The proposed model considers the above of near-canopy as shown in Figure 1. Net radiation moves from the air to the canopy and soil surface, and it is portioned into sensible, latent and underground fluxes. Ts is the canopy surface temperature including plant zone, Tz is the air temperature above the canopy at height z, q (Tz) is the specific moisture at height z, rehz is the relative humidity in air at height z, q (Ts) is the unsaturated specific moisture on the canopy surface including plant zone, and q_sat (Ts) is the saturated specific moisture at the same height of q (Ts). In addition, the meaning of observed Tz and rehz in the method describe in discussion section.

The fundamental formulae of the model satisfy the following well known heat balance relationship [3] . The relationship, i.e., energy conservation theorem, is a fundamental concept in the natural world that must be guaranteed at anywhere and anytime.

Heat balance relationship:

Here, Rn is the net radiation flux (W∙m⁻²), G is the heat flux into the canopy and ground (W∙m⁻²), H is the sensible heat flux (W∙m⁻²), and lE is the latent heat flux (W∙m⁻²). In addition, although there is heat flux stored in the canopy and plant zone, the effect of stored heat flux appeared on G, Ts and rehs.

On the other hand, the Bowen ratio (H∙lE⁻¹) is defined as follows with assuming continuity relationship of the H and lE between two heights [3] :

We apply the relationship on the canopy zone (including plant zone) as in Figure 1, i.e., the Bowen ratio con- cept is applied to the layer between the canopy zone and the observation height of the air temperature and humidity by assuming Ts and q (Ts). The reason is as follows: The Ts and q (Ts) in the canopy zone are usually

unknown and difficult to observe. If we try to observe, the observation position usually can’t be specify. This application results in the following:

The specific moisture on the canopy zone is expressed as follows as a function of Ts:

Here, l is the latent heat flux of evaporation (kJ∙kg⁻¹), Cp is the specific heat of the air at a constant pressure (1.004 kJ∙kg⁻¹∙K⁻¹).

According to the above definition of Ts and rehs, the two items are somewhat symbolic and comprehensive concept that did not specify the position. The other variables in Equation (3) and Equation (4) can be expressed by the following well-known equations:

Saturated specific moisture:

Saturated vapor pressure:

Latent heat flux of evaporation

where P is the atmospheric pressure (hPa).

The purpose of the optimization is to determine the unknown variables Ts and q (Ts) in Equation (3) without measurements, but with Ts sometimes observed. The governing equation to be solved is obtained by inserting Equation (3) into Equation (1). Initially, if Ts was observed, rehs is only assumed because

. Equation (1) is not as closed as Equation (8) because of the assumptions for rehs. The equation is expressed as follows:

The objective function is ε_i that goes to a minimum by repeating calculation using Equation (8) and Equation (3) in the optimization process.

The rehs can be unified mathematically because a governing Equation (8) determined a variable rehs.

After optimization completed, i.e., B_est in Equation (3) goes to B₀, lE and H can be obtained as follows.

The equation is nonlinear for Ts and q (Ts). Thus, an analytical solution is not available. Therefore, a numerical method was applied. Note that the other factors were obtained from observations or calculated independently using the aforementioned relationships. In addition, the analysis was conducted essentially using hourly data and summarized daily because the climate element change remarkably throughout a day.

The rehs in Equation (4) for estimating q (Ts) was assumed initially to be rehz because the humidity on the canopy has not remarkably different. The rehs was automatically modified.

The calculation follows a non-linear optimization procedure that employs a General reduced Gradient (GRG) algorithm, which can be applied with the Excel Solver on a personal computer (Appendix 1 and Appendix 2).