#### 2.1. Overview of Two-Source Model Adaptations

[11] Our distributed canopy resistance and evapotranspiration mapping method is based on the two-source evapotranspiration model by *Shuttleworth and Wallace* [1985]. We also test an alternative two-source model combining the canopy-airstream decoupling method of *Jarvis and McNaughton* [1986] and an estimate of soil evaporation [*Priestley and Taylor*, 1972]. Both two-source models are based on the premise that the transpiring canopy surface does not directly communicate with the above- and below-canopy air masses, i.e., the leaf surface and atmosphere are at least partially decoupled.

[13] Our modeling approach maintains the same overall structure as this conventional, canopy-scale, two-source model framework. We add three features in a submodel that represents finer-scale details of the canopy energy balance (Figure 2b), such as may occur at the cm-scale of the TIR canopy surface imagery examined in this study.

[14] 1. Rather than assuming that the canopy functions homogenously with one average canopy temperature and one canopy stomatal resistance as in conventional models, our submodel considers the heterogeneity exposed by high-resolution thermal imaging. This thermal heterogeneity represents spatial variability in the surface energy balance. The submodel allows for many individual canopy leaf surfaces, represented by the TIR pixels, to simultaneously communicate with the mean canopy airstream. The submodel also allows each pixel-sized leaf surface to be represented according to its own local leaf temperature and stomatal conductance (Figure 2b).

[15] 2. The submodel maintains continuity with conventional coarser-scale model frameworks by calculating aerodynamic resistances and radiation partitioning at the canopy scale precisely as in the conventional frameworks. These values are then passed to the submodel. We require that the sum of the evapotranspiration fluxes for all TIR pixels calculated by the submodel (the aggregate evapotranspiration) equals the bulk evapotranspiration calculated using the conventional two-source modeling approach applied at the canopy scale. This constraint mathematically conserves energy and water vapor across scales.

[16] 3. The submodel incorporates the nonlinear biophysical control of leaf stomatal aperture into the modeling framework via a simple quadratic relationship of stomatal resistance to temperature. The relationship is not specified a priori: the submodel iteratively derives it. The relationship's derivation requires only one model parameter beyond those in the conventional two-source models, the temperature of maximum stomatal conductance, which can typically be obtained from the literature. The derived stomatal conductance-temperature relationship then provides each pixel location its own stomatal resistance value that corresponds to the observed temperature variability.

#### 2.2. Submodel Procedure

[17] The five-step submodel procedure is illustrated in Figure 3, in which the circled numbers correspond to the following steps. The variables are collected in the notation list, for reference.

[18] Step 1: Select an evapotranspiration model.

[21] The second evapotranspiration model we tested was the “decoupling coefficient” canopy model of *Jarvis and McNaughton* [1986]. We chose the approach of *Jarvis and McNaughton* [1986] because of their leaf versus canopy scaling analysis, their recommendation of the model as appropriate across a range of canopy scales, and their model's strong contrast with the S&W framework. The Jarvis and McNaughton model is very similar to the Penman-Monteith combination equation [*Monteith*, 1965]. The decoupling approach of Jarvis and McNaughton is also amenable to having our canopy surface submodel nested within its broader, canopy-scale model framework. Unlike in the S&W model, the canopy temperature dependence of the Jarvis and McNaughton model occurs via the slope of the saturation vapor pressure curve (Δ_{j}) at the specified leaf temperature (*T*_{j}) [*Allen et al.*, 1998]. The other parameters are as in the S&W model. To enhance comparison of Jarvis and McNaughton's one-source canopy model with the two-source approach of S&W, we used only the canopy-level available radiation fraction (*A*_{c}) in the *Jarvis and McNaughton* [1986] canopy transpiration model:

We supplemented the canopy transpiration with additional soil evaporation calculated using the *Priestley and Taylor* [1972] model for a soil surface at air temperature (subscript “a”):

The Priestley-Taylor model is appropriate for the largely saturated wetland surface of our study. The support area of our demonstration examples was very similar to that of the lysimeters used by Priestley and Taylor in their original model verification [*Priestly and Taylor*, 1972]. Other soil evaporation equations could be used in other cases. The combined Jarvis and McNaughton/Priestley and Taylor model is hereafter referred to as J&M.

[22] Step 2: Calculate the bulk evapotranspiration of the canopy using the unmodified, conventional two-source model.

[23] Calculating bulk canopy evapotranspiration (*E*_{j} = E_{bulk}) using a conventional two-source model [e.g., *Shuttleworth and Wallace*, 1985; *Norman et al.*, 1995] uses the bulk radiometric temperature of the canopy (*T*_{j} = T_{bulk}) and a representative value of bulk canopy surface resistance provided by the user (e.g., from the literature or average porometer data) ( ). Our approach trusts that this bulk evapotranspiration value (*E*_{bulk}) is accurate, as calculated by the unmodified, well-established evapotranspiration models. To calculate the bulk radiometric surface temperature *T*_{bulk} at the canopy scale, one averages the radiances of all the TIR pixels in the selected canopy image area (pixels *i* = 1 to *N*) using the fourth-power mean Stefan-Boltzmann law:

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 surface resistance ( ) from a representative value of bulk stomatal resistance (*r*_{st,rep}) and the canopy's projected leaf area index (*LAI*_{p}) and stomatal ratio (*s*) [*Shuttleworth and Wallace*, 1985]:

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

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

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

[27] The submodel estimates a concave-down stomatal conductance-temperature relationship for the observed canopy conditions using as few parameters as possible by assuming a quadratic approximation. The quadratic relationship for the two-sided pixel-level leaf stomatal conductance (*g*_{st,i}) and its reciprocal, stomatal resistance (*r*_{st,i}), is

The unknown parameters in (8) are: the maximum stomatal conductance (*g*_{st,m}) and the parabola shape parameter (*ω*). The canopy temperature at each pixel location (*T*_{i}) is provided by the TIR imagery and the temperature at which stomatal conductance is maximized (*T*_{m}) is provided by the user from ancillary data or literature. Calculating the parabola shape parameter (*ω*) is the objective of the submodel. A larger value of the shape parameter results in the more peaked, narrower relationship illustrated by the dashed lines in Figures 1c and 3 (box 3).

[28] The temperature (*T*_{m}) at which stomatal conductance is maximized (*g*_{st,m}) is the only parameter required by the submodel, in addition to those in the conventional two-source approach. Since stomatal conductance and photosynthetic carbon assimilation are approximately proportional [*Ball*, 1988], *T*_{m} is similar to the temperature at which assimilation is maximized. This temperature value is readily obtained for most plant species and land cover classes of interest from assimilation-temperature curves in the literature [e.g., *Antlfinger and Dunn*, 1979; *Giurgevich and Dunn*, 1979; *Berry and Björkman*, 1980; *Pearcy and Ustin*, 1984; *Sage and Sharkey*, 1987; *Sellers et al.*, 1996; *Kim and Lieth*, 2003; *Yamori et al.*, 2006]. A representative value of *T*_{m} may also be obtained from laboratory gas flux measurements of the canopy of interest, which was the approach used in our demonstration examples.

[29] The modeler already knows one solution to (8): the bulk stomatal resistance (*r*_{st,rep}) and its corresponding temperature (*T*_{rep}) used to parameterize the conventional bulk flux model (see step 2 and box 3 in Figure 3). Although *T*_{rep} was not a parameter explicitly used in the conventional model framework, the *r*_{st,rep} value selected by the user in step 2 must implicitly have a corresponding *T*_{rep} value: for example, the leaf temperature recorded by a porometer used to measure *r*_{st,rep}. Our approach simply requires that the user explicitly acknowledge this assumed “representative” temperature. Also, in the context of the quadratic model (8), the influence of the one specific (*r*_{st,rep}, *T*_{rep}) parameter pair supplied by the user is reduced compared to the conventional approach, which is a strength of the new method given likely uncertainty in these values. We substitute the known point (*r*_{st,rep}, *T*_{rep}) into (8) and algebraically solve for the unknown maximum conductance *g*_{st,m}. (*T*_{rep} must be distinct from *T*_{m}.) Substituting for *g*_{st,m} in (8), we calculate stomatal resistance (*r*_{st,i}) values for each pixel:

Each stomatal resistance (*r*_{st,i}) is scaled up to a local, pixel-level surface resistance ( ) using (7). Thus, (7) and (9) relate the pixel-scale canopy surface resistances ( ) to the TIR pixel surface temperatures (*T*_{i}) via a shape parameter (*ω*) and five known scalars (*r*_{st,rep}, *T*_{rep}, T_{m}, s, and *LAI*_{p}). The shape parameter (*ω*) is derived in the next steps of the submodel.

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

[31] An initial estimate of each pixel's evapotranspiration (*E*_{i}) is calculated from the evapotranspiration model (equations (1)–(3) or (4) and (5)), using the TIR pixel temperature (*T*_{i}) and its corresponding canopy resistance value ( , estimated with (7) and (9)). Note that the parameters , , *A*, *A*_{c}, *A*_{s}, *ρ*_{a}, *c*_{p}, and *γ* remain at the bulk canopy scale exactly as in conventional two-source models [*Shuttleworth and Wallace*, 1985]. One applies these parameters uniformly across the whole TIR-imaged canopy to each pixel in the submodel, exactly as in conventional model applications.

[32] Step 3c: Aggregate the pixel scale evapotranspiration values by averaging.

[33] The initial pixel-level evapotranspiration estimates are aggregated to the canopy (TIR image) level by arithmetic averaging [*Raupach*, 1995]:

[34] Step 4: Reconcile bulk and aggregate evapotranspiration values.

[35] The numerical objective of the submodel is to minimize the difference between the bulk (conventional two-source canopy) and aggregate (fine-scale submodel) evapotranspiration rates by optimizing the *ω*-shape parameter that defines a realistic stomatal conductance-temperature relationship for the canopy of interest for observed conditions.

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

[37] The discrepancy (*δ*) between the aggregate evapotranspiration (*E*_{aggregate}) estimated by the submodel in step 3 and the bulk evapotranspiration for the canopy (*E*_{bulk}) known from step 2 is

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

[38] Step 4b: Minimize aggregate evapotranspiration discrepancy by iteratively adjusting the biophysical relationship in the submodel.

[39] Slight adjustments to the stomatal conductance-temperature relationship (e.g., solid versus dotted curves in Figure 1c) result in notable differences in the aggregate evapotranspiration of the imaged canopy (e.g., integral of solid curve versus integral of dotted curve in Figure 1f). Taking advantage of this sensitivity, the submodel's biophysical relationship, pixel-level evapotranspiration values, and aggregate evapotranspiration are progressively refined by automated numerical iteration until the aggregate and bulk evapotranspiration values match. Many solution methods could be employed; we used and suggest the Newton method. An initial value of *ω* on the order of 10^{−6} is suggested, but sensitivity analysis should be conducted for each application and an absolute minimum of *δ* sought.

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

[41] Once an optimal solution for *ω* has been obtained by minimizing *δ*, the spatially distributed, pixel-scale evapotranspiration values (*E*_{i}) are mapped from the final results of the submodel. The optimized *ω* value provides an estimate of the biophysical stomatal conductance-leaf temperature relationship for the canopy under observed conditions, which is used to convert the remotely sensed temperature field (*T*_{i}) into a high-resolution canopy surface resistance map. If the model separately represents canopy transpiration and soil evaporation components of total evapotranspiration, these variables are also mapped at the scale of the TIR data [e.g., *Shuttleworth and Wallace*, 1985; *Norman et al.*, 1995].