2.1.2. Priestley-Taylor Two-Source Energy Balance Model, TSEB-PT

Two-source energy balance models treat the land surface as two layers, soil and canopy, contributing to the energy and water fluxes (Equation (4))

$$R\_{\mathfrak{N},\mathbb{C}} = H\_{\mathbb{C}} + \lambda E\_{\mathbb{C}} \tag{4a}$$

$$R\_{\rm n\downarrow} = H\_{\rm S} + \lambda E\_{\rm S} + G \tag{4b}$$

where soil (canopy) sensible heat flux is computed from the gradient between the soil (canopy) temperature (*T<sup>S</sup>* and *T<sup>C</sup>* respectively) and the air temperature at the sink-source height (equivalent to *T*0). In the TSEB-PT model [28,36,53], an electrical circuit analogy is used in which *H* from soil and canopy are estimated based on three aerodynamic resistances to heat transport arranged in a series network. Since *T<sup>C</sup>* and *T<sup>S</sup>* are unknown *a priori*, they are estimated in an iterative process in which it is first assumed that green canopy (expressed as the fraction of LAI that is green, *fg*) transpires a potential rate based on Priestley–Taylor formulation [36]:

$$
\lambda E\_{\mathbb{C}} = \mathfrak{a}\_{PT} f\_{\mathbb{S}} \frac{\Delta}{\Delta + \gamma} \mathcal{R}\_{\mathfrak{n}, \mathbb{C}'} \quad \mathfrak{a}\_{PT} = 1.26\tag{5}
$$

where *αPT* is the Priestley and Taylor [57] coefficient, ∆ is the slope of the vapour pressure to air temperature curve and *γ* is the psychrometric constant. Then the canopy transpiration is sequentially reduced (i.e., *αPT* < 1.26) until realistic fluxes are obtained (*λE<sup>C</sup>* ≥ 0 and *λE<sup>S</sup>* ≥ 0).

TSEB-PT probably is the model that requires most accurate retrievals of physical inputs (*LAI* and *Trad*), and studies already reported larger uncertainty in senescent vegetation (i.e., *f<sup>g</sup>* < 1) and very dense (high *LAI*) or tall vegetation [43,58]. It is more complex than METRIC and therefore has a large number of parameters and modelling options. Finally, the Priestley–Taylor formulation was shown to produce larger uncertainty in high advection conditions, cases in which initializing *λE<sup>C</sup>* with a Penman-Monteith formulation showed better results [37]. Combining TSEB-PT model with the disaggregation approach (described in Section 1) results in a disTSEB model [23].

#### 2.1.3. End-Member-Based Soil and Vegetation Energy Partitioning, ESVEP

ESVEP is based on a trapezoid *Trad* − *fcover* framework, in which it considers fluxes acting in a "parallel" soil and canopy system [44]. As in TSEB-PT, ESVEP partitions *Trad* as a linear weight of emitted radiance. Other similar models to ESVEP are HTEM [59] and TTEM [60], but ESVEP solves the trapezoid in a pixel-per-pixel basis overcoming the need for homogeneous weather forcing and roughness (Equation (6a)).

$$T\_{S,max} = \frac{r\_a \left(R\_{\rm nS} - G\right)}{\rho\_a \mathcal{C}\_p} + T\_A \tag{6a}$$

$$T\_{\mathbb{C},\max} = \frac{r\_{\mathbb{A}}R\_{\mathbb{R},\mathbb{C}}}{\rho\_{\mathbb{A}}\mathbb{C}\_{p}} \frac{\gamma\left(1 + r\_{\mathbb{B},\text{dry}}/r\_{\mathbb{A}}\right)}{\Delta + \gamma\left(1 + r\_{\mathbb{B},\text{dry}}/r\_{\mathbb{A}}\right)} - \frac{vpd}{\Delta + \gamma\left(1 + r\_{\mathbb{B},\text{dry}}/r\_{\mathbb{A}}\right)} + T\_A \tag{6b}$$

$$T\_{\rm S,min} = \frac{r\_a \left(R\_{\rm n,S} - G\right)}{\rho\_a \mathcal{C}\_p} \frac{\gamma}{\Delta + \gamma} - \frac{vpd}{\Delta + \gamma} + T\_A \tag{6c}$$

$$T\_{\mathbb{C},\text{min}} = \frac{r\_a \mathbb{R}\_{n,\mathbb{C}}}{\rho\_a \mathbb{C}\_p} \frac{\gamma \left(1 + r\_{b,\text{net}}/r\_a\right)}{\Delta + \gamma \left(1 + r\_{b,\text{net}}/r\_a\right)} - \frac{vpd}{\Delta + \gamma \left(1 + r\_{b,\text{net}}/r\_a\right)} + T\_A \tag{6d}$$

where *r<sup>a</sup>* is the aerodynamic resistance, *rb*,*dry* and *rb*,*wet* are resistances of dry and wet canopy respectively, *ρa* is the density or air, *Cp* is specific heat capacity at constant pressure, *γ* is psychrometric constant and *vpd* is vapour pressure deficit of the air.
