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Application of the Reciprocal Analysis for Sensible and Latent Heat Fluxes with Evapotranspiration at a Humid Region ()

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*Rn*) and heat flux into the ground (

*G*). The method uses air temperature and humidity at a single height by reciprocally determining the soil surface temperature (

*Ts*) and the relative humidity (

*rehs*). This feature can be remarkably extended to the utilization. The validity of the method is confirmed by comparing of observed and estimated latent (

*lE*) and sensible heat flux (

*H*) using 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. Furthermore, monthly evapotranspiration is estimated from the

*lE*. The research was conducted using hourly data of FLUXNET at a site of Japan, three sites of the United States and two sites of Europe in humid regions having over 1000 mm of annual precipitation.

KEYWORDS

Cite this paper

*Open Journal of Modern Hydrology*,

**6**, 230-252. doi: 10.4236/ojmh.2016.64019.

1. Introduction

In the natural world, the air temperature and humidity are determined by H and lE from the net radiation (Rn) and heat flux into the ground (G). Therefore, our research attempts the reciprocal analysis of H and lE from the air temperature (Tz) and humidity (rehz) at single height while satisfying the heat balance relationship. The concept can’t find the other relevant methods, and it only requires Rn, G, Tz and rehz. This feature is remarkably widened a utilization purposes.

Recently, we reported the reciprocal analysis of sensible and latent heat fluxes in a forest region [1] . However, “humid region” is quite different from “forest region” because of no canopy. This paper described “humid region” instead of “forest region”, although there was similar concept in previous research.

The main different point is: the present paper contains two unknown variables, i.e., relative humidity (rehs) and temperature on the soil surface (Ts) while the previous paper contains only one variable, i.e., rehs, on the canopy surface. Therefore, the analysis has differences in that the present paper has to solve two simultaneous equations while the previous paper solved only one equation. In the analytical process, various new points arisen. Addition, this paper describes the comparison of the Penman method with our method because of humid region.

In the proposed method, the unknown variables, Ts and rehs were determined by the non-linear optimization technique known as the general reduced gradient (GRG) using the Excel Solver (Appendix 1).

2. Materials

We proposed a general method for estimating sensible and latent heat flux using single height temperature and humidity. The method contains two unknown variables: soil surface temperature, Ts, and humidity, rehs. This chapter describes the theoretical back- ground for estimating Ts and rehs, the practical procedure, data correction, the details of test sites and measurement instruments.

2.1. Method of Analysis

2.1.1. Fundamental Concept of the Model

A proposed model is somewhat similar to previous research [1] . Therefore, briefly the outline is described. The proposed model considers the near-soil surface as shown in Figure 1.

Figure 1. Components of the model and the relevant symbols.

Here, Rn is net radiation which is portioned into sensible, latent and underground heat fluxes. Ts is the soil surface temperature, Tz is the air temperature at height z, is the specific moisture at height z, rehz is relative humidity in air at height z, is the unsaturated specific moisture on the soil surface, and is the saturated specific moisture on the soil surface.

The fundamental formulae of the model satisfy the following well-known heat balance relationship [2] .

. (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 [2] :

. (2)

We apply the concept of Bowen ratio to the layer between the soil surface and observation height of Tz and rehz. But, the Ts and are just on the surface and usually unknown.

2.1.2. Governing Equation for Estimating the Unknown Variables Ts and rehs

The governing equation to be solved is obtained by heat balance relationship [1] . The unknown variables Ts and rehs are estimated as follows: The Ts and the, i.e., rehs × e_{sat} (Ts) are assumed initially; thus, the heat balance relationship has not closed as Equation (3):

(3)

(4)

and. (5)

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

The approximated Ts and rehs putting in Equation (4), lE and H of next order approximated values obtained by Equation (5).

By repeating the above calculation from Equation (3) to Equation (5), the B_{app} converged to B_{0} according to objective function ABS (ε_{i}) conversed to a minimum.

After optimization, B_{app} is conversed to B_{0}. Then, lE_{est} and H_{est} can be obtained as follows:

and. (6)

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

. (7)

Here, T_{0} is the observed soil temperature (˚C), D_{To} is the depth of the temperature observation (cm), Kt is the assumed thermal conductivity (W∙m^{−1}∙˚C^{−1}).

Equation (7) describes how to obtained Ts by extrapolating T_{0} using G, D_{To} and Kt. The calculation follows General Reduced Gradient (GRG) algorithm, which can be applied with the Excel Solver on a personal computer (Appendix 1 and Appendix 2).

2.1.3. General Solution

To uniquely determine the two unknown variable Ts and rehs, two equations are required mathematically. We set the two equations as follows assuming Ts and rehs has no remarkable difference between two unit hours:

(8)

. (9)

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 (8) and Equation (9) simultaneously under T^{j} = Ts^{j}^{+1} and reh^{j} = rehs^{j}^{+1} conditions: conversed to minimum.

In addition, to prevent abnormal fluctuation of H_{est} versus lE_{est} in optimization process, constraints Rn − G < H, lE are applied as follows (Equation (10)):

. (10)

Equation (8) and Equation (9) are nonlinear two element simultaneous equations. 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 were obtained from observations or were calculated independently.

2.1.4. Correction of the Heat Imbalance Based on Multiple Regression Analysis

The heat imbalance is observed in actual data, which is well known as a “closure issue” [3] [4] . Therefore, the data was corrected conventionally according to Allen’s procedure by multiple regression analysis [5] :

. (11)

Here: Rn, G, lE and H are described earlier. A, B are the regression coefficient for lE, H.

To guarantee the heat balance relationship, all sites used the corrected data. In addition, the correction is conducted using the daily basis.

2.1.5. Constraint to Improve the Underestimation of lE

To improve the under or overestimation of lE i.e., over or underestimation of H, we set the following constant defined as Equation (12):

(12)

b is a constant passing through straight line at T = 0˚C with slope

. In Equation (12) the and are converted from and using and relationship.

The constraint for optimization process set as follows:

or. (13)

The constraint is expected increasing of lE_{est}, whereas decrease H_{est} at high humidity area or vice versa. General analysis applied the constraint of Equation (13).

In addition, the constraints of Equation (13) have a similar role of rehs > rehz or rehs < rehz depending on initial values of rehs = rehz or rehs = 1.0 that is expected in humid region.

2.1.6. Initial Values for Optimization and Constraints

The initial values of Ts and rehs are key factors for obtaining reliable results. The value of Ts is chosen as T_{0} because the T_{0} is observed at near the soil surface. The initial value of rehs chosen as rehs = 1.0 because humid region or rehs = rehz depending on site specific conditions. Then, RTs was assumed to be 0, The RTs was automatically improved to satisfy the optimum value of Ts and rehs.

The ε has very small values on the order of 10^{−15} W∙m^{−2} initially, because B_{app} nearly satisfies the heat balance relationship. Therefore, the objective function is multiplied by 10^{15}. To avoid abnormal fluctuation of H and lE, in the optimization process, constraints on those are set as less than (Rn − G) as mentioned earlier. Additionally, B_{app} is constrained as −100 < B_{app} < 100 by referring to the actual data and optimization process [1] . The reason is described in the discussion section. We set the precision: =0.000001 and convergence: =0.0001 in Solver option.

2.2. Investigation Sites and Equipment

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 T_{0} was observed by thermometer at the depth of 2 ~ 5 cm.

2.3. Heat Balance Relationship of Observed Sites and Data Gap

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.

3. Result

The general solution determines two variables, Ts and rehs, using two equations simultaneously. Therefore, Ts and rehs can be uniquely determined mathematically. The initial value is set as aforementioned. Furthermore, the heat balance is not achieved instantaneously; it requires a few hours [5] . Thus, the hourly figure adjusts to a five-hour moving average.

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

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.

3.2. Comparison of the Hourly Change of the lE and H at all Sites

To confirm the validity, Figure 3 compares the hourly changes in lE_{obs} with lE_{est} and

Table 1. Features of the tested sites.

Table 2. Measurement instruments of the tested sites including D_{To}.

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

Table 3. Heat balance of the sites including data gap and annual precipitation (unit: heat flux).

Note: Data gap is not available data for analysis, i.e., lacked one of which G, T_{z}, T_{0}, 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.

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

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).

3.4. Comparison of the Observed and Estimated lE and H

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. R^{2} (R is corrected determination coefficient) of lE_{est} shows underestimated at US-Kon (>60%) and R^{2} 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).

3.5. Relationship of the rehz and T_{0} and Estimated rehs and Ts

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

Figure 2. Comparison of (Rn − G) with (H + lE) observed and corrected (W∙m^{−2}).

Table 4. Regression coefficient for lE and H.

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

Figure 3. Hourly changes of lE and H observed and estimated (W∙m^{−}^{2}) (general solution). Note: 1) Initial condition 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.

Figure 4. Yearly change of lE and H observed and estimated (W∙m^{−2}) (general solution). Note: Initial condition and constraints are the same with Figure 3.

Figure 5. Comparison of lE and H observed and estimated (W∙m^{−2}). Note: Initial condition and constraints are the same with Figure 3.

Figure 6. Hourly change of rehs and Ts (general solution). Note: Initial condition and constraints are the same with Figure 3.

shows the change of Ts − T_{0} and Ts − Tz. The Ts changed remarkably from initial value T_{0}. The Ts − T_{0} 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.

3.6. Slope of Estimated against Observed in All Analyzed Data

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 T_{0} 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 T_{0} 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

Table 5. All data analyzed by general method (general solution).

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.

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 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.

Figure 7. Comparison of observed (ETa_{obs}) and estimated (ETa_{est}) monthly evapotranspiration (mm∙month^{−1}). Note: Initial condition and constraints are the same with Figure 3. Observed data attached as reference.

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.

4. Consideration

4.1. Relationship of Penman Method with Proposed Method

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]

. (14)

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}), U_{10} 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.

Table 6. Total amount of evapotranspiration estimated and observed including correction (mm∙year^{−1}).

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%.

Figure 8. Comparison of Penman method with our method (W∙m^{−2}).

4.2. Comparison of Bulk Transfer Method at Wetted Soil Surface with our Method

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.

. (15)

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.

4.3. Comparison of Observed Ts with Estimated by Radiometer Ts

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.

5. Discussion

5.1. Initial Values and Constraints

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

Figure 9. Comparison of Ts observed by radiometer and estimated (˚C).

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.

5.2. Abnormal Fluctuation of B_{app} (Singularity of B_{app})

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] .

6. Conclusions

In the natural world, the air temperature and humidity reflect the partitioning of sensible and latent heat flux from Rn and G. Based on this concept, we attempt to estimate H

Figure 10. Relationship between B_{app} and temperature Ts when Ts − Tz = 1.0˚C [1] .

and lE reciprocally using single height temperature and humidity, and Rn and G by applying the Bowen ratio concept on the soil surface. This feature can be remarkably extended to the field of utilization. The unknown variables Ts and (i.e., rehs) are estimated by an optimization procedure as satisfying heat balance relationship. The validation of the method was achieved by the six sites in the humid regions of Japan, USA and Europe. lE and H were observed by the eddy covariance method at these sites, except lE in Japan site. Analysis is conducted on an hourly basis and summarized daily. The main results are as follows:

1) The hourly and yearly change of the estimated lE and H very well coincided with the observed values at all sites.

2) The estimated lE and H versus corrected lE and H or observed lE and H are satisfactory coincided.

3) The hourly change of Ts and rehs can be estimated by the method that is very difficult to observe at actual site.

4) The estimated evaporation ETa satisfactorily coincided with corrected and observed ETa not only monthly change but also annual amount.

5) The method compared with penman method and confirmed the validity.

The estimated results have not completely reproduced the observations, but the results are mostly satisfactory. This fact shows that the method is useful for the estimation of lE and H. The remarkable feature of the new method is that it is applicable for the approximate of lE and H using a single height of Tz and rehz with Rn and G. For estimation of ETa, this method will be applicable to various local areas because of required data easily obtained.

But, there are problems that still remain. The error plain i.e., ε_{i} in Equation (3) related to Ts and rehs, is very complicated because of nonlinear simultaneous equation having many local minimum. Therefore, the selection of initial values of Ts and rehs is important issue to be solved in future. On the other hand, this research is restricted at humid region but analysis of sensible and latent heat flux at arid and semi-arid region is also very important. This is also another big problem to be solved in future.

We conclude that the partitioning of lE and H is 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, the lE and H values are reciprocally approximated by the optimized techniques.

Acknowledgements

We sincerely thanks for providing the AmeriFlux and EuroFlux principal investigation for data accessed on July 5, 2015. We sincerely thank Dr. Asanuma Jun, a professor at Tsukuba University, for providing valuable data for the eddy covariance method; Dr. Kuwagata Tsuneo, Dr. Fujihara Yoichi and Dr. Takimoto Hiroshi for providing valuable comments on the optimization procedure. We also thank Dr. Yoshida Masashi and Dr. Noto Fumikazu, who are staff members at Ishikawa Prefectural University, for recording the data. We also thank the staff at the Ishikawa Forest Experiment Station.

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

Estimated 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

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

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

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

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

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

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

SolverAdd CellRef:="$ 〈H estimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:= "$ 〈Rn-G observed’ s Column Alphabet〉$ &r

SolverAdd CellRef:="$ 〈H estimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－100

SolverAdd CellRef:="$ 〈LE estimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:= "$ 〈Rn-G observed’ s Column Alphabet〉$ &r

SolverAdd CellRef:="$ 〈LE estimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=－100

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

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

※in case of b>0

SolverAdd CellRef:="$ 〈b estimated’s Column Alphabet〉" & r, Relation:=3, FormulaText:=0

※in case of b<0

SolverAdd CellRef:="$ 〈b estimated’s Column Alphabet〉" & r, Relation:=1, FormulaText:=0

SolverSolve UserFinish:= True, ShowRef:="DummyMacro"

Next

End Sub

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Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
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http://dx.doi.org/10.4236/jwarp.2016.87059 |

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