PET Using Measured RN at Four AmeriFlux Sites. R2 is the coefficient of determination; RMSE is theroot mean square error (mm); dark symbols are PET estimates for rain-free days and light symbols are for days with rain.



only Hargreaves and Hamon models are shown,Oudin was similar to Hamon model, data not shown).In addition, in all four sites the slope of the regression

was higher using radiation-based methods to com-pare PET to ET relative to the empirical methods.The Boreal Forest site had a strong correlationbetween the P-T PET and measured ET, althoughthe actual ET was approximately one-third of the cal-culated P-T PET. This is consistent with previousresearch in coniferous ecosystems, which show thatET in these ecosystems tends to be very low due tostomatal resistance, but tightly coupled to radiationirrespective of water availability (e.g., Br€ummeret al., 2012).

As we expected, the Mead-irrigated site had goodmatch between RN-based P-T PET and ET, as thissite was manipulated to minimize water stress, i.e., itis an ideal case of abundant water availability. TheMead site also experienced days when ET valuesexceeded measured-RN-based P-T modeled PET,which is consistent with the advection-aridity theoryas irrigated areas surrounded by less humid areaswould be expected to have the additional dryingpower driven by the exchange of dry air over the irri-gated surface, and would approach the Penman PETestimation which incorporates the difference in vapor

02

46

8 Mead

RMSE R2

OudinHamon

Hargreaves 1.10 0.82 0.98 0.84

0.92 0.88

Morgan Monroe

RMSE R2

OudinHamon

Hargreaves 1.22 0.70 1.05 0.77

0.95 0.79

02

46

8

0 2 4 6 8

Bartlett

RMSE R2

OudinHamon

Hargreaves 1.70 0.59 1.41 0.67

0.97 0.76

0 2 4 6 8

Boreal

RMSE R2

OudinHamon

Hargreaves 1.33 0.67 1.33 0.67

0.83 0.82

OudinHamonHargreaves

P−T PET from Measured RN (mm/day)

Em

piric

al P

ET

(m

m/d

ay)

FIGURE 3. Comparison of Priestley-Taylor (P-T) Potential Evapotranspiration (PET) Calculated Using Empirical T-BasedPET Models vs. P-T PET Using Measured RN at four AmeriFlux Sites. R2 is the coefficient of determination; RMSE is the

root mean square error (mm). Black triangles are PET estimates using the Oudin equation (A3), dark gray open squares areestimates using the Hamon equation (A2), and light + symbols are PET estimates using the Hargreaves equation (A1).

TABLE 2. Slopes, Intercepts, and R2 of the T-BasedPotential Evapotranspiration (PET) Equationsand RN-Based Priestley-Taylor (P-T) PET.

Slope MeadMorganMonroe Bartlett Boreal Average

SlopeT-based P-T 1.05 1.00 0.94 0.89 0.97Oudin 0.74 0.74 0.57 0.58 0.66Hamon 0.63 0.59 0.44 0.49 0.54Hargreaves 0.84 0.71 0.71 0.91 0.79

InterceptT-based P-T 0.09 0.02 0.11 0.08 0.08Oudin 0.38 0.36 0.61 0.19 0.39Hamon 0.63 0.68 0.84 0.49 0.66Hargreaves 0.85 0.72 0.69 0.37 0.66

R2

T-based P-T 0.89 0.85 0.76 0.83 0.83Oudin 0.84 0.77 0.67 0.67 0.74Hamon 0.82 0.70 0.59 0.67 0.70Hargreaves 0.88 0.79 0.76 0.82 0.81



pressure between the surface and the air (Brutsaertand Stricker, 1979).

One important concern when using empirical, tem-perature-based models is the uncertainty of their per-formance under climate change. PET determined

using measured RN is much less sensitive toincreased temperatures than the empirically basedPET estimates — in particular the Hamon model(Figure 5). The P-T PET determined by estimatingthe components of the radiation budget through

02

46

8 Mead

slope R2

MeasuredRN

EstimatedRN

0.89 0.76

0.63 0.49

Mead

slope R2

Hamon

Hargreaves

0.34 0.33

0.45 0.50

02

46

8 Morgan Monroe

slope R2

MeasuredRN

EstimatedRN

0.91 0.72

0.74 0.38

Morgan Monroe

slope R2

Hamon

Hargreaves

0.18 0.06

0.39 0.20

02

46

8 Bartlett

slope R2

MeasuredRN

EstimatedRN

0.85 0.39

0.64 0.21

Bartlett

slope R2

Hamon

Hargreaves

0.25 0.09

0.42 0.18

02

46

8 Boreal

MeasuredRNTemp−approximatedRN

0 1 2 3 4 5 6 7

slope R2

MeasuredRN

EstimatedRN

2.98 0.74

2.58 0.56

Boreal

slope R2

Hamon

Hargreaves

0.90 0.31

2.09 0.53

0 1 2 3 4 5 6 7 8

Hamon Hargreaves

Measured ET (mm/day)

Mod

eled

PE

T (

mm

)

Empirical T−based PETPriestley−Taylor PET

FIGURE 4. Measured Evapotranspiration (ET) Compared to Four Potential Evapotranspiration (PET) Models during Peak Growing Season.Data for Mead-irrigated corn was between 60 and 120 days after first planting of corn, when irrigation occurred. For Morgan Monroe Forest,Bartlett Forest, and Boreal Forest, data are from the period between June 1 and August 31, when ET rates are expected to be closest to thepotential rate. In the Priestley-Taylor (P-T) PET graphs, the dotted line is the linear regression for the P-T PET using measured RN, and thedashed line is for P-T PET using estimated RN. In the empirical PET graphs, the dotted line corresponds to the Hamon model (Equation A2)and the dashed corresponds to Hargreaves (Equation A1). All regressions are significant at the p < 0.01 level.



temperature is somewhat more sensitive to increasingtemperatures than the measured RN-based PET butsubstantially less sensitive than the Hamon andOudin equations (Figure 5). The Hargreaves equationshowed slightly lower sensitivity to increases in tem-perature than the temperature-based P-T method. Inour analysis, the P-T PET determined using mea-sured RN implicitly assumed no changes in RN withincreased temperatures whereas our RN estimateschanged with increasing temperatures. Indeed, onewould expect longwave radiation to be higher underhigher temperatures. Thus, the P-T PET based onmeasured RN in Figure 5 should probably be consid-ered as a lower bound of PET under increasingtemperatures.

DISCUSSION

One of the dangers of empirical models is theirapplication in situations for which they were notdeveloped or tested. This is a particular problem forwatershed modelers who need to estimate ET fromPET, as watersheds as a rule generally contain multi-ple land cover types, most of which will likely be dif-

ferent from that used to develop an empirical PETmodel. With this in mind, it is interesting that theHargreaves model performed so well for the studysites as it was developed in the context of a referencecrop, not a forest. The Hamon and Oudin methodswere developed for whole watersheds, so they maynot have been appropriate for the eddy-flux-towerscale measurements used here. Because the P-T PETmodel is more physically based, it does not have thesesame scale or land cover issues.

The model presented here is based on a well-understood physical relationship between availableenergy and equilibrium ET, and matched radiation-estimated and measured summer ET as well or betterthan the other models. Of the empirical models, theHargreaves model performed the best, with correla-tions to observed ET during the summer similar tothe T-based P-T method we described. However,despite reasonably strong correlation with summerET values, for all sites except the Boreal Forest, theslope of the relationship for Hargreaves was consider-ably below one (0.39-0.45), with estimated PET val-ues at the three non-coniferous sites consistentlybelow the 1-1 line at the high end (Figure 4). This isproblematic because ET is generally estimated formodeling purposes by scaling down PET estimatesbased on water availability and leaf area index (e.g.,Shuttleworth 1993; Neitsch et al., 2011). Thus, inmost situations, except for irrigated crops in highlyarid areas, where advection would play a large rolein evaporation, we should expect estimated PET to begreater than measured ET.

In the proposed T-based P-T methodology, themost difficult part of the energy budget to modelwas estimating atmospheric transmissivity to solarradiation, for which we do not have a physicallybased equation. However, Bristow and Campbell(1984) did provide some physical meaning to theexponential equation used to model this. Using mea-sured incoming solar radiation, we calculated theactual daily transmissivity of a site by dividing themeasured solar radiation by the potential solar radi-ation at the edge of the atmosphere (Equation 4).Our modeled transmissivity was correlated with thismeasured transmissivity with R2-values between0.26 (Bartlett Forest) to 0.48 (Mead irrigated). Themodeled transmissivity matched the measuredvalues more accurately between June-August(R2 = 0.46 at Morgan Monroe to 0.61 at Bartlett For-est), i.e., this period is when transmissivity wouldhave its greatest impact on RN as solar radiationand atmospheric temperature are highest at thistime. Interestingly, Hargreaves and Allen (2003)note that the Bristow-Campbell transmissivity equa-tion was based on a relationship built into the Harg-reaves methods.

0 2 4 6 8 10

020

4060

80

T increase (degrees C)

Per

cent

Incr

ease

in P

ET

P−T assuming unchangingRN

P−T using temp to estimateRN

HamonOudinHargreaves

FIGURE 5. The Sensitivity of the Five Potential Evapotranspiration(PET) Models to Increasing Temperature. The classic Priestley-Taylor (P-T) estimate (solid line) which is based on measured RN

assumes that net radiation would not change under warmer scenar-ios. The black dashed line represents the methodology proposed here(P-T PET using estimated RN), the dotted dark gray line representsHamon, the light gray dot-dash line represents Oudin, and the inter-mediate gray large dashed line represents Hargreaves. Adapted fromShaw and Riha (2011).



In addition, transmissivity modifies two energyfluxes in opposite directions, which minimizes theerror from this term. Atmospheric longwave radia-tion decreases with increasing transmissivity, whilesolar radiation increases. Thus, on days that weoverestimate solar radiation due to inaccuratelymodeled transmissivity, we would also expectto underestimate downward longwave radiation,and vice versa. This helps to explain how RN

was able to be reasonably approximated despitedepending on transmissivity, which was difficult tomodel.

Empirical models become even more problematicwhen modelers attempt to consider PET in the con-text of climate change, both considering pastchanges and projections into the future climate. Inboth cases, we have no way of knowing if the empir-ical relationship between temperature and PET willbe valid under these conditions. An interestingresult found here is that the empirical Hargreavesequation is slightly less sensitive to increasing tem-peratures, assuming an unchanging diurnal tempera-ture range, than the T-based P-T formulation. TheHargreaves equation is dependent on both the diur-nal temperature range, which we did not change forthis analysis, and average daily temperature, whichincreased. In addition, the equation incorporatespotential solar radiation, which moderates theimpact of an increase in temperature on the calcu-lated PET. Likewise, the T-based P-T methodologyproposed here also incorporates the diurnal tempera-ture range in calculating transmissivity using theBristow-Campbell formulation. However, atmo-spheric emissivity is also dependent on temperature,and will increase due to increasing estimated vaporpressure under higher daily minimum temperatures.Thus, both the slope of the saturation vapor pres-sure term (D) and the estimated net radiation (viaincreasing proportion of downward longwave radia-tion) are sensitive to increasing temperatures inEquation (1), making this formulation more sensitiveto increasing average (and minimum) temperatures,when everything else is held constant. We hope thatthis study at least challenges the notions held bymany watershed modelers that they are limited toill-suited, temperature-based models because of themisconception that more physically based, energy-budget models require input data that are unavail-able (e.g., Frankenberger et al., 1999; Burns et al.,2007; Bae et al., 2011).

In this study, we used the standard Priestley-Taylor constant of 1.26, which has been supported bya number of studies of wet environments (e.g., Priest-ley and Taylor, 1972; Davies and Allen, 1973) andhas some physical justification (Eichinger et al.,1996). However, the results from the Boreal Forest

illustrate the importance of a more complete under-standing of ecosystem controls over actual ET. Anumber of studies of ET in coniferous boreal ecosys-tems (e.g., Saugier et al., 1997; Br€ummer et al., 2012)have shown that coniferous Boreal Forests transpireat a markedly reduced rate compared to the availableenergy for PET, irrespective of water availability.Specifically, Komatsu (2005) found an average calcu-lated P-T constant of 0.55 in evergreen Boreal Forestsin 10 sites due to decreased surface conductance fromstomatal control. Zha et al. (2010) also found a mark-edly lower P-T constant for boreal coniferous forests,and found net radiation and temperature, not waterlimitations, were driving variations in ET at thesesites. These results underline the importance of athorough understanding of biological controls exert-ing an influence on actual ET in addition to the mete-orological controls over PET. In the particular case ofa coniferous Boreal Forest ecosystem, it would be use-ful to employ a modification in the P-T method toaccount for the stomatal control over actual ET, orfurther modify the resulting PET using crop coeffi-cients which are generally employed in watershedmodels to incorporate PET values into the waterbudget.

We also applied the Penman equation to thesestudy sites using our temperature-estimated RN andassuming a constant wind speed of 2 m/s forthe advection term, as suggested by Allen et al.(1998). The results were similar to our P-T PETresults and, in some cases, a little better (data notshown). This suggests, although speculatively atthis point, that more mechanistic PET equationsthat include advection may also be usable evenwhen measurements of wind and vapor pressureare unavailable.

CONCLUSION

Hydrologic modelers may not be justified in usingempirical temperature-based PET equations based onthe premise that more physically based energy-budget models require too many input data that arenot readily available. We present a suite of approxi-mations for estimating the net radiation that gener-ally have well-founded physical basis. Of course, weencourage modelers to consider many of the methodsthat have been proposed for various energy-relatedparameters, many of which are summarized in Allenet al. (1998) and ASCE-EWRI (2005). Using physi-cally based approaches may be especially importantwhen considering PET in the context of climatechange.



APPENDIX

Hargreaves PET method (Hargreaves and Samani,1985):

PET ¼ KETSPDT0:5ðT þ 17:8Þ ðA1Þ

where PET is in mm/day; SP is the potential solarradiation (kJ/m2/day); KET is the calibration coeffi-cient, here 9.38E-7 m2/MJ; DT is the daily tempera-ture range (Tmax-Tmin) (°C); T is the temperature (°C).

Hamon PET method (1963):

PET ¼ 29:8DesatðTÞ

ðT þ 273:2Þ ðA2Þ

where PET is in mm/day; D is the number of daylighthours; T is the temperature (°C); esat is the saturationvapor pressure (kPa) at the air surface temperature.

Oudin PET method (2005):

PET ¼SP

qkTþ5100

� �; if T[ � 5�C

0; otherwise

�ðA3Þ

where PET is in mm/day; SP is the potential solarradiation at the edge of the atmosphere (MJ/m2/day);T is the temperature (°C); q is the density ofwater (1,000 kg/m3); k is the latent heat flux (2.5 MJ/kg).

ASCE Radiation Methods:ASCE transmissivity (ASCE-EWRI, 2005):

TrASCE ¼ KRs

ffiffiffiffiffiffiffiDT

pðA4Þ

where TrASCE is the transmissivity (�); KRs is theadjustment coefficient, here 0.16 (°C�0.5); DT is thedaily temperature range (Tmax-Tmin) (°C).

ASCE net longwave radiation (ASCE-EWRI, 2005):

LWN;ASCE ¼rð1:35TrASCE � 0:35Þð0:34� 0:14ffiffiffiffiffiffiffiVP

pÞ

� T4max þ T4

min

2

�ðA5Þ

where LWN,ASCE is the net long-wave radiation (kJ/m2/day); r is the Stefan-Boltzmann constant (4.89 9

10�6 kJ/m2/K4/day); TrASCE is the transmissivity,calculated above (Equation A4); VP is the vapor pres-sure (kPa) calculated from minimum temperature(Equation 8); Tmax, Tmin are the temperature (°K).

ASCE Net Radiation (ASCE-EWRI, 2005):

RN;ASCE ¼ ð1� aÞTrASCESP � LWN;ASCE ðA6Þ

where RN,ASCE is the net radiation using the ASCEmethod (kJ/m2/day); a is the albedo, here 0.23 (unit-

less); SP is the potential solar radiation (Equation 4,kJ/m2/day).

ACKNOWLEDGMENTS

We thank the FluxNet network and the many scientists whohave worked to put their data online to be available for others touse. In addition, we thank the funding sources that support thework of scientists to gather eddy-covariance measurements andmake them available publicly, including DOE-Terrestrial Ecosys-tem Sciences. We also gratefully acknowledge the two anonymousreviewers whose thoughtful comments helped to greatly improvethe content of this paper.

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