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Evapotranspiration in forests has been researched for a long time because it serves an important role in water resource issues and biomass production. By applying the reciprocal analysis based on the Bowen ratio concept to the canopy surface, the sum result of sensible and latent heat fluxes, i.e., actual evapotranspiration ( ET), is estimated from engineering aspect using the net radiation ( Rn) and heat flux into the ground ( G). The new method uses air temperature and humidity at a single height by determining the relative humidity ( rehs) using the canopy temperature ( Ts). The validity of the method is confirmed by the latent heat flux ( lE) and sensible heat flux ( H) observed by mean of eddy covariance method. The heat imbalance is corrected by multiple regression analysis. The temporal change of lE and H at the canopy surface is clarified using hourly and yearly data. Furthermore, the observed and estimated monthly evapotranspiration of the sites are compared. The research is conducted using hourly data and the validation of the method is conducted using observed covariance at five sites in the world using FLUXNET.

Reasonable water resource planning requires estimation of actual evapotranspiration. Many relevant research projects have provided useful results; however, the research is still incomplete. Existing research uses theoretical, observational (eddy covariance and Bowen ratio methods), and experimental approaches (the complementary relationship method). All of the above approaches are based on aerodynamic theory, including the heat balance approach, but excluding the complementary relationship.

The eddy covariance method [

In contrast, the Bowen ratio method [

The complementary relationship method is based on the hypothesis that the actual plus potential evapotranspiration is twice the equilibrium evaporation [

In the natural world, the air temperature and humidity is determined by H and lE from the net radiation (Rn) and heat flux into the ground (G). Thus, our research attempts the reciprocal estimation of H and lE from the not observed humidity (rehs) while satisfying the heat balance relationship using the canopy surface temperature (Ts). The concept is different from that of the other relevant methods, and it only requires Rn, G and common climate measurements, including the air temperature and humidity at a single height.

In the proposed method, the unknown variables, relative humidity at the canopy surface (rehs) were determined reciprocally with keeping heat balance relationship by the non-linear optimization technique known as the general reduced gradient (GRG) attached in the Excel Solver (Appendix 1).

The proposed model considers the above of near-canopy as shown in _{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 [

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 [

We apply the relationship on the canopy zone (including plant zone) as in

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

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 B_{0}, 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).

To examine the validity of proposed method, five sites were chosen throughout the world as identified in

To investigate the accuracy of observed data, _{imb} ranged from −0.05 (US-Slt) to 0.37 (CN-Cha) with an average of 0.16 (the upper low of

Sitename/FLUXNET ID: | FR-Pue | JP-Tom | CN-Cha | US-Slt | US-WCr |
---|---|---|---|---|---|

Country: | France | Japan | China | USA | USA |

State/Province: | Herault | Hokkaido | Antu County, Jilin Province | New Jersey | Wisconsin |

Latitude (+N/−S): | 43.7414 | 42.737 | 42.4025 | 39.9137 | 45.806 |

Longitude (+E/−W): | 3.5958 | 141.5186 | 128.0958 | −74.5960 | −90.0798 |

Elevation: | 211 m | 140 m | 736 m | 30 m | 515 m |

Vegetation (IGBP): | Evergreen Broadleaf Forests | Japanese larch forest | Pinus-koraiensis-dominanted Pinus koraiensis broad-leafed mixed forest | Deciduous broadleaf forest | Deciduous broadleaf forest |

Tower height: | 10 m | about 42 m | about 40 m | 19 m | 30 m |

Canopy height: | - | 15m | - | 9.52 ± 2.28 | 24.2 |

Data available | 2008 1/1-12/31 | 2003 1/1-12/31 | 2005 1/1-12/31 | 2012 1/1-12/31 | 2005 1/1-12/31 |