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Estimation of the Sensible and Latent Heat Fluxes by Reciprocal Analysis at an Arid and Semi-Arid Region

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*i.e.*, evapotranspiration, has been researched for long time because it serves an important role for water resource issues. However, the issues have not solved completely yet. Accordingly, by applying the Bowen ratio concept on the soil surface, the sensible and latent heat fluxes are reciprocally estimated using single height temperature (

*Tz*) and humidity (

*rehz*) with the net radiation (

*Rn*) and heat flux into the ground (

*G*). The procedure proposed by authors initially estimates the soil surface temperature (Ts) and the relative humidity (

*rehs*) using optimization techniques. The method is remarkably effective to expand for estimating evapotranspiration at various regions. The validity of the method is confirmed by the latent heat flux (

*lE*) and sensible heat flux (

*H*) observed by the eddy covariance method. The hourly change of the

*lE*,

*H*,

*Ts*and

*rehs*on the soil surface, yearly change of

*lE*and

*H*and relationship of estimated

*lE*and

*H*versus observed are clarified. Yearly change of evapotranspiration is also estimated. The analysis is performed by general method (1), conventional method and general method (2). Above results are very useful for water resources issue and irrigation planning. The research is conducted using hourly data at eight globally dispersed sites using FLUXNET.

KEYWORDS

1. Introduction

Recently, we firstly reported reciprocal analysis of sensible and latent heat fluxes in a forest and humid region [1] [2] . By applying the procedure, this research highlighted the same problems in “an arid and semi-arid regions”. As is well known, the region usually faced to very severe water shortage. Therefore, reasonable estimation of latent heat flux (evapotranspiration) is very important issues. But the analysis in the local area still has a big problem because the flux severely affects local climate conditions.

Our proposed method is reciprocal analysis of the sensible and latent heat fluxes using single height temperature (Tz) and humidity (rehz) with net radiation (Rn) and heat flux into the ground (G). This method can’t be found out in the previous research except ours [1] [2] . This method is remarkably effective to extend the utilization for various regional areas.

One of the problems in the method is how to determine the initial values and constraints. This paper describes mainly focusing on this problems including estimation of those fluxes in arid and semi-arid region.

The results compare general solution (1) (solved two simultaneous equations that mathematically unified soil surface temperature (Ts) and humidity (rehs)), general solution (2) (solved one equation using Ts observed by radiometer that unified rehs mathematically) and conventional solution (solved one equation that not unified Ts and rehs mathematically). Furthermore, this research describes relationship between annual evapotranspiration and precipitation from the aspect of water balance.

Actually, the unknown variables in the proposed method, Ts and rehs, were estimated by the non-linear optimization technique known as the General Reduced Gradient (GRG) using the Excel Solver (Appendix 1) [1] [2] .

2. Methods

2.1. Method of Analysis

2.1.1. Fundamental Concept of the Model

The fundamental concept of the method is quiet similar with previous research. Here, we describe the outline briefly. The proposed model considers the near-soil surface as shown in Figure 1. Net radiation moves from air to the soil surface, and it is portioned into sensible, latent and underground heat fluxes. Ts is the soil surface temperature, Tz is the air temperature at height z, q(Tz) is the specific moisture at height z, rehz is relative

humidity in air at height z, q(Ts) is the unsaturated specific moisture on the soil

Figure 1. Schematic presentation of considered domain and used symbols [2] . (Symbols used in research defined as this figure. The net radiation Rn partitioned into ground G, sensible heat H and latent heat lE. Thus, Heat balance relationship Rn = H + lE + G is always guaranteed).

surface, and q_{sat}(Ts) is the saturated specific moisture on the soil surface [1] [2] .

The fundamental formulae of the model satisfy the following well-known heat balance relationship [3] .

Heat balance relationship:

(1)

Here, Rn is the net radiation flux (W・m^{−2}), G is the heat flux into the 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, the Bowen ratio (H/lE) is defined as follows [1] [2] [3] :

(2)

We apply the concept to the layer between the soil surface and observation height of Tz and rehz. However, the Ts and q(Ts) just on the surface are usually unknown.

2.1.2. Equation for Determining the Unknown Variables Ts and rehs

The purpose of the optimization is to determine the unknown variables Ts and q(Ts) without measurements. Initially, Ts and q(Ts), i.e., rehs ´ q_{sat}(Ts), are assumed. Then it’s gradually improved according to the following optimization procedure [1] [2] :

(3)

(4)

and (5)

Here, i is number of iterations. H_{est,i} is estimated sensible heat flux in i times iteration, lE_{est,i} is estimated latent heat flux of i times iteration, ε_{i} is residual of heat balance relationship of i times iteration, T_{sass} is assumed soil surface temperature, q(T_{sass}) is assumed specific moisture at T_{sass}, B_{app} is apparent ratio of sensible and latent heat flux under convergence process

Repeating the above calculation, the B_{app} goes to B_{0} according to the objective function where ABS (ε_{i}) reaches a minimum. Symbols used here presented in Figure 1.

Using the determined Bowen ratio (B_{0}) after optimization, lE and H can be obtained as follows:

and (6)

Symbols used here presented in Figure 1.

To estimate Ts, an adjustment factor RTs was introduced using T_{0} as follows:

(7)

Here, T_{0} is the observed soil temperature at a depth of x cm, D_{To} is the depth of the soil temperature observation, Kt is the assumed temperature conductivity (W・cm^{−1}・˚C), Equation (7) describes how to estimate Ts by extrapolating T_{0} using G, D_{To} and Kt [1] [2] .

2.1.3. Correction of the Heat Imbalance by Multiple Regression Analysis

The observed data have a heat imbalance that is well known as a “closure issue”. Twine, et al. [4] indicated an approximately 10% - 30% imbalance in H and lE. Wilson, et al. [5] also indicated that approximately 20% of the imbalance is due to heat flux. Allen [6] suggested that the under-measurement of lE is sometimes approximately 40%.

Table 1 describes the imbalance in the heat energy. Therefore, the data should be correct. According to Allen’s procedure, we corrected the data by multiple regression analysis using the observed data as follows [1] [2] :

(8)

Here, Rn is the net radiation, G is the heat flux into the ground, lE is the latent heat flux observed, and H is the sensible heat flux observed. A is the regression coefficient for lE, and B is the regression coefficient for H. The corrected data are expressed as H_{cor} for H and lE_{cor} for lE. The observed data are expressed as H_{obs} and lE_{obs}, and the imbalance data are also defined as H_{imb} () for H and lE_{imb} () for lE.

2.1.4. Method of General Solution (1)

To uniquely estimate the unknown variables Ts and rehs, pair of two equations is required mathematically. We set the two equations as follows [1] [2] :

(9)

(10)

goes to minimum. (11)

Here, j is the order of hours from 1 to the end of the analyzed hours and i is the number of iterations.

The calculation is performed by solving Equation (9) and Equation (10) simultaneously under and conditions.

In addition, to prevent abnormal fluctuation of H_{est} and lE_{est} in the optimization process, constraints, lE are applied. The reason described in previous research [1] [2] . The constraints for Equation (9) and Equation (10) are follows:

(12)

Equation (9) and Equation (10) are nonlinear two element simultaneous equation. The two unknown variables can be estimated for the limit to which ε is minimized, allowing H and lE to be estimated. Note that the other factors are obtained from observations or calculations independently using the aforementioned relationships.

2.1.5. Initial Values for Ts and rehs

Setting of the initial value of Ts and rehs are the most important. Figure 2 describes the concept of the initial values in the plane of vapor pressure and temperature [e(T)~T] relationship. The dotted line indicates a partial vapor pressure for a given relative humidity. The saturated vapor pressure calculated by Tetens Formula.

On the plane, we can select two point as a[(Tz, e(Tz)] and b[T_{0}, e(Tz)] using observed data. If we assume rehs = rehz, c[T_{0}, e(T_{0})] point can be determined, thus, the hatched triangle can be defined as the initial values. The ratio of lE_{est}/H_{est} is similar to [refer Equation (2) or Equation (3) if e(T) converted to q(T)]. We consider the optimized value of b’[Ts, e(Tz)] and c’[Ts, e(Ts)] as triangle with dotted line on the figure. We searched the optimal value b’[Ts, e(Tz)] and c’[Ts, e(Ts)] at the vicinity of the initial value b[T_{0}, e(Tz)] and c[T_{0}, e(T_{0})]. Based on the above concept, we set the initial value of and rehs = rehz.

To guarantee better reproducibility, we set the constraints as Equation (13) as indicating in Figure 2 [2] :

Figure 2. Schematic expression of initial value of on the T~e(T) plane (The saturated vapor pressure T~e_{sat}(T)-relationship obtained by Tetens Formula (a). The unsaturated (partial) vapor pressure T~e(T)-line calculated by multiplying relative humidity (reh). lE_{est}/H_{est} is similar to the relationship Ts − Tz over [q(Ts) − q(Tz)] from Equation (c). Specific humidity q(T) is proportional to vapor pressure (T). Thus, the hatched triangle is similarity to lE_{est}/H_{est}. The ratio lE_{est}/H_{est} is set as the initial value of optimization. The dotted triangle is a schematic presentation of converged ratio of lE_{est}/H_{est}. The initial triangle point c[T_{0}, e(T_{0})] should be set as possible as the vicinity of the final triangle point c’[Ts, e(Ts)]. That is very important things).

(13)

b is a constant passing through a straight line at T = 0˚C with gradient. The b is experimental constant to compensate the weakness of sensitivity in the optimization process. The maximum of b is defined as follows:

(14)

Using b and b_{max}, the constraint set as follows:

(15)

2.1.6. Actual Process of the Analysis

The estimate of H_{est} and lE_{est} is performed as follows: First, the value of Ts is chosen from observed values (T_{0}) collected near the soil surface. The rehs value is set to be the same as the rehz observed value. Second, the Ts and rehs are optimized together to satisfy the heat balance relationship using Equation (9) and Equation (10).

The values of ε_{i} are initially very small, on the order of 10^{−15} (W・m^{−2}). Therefore, the objective function is multiplied by 10^{15}. To avoid abnormal fluctuations of H_{est} and lE_{est}, constraints on those variables are set as less than ABS (Rn − G). The constraints for B_{app} are set at by referring to the singularity of convergence process [1] [2] and actual data. And also applied the constraint as. The reason for these constrains is described in the discussion section.

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) [1] [2] .

2.2. Investigation Site and Equipment

To examine the proposed method, eight sites were chosen throughout the world as listed in Table 1: three sites are in the USA and Australia and one site is in Europe and China. The data of all sites were prepared by FLUXNET (Beringer, et al. [7] : Sturt Plains (AU-Stp) Ozflux L2 Data; Billesbach, et al. [8] : Woodward (US-AR2) Data; Ceschia and Tallec [9] : Lamasquere (FR-Lam) Data; Eamus and Cleverly [10] : Ti Tree East (AU-TTE) Ozflux L2 Data; Beringer J. and L. Hutley [11] : Dry River (AU-Dry) Ozflux L2 Data; Scott [12] : Lucky Hills Shrubland (US-Whs) AmeriFlux L2 Data; Wharton [13] : Diablo (US-Dia) AmeriFlux L2 Data; Yanhong, et al. [14] : Qinghai Flux Research Site (CN-QHB) AsiaFlux Data).

Table 1 shows the name of the sites, country, state/province, location, elevation, vegetation, tower height, plant height and year of examined data. The tested year ware chosen as well as minimize the data gaps.

Table 2 describes the type of instruments applied with the variables of the heat balance components, unit and description of those measurements, including temperature T_{0} near the soil surface at depth d_{T}_{0} cm.

To investigate the accuracy of the observed data, Table 3 describes the heat balance relationship of observed at the tested sites that expressed in the heat flux. The

Table 1. Feature of the tested sites (all sites active core measurement presently being made).

Table 2. Measurement instruments of the tested sites including d_{T}_{0}.

imbalance was estimated by using observed annual data and the imbalance ratio defined as. There are numerous data gaps at all sites on an hourly basis.

3. Result

3.1. Conversion of Observed Data (H_{obs} and lE_{obs}) into Corrected Data (H_{cor} and lE_{cor})

The observed data do not achieve the heat balance relationship shown in Table 3. To maintain the heat balance, multiple regression analysis is applied using Equation (8) as aforementioned. Figure 3 describes the relationship (Rn − G) versus (H + lE) of the original and corrected data at all tested sites. The observed data are shown in the red circle, whereas the corrected data are shown in the blue circle. The slope of the corrected data at the tested sites mostly increased and approached 1.0 [1] [2] .

The regression coefficients for all tested sites are described in Table 4 that A is for lE and B is for H. The correction coefficient A ranged from 0.836 to 1.495 for H, with an average of 1.130, and the coefficient B ranged from 0.950 to 1.312 for H, with an average of 1.157. The observed data are corrected using these coefficients for all tested sites. The imbalance ratio Rai_{mb} before and after correction is described in Table 3.

Table 3. Annual heat balance relationship of the tested sites (except no observed data): Data gap, one of which G, Tz, T_{0}, P (atmospheric pressure), rehz, Rn, H_{obs} and lE_{obs}, is do not use for analysis. Imbalance is calculated by using yearly observed data and the imbalance ratio defined as.

Table 4. Regression coefficients A for lE and B for H: Regression coefficient is the same with correction coefficient of observed H and lE.

Figure 3. Relationship between (Rn − G) versus (H + lE) (W・m^{−2}): According to the heat balance relationship, Rn + G = H + lE must be guaranteed by actual data. Therefore, Rn + G versus H + lE compared. But Rn + G does not coincide with H + lE as red circle in the figure, to improve the relationship, regression analysis performed. The blue circle indicates the result of the regression analysis at eight tested sites.

3.2. Result of the General Solution by Equation (9) and Equation (10)

We set the initial value as Ts = T_{0} and rehs = rehz based on the triangle concept in Figure 2 as aforementioned. According to the initial values, the heat balance relationship is initially almost satisfied. The constraints Equation (15) applied for all sites except FR- Lam and CN-QHB because these two sites having relatively large amount of precipitation. This reason describe in the discussion section.

The estimation of Ts and rehs is conducted with precision = 0.000001 and convergence = 0.0001 in Solver option. In addition, heat balance is not achieved instantaneously; it requires a few hours [3] . Thus, the figure adjusts to a five-hour moving average. The comparison procedure is similar to the previous research [1] [2] .

3.3. Comparison of the Hourly Change of lE and H Estimated and Corrected

To confirm the validity of the general method, Figure 4 compares the hourly change in lE_{cor} with lE_{est} and H_{cor} with H_{est} at all sites in summer. All sites are very well consisted with each other. However, in detail, lE_{es}_{t} is not consistent with lE_{cor} at US-Dia and H_{est} is

Figure 4. Hourly change of lE and H observed and estimated at the tested sites by general method (1) (W・m^{−2}): The left hand sites describe the comparison of estimated and corrected lE and right hand side describe the estimated and corrected H at all tested sites.

not consistent with H_{cor} at FR-Lam. In the former case, lE_{est} is overestimated, while H_{est} is underestimated. The other items, such as lE_{obs} and lE_{imb}, indicate almost similar trends, and H_{obs} and H_{imb} also show the same trend but with small differences (not shown).

3.4. Annual Change of the Estimated and Observed lE and H

To investigate annual change in the estimated and observed lE and H, Figure 5 describes the changes of those variables throughout the year. All sites describe the trend is relatively well reproduced.

However, in detail, at AU-Dry and AU-Stp, lE_{est} indicate underestimate at former part of the year while H_{est} the overestimate at the same period. At FR-Lam, lE_{est} indicate a little underestimate at the year while H_{est} are a little overestimate. The other items of lE_{obs} and lE_{imb} exhibit similar trends, and H_{obs} and H_{imb} also display the same trend but with small differences (not shown).

3.5. Comparison of the Observed and Estimated lE and H

To confirm the validity of the general solution (1), Figure 6 compares the observed versus estimated data of lE and H on a daily basis. If the slope (slope of the straight line) is 1.0, the observed value coincides well with the estimated value. For H, the slopes of the observed versus estimated relationship nearly coincide (<15%) at the all sites except AU-Dry. For H, all sites indicate underestimate (>15%). R^{2} values (determination coefficient) shows a small values (<60%) for all sites except US-Dia (0.815) and AU- TTE (0.822), US-Whs (0.745). The other items of lE_{obs} and lE_{imb} are exhibited similar trends, and H_{obs} and H_{imb} show the same trend but with small differences (not shown). In addition, boundary of estimation accuracy is selected as 15% by referring the heat imbalance of original data.

Figure 5. Annual change of lE and H estimated and observed by general method (1) (W・m^{−2}): The left hand sites describe the comparison of estimated and corrected lE and right hand side describe the estimated and corrected H at all tested sites.

3.6. Comparison of the Observed rehz with Estimated rehs and T_{0} with Ts

The relationship between the estimated rehs and the observed rehz, i.e., the initial values and converged values, is of great concern. The left hand side of Figure 7 describes the hourly changes of those two items in summer. The parameters exhibit quite similar trends, i.e., the difference between rehs and rehz is quite small. The right hand side of Figure 7 describes the change of Ts − T_{0} and Ts − Tz. T_{0} is the observed temperature near the soil surface, and Tz is the air temperature, as previously mentioned. Ts − T_{0} changed periodically with daily changes about ±3˚C with site specific but sometimes shows a large difference. The difference between Ts − Tz is approximately within 8˚C.

The above trend of rehs and Ts changes is quite similar in the other seasons and at the other sites, although small differences are observed (not shown). In addition, the discontinuous portion is originated from data gap.

3.7. Comparison of Estimated and Observed Evapotranspiration Rate (ETa)

Using observed and estimated lE, monthly evapotranspiration was obtained at the all sites, as shown in Figure 8 by assuming 100 W・m^{−2} equivalents for 3.53 mm・day^{−1} [15] . The initial value of rehse quals rehz and constrains as Equation (12) and Equation (15) was applied. In addition, if there are data gap in a given month, the average ETa in a day multiplied the number of days of the month.

All sites describe satisfactory well reproduced Ha and Eta except AU-Dry. In detail, although there are small differences between Ha_{ob}_{s}, Ha_{cor} and Ha_{est}_{,} and also ETa_{obs}, ETa_{cor}, and ETa_{est} at all sites, the difference was relatively small.

Figure 6. Comparison of lE and H estimated and observed at the tested sites by general method (1) (W・m^{−2}).

Relative humidity at US-AR2 (2010) Ts-Tz and Ts-To at US-AR2 (2010)

Figure 7. Hourly change of relative humidity and temperature observed and estimated at the tested sites by general method (1) (˚C): The left hand sites describe the comparison of rehs and rehz and right hand side describe the temperature difference Ts-Tz and Ts-T_{0} at all tested sites.

Figure 8. Monthly change of Ha and ETa (evapotranspiration) at the tested sites (mm・month^{−1}).

4. Consideration

4.1. Comparison of General Method (1) with Conventional Method

The conventional method using Equation (3) cannot uniquely determine both Ts and rehs because two variables can’t determine mathematically by one equation. However, our aim of this analysis is to determine reasonable lE and H values for conserving the heat balance relationship. If we attempt to determine reasonable lE and H values using one equation, optimization can be performed by adjusting either Ts or rehs to satisfy the relationship. Based on the concept, the following analysis is conducted under initial values of Ts = Tz and rehs = rehz with the constraint of 0 < b < b_{max}. Table 5 compares the slope of lE_{est} and H_{est} against lE_{cor} and H_{cor} (as Figure 6) for general method (1) as well as conventional method.

For H_{cor}, the general method (1) is almost the same accuracy with the conventional method in that the slopes of all sites are adequate range (1.0 ± 0.15), while for lE_{cor}, all sites under estimate (<0.85). However, conventional method more balanced from the slope of lE_{cor} and LE_{cor}. In addition, the figures as Figure 4 to Figure 8 in general method (1) that previously described are quite similar those of the conventional method, therefore, the figures abbreviated because of space limitation. In addition, the accuracy criteria (1.0 ± 0.15) determine conventionally referring to observed data accuracy (Table 3).

Table 5. Comparison of the slope with corrected against estimated for general method (1), conventional method and general method (2): General method (1) estimates Ts and rehs using simultaneously two equations as Equation (9) and Equation (10), which guaranteed the uniqueness of variable determination, while conventional method estimate Ts and rehs using Equation (3) only which not guaranteed the uniqueness of variable determination. General method (2) used Ts observed by radiometer and rehs determined by Equation (13), thus, guaranteed the uniqueness of variable determination.

Note: *indicate accuracy; (1.0 ± 0.15).

4.2. Comparison of General Method (1) with General Method (2) Using Ts Observed by Radiometer

To verify the validity of our method, estimation lE and H using observed Ts (Ts_{rad}) by radiometer is performed. By using the Ts, the lE and H can be determined uniquely by Equation (1) to Equation (3) using T_{srad} instead of T_{sass} because the unknown variable is only one. The result described in Table 5 in that this method almost the same accuracy those of conventional method and more balanced H_{cor} and lE_{cor}. In addition, the result as Figures 4-8 in general method (1) previously described are quite similar those of this method. Therefore, the figures abbreviated because of space limitation.

4.3. Comparison of Total Amount of Yearly Ha and ETa of Three Methods

To compare the total amount of sensible heat (Ha) and latent heat flux (ETa) estimated and corrected expressed in mm・year^{−1} base. The estimation of Ha and ETa performed three ways as general method (1), conventional method and general method (2). The results summarized in Table 6.

Table 6 describes that the estimated Ha and ETa by the three methods mostly reproduced the corrected Ha and ETa. But there are site specific and method specific difference. Among this, general method (2) is seemed to be mostly suitable because which more coincided with the Ha_{cor} and ETa_{cor}. However, the data of T_{srad} is rarely at common climate observation, thus, we focused on general method (1) and conventional method.

Table 6. Comparison of corrected Ha and Eta versus estimated Ha and Eta of various method (mm・year^{−1}): General meth. (1), conventional meth. and general meth. (2) refer to the caption of Table 5. Unit of heat flux (W・m^{−2}) converted to (mm・day^{−1}) as ratio of 100 (W・m^{−2}) is equivalent for 3.53 (mm・day^{−1}) [15] . FR-Lam and CN-QHB does not apply b > 0.

Note: second row of each sites indicate the ratio against observed (corrected) data. Note: *indicate the accuracy; (1.0 ± 0.15). Note: observed data obtained by general solution (1).

To make clearer the accuracy, the annual Ha and ETa, compared as the ratio of estimated Ha_{est} and ETa_{est} against corrected Ha_{cor} and ETa_{cor}. Which shown in second row of each sites in Table 6 because the corrected data is a standard of reasonable estimation.

Attached star indicate the accuracy range (1.0 ± 0.15). For general method (1), most of Ha_{est}/Ha_{cor} indicated (1.0 ± 0.15) except FR-Lam while over half sites of ETa_{est}/ETa_{cor} indicate out of (1.0 ± 0.15). For conventional method, most of Ha_{est}/Ha_{cor}, indicate under 0.15 except AU-dry and FR-Lam while ETa_{est}/ETa_{cor} all sites indicate under 0.15 except US-Dia and AU-TTE where are belong to arid region. For general method (2), Ha_{est}/Ha_{cor} and ETa_{est}/ETa_{cor} at three sites indicate (1.0 ± 0.15) but AU-TTE and AU-Dry and FR-Lam of Ha_{est} are out of (1.0 ± 0.15). All of the above data describe mostly reasonable against our proposed method, although there are some exceptions.

4.4. Comparison of ETa versus Precipitation from the Aspect of Water Balance

It is considered that annual precipitation almost consumed as ETa in arid and semi- arid region. Figure 9 describes the relationship of annual precipitation versus ETa_{cor} and ETa_{est} by general method (1). Both figures describe mostly consistent each other. The facts indicate the reasonability of our proposed method from the aspect of water balance.

Figure 9. Comparison of ETa_{cor} and ETa_{est} versus precipitation.

5. Discussion

5.1. Effect of Constraints “b” on lE/H Ratio

The constraint expressed by Equation (15) is very well functioned for estimation of lE and H in arid region as US-Dia and AU-TTE having annual precipitation 297 mm and 180 mm. If the constraint does not apply at AU-TTE, for example, slope of lE_{est}/lE_{cor} is 1.552, while H_{est}/H_{cor} is 0.586 (not shown in figure or table). But if it applied the constraint at the same site, the slope of lE_{est}/lE_{cor} is 0.721 while H_{est}/H_{cor} is 0.904, resulting in remarkably improved. Apparent meaning of the constraint is almost rehs < rehz, but if the constraint applied directly the H_{est} and lE_{est} do not achieve optimal value smoothly.

Moreover, this constraint does not well functioned at semi-arid region having relatively large amount of precipitation as FR-Lam and CN-Hab. Therefore, the two sites do not apply the constraints. The other sites as AU-Dry, AU-Stp, US-Ar2 and US-Whs are intermediate of above sites. If the constraint applied, the H_{est} are well estimated while if do not applied the lE_{est} are well estimated.

5.2. Exceptions of the Abnormal Fluctuation in Convergence Process

In convergence process, The H_{est} and lE_{est} sometimes fluctuate plus or minus abnormally because sum of those is limited (=Rn + G). For example, the former estimate abnormally large (plus) while the latter abnormally small (minus). To avoid the abnormal fluctuation, constraint as ABS (Rn + G) < H or lE is applied. On the other hand, observed data contain sometime the abnormal data that appear at near 0˚C in winter season or early morning. The analyses were conducted without the abnormal data [1] [2] .

6. Conclusions

In the natural world, air temperature and humidity reflect the partitioning of the sensible and latent heat flux from Rn and G. Based on the concepts, we attempt to reciprocally estimate the H and lE from Rn and G. By applying the Bowen ratio concept to the soil surface, the unknown variables Ts and q(Ts), i.e., rehs, are estimated by an optimization procedure satisfying the heat balance relationship [1] [2] .

This method is very effective to expand the utility of the water recourses issues through estimation of evapotranspiration for various areas because it requires only single height Tz and rehz that are very popular climate elements.

The validation of the method was conducted using eight globally dispersed sites, where observed lE and H by the eddy covariance using FLUXNET. The analysis was conducted on an hourly basis and was summarized as daily averages.

The analysis was conducted using three methods, namely, the general (1), conventional and general (2) methods. The general method (1) is based on the unified determination of Ts and rehs by solving two heat balance equations simultaneously, whereas the conventional method is based on solving a single heat balance equation that does not unify Ts and rehs. The general method (2) is based on Ts observed by radiometer.

The general method (1) is mathematically reasonable, but the data of two unit times are required. The conventional method does not allow for the uniqueness of Ts and rehs, but reasonable partitioning of H and lE can be achieved by adjusting either Ts or rehs, depending on the GRG algorithm. The main purpose of our research is to accurately estimate lE and H rather than to accurately estimate Ts and rehs; thus, the conventional method can be used as well as the general method (1). If Ts observed by radiometer, application of the general method (2) is reasonable.

The main validated results are as follows:

1) The observed data are corrected by regression analysis because it does not guarantee the heat balance relationship.

2) The general and conventional solutions are nearly consistent for hourly changes of lE_{est} with lE_{cor} as well as H_{est} with H_{cor} at eight sites. However, there are some site- specific differences.

3) The general and conventional solutions are nearly consistent for the annual changes of lE_{est} with lE_{cor}, and H_{est} with H_{cor} at eight sites. However, there are some seasonal and site-specific differences.

4) Analysis of annual change of Ha and ETa are performed. The results describe that Ha_{est} is consistent with Ha_{cor} and ETa_{est} consistent with ETa_{cor}.

5) To confirm the reasonability of general method (1), conventional method and general method (2), the ratio of ETa_{es}_{t}/ETa_{cor} is compared. The result indicates that there is no remarkable difference among the two methods.

6) Annual precipitation is mostly consistent with ETa_{est} and ETa_{cor} from the aspect of water balance.

The estimated results do not completely reproduce the observed data, but the results are mostly satisfactory for the estimation of lE and H. The remarkable feature of this method is that it is applicable for single height of temperature and humidity with Rn and G. This feature shows the method would be widely applicable for estimation of lE. If this procedure is approved widely, the resolution of water resources problem and reasonable irrigation planning will be more advanced.

However, there are some issues to be solved in future: 1) Error plain of ε in Equation (3) or Equation (9) and Equation (10), i.e., relationship of ε with Ts and rehs, is very complicated, having many local minimum. Therefore, the importance of determination of initial values is still important; 2) Applying constraint is also important issues because it strongly affects the results; 3) Snow and frozen problems are not considered in this research so that these problems should be solved in future and 4) Accuracy of the observed data is also very important issue which has been improved nowadays by many researchers and technologists.

We conclude that ET and H are controlled by energy conservation in nature. Realistically, the observed temperature and humidity are strongly affected by the partitioning of H and lE and vice versa. Therefore, using the observed temperature, humidity and common climate elements, lE and H values are reciprocally approximated by the optimized techniques [1] [2] .

Acknowledgements

We express sincere thanks to the AmeriFlux，EuroFlux and AsiaFlux principal investigation for data accessed July 5, 2015. We thank Dr. Fujihara Yooich and Dr. Takimoto Hiroshi for providing valuable comments for the optimization procedure. We acknowledge the following AmeriFlux sites for their data records: site IDs. In addition, funding for AmeriFlux data resources was provided by the U.S. Department of Energy’s Office of Science.

Appendix 1

The GRG Nonlinear Solving Method for nonlinear optimization: developed by Leon Lasdon (University of Texas at Austin) and Alan Waren (Cleveland State University) and enhanced by Frontline Systems, Inc.

For more information about the other solution algorithms, advice on building effective solver models, and solving larger scale problems, contact: Frontline Systems, Inc.

Web site: http://www.solver.com, E-mail: info@solver.com

Eestimated results have not completely reproduced the observations, but the results are mostly satisfaction

Appendix 2

Using modules of Visual Basic for Applications (VBA) in the manuscript

Sub Macro “Number1 ( )

' Macro ”Number 1”：GRG method

Dim r As Long

Dim lastRow As Long

lastRow = Range(“〈Column Alphabet〉” & Rows Count).End (xlUp).Row

SolverReset

For r = 〈Start row number〉To〈End row number〉

SolverReset

SolverOptions Precision:=0.000001, Convergence:=0.0001, StepThru:=False, Scaling:=False _

, AssumeNonNeg:=False, Derivatives:=2

SolverOkSetCell:= "Row" & r, MaxMinVal:=2, ValueOf:=0_

, ByChange:=Range(Cells(r, 〈First column number〉), Cells(r, 〈Last column number〉))

SolverAddCellRef:="$ 〈rehs’s Column Alphabet〉" & r, Relation:=1, FormulaText:=1

SolverAddCellRef:="$ 〈rehs’s Column Alphabet〉" & r, Relation:=3, FormulaText:=0

SolverAddCellRef:="$ 〈RTs’s Column Alphabet〉" & r, Relation:=1, FormulaText:=5

SolverAddCellRef:="$ 〈RTs’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－5

SolverAddCellRef:="$ 〈Hestimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:= "$ 〈Rn-G observed’ s Column Alphabet〉$ &r

SolverAddCellRef:="$ 〈Hestimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－100

SolverAddCellRef:="$ 〈LEestimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:= "$ 〈Rn-G observed’ s Column Alphabet〉$ &r

SolverAddCellRef:="$ 〈LEestimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－100

SolverAddCellRef:="$ 〈B_{app}’s Column Alphabet〉" & r, Relation:=1, FormulaText:=100

SolverAddCellRef:="$ 〈B_{app}’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－100

※in case of 0
**
max
**

SolverAddCellRef:="$ 〈bestimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=0

SolverAddCellRef:="$ 〈bestimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:= "$ 〈b_{max}’ s Column Alphabet〉$ &r

SolverSolveUserFinish:= True, ShowRef:="DummyMacro"

Next

End Sub

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Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Open Journal of Modern Hydrology*,

**7**, 38-64. doi: 10.4236/ojmh.2017.71003.

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