[64] The surface boundary conditions in ALEXI are provided by the TSEB model [*Norman et al.*, 1995], which extracts soil and canopy temperatures (*T*_{s} and *T*_{c}) from composite directional surface radiometric temperature (*T*_{RAD}(*θ*)) measurements acquired by satellite:

where *f*(*θ*) is the fractional cover:

*F* is the leaf area, and Ω(*θ*) is the vegetation clumping factor apparent at view angle *θ* [*Anderson et al.*, 2005].

[65] In TSEB, equations A1 and A2 are solved simultaneously with a set of equations describing the surface energy budget for the soil, canopy, and composite land-surface system: *System, soil, and canopy energy budgets:*

*Net radiation:*

*Sensible heat:*

*Latent heat:*

*Soil conduction heat:*

Here *RN* is net radiation, *H* is sensible heat, *λE* is latent heat, *G* is the soil heat conduction flux, *T* is temperature, *R* is a transport resistance, *ρ* is air density, *c*_{p} is the heat capacity of air at constant pressure, *γ* is the psychometric constant, and Δ is the slope of the saturation vapor pressure vs. temperature curve. The subscripts ‘A’, ‘AC’, and ‘X’ signify properties of the air above and within the canopy, and within the leaf boundary layer, respectively, while ‘S’ and ‘C’ refer to fluxes and states associated with the soil and canopy components of the system. The soil heat conduction flux is computed as a fraction *α*_{g} of the net radiation below the canopy, at the soil surface [equation A15; *Choudhury et al.*, 1987], with *α*_{g} = 0.31 typical of values derived from mid-morning flux observations, the period when ALEXI is applied [*Kustas et al.*, 1998]. In equation A14, transpiration is tied to the net radiation divergence in the canopy (RN_{C}) through a modified Priestley-Taylor relationship [*Priestley and Taylor*, 1972], where *α*_{c} is a coefficient with a nominal value of 1.3 that is downward-adjusted if signs of vegetative stress are detected (see main text) and *f*_{g} is the fraction of green vegetation in the scene. Justification for this parameterization of *λE*_{C} is provided by *Norman et al.* [1995].

[66] The series resistance formalism described here allows both the soil and the vegetation to influence the microclimate within the canopy air space, as shown in Figure 1. The resistances considered include *R*_{A}, the aerodynamic resistance for momentum between the canopy and the upper boundary of the model (including diabatic corrections); *R*_{X}, the bulk boundary layer resistance over all leaves in the canopy; and *R*_{S}, the resistance through the boundary layer immediately above the soil surface. Mathematical expressions for these resistance terms are given by *Norman et al.* [1995].

[67] In equations A1–15, *RN* is the net radiation above the canopy, RN_{C} is the component absorbed by the canopy, and RN_{S} is the component penetrating to the soil surface. The longwave components of *RN*and RN_{S} are a function of the thermal radiation from the sky (*L*_{d}), the canopy (*L*_{c}) and the soil (*L*_{s}), and the coefficient of diffuse radiation transmission through the canopy (*τ*_{c}). The shortwave components depend on insolation values above the canopy (*S*_{d}) and above the soil surface (*S*_{d,s}), and the reflectivity of the soil-canopy system (*A*) and the soil surface itself (*ρ*_{s}). Based on the work of *Goudriaan* [1977], *Campbell and Norman* [1998] provide analytical approximations for *τ*_{c} and *A* for sparse to deep canopies, depending on leaf absorptivity in the visible, near-infrared and thermal bands, *ρ*_{s}, and leaf area index [see Appendix B in *Anderson et al.*, 2000 for further information].

[68] The ALEXI model uses an atmospheric boundary layer (ABL) closure technique to evaluate the morning evolution of air temperature, *T*_{A}, in the surface layer. Using radiometric temperature data at times *t*_{1} and *t*_{2} (about 1.5 and 5.5 h past local sunrise) and initial estimates of air temperature, the TSEB surface model component of ALEXI (equations A1–A15) computes instantaneous sensible heat flux estimates *H*_{1} and *H*_{2}. Assuming a linear functional form for *H*(*t*) during this morning interval, a time-integrated heat flux can be obtained:

*McNaughton and Spriggs* [1986] give a conservation equation relating the rise in height (*z*) and potential temperature (*θ*_{m}) of the mixed layer to the time-integrated sensible heating from the surface:

where *θ*_{s}(*z*) represents an early morning ABL potential temperature sounding. Near the land surface, the mixed layer potential temperature and the air temperature are related by

where *p* is the atmospheric pressure (in kPa) and *R*/*c*_{p} = 0.286. Because differential surface temperature measurements are more reliable than absolute temperature measurements, in practice *z*_{1} (the ABL height at time *t*_{1}) is fixed at some small value (50 m), and the *change* in modeled *θ*_{m} is to allowed to govern the ABL growth based on the lapse rate profile above *z*_{1} [as opposed to diagnosing both *z*_{1} and *z*_{2}; see *Anderson et al.*, 1997]. While this equation A17 represents a very simplified treatment of entrainment, *McNaughton and Spriggs* [1986] found that it produces reasonable values of simulated ET, although boundary layer height *z*_{2} is sometimes greatly overestimated. The surface and boundary layer components of the model iterate until the time-integrated sensible heat flux estimates from both components converge. *Anderson et al.* [1997] provide further details concerning the solution sequence used in the ALEXI model.