To examine the proposed method, six sites were chosen in humid regions having annual precipitation over 1000 mm (Table 1), including a site in Japan, three sites in the USA and two sites in Europe. Site2-Jap data in Japan were prepared by Tukuba University (2006) [6] . Three sites in USA data were prepared by AmeriFlux (Brooks Field Site 11 of US-Br3 [7] , Konza Prairie of US-Kon [8] , Goodwin Creek of US-Goo [9] ). And two sites of Europe data prepared by European Fluxes Database Cluster (Vall dAlinya of ES-VDA [10] and Dripsey of IE-Dri] [11] .

H was observed by eddy covariance at all sites (H_obs). lE was also observed by eddy covariance at five sites (lE_obs) excluding site2-Jap. The lE_obs at site2-Jap was estimated by imbalance (lE_imb = Rn ? G − H_obs). Rn and G were observed at all sites. As shown in Table 2, the soil temperature T₀ was observed by thermometer at the depth of 2 ~ 5 cm.

Table 3 describes the accuracy and data gap of the observed data at the tested sites expressed in heat flux. The imbalance was observed at USA and European sites because directory observed lE by the eddy covariance. US-Kon, IE-Dri and ES-VPA has remarkable large imbalance of 18%, 31% and 19%. The imbalance is zero at the site2-Jap because no observed of lE.

Site2-Jap, US-Br3, IE-Dri and ES-VPA have relatively small data gap while US-Kon and US-Goo have remarkable. The time of having data gap is avoided in the analysis. The annual precipitation of the examined year is shown.

Observed data do not achieve the heat balance relationship, as shown in Table 3. To maintain the relationship, multiple regression analysis is applied using Equation (11). Figure 2 describes the relationship (Rn − G) versus (H + lE) of the original and corrected data in which the observed data are shown in the red circle while the corrected data are shown in the blue circle. The slope of the five tested sites increased and approached to 1.0. The regression coefficients described in Table 4 are A for H and B for lE. The observed data are corrected by these coefficients for all of the tested sites.

To confirm the validity, Figure 3 compares the hourly changes in lE_obs with lE_est and

Data store: every 30 minutes, hourly.

H_obs or H_est at the six sites in summer. All sites data are reproduced well.

However, in detail, lE_est is coincided very well with lE_cor excluding IE-Dri whereas H_est also very well coincided with H_cor without US-Kon. The small differences of H_est may have a little reflected to the lE_est. The other terms, such as lE_obs and H_obs describe almost similar trends but have small site specific differences. In addition, the initial values of

Note: Data gap is not available data for analysis, i.e., lacked one of which G, T_z, T₀, P, erhz, Rn, H_obs and lE_obs. Imbalance is estimated by Imb = Rn ? G ? lE − H using yearly observed data and the imbalance ratio defined as Ra_imb = Imb/(Rn − G). 100 W∙m⁻² = 3.53 mm∙day⁻¹ [12] .

rehs set as follows: US-Kon and US-Goo are rehs = rehz with constrains b < 0 and the other sites uses rehs = 1.0 with constrains b > 0.

Figure 4 describes the yearly changes of the estimated and observed lE and H for the six sites. All sites describe that the trend relatively well reproduced. However in detail, the results show small differences at spring of lE_est at US-Kon. It shows overestimate for lE_est while shows underestimate for H_est. The other terms of lE_obs exhibits similar trends and H_obs also display the same trend but with small differences (not shown).

Figure 5 compares the relationship of lE and H on daily basis to confirm the validity of the general solution. If the slope (slope of the straight line) is 1.0, the observed value coincides are well with the estimated values. For lE_est, all sites well reproduced (±15%) whereas lE_est are underestimated (>15%) excludes US-Goo. R² (R is corrected determination coefficient) of lE_est shows underestimated at US-Kon (>60%) and R² for H_est show remarkably small values excludes ES-VDA. In addition, the criteria of accuracy (±15%) were determined referring to observed data (Table 3).

The relationship between estimated rehs and observed rehz, i.e., the initial values, is a great concern to obtain the reliable results. The left hand side of Figure 6 shows hourly

A is regression coefficient for lE, B is regression coefficient for H.

change of rehs and rehz in summer. The figure describes the well functioned optimization process because the rehs changed remarkably from initial values of 100% of rehz. Difference of rehs and rehz is quite small at all sites. The right hand side of Figure 6

shows the change of Ts − T₀ and Ts − Tz. The Ts changed remarkably from initial value T₀. The Ts − T₀ changes a difference ranging from −10˚C to +10˚C at site1-Jap and ES-VDA while −3˚C to +2˚C at US-Kon, US-Goo and IE-Dri, and from −3˚C to +12˚C at US-Br2. The difference Ts and Tz is about −10˚C to +12˚C, which has no site specific trends. The above features of rehs and Ts changes are quite similar to the other that in season although they have a small difference.

Seasonal change of the lE and H at the all sites is also investigated. The feature has not remarkable difference among February, May, Jun-July, September and November, although the quantity has season specific changes.

Table 5 describes all analyzed daily data at tested six sites including observed and corrected versus estimated using the proposed method for lE and H as well as Ts versus T₀ with rehs versus rehz. The feature is site specific. For corrected against estimated lE and H, the relationship is already described by Figure 5.

For lE_obs versus lE_est, IE-Dri and ES-VPA are overestimated (>15%). For H_obs versus H_est, US-Goo and ES-VPA are overestimated while the other sites are underestimated. (<±15%).

The Ts versus T₀ relationship are strongly correlated for all sites. The relationship of rehs versus rehz is also strong randomized at site-Jap and US-Br3, US-Kon remarkably

Slope express the gradient of estimation (lE_est, H_est) against correction (lE_cor, H_cor) and observation (lE_obs, H_obs), Initial condition rehs = rehz, b > 0. Note * indicates ±15%. Note: initial cindition at site2-Jap, US-Br3, IE-Dri and ES-VDA are rehs = 1.0. US-Kon and US-Goo are rehs = rehz. Constraints: at site2-Jap, US-Br3, IE-Dri and ES-VDA are b > 0. US-Kon and US-Goo are b < 0.

randomized.

Using observed and estimated lE, monthly evapotranspiration was obtained at the all sites, as shown in Figure 7 by assuming 100 W∙m⁻² equivalents for 3.53 mm∙day⁻¹ [12] . The initial value of rehs and constrains are chosen as aforementioned. If there are data gap in a given month, the monthly average ETa obtained as follows: The average ETa in a day multiplied the number of days of the month.

All sites describe very well reproduced the monthly change of ETa. In detail, although there are small differences between ETa_obs, ETa_cor, and ETa_est at all sites, the difference was relatively small.

Besides the pattern of monthly changes, the total amount of the ETa is summarized in Table 6. The amount of annual lE_est and H_est are satisfactorily consistent with lE_cor and H_cor or lE_obs and H_obs, i.e., ETa_est/ETa_cor (<±15%) excluding US-Kon. US-Kon has big imbalance 140 mm∙year⁻¹ even if after correction by regression analysis. The other sites have a relatively small imbalance. The facts describe that ETa can be estimate by our method within 85% accuracy.

To verify the validity of our method, our method was compared with penman method. Penman method is used to evaluate evaporation from the saturated or wet soil surface that corresponding to our proposed method as rehs equals to 100%.

Penman evaporation evaluated by Equation (14) [13]

Here, Δ is the slope of saturated vapor pressure curve (hP∙˚C⁻¹) at Tz, γ is hygroscopic constant (hP∙˚C⁻¹), λ is latent heat flux (MJ∙kg⁻¹), U₁₀ is wind speed at 10 m height (m∙sec⁻¹), another variable already described.

Figure 8 describes the comparison of evaporation estimated by Penman method with our proposed method using daily data of Ishikawa Prefectural Forest Experimental Station (Latitude (+N/−S): 36.4309, Longitude (+E/−W): 136.6424) (2014). The result by our method obtained using Equation (3) that optimized Ts at 100% of rehs reproduced well Penman’s result even though a little scattered. The scattered point may produce with observation quality by related climate elements. Our method does not require the wind speed correction that appeared in the second term of right hand side in Penman Equation (14), which was already pointed out by Urano [13] . In addition, constraint of Rn ? G > lE and H is applied.

Note 1) Initial cindition at site2-Jap, US-Br3, IE-Dri and ES-VDA are rehs = 1.0. US-Kon and US-Goo are rehs = rehz. 2) Constraints: at site2-Jap, US-Br3, IE-Dri and ES-VDA are b > 0. US-Kon and US-Goo are b < 0. 3. Imbalance: (H_est + lE_est) − (H_cor + H_obs). Note * indicates ±15%.

Furthermore, to obtain more reasonable result, we applied the Bulk Transfer Concept (BTC). The heat balance equation of the BTC can be expressed as Equation (15) [14] . The third term of left hand of the equation expressed the sensible heat flux and the fourth term expressed the latent heat flux. Before optimization, Equation (15) is not closed because C_H, C_E and Ts are assumed. The optimization conducted as the ε goes to minimum.

Here, C_H is bulk transfer coefficient of sensible heat flux, C_E is bulk transfer coefficient of latent heat flux, Uz is wind speed, other variables already described.

As described in Figure 8, our method using Equation (15) with the condition of C_H = C_E, that is the same of Penman method’s assumption [14] , is very well reproduced, although the procedure does not unified the variables C_H = C_E and Ts mathematically because one equation determine two variables.

To verify the reasonability of estimated Ts, Figure 9 compares the estimated Ts with observed Ts by radiometer at three sites. The sites almost indicate well coincident with each other, thus, the data shows the validity of the Ts estimation.

There are plural results i.e., local minimum, as satisfying Equation (8) and Equation (9) at different initial values because of nonlinear simultaneous solution. One of the technical points of our research is how to find out the reasonable initial values of Ts and

rehs with constrains. We approach the final values of rehs and Ts from both sides saturated and observed rehz with constraints of b < 0 or b > 0. The results obtained by this procedure are mostly successful. One important thing is that the initial values Ts and rehs to be set as possible as vicinity to the final values.

If Ts approaches zero in convergence process, B_app is remarkably increased according to approaching zero from the opposite side, positive and negative, as shown in Figure 10. This tendency is almost independent of, although there are small differences. Actually, when denominator of Equation (4) approaches zero i.e., rehs approaches to, the abnormal B_app appeared. To avoid this conflict, B_app is limited to (−100 < B_app < 100) as aforementioned, referring to the observed and calculated data approximately [1] .