A method to calculate heterogeneous evapotranspiration ...Citation: Moffett, K. B., and S. M....

A method to calculate heterogeneous evapotranspirationusing submeter thermal infrared imagery coupled to astomatal resistance submodel

Kevan B. Moffett1 and Steven M. Gorelick1

Received 12 January 2011; revised 6 December 2011; accepted 16 December 2011; published 31 January 2012.

[1] Thermal infrared (TIR) remote sensing of vegetation temperature, combined withsurface energy balance modeling, allows efficient estimation of spatially distributedevapotranspiration (ET). Many ET models are sensitive to the parameterization ofstomatal control; yet, modelers often employ spatially uniform stomatal resistancevalues, even in distributed applications. Unfortunately, assuming uniform resistanceacross a canopy with large temperature variance is physically unrealistic and may produceartifacts in ET magnitude. To account for spatial variations in stomatal control that likelyaccompany temperature variations, we propose nesting a new submodel within somewell-established ET models. The submodel derives, for the canopy patch of interest, aconcave-downward relationship between stomatal conductance and temperature, asexpected from plant biology. Using the submodel, each pixel’s contribution to the totalcanopy patch ET is influenced both by its observed temperature and by its location-specific estimated stomatal resistance. The submodel requires only one more parameterthan the unmodified ET models, which can be obtained from the literature; it conservesenergy between the pixel and image scales, unlike single-valued resistance approaches;it produces realistic ET values at extreme temperature locations; and provides a remotesensing-based way to estimate the in situ canopy stomatal conductance-temperaturerelationship, which otherwise must be measured under controlled conditions. Since veryhigh-resolution TIR data provide one means to observe large temperature variance, thesubmodel was tested using data with cm-scale pixels collected over 1.5 m2 patches of twovegetation types. The biophysical relationships derived by the submodel were successfullyverified against laboratory data.

Citation: Moffett, K. B., and S. M. Gorelick (2012), A method to calculate heterogeneous evapotranspiration using submeter thermal

infrared imagery coupled to a stomatal resistance submodel, Water Resour. Res., 48, W01545, doi:10.1029/2011WR010407.

1. Introduction[2] Thermal infrared (TIR) remote sensing of land sur-

face temperature enables spatially distributed evapotrans-piration rates to be calculated across vegetation canopiesat various scales, according to the extent of the remotesensing data. Combination evapotranspiration models inthe tradition of Penman–Monteith [Monteith, 1965] andtwo-source models after Shuttleworth and Wallace [1985]are some of the most prevalent approaches to do so.Applied to remote sensing data, these models solve thesurface energy balance and turbulent heat transport equa-tions for each pixel in the TIR image. The TIR pixels oftenrange from submeter scale [Jones et al., 2002; Loheideand Gorelick, 2005; Shimoda and Oikawa, 2008; thisstudy] to subfield scale [e.g., Blonquist et al., 2009] to 60m for Landsat-7 data [e.g., Anderson et al., 2004], and

even larger for other satellite platforms [e.g., Kustas et al.,2004]. Although TIR-based evapotranspiration models areoften applied to coarse scale, low-resolution satellite data,the model adjustments proposed by this study will bedemonstrated using submeter, high-resolution TIR data.The rationale for examining this fine scale and the poten-tial applicability to coarser scales of general interest aredescribed in due course.

[3] Because one generally cannot measure all evapo-transpiration model parameters at the same resolution as theTIR imagery, one typically assumes coarse scale homoge-neity in many parameters (e.g., total radiation, ground heatflux, humidity, wind speed, canopy height, and stomatal re-sistance). Unfortunately, this assumed homogeneity may becontrary to real and important land surface heterogeneity inone or more of these parameters. For example, if TIR datareveal substantial heterogeneity in canopy surface tempera-ture, it is then desirable to account for the likely concurrentheterogeneity in temperature-related components of the sur-face energy balance.

[4] In practice, spatially distributed TIR-based evapo-transpiration estimates are usually based on only one valueof stomatal resistance for all TIR pixels in a canopy,

1Department of Environmental Earth System Science, Stanford Univer-sity, Stanford, California, USA.

Copyright 2012 by the American Geophysical Union0043-1397/12/2011WR010407

W01545 1 of 18

WATER RESOURCES RESEARCH, VOL. 48, W01545, doi:10.1029/2011WR010407, 2012

http://dx.doi.org/10.1029/2011WR010407

regardless of pixel temperature variations. This practiceresults in local evapotranspiration rates that vary monotoni-cally with temperature. In some combination models, suchas the Penman-Monteith model (Figure 1a [Monteith,1965]) and the Jarvis-McNaughton/Priestley-Taylor com-bined model (introduced presently [Priestley and Taylor,1972; Jarvis and McNaughton, 1986]), evapotranspirationincreases monotonically with increasing temperature, allelse being equal. This model behavior contrasts with one’sgenerally correct intuition that higher-surface temperaturesshould result in higher sensible heat flux and lower evapo-transpiration. In other models, such as the Shuttleworthand Wallace [1985] model (also introduced presently),evapotranspiration decreases monotonically with increas-ing temperature, sometimes achieving anomalously nega-tive evapotranspiration values at high temperatures. Ineither case, the combination models fail to capture the non-linear relationship between evapotranspiration and temper-ature expected from plant biophysics when they areevaluated using one value of stomatal resistance to calcu-late surface evapotranspiration across a range of surfacetemperatures.

[5] Plant biophysics indicates that stomatal conductance(the inverse of stomatal resistance) and temperature covaryin a predictably nonlinear, concave-down manner, withlower stomatal conductance values occurring at high- andlow-canopy temperatures (Figure 1c) [Jarvis, 1976; Armondet al., 1978]. Because canopy temperature is significantlyeasier to measure than stomatal conductance, determiningthe shape of this concave-down biophysical relationshipfrom the TIR data alone would usefully inform spatially dis-tributed evapotranspiration calculations: this is the aim ofthis study. If one accounts for this nonlinearity, leaf temper-ature and modeled evapotranspiration should covary in amore biophysically realistic manner (Figure 1d).

[6] Conventional evapotranspiration models are highlysensitive to the canopy stomatal resistance parameter [Beven,1979; Raupach, 1998]. This sensitivity is especially pro-nounced for the low resistance, moderate leaf temperatureconditions characteristic of efficiently functioning leaves(Figure 1b) and when the other resistances to vapor trans-port, the aerodynamic and boundary layer resistances, arelow [Jarvis and McNaughton, 1986]. Under these conditions,given observed spatial variation in canopy temperature, it isespecially desirable to account for concurrent spatial varia-tions in stomatal resistance. Unfortunately, stomatal resist-ance measurement methods are labor-intensive and limitedto leaf scale, typically environmentally controlled conditions[LI-COR, 2005; Leinonen et al., 2006; Guilioni et al.,2008], so it is impractical to measure the stomatal resistanceof every pixel location using current technology.

[7] Stomatal conductance may covary with many otherenvironmental variables besides leaf temperature and evap-otranspiration rate, such as total incident radiation, vaporpressure deficit, soil moisture, ambient CO2 concentration,leaf age and nutrient status, and plant acclimation history.Yet, nonlinear coupling among stomatal resistance, leaftemperature, evapotranspiration, and canopy energy bal-ance is an intrinsic aspect of canopy physiological function[Jarvis, 1976; Farquhar and Sharkey, 1982; Collatz et al.,1991]. Leaf temperature observations integrate all the con-tributing variables into one final symptom of the local

Figure 1. (a) Evapotranspiration over a range of canopytemperatures, assuming constant stomatal conductance of gst

¼ 0.0012 m s�1. Evapotranspiration curves in (a–d) arebased on the Penman-Monteith combination model [Mon-teith, 1965] using parameters: Ta ¼ 25�C, RH ¼ 80%, u ¼2.5 m s�1, P ¼ 101325 Pa, A ¼ 600 W m�2, � ¼ 2.45 � 106

J kg�1, cp ¼ 1013 J/kgC, �w ¼ 998.2071 kg m�3. (b) Evapo-transpiration (E) rates over a range of stomatal conductancevalues, assuming constant canopy temperature of T ¼ 20�C.(c) Concave-down variation of stomatal conductance withtemperature [Jarvis, 1976] for two canopy types: solid linebased on Salicornia virginica, gst,solid (T) ¼ �(4.3 �10�6)(T)2 þ (2.15 � 10�4)(T) – (1.17 � 10�3); dotted linegiven by gst,dotted (T) ¼ �(6.4 � 10�6)(T)2 þ (3.2 �10�4)(T) – (3.15 � 10�3). (d) Evapotranspiration over arange of canopy temperatures if stomatal conductance varieswith temperature as in Figure 1c compared to if stomatalconductance is fixed as in Figure 1b. (e) Example distributionof pixel-level canopy surface temperatures within a TIRremote sensing image. (f) Distributions of pixel-level evapo-transpiration values given the distribution of temperatures inFigure 1e used as input to the three different models inFigure 1d.

W01545 MOFFETT AND GORELICK: MAPPING ET AND STOMATAL RESISTANCE WITH TIR W01545

2 of 18

canopy energy balance. For example, low midday soil mois-ture may hydraulically limit the supply of water for transpi-ration, induce stomatal closure, or both. Such limitationswill prevent efficient canopy cooling via transpiration, soleaf temperatures will rise. However, if we know that leaftemperatures are elevated, we know that the leaves are notbeing adequately cooled, and so we do not need to knowprecisely how much soil moisture is available to understandthat transpiration is reduced. It is in this sense that leaf tem-perature is a symptom of the surface energy balance thatintegrates over other contributing factors. Other causes ofhigh apparent leaf temperatures include partial leaf senes-cence, standing dead plant matter, and elevated canopy posi-tions incurring high radiation loads. However, leaftemperature is known to be more sensitive to conductancethan to other variables such as leaf size or absorptivity forboth sunlit and shaded leaves [Smith and Nobel, 1977].

[8] The severity of the discrepancies in pixel-level evap-otranspiration estimates caused by not accounting for thenonlinear stomatal conductance-temperature relationshipare schematically represented at different temperatures bythe vertical distance between the solid curve and dashedline in Figure 1d. To reduce these discrepancies, in thisstudy we impose two new constraints on some existingevapotranspiration models. First, stomatal conductance andtemperature should vary in a biophysically realistic man-ner, as in Figure 1c. Second, total evapotranspiration flux(energy) should be conserved across scales: this would beschematically represented by the area under one of thecurves in Figure 1f matching the total evapotranspirationcalculated by an unmodified, conventional evapotranspira-tion model for the same canopy area.

[9] A heterogeneous TIR image, containing thousands tomillions of pixels, will exhibit some variance among itstemperature values (e.g., Figure 1e). Because variance nat-urally increases as resolution increases and more extremevalues are resolved rather than averaged out [Journel andHuijbregts, 1978; Isaaks and Srivastava, 1989], we shouldparticularly seek to account for spatial variations in surfaceconditions when studying the surface energy balance atfine spatial scales. Hence, as a first step in this direction,this study develops a modeling methodology incorporatingthe above constraints using example data collected at 1-cmpixel resolution over small, 1.5 m2 canopy patches of twovegetation types. This demonstration scale also permittedverification of the stomatal conductance-temperature rela-tionships derived by the new method against laboratorydata collected at a comparable (cm) scale.

[10] We acknowledge that the coupling of evapotranspira-tion, stomatal conductance, and leaf temperature can besimulated in great detail by biochemical photosyntheticassimilation-stomatal conductance models [e.g., Farquharet al., 1980; Collatz et al., 1991]. However, applying bio-chemical models over large land areas still requires assum-ing homogeneity in the many parameters used tocharacterize photosynthetic assimilation rates, many moreparameters than are required by combination models or byour method. In this study, our intention is only to improveupon the erroneous assumption of homogenous canopy sto-matal resistance in the face of observed canopy temperaturevariations. Our method provides a hybrid approach betweencomplex biochemical canopy models and more approximate,

but generally useful, homogenous-canopy combination mod-els [Raupach and Finnigan, 1988].

2. Method: Spatially DistributedEvapotranspiration Calculations Via CanopyResistance Mapping2.1. Overview of Two-Source Model Adaptations

[11] Our distributed canopy resistance and evapotranspi-ration mapping method is based on the two-source evapo-transpiration model by Shuttleworth and Wallace [1985].We also test an alternative two-source model combiningthe canopy-airstream decoupling method of Jarvis andMcNaughton [1986] and an estimate of soil evaporation[Priestley and Taylor, 1972]. Both two-source models arebased on the premise that the transpiring canopy surfacedoes not directly communicate with the above- and below-canopy air masses, i.e., the leaf surface and atmosphere areat least partially decoupled.

[12] In the conventional two-source model designed forapplication at the bulk canopy scale (Figure 2a) [Shuttle-worth and Wallace, 1985], the total transpiration (Ec) from ahomogenous ‘‘big-leaf’’-like canopy is coupled to the meancanopy airstream. Transpiration is governed by the gradientbetween the mean canopy airstream vapor pressure (e0) andthe saturated vapor pressure within the canopy ‘‘big-leaf’’(e�c). This gradient is scaled by the sum of the average can-opy surface resistance (rc

s , proportional to stomatal resist-ance) and the in-canopy aerodynamic resistance (rc

a). Soilevaporation (Es) is driven by the gradient between the meancanopy airstream vapor pressure (e0) and the saturated vaporpressure at the soil pore surface (e�s ), scaled by the sum ofthe soil surface resistance (rs

s) and the below-canopy aerody-namic resistance (rs

a). Solar radiation is partitioned betweenthe mean canopy level and the soil substrate via exponentialradiation absorption in the canopy [Shuttleworth and Wal-lace, 1985]. The mean canopy airstream connects thebelow-canopy soil fluxes and in-canopy vegetation fluxes tothe above-canopy airstream using an aerodynamic resistancemodel based on Monin-Obukhov similarity theory. The sumof the canopy transpiration and soil evaporation is the bulkcanopy evapotranspiration (E ¼ Ec þ Es), which is drivenby the gradient between the in-canopy airstream vapor pres-sure (e0) and the ambient airstream (ea), scaled by theabove-canopy aerodynamic resistance (ra

a). The variablescomprising this two-source model framework are listed, forreference, in the notation list.

[13] Our modeling approach maintains the same overallstructure as this conventional, canopy-scale, two-sourcemodel framework. We add three features in a submodelthat represents finer-scale details of the canopy energy bal-ance (Figure 2b), such as may occur at the cm-scale of theTIR canopy surface imagery examined in this study.

[14] 1. Rather than assuming that the canopy functionshomogenously with one average canopy temperature andone canopy stomatal resistance as in conventional models,our submodel considers the heterogeneity exposed by high-resolution thermal imaging. This thermal heterogeneityrepresents spatial variability in the surface energy balance.The submodel allows for many individual canopy leafsurfaces, represented by the TIR pixels, to simultaneouslycommunicate with the mean canopy airstream. The


3 of 18

submodel also allows each pixel-sized leaf surface to berepresented according to its own local leaf temperature andstomatal conductance (Figure 2b).

[15] 2. The submodel maintains continuity with conven-tional coarser-scale model frameworks by calculating aero-dynamic resistances and radiation partitioning at the canopyscale precisely as in the conventional frameworks. Thesevalues are then passed to the submodel. We require that thesum of the evapotranspiration fluxes for all TIR pixels cal-culated by the submodel (the aggregate evapotranspiration)equals the bulk evapotranspiration calculated using the con-ventional two-source modeling approach applied at the can-opy scale. This constraint mathematically conserves energyand water vapor across scales.

[16] 3. The submodel incorporates the nonlinear biophys-ical control of leaf stomatal aperture into the modelingframework via a simple quadratic relationship of stomatalresistance to temperature. The relationship is not specified a

priori: the submodel iteratively derives it. The relation-ship’s derivation requires only one model parameter beyondthose in the conventional two-source models, the tempera-ture of maximum stomatal conductance, which can typicallybe obtained from the literature. The derived stomatal con-ductance-temperature relationship then provides each pixellocation its own stomatal resistance value that correspondsto the observed temperature variability.

2.2. Submodel Procedure

[17] The five-step submodel procedure is illustrated inFigure 3, in which the circled numbers correspond to thefollowing steps. The variables are collected in the notationlist, for reference.

[18] Step 1: Select an evapotranspiration model.[19] The first evapotranspiration model we tested was

the ‘‘two-source’’ model of Shuttleworth and Wallace[1985] (hereafter S&W), explained in section 2.1. We

Figure 2. (a) The conventional two-source surface energy balance, evapotranspiration flux (E), andavailable energy (A) partitioning framework by Shuttleworth and Wallace [1985]. (b) The proposed sub-model is nested within the conventional framework and maintains the original vertical canopy structure,energy balance, flux, and radiation partitioning assumptions. The submodel permits many individual can-opy leaf surfaces (Ti) to communicate simultaneously with the mean canopy airstream (T0), subject toindividual leaf surface resistances (rc

si). The individual canopy leaf surfaces are represented by thermalremote sensing (TIR) image pixels. The submodel also constrains fluxes to conserve energy across scalesand constrains rc

si to be drawn from a biophysically realistic function rcsi(Ti), which the submodel derives.


4 of 18

algebraically reduced the S&W set of equations to oneequation for total evapotranspiration (Ej) :

EðS&WÞj ¼

ðraaAþ rc

aAcÞðrca þ rc

sjÞra

arcsj

��acpðTj � TaÞðrc

a þ rcsjÞ

raarc

sj

�rc

að�acp=�Þðe�j � eaÞra

arcsj

:

(1)

Using the calculated value of evapotranspiration (Ej), thecanopy transpiration (Ecj) and soil evaporation (Esj) werecalculated as

EðS&WÞcj ¼

ð�acp=�Þðe�j � eaÞðrc

a þ rcsjÞ

� raaEj

ðrca þ rc

sjÞ; (2)

EðS&WÞsj ¼ Ej � Ecj: (3)

Additional parameter definitions not in section 2.1 orFigure 2 were the radiation available at the canopy surfaceAc ¼ Rn � Rn exp ð�� LAIpÞ, the radiation available atthe soil surface As ¼ Rn exp ð�� LAIpÞ � G, the totalavailable radiation A ¼ Rn � G ¼ As þ Ac, the density ofair �a ¼ ð3:486� PÞ=ð275þ TaÞ=1000, the specific heatof air at constant pressure cp ¼ 1005 [J/kg/C], and thepsychometric constant � ¼ 0:000665� P:

[20] The flux resistances (r) were calculated usingthe approaches of Shuttleworth and Wallace [1985].Calculation of the available radiation components (A,Ac, As) from the net radiation (Rn) was modified fromShuttleworth and Wallace [1985] to include the groundheat flux (G). The vapor pressure deficit (e�j � ea) andparameters �a, cp, and � were calculated based on theapproaches of Shuttleworth [1993] and Allen et al.[1998].

[21] The second evapotranspiration model we tested wasthe ‘‘decoupling coefficient’’ canopy model of Jarvis andMcNaughton [1986]. We chose the approach of Jarvis andMcNaughton [1986] because of their leaf versus canopyscaling analysis, their recommendation of the model asappropriate across a range of canopy scales, and theirmodel’s strong contrast with the S&W framework. TheJarvis and McNaughton model is very similar to the Pen-man-Monteith combination equation [Monteith, 1965].The decoupling approach of Jarvis and McNaughton isalso amenable to having our canopy surface submodelnested within its broader, canopy-scale model framework.Unlike in the S&W model, the canopy temperature de-pendence of the Jarvis and McNaughton model occurs viathe slope of the saturation vapor pressure curve (Dj) at thespecified leaf temperature (Tj) [Allen et al., 1998]. Theother parameters are as in the S&W model. To enhancecomparison of Jarvis and McNaughton’s one-source can-opy model with the two-source approach of S&W, weused only the canopy-level available radiation fraction(Ac) in the Jarvis and McNaughton [1986] canopy transpi-ration model:

EðJ&MÞcj ¼ Ac�j þ �acpðe�a � eaÞ=ra

a

�j þ �ð1þ ½rcsj=ra

a�Þ: (4)

We supplemented the canopy transpiration with additionalsoil evaporation calculated using the Priestley and Taylor[1972] model for a soil surface at air temperature (subscript‘‘a’’) :

EðJ&MÞsj ¼ 1:26

�a

�a þ �As: (5)

The Priestley-Taylor model is appropriate for the largelysaturated wetland surface of our study. The support area ofour demonstration examples was very similar to that of thelysimeters used by Priestley and Taylor in their originalmodel verification [Priestly and Taylor, 1972]. Other soilevaporation equations could be used in other cases. Thecombined Jarvis and McNaughton/Priestley and Taylormodel is hereafter referred to as J&M.

Figure 3. Schematic of submodel procedure. Pixel-levelstomatal resistances (rst,i ¼ 1/gst,i) are estimated based onobserved heterogeneity in canopy surface temperature (Ti)and an unknown biophysical relationship between stomatalresistance and temperature, which is derived by the submo-del. Circled numbers correspond to procedure stepsdescribed in section 2.2.


5 of 18

[22] Step 2: Calculate the bulk evapotranspiration of thecanopy using the unmodified, conventional two-sourcemodel.

[23] Calculating bulk canopy evapotranspiration (Ej ¼Ebulk) using a conventional two-source model [e.g., Shuttle-worth and Wallace, 1985; Norman et al., 1995] uses thebulk radiometric temperature of the canopy (Tj ¼ Tbulk) anda representative value of bulk canopy surface resistanceprovided by the user (e.g., from the literature or averageporometer data) (rc

sj ¼ rcs rep). Our approach trusts that this

bulk evapotranspiration value (Ebulk) is accurate, as calcu-lated by the unmodified, well-established evapotranspira-tion models. To calculate the bulk radiometric surfacetemperature Tbulk at the canopy scale, one averages theradiances of all the TIR pixels in the selected canopy imagearea (pixels i ¼ 1 to N) using the fourth-power meanStefan-Boltzmann law:

Tbulk ¼1

N

XN

i¼1

T 4i

!14

: (6)

The relationship (6) is similar to that used by Su et al.[1999, equation 32], Anderson et al. [2004, equation 1],and Liu et al. [2006, equation 3b] for flat terrain.

[24] Consistent with the conventional two-source approach,we account for potential canopy layering beneath each TIR-imaged pixel by calculating the representative canopy surfaceresistance (rc

s rep) from a representative value of bulk stomatalresistance (rst,rep) and the canopy’s projected leaf area index(LAIp) and stomatal ratio (s) [Shuttleworth and Wallace,1985]:

rcs ¼

rst

s� LAIp: (7)

The stomatal ratio is 1 for hypostomatous leaves and 2 foramphistomatous leaves.

[25] Step 3: Calculate the aggregate evapotranspirationof the canopy using the pixel-level submodel.

[26] Step 3a: Initialize the stomatal conductance-temper-ature relationship.

[27] The submodel estimates a concave-down stomatalconductance-temperature relationship for the observedcanopy conditions using as few parameters as possibleby assuming a quadratic approximation. The quadraticrelationship for the two-sided pixel-level leaf stomatalconductance (gst,i) and its reciprocal, stomatal resistance(rst,i), is

gst;i ¼ gst;m � !ðTi � TmÞ2 ¼1

rst;i

: (8)

The unknown parameters in (8) are: the maximum stomatalconductance (gst,m) and the parabola shape parameter (!).The canopy temperature at each pixel location (Ti) is pro-vided by the TIR imagery and the temperature at whichstomatal conductance is maximized (Tm) is provided by theuser from ancillary data or literature. Calculating the parab-ola shape parameter (!) is the objective of the submodel. Alarger value of the shape parameter results in the more

peaked, narrower relationship illustrated by the dashedlines in Figures 1c and 3 (box 3).

[28] The temperature (Tm) at which stomatal conduct-ance is maximized (gst,m) is the only parameter required bythe submodel, in addition to those in the conventional two-source approach. Since stomatal conductance and photo-synthetic carbon assimilation are approximately propor-tional [Ball, 1988], Tm is similar to the temperature atwhich assimilation is maximized. This temperature value isreadily obtained for most plant species and land coverclasses of interest from assimilation-temperature curves inthe literature [e.g., Antlfinger and Dunn, 1979; Giurgevichand Dunn, 1979; Berry and Björkman, 1980; Pearcy andUstin, 1984; Sage and Sharkey, 1987; Sellers et al., 1996;Kim and Lieth, 2003; Yamori et al., 2006]. A representativevalue of Tm may also be obtained from laboratory gas fluxmeasurements of the canopy of interest, which was theapproach used in our demonstration examples.

[29] The modeler already knows one solution to (8): thebulk stomatal resistance (rst,rep) and its corresponding tem-perature (Trep) used to parameterize the conventional bulkflux model (see step 2 and box 3 in Figure 3). AlthoughTrep was not a parameter explicitly used in the conventionalmodel framework, the rst,rep value selected by the user instep 2 must implicitly have a corresponding Trep value: forexample, the leaf temperature recorded by a porometerused to measure rst,rep. Our approach simply requires thatthe user explicitly acknowledge this assumed ‘‘representa-tive’’ temperature. Also, in the context of the quadraticmodel (8), the influence of the one specific (rst,rep, Trep) pa-rameter pair supplied by the user is reduced compared tothe conventional approach, which is a strength of the newmethod given likely uncertainty in these values. We substi-tute the known point (rst,rep, Trep) into (8) and algebraicallysolve for the unknown maximum conductance gst,m. (Trep

must be distinct from Tm.) Substituting for gst,m in (8), wecalculate stomatal resistance (rst,i) values for each pixel :

rst;i ¼1

1rst;rep

� �þ !ðTrep � TmÞ2 � !ðTi � TmÞ2

: (9)

Each stomatal resistance (rst,i) is scaled up to a local, pixel-level surface resistance (rc

si) using (7). Thus, (7) and (9)relate the pixel-scale canopy surface resistances (rc

si) to theTIR pixel surface temperatures (Ti) via a shape parameter(!) and five known scalars (rst,rep, Trep, Tm, s, and LAIp).The shape parameter (!) is derived in the next steps of thesubmodel.

[30] Step 3b: Estimate initial pixel-scale evapotranspira-tion values.

[31] An initial estimate of each pixel’s evapotranspira-tion (Ei) is calculated from the evapotranspiration model(equations (1)–(3) or (4) and (5)), using the TIR pixeltemperature (Ti) and its corresponding canopy resistancevalue (rc

si, estimated with (7) and (9)). Note that the pa-rameters ra

a, rca, A, Ac, As, �a, cp, and � remain at the bulk

canopy scale exactly as in conventional two-source mod-els [Shuttleworth and Wallace, 1985]. One applies theseparameters uniformly across the whole TIR-imaged can-opy to each pixel in the submodel, exactly as in conven-tional model applications.


6 of 18

[32] Step 3c: Aggregate the pixel scale evapotranspira-tion values by averaging.

[33] The initial pixel-level evapotranspiration estimatesare aggregated to the canopy (TIR image) level by arithme-tic averaging [Raupach, 1995]:

Eaggregate ¼ meanðEiÞ: (10)

[34] Step 4: Reconcile bulk and aggregate evapotranspi-ration values.

[35] The numerical objective of the submodel is tominimize the difference between the bulk (conventionaltwo-source canopy) and aggregate (fine-scale submodel)evapotranspiration rates by optimizing the !-shape parame-ter that defines a realistic stomatal conductance-tempera-ture relationship for the canopy of interest for observedconditions.

[36] Step 4a: Subtract bulk and aggregate evapotranspi-ration values to calculate discrepancy.

[37] The discrepancy (�) between the aggregate evapo-transpiration (Eaggregate) estimated by the submodel in step 3and the bulk evapotranspiration for the canopy (Ebulk)known from step 2 is

� ¼ Ebulk � Eaggregate: (11)

To conserve energy (evapotranspiration) across scales, �should be zero.

[38] Step 4b: Minimize aggregate evapotranspirationdiscrepancy by iteratively adjusting the biophysical rela-tionship in the submodel.

[39] Slight adjustments to the stomatal conductance-temperature relationship (e.g., solid versus dotted curvesin Figure 1c) result in notable differences in the aggregateevapotranspiration of the imaged canopy (e.g., integral ofsolid curve versus integral of dotted curve in Figure 1f).Taking advantage of this sensitivity, the submodel’s bio-physical relationship, pixel-level evapotranspiration val-ues, and aggregate evapotranspiration are progressivelyrefined by automated numerical iteration until the aggre-gate and bulk evapotranspiration values match. Many so-lution methods could be employed; we used and suggestthe Newton method. An initial value of ! on the order of10�6 is suggested, but sensitivity analysis should be con-ducted for each application and an absolute minimum of� sought.

[40] Step 5: Map evapotranspiration values and canopyresistances at the pixel scale.

[41] Once an optimal solution for ! has been obtainedby minimizing �, the spatially distributed, pixel-scale evap-otranspiration values (Ei) are mapped from the final resultsof the submodel. The optimized ! value provides anestimate of the biophysical stomatal conductance-leaf tem-perature relationship for the canopy under observed condi-tions, which is used to convert the remotely sensedtemperature field (Ti) into a high-resolution canopy surfaceresistance map. If the model separately represents canopytranspiration and soil evaporation components of totalevapotranspiration, these variables are also mapped at thescale of the TIR data [e.g., Shuttleworth and Wallace,1985; Norman et al., 1995].

3. Demonstration Examples[42] We demonstrate our stomatal resistance-mapping

submodel for two different vegetation types using cm-reso-lution TIR imagery of 1.5 m2 canopy patches. Testing thesubmodel with images at approximately leaf resolution(�cm) made conceptual interpretation in terms of leaf bio-physical processes straightforward and permitted valida-tion against laboratory measurements of leaf biophysicalfunction. Although the submodel is, in principle, validacross many scales since the physical requirement ofenergy (evapotranspiration) conservation is scale inde-pendent [Raupach, 1995], the submodel is likely to bemost useful in cases of high-canopy surface temperaturevariance. High-temperature variance may be produced byvarious circumstances differentially affecting canopy tem-perature (e.g., heterogeneous stomatal control, soil mois-ture, leaf nutrient status) and revealed by high-resolutionthermal remote sensing. Our cm-resolution, m-extent, dem-onstration examples follow others’ success in using TIRimagery to better understand fine-scale heterogeneous soilevaporation [Shahraeeni and Or, 2010] and plant transpira-tion relative to reference surfaces [Leinonen et al., 2006],and are consistent with the informative patch scale analysisof local land-air exchange feedbacks by Raupach [1998].

3.1. Data Collection

[43] The TIR remote sensing imagery was collected in anintertidal salt marsh in southern San Francisco Bay duringlow tide periods. The surface energy balance of the marshbehaves during low tide as if the marsh were a wet grass-land [Moffett et al., 2010]. Repeated TIR images of nearlymonospecific and closed canopy surfaces were collected:on 24 September 2008 for the C4 grass Spartina foliosa(cordgrass, Figure 4a) and on 26 September 2008 for the C3

succulent forb Salicornia virginica (also known as Sarco-cornia pacifica, pickleweed, Figure 4b). Imaging was froma ground-based tower using a MobIR M4 thermal camera(8–14 mm; Wuhan Guide Infrared Technology Co., Ltd.,Wuhan, China) inside a radiation shield at 3 m height. Thecamera field of view was 1.05 m � 1.40 m with 120 � 160pixels, for a pixel size of 0.87 cm � 0.87 cm (rounded to 1cm in this paper, for convenience). A metal ruler 32 mmwide was included in some images to verify the resolution(Figure 4c). TIR image brightness temperatures were con-verted to radiometric temperatures by adjusting for thereflected long wave radiation from the sky [Norman andBecker, 1995], as measured by a four-component net radi-ometer located �23 m from the imaged marsh locations(CNR1, Kipp and Zonen, Delft, Netherlands). Soil heat fluxwas measured at the same location as net radiation(HFP01SC, TCAV, and CS616, from Campbell Scientific,by Hukseflux, Delft, Netherlands) [Moffett et al., 2010].Meteorological data were collected every 10 min at aweather station 210 m away in an adjacent marsh: windspeed and precipitation at a height of 3 m and air tempera-ture, relative humidity, and barometric pressure at a heightof 2 m (HOBO weather station, Onset, Cape Cod, MA,USA).

[44] In principle, the greater the quantity and variance ofthe TIR data, the better the desired stomatal conductance-temperature relationship can be mathematically con-strained using our submodel. To increase the quantity of


7 of 18

data on which our solutions were based, we used multipleTIR images from different times of day. TIR data were col-lected in replicates every 30–40 min during each imagingday and a representative subset of images extracted for

analysis: 22 images of Spartina foliosa and 15 of Salicorniavirginica. For each canopy, we sought the stomatalconductance-temperature relationship that best fit all of the15 or 22 aggregate fluxes to their corresponding bulk fluxes.This resulted in the final derived relationships being basedon 422,400 data for Spartina foliosa (22 images � 19,200pixels) and 288,000 data for Salicornia virginica (15 images� 19,200 pixels). We constrained the pixel-level surfaceresistances (rc

si) to be positive while solving for !. Themodel parameters used in each case are listed in Table 1.

[45] To validate the stomatal conductance-temperaturerelationships derived by the submodel, we collected labora-tory gas flux chamber measurements of plant leaf stomatalresponse for each plant species. An individual of each spe-cies was taken from the field site with intact roots in August2008, shortly before the field study. The plants were kept inan outdoor greenhouse under natural light and wateredperiodically with water from the field site. Prior to meas-urements, the selected leaves/stems were wiped clean witha damp cloth and allowed to air dry. Leaves were subjectedto different air temperatures while photosyntheticallyactive radiation (PAR), influent CO2, and airflow rate wereheld constant. Measurement conditions for Spartina foliosawere (� 6 1�) : PAR 1500 6 1 mmol m�2 s�1, relative hu-midity 72% 6 19%, airflow rate 200 mmol s�1, and airtemperatures 18�C–34�C, resulting in leaf temperatures21�C–33�C. Measurement conditions for Salicornia virgin-ica were: PAR 1500 6 1 mmol m�2 s�1, relative humidity55% 6 18%, airflow rate 400 mmol s�1, and air tempera-tures 16�C–37�C, resulting in leaf temperatures 19�C –32�C. Leaf carbon dioxide assimilation and transpirationwere recorded using an open-path system (Licor LI-6400)once the fluxes had equilibrated to each change in tempera-ture, after 30–50 min. Stomatal conductance values werecalculated from these data [LI-COR, 2005].

3.2. Demonstration Results

[46] The canopy temperatures indicated by the TIR dataexhibited diurnal signals (Figure 5). Despite very similarweather and incident radiation on the two measurementdays, the temperature range within images of the succulentSalicornia canopy was less than the range within images ofthe Spartina grass canopy. The bulk evapotranspiration val-ues (Ebulk) simulated by the unmodified S&W and J&Mflux models responded to these diurnal signals in surfacetemperature and incident radiation.

[47] The biophysical relationships between stomatal con-ductance and temperature derived by the submodel werecomparable to those estimated from laboratory gas fluxchamber measurements (Figure 6). Scatter in the laboratorydata was due to noise among about 600 stomatal conduct-ance values for each plant species, obtained every 30 s. Therelationships derived by the submodel were similarly accu-rate, compared to the laboratory results, for the tall C4 grasscanopy (Spartina foliosa) and for the short C3 succulent forbcanopy (Salicornia virginica), providing evidence that themethodology is not dependent upon plant morphology orphysiology. The submodel converged to nearly identical val-ues of the shape parameter !, whether run within the S&Wor J&M model framework: ! � 3.15 � 10�6 for the Spar-tina foliosa canopy and ! � 2.45 � 10�6 for the Salicorniavirginica canopy (an average of S&W and J&M results).

Figure 4. Examples of high-resolution radiometric sur-face temperatures (Ti) of: (a) Spartina foliosa and (b) Sali-cornia virginica. Image dimensions: 160 � 120 pixels with� 1 cm pixel size. (c) Example raw TIR image of Spartinafoliosa and corresponding digital camera image (taken at10:34 A.M., shortly before image in Figure 4a). Metal ruler3.2 cm wide for scale, at temperature 28.3�C to 29.6�C.


8 of 18

The derived ! values were approximately half of those sug-gested by the quadratic fits to the laboratory data, but thederived curves closely matched the laboratory data over thecommon leaf temperature range of �22�C to 32�C. Withinthis temperature range spanned by the laboratory data, thecorrelation coefficients between the mean values of eachcluster of laboratory data and the modeled stomatal conduct-ance values were r ¼ 0.97 for Spartina foliosa (p ¼ 0.0062)and r > 0.99 for Salicornia virginica (p ¼ 0.00036).

[48] Applying the biophysical relationships derived bythe submodel to the TIR data, we made maps of spatiallyvariable estimated canopy surface resistance (rc

si) at thesame resolution as the observed variations in canopy sur-face temperature (Figure 7). The other model results weremaps of evapotranspiration (Ei), partitioned canopy tran-spiration (Eci), and soil evaporation (Esi) (Figure 8). Theanomalous negative evapotranspiration values calculatedby the S&W model for Spartina foliosa were due to

Table 1. Model Parameters for Demonstration Examples

Symbol Description Source

Value

Spartina foliosa Salicornia virginica

G Soil heat flux (W m�2) Empirical regression between Rn and Gmeasured by a net radiometer and a soilheat flux plate system in the field

G ¼ 0.26 Rn � 64

� In-canopy light extinction coefficient Standard assumption 0.5LAIp Projected leaf area index (LAI/2) Averages from Zhang et al. [1997] 1.2 2.2emc Canopy emissivity (used in sky radiation

correction)Standard assumption 0.98

zu Height of wind speed measurements Field measurement 3 md Roughness dimension of canopy (leaf width) Average field estimate 0.02 m 0.01 mhc Canopy height Average field estimate 0.5 m 0.35 mrb Total (two-sided) leaf boundary layer resist-

ance (s m�1)Laboratory data 29 s m�1 24 s m�1

rrep Total (two-sided) leaf stomatal resistance attemperature Trep

Laboratory data 682 s m�1 1135 s m�1

s Stomatal ratio [LI-COR, 2005] 1 2Trep Leaf temperature of ‘‘representative’’ stoma-

tal resistance measurement rst,rep

Laboratory data 23.4�C 20.0�C

Tm Temperature of maximum stomatalconductance

Laboratory data 29.3�C 25.0�C

Figure 5. Diurnal progression of canopy temperatures of (a) Spartina foliosa and (b) Salicornia virgin-ica. Vertical sets of gray dots mark the centers of 30 histogram bins spanning the range of temperaturesin each TIR image; the shades of the dots indicate the relative frequency of temperatures in the bins.Apparent data gaps are because of variable timing between TIR data acquisitions and are inconsequentialin this study. Dark line behind dots: average radiometric temperature of whole image (Tbulk). Meteoro-logical conditions illustrated: net radiation (Rn/10, solid line), air temperature (Ta, dashed line), relativehumidity (RH/10, dotted line), and wind speed (u, dashed-dotted line).


9 of 18

fundamental issues with the S&W conceptual model atextreme-temperature locations, not due to the canopy re-sistance mapping submodel, as discussed presently in sec-tion 4.2.

4. Discussion4.1. Significance of Using Stomatal ResistanceMapping Submodel in Combination Models

[49] The submodel proposed in this study was designedto ensure conservation of energy (evapotranspiration)between the pixel (leaf) and TIR image (canopy) scales andto modulate the evapotranspiration contributions ofextreme-temperature pixels via lower stomatal conductancevalues. But how important were these model revisions? Wequantitatively assessed the discrepancies in evapotranspira-tion estimates incurred by neglecting these considerationsvia a scaling analysis. We found that the discrepancy intotal evapotranspiration calculated using a conventionaltwo-source approach with one value of stomatal resistancewas scale dependent and proportional to the change in tem-perature variance between scales. The scale-dependent dis-crepancy in aggregate evapotranspiration was largely

eliminated by the submodel, since its specified mathemati-cal objective was to minimize such discrepancy.

[50] The scaling analysis used the bulk evapotranspira-tion for each TIR image as the basis for comparison (Ebulk),consistent with the expectation that the conventional mod-els perform accurately at the coarser (whole-image) scale.The finest scale of comparison was that of the original TIRdata. For this scaling analysis, pixel-level evapotranspira-tion values were calculated using the same spatially uni-form stomatal resistance value as in the bulk model, thenaveraged over all of the pixels in the canopy image toobtain Eaggregate, as in (10). We also recursively dividedeach TIR image in half to produce comparable data setsat eight intermediate resolutions. We calculated an evapo-transpiration flux for each subdivision of each imageusing average subdivision temperatures, similar to (6), andthe same spatially uniform stomatal resistance value aswas used at the coarse and fine scales. We aggregated thesubdivision fluxes for each image at each scale by arithme-tic averaging, as in (10).

[51] The scale-dependent discrepancies in total evapo-transpiration that were produced by applying the conven-tional, uniform-resistance models to high-variance TIRdata were more pronounced at finer scales and for TIR

Figure 6. Leaf stomatal conductance–temperature relationships. Quadratic relationships between leafstomatal conductance and leaf temperature are derived from laboratory measurements (solid lines, solidsymbols) and from the remote sensing method (dashed lines; combined Jarvis and McNaughton [J&M]and Shuttleworth and Wallace [S&W] model results nearly identical and plot together). Parameter valuesused in the method (Trep, rst,rep, Tm) indicated by open symbols.


10 of 18

images taken around mid-day (Figure 9). The sign differencebetween the S&W and J&M models suggested that, as the re-solution and variance of TIR data are increased, the aggre-gate evapotranspiration calculated from a J&M-like model

may appear to increase but the aggregate evapotranspirationcalculated from a S&W-like model may appear to decreaseunder the same conditions. In either case, the change in theevapotranspiration rates with changing resolution (when

Figure 7. Spatial variability in estimated canopy surface resistance (rcsi) among (a, c) Spartina foliosa

and (b, d) Salicornia virginica, derived by the submodel from the temperature fields in Figures 4a and 4b.Higher simulated rc

si corresponded with comparatively extreme (warm or cool) canopy temperatures(compare with Figure 4) as expected. Color scales differ between species to adequately illustrate rc

si variations.

Figure 8. Evapotranspiration fields for Spartina and Salicornia canopies calculated using the spatiallyvariable canopy surface resistances from the submodel in Figure 7 and temperature fields in Figures 4aand 4b. Pixel-level variables are: total evapotranspiration (Ei), canopy transpiration (Eci), and soil evapo-ration (Esi). Color scales differ to adequately illustrate variations. Soil Esi is single-valued in the J&Mmodel due to the single-valued modeling approach after Priestley and Taylor [1972]. Causes of verylarge and very small (negative) values calculated using the S&W model are assessed in section 4.2.


11 of 18

calculated using the same uniform resistance value) violatedthe conservation of energy across scales.

[52] The scale-dependency of these discrepancies in ag-gregate evapotranspiration was caused by the increasinginconsistency at finer scales between the single resistancevalue used and the more pronounced heterogeneity of finer-scale temperature fields. This conclusion was evidenced bythe strict proportionality between the aggregate evapotrans-piration discrepancy and the surface temperature varianceof each TIR image at different levels of subdivision (Figure10). The reduction in temperature variance at coarser reso-lutions scaled with the subdivision averaging dimension(inversely with n) as expected from regularization of ran-dom values [Journel and Huijbregts, 1978; Isaaks andSrivastava, 1989].

[53] The spatially variable stomatal resistance valuesderived by the submodel reduced the overall variance of themodeled evapotranspiration field, especially for the J&Mmodel. This variance reduction effect was mostly due to thesuppression of anomalously extreme evapotranspiration val-ues by lower conductance values imposed at warm tempera-tures (see Figure 1d). High-variance evapotranspiration

fields produced by the S&W model were partially due toanomalous negative values, not due to the submodel. (Seesection 4.2 regarding difficulties with applying the S&Wmodel to high-variance temperature fields.) Since evapo-transpiration should be relatively low when sensible heatflux is high, such as at locations with surface temperaturesgreatly elevated above air temperature, we deemed thesmaller evapotranspiration values calculated by the submo-del at such locations more realistic than the extreme evapo-transpiration values calculated by the conventionalapproach. The ranges of evapotranspiration values calcu-lated using the new submodel compared favorably with theranges of latent heat flux measurements made at a nearbyeddy covariance station on the same days [Moffett et al.,2010], although the great difference in support areas andmeasurement methods precluded direct comparison. Theevapotranspiration variance compression produced by oursubmodel within the J&M framework was also consistentwith the rapid, fine-scale regulation of water loss describedby stomatal optimization theory [Katul et al., 2010] in thatsuch regulation among a small, relatively homogeneous can-opy patch may be expected to homogenize water loss rates.

Figure 9. Scale dependence of aggregate evapotranspiration values for each model ((a, b) J&M modeland (c, d) S&W model) and each plant species (Spartina (Figures 9a and 9c) and Salicornia (Figure 9band 9d)). The magnitude of the discrepancy (�) in the aggregate evapotranspiration increased as thedimension of the scaled TIR input data decreased (i.e., resolution of the TIR pixels increased). Each linetraces the trend in � for one TIR image across 10 scales. The lines are color-coded by image time of day.Y-axes differ to maximize display.


12 of 18

4.2. Additional Model Considerations: Problems Witha Common Two-Source Surface Energy Balance Modelat Extreme Temperatures, High Resolutions

[54] According to our results, the two-source modelframework of Shuttleworth and Wallace [1985] results inanomalously extreme evapotranspiration estimates whenapplied to high-variance canopy temperature fields.Although one of the most widely applied frameworks forremote sensing-based surface energy balance calculations[Kustas and Anderson, 2009; Overgaard et al., 2006], weidentify two features of its mathematical implementationthat are poorly suited for analysis of especially warm orcool land surfaces. The two difficulties are intrinsic to themathematical model structure, and so do not depend on thescale of TIR imagery, only on the presence of especiallycool or warm pixels among the data. The difficulties werefeatures of the unmodified S&W model, not the result ofour submodel.

[55] The first difficulty with the S&W model was its esti-mation of negative evapotranspiration (condensation) atwarm canopy locations. This phenomenon has been noted

previously and has been overcome by reassigning negativefluxes a value of zero [Anderson et al., 2008; Normanet al., 1995], justified by the idea that very warm surfacesare likely dry and so evaporation and transpiration are neg-ligible. However, this justification was not applicable inour study of nearly saturated wetland soils and nonwater-limited vegetation. The model itself provides a mathemati-cal explanation for the negative simulated evapotranspira-tion values. Very-high canopy-surface temperatures (Tj)applied in the second term of (1) may easily drive the evap-otranspiration to negative values, given typical values ofother parameters. Even if high observed temperatures doindicate dry surfaces, for example, if there were partiallysenesced or dead portions of the imaged canopy, the con-cave-down stomatal conductance-temperature relationshipimposed by our submodel provides a new means to tuneevapotranspiration at warm canopy locations toward zero,as an alternative to ad hoc assignments of zero values.

[56] We suggest that this potentially problematic modelbehavior has not hindered the successful application ofS&W-like models in previous studies because such studies

Figure 10. Scaling of aggregate evapotranspiration discrepancy and surface temperature variance foreach model ((a, b) J&M model and (c, d) S&W model) and each plant species (Spartina (Figures 10aand 10c) and Salicornia (Figures 10b and 10d)). The magnitude of the discrepancy (�) in the evapotrans-piration of the canopy increased in proportion to the increase in the variance of the radiometric tempera-tures (Tj) within the canopy at finer scales. Each line traces the trend in � for one TIR image across 10scales. The lines are color-coded by image time of day. Axes differ to maximize display.


13 of 18

are mostly based on coarse-scale satellite TIR imagery.Coarse-scale observation captures less surface temperaturevariance by naturally averaging out extreme values, makingobservation of extremely warm surface temperatures atcoarse scales relatively rare. However, this difficultyappears to render this type of two-source surface-energybalance calculation unsuitable for applications in whichvery warm surface temperatures are resolved in the TIRdata. Problematically warm surface temperatures may beparticularly likely among fine-scale TIR imagery such asexplored in this study. The specific limits of applicabilityof the S&W modeling framework are an open question,especially the threshold conditions combining specific tem-perature extremes and concurrent meteorological andboundary layer influences.

[57] The second difficulty that we observed with theS&W model framework was that it can produce very largesoil evaporation values beneath especially cool portions ofthe canopy. This occurs in the model because the evapo-transpiration calculation can easily be dominated by thecanopy-air temperature difference (Tj � Ta in the secondterm of (1), which is magnified by a combination of resis-tances), rather than by the canopy-air vapor pressure gradi-ent (e�j � ea in the third term of (1)). Since the canopytranspiration fraction (2) is largely set by the vapor pressuregradient (e�j � ea), the soil evaporation fraction (3) canremain strongly dominated by the canopy-air temperaturedifference, leading to extremely large soil evaporation val-ues being calculated for especially cool canopy locations.This model result is problematic because the canopy islikely to be cool either due to a small supply of radiantenergy or because the incident radiation is very effectivelydissipated by latent and sensible heat fluxes from the can-opy; in either case, there would seem to be little energyavailable to drive exceptionally large soil evaporation. Thisdifficulty is implicit in the model’s design, as if an electri-cal circuit of two joined loops. Analysis of the circuit ana-log reveals that, for fixed resistance and vapor pressurevalues, if the canopy transpiration (Ec) becomes small, thesoil evaporation (Es) must increase to satisfy Ohm’s Lawand Kirchhoff’s Voltage Law. Soil evaporation may trulyincrease as indicated by the model if the canopy transpirationconcurrently tends toward zero due to canopy senescence(reduced LAI) or another exogenous factor, but there was noevidence for such conditions in our demonstration examples.We conclude that, irrespective of the scale of analysis, ifsimulated canopy transpiration is small due to a small vaporpressure gradient between the canopy and the air (e�j � ea),conceptual models similar to S&W may produce erroneouslyhigh values of soil evaporation at especially cool canopylocations, for some input parameter combinations.

4.3. Discussion of the Stomatal Resistance MappingSubmodel

[58] The evapotranspiration submodel developed in thisstudy produced three results not previously availablefrom TIR image analysis: pixel-level maps of evapotrans-piration downscaled from coarse-scale bulk values whileconserving energy across scales, biophysically realisticestimates of heterogeneous canopy surface resistance, andan estimate of the stomatal conductance-temperature rela-tionship of the canopy for observed conditions. The con-

servation of energy across scales and the derivation ofthis biophysical relationship are the major differencesbetween this method and other methods that have derivedhigh-resolution canopy resistances from TIR data [Boeghet al., 2002]. Notably, the new submodel derives the can-opy stomatal conductance-temperature relationship for insitu conditions and does not require one to assume thatthe in situ canopy behaves identically to leaves observedin the laboratory. A quadratic function was used in thisstudy for simplicity, but a cubic or quartic relationship(e.g., after Tenhunen and Westrin [1979] or Farquharet al. [1980], respectively) could be used with simpleadjustments to the numerical solution scheme: either bysupplying one or two additional pieces of information tothe relationship, or by optimizing the model (minimizing�) to fit two or three parameters controlling the shape ofthe stomatal conductance-temperature relationship.

[59] The overall accuracy and generality of the stomatalconductance-temperature relationship derived by the sub-model will depend on the soil moisture conditions and planthydraulic maintenance strategy at the time of imaging. Soilmoisture can influence evapotranspiration and leaf temper-ature and there are multiple plant strategies for managingwater pressure and stomatal aperture [Jones, 1998; Frankset al., 2007]. Anisohydric plants maintain low stomatalresistances, regardless of soil water potential and meteoro-logical conditions, until near plant hydraulic death. Isohy-dric plants maintain constant internal water pressure bydynamically adjusting stomata in response to atmosphericand soil moisture changes. In both cases, however, elevatedleaf temperature will be symptomatic of locally inefficientheat dissipation by sensible heat flux and transpiration. Inthis study, we assumed that observed spatial variability inleaf temperature at the cm scale of our TIR data was morelikely to be due to patchy stomatal control, which can varyat leaf-to-subleaf scales [Mott and Buckley, 2000], than tobe due to spatial variations in soil moisture. This assump-tion is justified for wet soil conditions, such as in ourexamples. However, heterogeneous or water-limited soilconditions may provide additional, nonstomatal controls ontranspiration and leaf temperature [Turner et al., 1985;McDowell et al., 2008]. Under these conditions, the submo-del would effectively combine soil moisture and stomatalcontrols into the derived ‘‘biophysical’’ relationship, reduc-ing the validity of the relationship’s interpretation instrictly botanical terms. The stomatal conductance-temper-ature relationship derived using the submodel thus repre-sents an effective relationship appropriate for the radiationand moisture patterns experienced by the canopy during theobservation period. However, we believe it will alsoinclude the effects of longer-term plant acclimation [Berryand Björkman, 1980; Matthews and Boyer, 1984], whichwe consider a useful feature of the method.

[60] The principle source of uncertainty in the submodelis the degree of variability in the TIR canopy surface tem-perature field. Greater observed temperature variance willmore tightly constrain the submodel’s numerical optimiza-tion. Relatively scarce extreme temperature values in ourTIR images likely account for some of the differencebetween the biophysical relationships derived in our dem-onstration examples and those from laboratory data, espe-cially at high and low temperatures (Figure 6). Without


14 of 18

increasing the TIR resolution, greater variance may beobtained by increasing the quantity of data: by time-lapseimaging, as in our examples, or by the combination ofregions of the same land cover or canopy type from differ-ent images. Combining images from different times of dayin the same analysis has the potential advantage of captur-ing some of the variations in canopy temperature that aredue to the three-dimensional canopy structure, but whichmay only be visible to the TIR sensor at certain solarangles. Of course, this same phenomenon of canopy struc-ture, solar angle, and occlusion or shading is also a sourceof potential evapotranspiration errors for any analysis ofhigh-resolution TIR data, which is no more mitigated byour approach than by existing methods. In combining datafrom more than one time or region, separate bulk evapo-transpiration values should be calculated using the appro-priate meteorological data for each contributing time orspatial region, then one stomatal conductance-temperaturerelationship may be sought that best matches all of the ag-gregate and bulk evapotranspiration values of each of thecontributing data blocks.

[61] In principle, we believe the submodel is not re-stricted to the submeter scale of our demonstration exam-ples. The demonstrations presented here are an initialproof-of-concept for the methodology at a scale permittingvalidation against laboratory data and straightforward con-ceptual interpretation of the derived conductance-tempera-ture relationship. Large temperature variance was impartedto our examples by high-imagery resolution, but we expectthe results would be mathematically equivalent at anyresolution given the same temperature field. Our scalinganalysis showed that discrepancies in evapotranspirationestimates are proportional to temperature variance across arange of scales (Figure 10). We expect that application ofthe method at coarser scales could result in a pseudo-bio-physical relationship of effective, coarse-scale canopy sur-face resistance versus coarse-scale radiometric temperaturefor the land cover of interest, but might also require atten-tion to possible spatial variations in variables such as soilmoisture and leaf-area index. Such coarse-scale tests of thesubmodel are the subjects of ongoing research.

[62] We have emphasized that our nested modelingapproach maintains continuity with well-established two-source surface energy balance models designed for applica-tion at the canopy scale or coarser levels, while accountingfor pixel-level spatial variations in surface temperature andstomatal conductance. One might suggest that a further de-parture from conventional model frameworks that alsoaccounts for spatial variations in radiation and meteorolog-ical variables would be desirable; however, it is not neces-sarily clear how each variable should be accounted for, ifnot already included in conventional two-source modelframeworks. Observed canopy surface temperatures aresymptoms of the balance of the dissipation of energy bysensible and latent heat fluxes and the total absorbed radia-tion, not just the shortwave photosynthetically active radia-tion [Pieruschka et al., 2010]. The total radiation within avegetation canopy may differ significantly from the illumi-nation by photosynthetically active radiation, especiallywhen one accounts for reradiation of thermal energyamong leaves within the canopy (e.g., see data by Teal andKanwisher [1970] for the Spartina genus of this study) and

for radiation feedback on the canopy energy balance [Rau-pach, 1998]. Considering canopy structure, deeper layersmay experience lower radiation loads, causing the modelsexamined in this study to overestimate evapotranspiration.However, deeper layers may also experience lower airtemperatures and wind speeds, causing the models to overes-timate sensible heat flux and underestimate evapotranspira-tion. To the degree that these opposing systematic errors maycancel, uncertainties due to assumptions of uniform meteoro-logical conditions within the canopy may be reduced.

[63] Some uncertainty may also be contributed to thesubmodel approach by the assumption that the bulk evapo-transpiration value (Ebulk) used to constrain the submodelwas calculated accurately by the unmodified, conventionalmodel. If Ebulk were uncertain because of insufficient confi-dence in the coarse scale model, it could instead be pro-vided by measurements. If Ebulk were uncertain because ofa poorly defined representative value of coarse-scale sto-matal resistance (rst,rep), this uncertainty could be mitigatedby adding to the submodeling approach an iteration loop atthe coarse scale. The additional loop would draw a newrst,rep value from the current state of the submodel and usethis value to re-estimate the Ebulk value used in the succes-sive coarse-scale model iteration.

5. Conclusion[64] In this study, we proposed that it is physically unre-

alistic to assume a uniform stomatal resistance throughouta canopy that exhibits large temperature variance and thatthis assumption may compromise TIR-based evapotranspi-ration calculations. We developed a submodel to estimatethe spatial variations in stomatal control that are likely toaccompany observed temperature variations. We nestedthis submodel within two well-established evapotranspira-tion models designed for use at the coarse canopy scale.Applying both the unmodified models and the new nestedsubmodel to two different vegetation types imaged at cmresolution over 1.5 m2 canopy areas, we showed that theunmodified, single-valued resistance approach failed toconserve energy between the pixel and image scales. Incontrast, the mathematical objective of the new submodelwas to conserve energy between scales, which it achievedsuccessfully in the demonstration examples. The modifiedmodels, which included the variable-resistance submodel,produced more moderate, physically realistic pixel-scaleevapotranspiration values at locations of extreme tempera-ture than the unmodified models. The submodel also pro-vided a new remote sensing-based means to estimate the insitu canopy stomatal conductance-temperature biophysicalrelationship, which otherwise must be measured under con-trolled experimental conditions. As thermal infraredremote sensing is now being applied at submeter resolu-tions [Jones, 1999; Jones et al., 2002; Jones and Leinonen,2003; Leinonen and Jones, 2004; Loheide and Gorelick,2005; Shimoda and Oikawa, 2008] and TIR imaging tech-nologies tend toward higher resolutions in the future, theavailability of high-variance temperature data is likely toincrease. Given such data, the utility of methods, such as thesubmodel proposed in this paper, may also increase to addressthe additional complexity introduced into evapotranspiration


15 of 18

calculations by the large thermal variance of imaged vegeta-tion canopies.

Notation

Variables defining the two-source evapotranspiration modelframework (see Figure 2a):

E total canopy evapotranspiration from area ofinterest

Ta ambient above-canopy air temperatureea ambient above-canopy airstream vapor pressureu ambient above-canopy airstream wind speed

raa above-canopy aerodynamic resistance

T0 mean canopy airstream temperaturee0 mean canopy airstream vapor pressurerc

a in-canopy aerodynamic resistanceEs total soil evaporation from area of interestrs

a below-canopy aerodynamic resistanceTs soil surface temperaturees vapor pressure at soil surfacers

s soil surface resistancee�s saturated vapor pressure at soil pore surfaceEc total canopy transpiration from area of interestrc

s average canopy surface resistanceTc canopy (leaf) surface temperaturee�c saturated vapor pressure at stomatal pore

surfaceA total available radiation

Ac radiation available at mean canopy levelAs radiation available at soil surfacehc mean canopy heightRn net incident radiation� in-canopy light extinction coefficient

LAIp projected leaf area indexs stomatal ratio (1 for hypostomatous leaves, 2

for amphistomatous leaves)P ambient atmospheric pressureG soil heat flux�a density of air (at Ta and P)cp specific heat of water at constant pressure

RH relative humidityDa slope of the saturation vapor pressure curve

at Ta

� psychometric ConstantModel variables calculated at a specific scale (j). Exceptexcept in scaling analysis, the scale is either the bulk can-opy level (whole TIR image) of conventional models (j ¼bulk) or the TIR pixel scale (j ¼ i).

Ej evapotranspiration, from bulk (Ebulk) or pixel(Ei) scale

Ecj canopy transpiration, from bulk (Ec,bulk) orpixel (Eci) scale

Esj soil evaporation, from bulk (Es,bulk) or pixel(Esi) scale

rcsj local canopy surface resistance, at bulk (rc

s bulk)or pixel (rc

si) scaleTj local canopy surface temperature, at bulk

(Tbulk) or pixel (Ti) scalee�j saturated vapor pressure at stomatal pore sur-

face (at Tj)Dj slope of the saturation vapor pressure curve at Tj

Variables defined for the submodel:rst,rep any representative value of stomatal resistance

(e.g., from porometer measurements orliterature)

Trep leaf temperature corresponding to rst,rep

rcs rep canopy surface resistance corresponding to

rst,rep (see (7))gst,i stomatal conductance at pixel location irst,i stomatal resistance at pixel location i

Ti canopy temperature at pixel location i (fromTIR imagery)

gst,m maximum stomatal conductance of canopy ofinterest (not required by submodel)

Tm temperature of maximum stomatal conductanceof canopy of interest (similar to temperature ofmaximum assimilation; typically availablefrom the literature)

! shape parameter defining with of gst,i(Ti) parab-ola (see (8)), derived by submodel

Eaggregate total (mean) evapotranspiration from all i ¼1:N pixels (see (10))

� the discrepancy between the total evapotranspi-ration values calculated at the bulk (Ebulk) andpixel (Eaggregate) scales (see (11)) : to conserveenergy between the pixel and bulk scales, �should be zero

[65] Acknowledgments. This work was supported by the NationalScience Foundation under grant EAR-0634709 to Stanford University andby Stanford University discretionary funds from A. G. Journel. Any opin-ions, findings, and conclusions or recommendations expressed in this mate-rial are those of the authors and do not necessarily reflect the views of theNational Science Foundation. We thank the City of Palo Alto and Bay-lands Nature Preserve for permitting the fieldwork, P. Matson for suggest-ing the investigation of within-canopy heterogeneity, B. Sabala forimproving the thermal camera mounting hardware based on a prototype, J.A. Berry for many helpful discussions of stomatal function and leaf tem-perature, and Dani Or and others for suggestions made based on a related2010 American Geophysical Union Fall Meeting poster. We also thankfour anonymous reviewers.

ReferencesAllen, R. G., L. S. Pereira, D. Raes, and M. Smith (1998), Crop evapotrans-

piration: Guidelines for computing crop water requirements, in FAOIrrig. Drain. Pap. 56, Food and Agric. Org., Rome, N. Y.

Anderson, M. C., J. M. Norman, J. R. Mecikalski, R. D. Torn, W. P. Kustas,and J. B. Basara (2004), A multiscale remote sensing model for disaggre-gating regional fluxes to micrometeorological scales, J. Hydrometeorol.,5, 343–363.

Anderson, M. C., J. M. Norman, W. P. Kustas, R. Houborg, P. J. Starks, andN. Agam (2008), A thermal-based remote sensing technique for routinemapping of land-surface carbon, water and energy fluxes from field to re-gional scales, Remote Sens. Environ., 112, 4227–4241, doi:10.1016/j.rse.2008.07.009.

Antlfinger, A. E., and E. L. Dunn (1979), Seasonal patterns of CO2 andwater vapor exchange of three salt marsh succulents, Oecologia, 43(3),249–260.

Armond, P. A., U. Schreiber, and O. Björkman (1978), Photosyntheticacclimation to temperature in the desert shrub, Larrea divaricata, PlantPhysiol., 61, 411–415.

Ball, J. T. (1988), An analysis of stomatal conductance, Ph.D. thesis, Stan-ford Univ., Stanford, Calif.

Berry, J. A., and O. Björkman (1980), Photosynthetic response and adap-tation to temperature in higher plants, Ann. Rev. Plant Physiol., 31,491–543.

Beven, K. (1979), A sensitivity analysis of the Penman-Monteith actualevapotranspiration estimates, J. Hydrol., 44, 169–190.


16 of 18

Blonquist, J. M. Jr., J. M. Norman, and B. Bugbee (2009), Automated mea-surement of canopy stomatal conductance based on infrared temperature,Agric. For. Meteorol., 149, 2183–2197, doi:10.1016/j.agrformet.2009.10.003.

Boegh, E., H. Soegaard, and A. Thomsen (2002), Evaluating evapotrans-piration rates and surface conditions using Landsat TM to estimateatmospheric resistance and surface resistance, Remote Sens. Environ.,79, 329–343.

Collatz, G. J., J. T. Ball, C. Grivet, and J. A. Berry (1991), Physiologi-cal and environmental regulation of stomatal conductance, photosyn-thesis and transpiration: A model that includes a laminar boundarylayer, Agric. For. Meteorol., 54, 107–136, doi:10.1016/0168-1923(91)90002-8.

Farquhar, G. D., and T. D. Sharkey (1982), Stomatal conductance and pho-tosynthesis, Annu. Rev. Plant. Physiol., 33, 317–345.

Farquhar, G. D., S. von Caemmerer, and J. A. Berry (1980), A biochemicalmodel of photosynthetic CO2 assimilation in leaves of C3 species,Planta, 149(1), 78–90, doi:10.1007/BF00386231.

Franks, P. J., P. L. Drake, and R. H. Froend (2007), Anisohydric but isohy-drodynamic: Seasonally constant plant water potential gradientsexplained by a stomatal control mechanism incorporating variable planthydraulic conductance, Plant Cell Environ., 30, 19–30, doi:10.1111/j.1365-3040.2006.01600.x.

Giurgevich, J. R., and E. L. Dunn (1979), Seasonal patterns of CO2 andwater vapor exchange of the tall and short height forms of Spartina alter-niflora Loisel in a Georgia salt marsh, Oecologia, 43, 139–156.

Guilioni, L., H. G. Jones, I. Leinonen, and J. P. Lhomme (2008), On therelationships between stomatal resistance and leaf temperatures in ther-mography, Agric. For. Meteorol., 148, 1908–1912, doi:10.1016/j.agrformet.2008.07.009.

Isaaks, E. H., and R. M. Srivastava (1989), Applied Geostatistics, 592 pp.,Oxford Univ. Press, N. Y.

Jarvis, P. G. (1976), The interpretation of the variations in leaf water poten-tial and stomatal conductance found in canopies in the field, Philos.Trans. R. Soc. London B, 273, 593–610.

Jarvis, P. G., and K. G. McNaughton (1986), Stomatal control of transpira-tion: Scaling up from leaf to region, Adv. Ecol. Res., 15, 1–49.

Jones, H. G. (1998), Stomatal control of photosynthesis and transpiration,J. Exp. Bot., 49, 387–398.

Jones, H. G. (1999), Use of thermography for quantitative studies of spatialand temporal variation of stomatal conductance over leaf surfaces, PlantCell Environ., 22, 1043–1055.

Jones, H. G., and I. Leinonen (2003), Thermal imaging for the study ofplant water relations, J. Agric. Meteorol., 59(3), 205–217.

Jones, H. G., M. Stoll, T. Santos, C. de Sousa, M. M. Chaves, and O. M.Grant (2002), Use of infrared thermography for monitoring stomatal clo-sure in the field: Application to grapevine, J. Exp. Bot., 53(378), 2249–2260, doi:10.1093jxb/erf083.

Journel, A. G., and C. J. Huijbregts (1978), Mining Geostatistics, 600 pp.,Academic Press, London, U. K.

Katul, G., S. Manzoni, S. Palmroth, and R. Oren (2010), A stomatal optimi-zation theory to describe the effects of atmospheric CO2 on leaf photo-synthesis and transpiration, Ann. Bot., 105, 431–442, doi:10.1093/aob/mcp292.

Kim, S.-H., and J. H. Lieth (2003), A coupled model of photosynthesis, sto-matal conductance and transpiration for a rose leaf (Rosa hybrida L.),Ann. Bot., 91, 771–781, doi:10.1093/aob/mcg080.

Kustas, W., and M. Anderson (2009), Advances in thermal infrared remotesensing for land surface modeling, Agric. For. Meteorol., 149, 2071–2081, doi:10.1016/j.agrformet.2009.05.016.

Kustas, W. P., F. Li, T. J. Jackson, J. H. Prueger, J. I. MacPherson, and M.Wolde (2004), Effects of remote sensing pixel resolution on modeledenergy flux variability of croplands in Iowa, Remote Sens. Environ., 92,535–547, doi:10.1016/j.rse.2004.02.020.

Leinonen, I., and H. G. Jones (2004), Combining thermal and visible im-agery for estimating canopy temperature and identifying plant stress,J. Exp. Bot., 55(401), 1423–1431, doi:10.1093/jxb/erh146.

Leinonen, I., O. M. Grant, C. P. P. Tagliavia, M. M. Chaves, and H. G.Jones (2006), Estimating stomatal conductance with thermal imagery,Plant Cell Environ., 29, 1508–1518, doi:10.1111/j.1365-3040.2006.01528.x.

LI-COR (2005), Using the LI-6400 Portable Photosynthesis System, Ver-sion 5, 1106 pp., LI-COR Biosciences, Lincoln, Nebr.

Liu, Y., T. Hiyama, and Y. Yamaguchi (2006), Scaling of land surface tem-perature using satellite data: A case examination on ASTER and MODIS

products over a heterogeneous terrain area, Remote Sens. Environ., 105,115–128, doi:10.1016/j.rse.2006.06.012.

Loheide, S. P. II, and S. M. Gorelick (2005), A local-scale, high-resolutionevapotranspiration mapping algorithm (EMA) with hydroecologicalapplications at riparian meadow restoration sites, Remote Sens. Environ.,98, 182–200, doi:10.1016/j.rse.2005.07.003.

Matthews, M. A., and J. S. Boyer (1984), Acclimation of photosynthesis tolow leaf water potentials, Plant Physiol., 74(1), 161–166.

McDowell, N., et al. (2008), Mechanisms of plant survival and mortalityduring drought: Why do some plants survive while others succumb todrought?, New Phytol., 178, 719–739, doi:10.1111/j.1469-8137.2008.02436.x.

Moffett, K. B., A. Wolf, J. A. Berry, and S. M. Gorelick (2010), Saltmarsh-atmosphere exchange of energy, water vapor, and carbon diox-ide: Effects tidal flooding and biophysical controls, Water Resour. Res.,46, W10525, doi:10.1029/2009WR009041.

Monteith, J. L. (1965), Evaporation and environment, Sym. Soc. Exp. Biol.,19, 205–224.

Mott, K. A., and T. N. Buckley (2000), Patchy stomatal conductance:Emergent collective behavior of stomatal, Trends Plant Sci., 5, 258–262.

Norman, J. M., and F. Becker (1995), Terminology in thermal infraredremote sensing of natural surfaces, Agric. For. Meteorol., 77, 153–166.

Norman, J. M., W. P. Kustas, and K. S. Humes (1995), Two-source approachfor estimating soil and vegetation energy fluxes in observations of direc-tional radiometric surface temperature, Agric. For. Meteorol., 77, 263–293.

Overgaard, J., D. Rosbjerg, and M. B. Butts (2006), Land-surface modelingin hydrological perspective––A review, Biogeosci., 3, 229–241.

Pearcy, R. W., and S. L. Ustin (1984), Effects of salinity on growth andphotosynthesis of three California tidal marsh species, Oecologia, 62,68–73.

Pieruschka, R., G. Huber, and J. A. Berry (2010), Control of transpirationby radiation, Proc. Natl. Acad. Sci. U. S. A., 107(30), 13372–13377,doi:10.1073/pnas.0913177107.

Priestley, C. H. B., and R. J. Taylor (1972), On the assessment of surfaceheat flux and evaporation using large-scale parameters, Mon. WeatherRev., 100, 81–92.

Raupach, M. R. (1995), Vegetation-atmosphere interaction and surfaceconductance at leaf, canopy and regional scales, Agric. For. Meteorol.,73, 151–179.

Raupach, M. R. (1998), Influences of local feedbacks on land-air exchangesof energy and carbon, Global Change Biol., 4, 477–494.

Raupach, M. R., and J. J. Finnegan (1988), ‘‘Single-layer models of evapora-tion from plant canopies are incorrect but useful, whereas multilayer mod-els are correct but useless’’: Discuss, Aust. J. Plant. Physiol., 15, 705–716.

Sage, R. F., and T. D. Sharkey (1987), The effect of temperature on theoccurrence of O2 and CO2 insensitive photosynthesis in field grownplants, Plant Physiol., 84, 658–664.

Sellers, P. J., D. A. Randall, G. J. Collatz, J. A. Berry, C. B. Field, D. A.Dazlich, C. Zhang, G. D. Collelo, and L. Bounoua (1996), A revisedland surface parameterization (SiB2) for atmospheric GCMs. Part I:Model formulation, J. Clim., 9, 676–705.

Shahraeeni, E., and D. Or (2010), Thermo-evaporative fluxes from hetero-geneous porous surfaces resolved by infrared thermography, WaterResour. Res., 46, W09511, doi:10.1029/2009WR008455.

Shimoda, S., and T. Oikawa (2008), Characteristics of canopy evapo-transpiration from a small heterogeneous grassland using thermalimaging, Environ. Exp. Bot., 63, 102–112, doi:10.1016/j.envexpbot.2007.12.006.

Shuttleworth, W. J. (1993), Evaporation, in Handbook of Hydrology, editedby D. R. Maidment, chap. 4, pp. 4.1–4.53, McGraw-Hill, N. Y.

Shuttleworth, W. J., and J. S. Wallace (1985), Evaporation from sparsecrops—An energy combination theory, Q. J. R. Meteorol. Soc.,111(469), 839–855.

Smith, W. K., and P. S. Nobel (1977), Temperature and water relations forsun and shade leaves of a desert broadleaf, Hyptis emoryi, J. Exp. Bot.,28(102), 169–183.

Su, Z., H. Pelgrum, and M. Menenti (1999), Aggregation effects of surfaceheterogeneity in land surface processes, Hydrol. Earth Syst. Sci., 3(4),549–563.

Teal, J. M., and J. Kanwisher (1970), Total energy balance in salt marshgrasses, Ecology, 51(4), 690–695, doi:10.2307/1934050.

Tenhunen, J. D., and S. S. Westrin (1979), Development of a photosynthe-sis model with an emphasis on ecological applications. Part IV.Wholephot––Whole leaf photosynthesis in response to four independentvariables, Oecologia, 41, 145–162.


17 of 18

Turner, N. C., E.-D. Schulze, and T. Gollan (1985), The responses of sto-mata and leaf gas exchange to vapour pressure deficits and soil watercontent. Part II. In the mesophytic herbaceous species Helianthusannuus, Oecologia, 65, 348–355.

Yamori, W., K. Noguchi, Y. T. Hanba, and I. Terashima (2006), Effects ofinternal conductance on the temperature dependence of the photosyn-thetic rate in spinach leaves from contrasting growth temperatures, PlantCell Physiol., 47(8), 1069–1080, doi:10.1093/pcp/pcj077.

Zhang, M., S. L. Ustin, E. Rejmankova, and E. W. Sanderson (1997), Moni-toring pacific coast salt marshes using remote sensing, Ecol. Appl., 7,1039–1053.

S. M. Gorelick and K. Moffett, Department of Environmental EarthSystem Science, Stanford University, MC 4216, 473 Via Ortega, Rm. 140,Stanford, CA 94305, USA. ([email protected])


18 of 18

A method to calculate heterogeneous evapotranspiration ...Citation: Moffett, K. B., and S. M....

Documents

Transcript of A method to calculate heterogeneous evapotranspiration ...Citation: Moffett, K. B., and S. M....