^{1}

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Evapotranspiration acts an important role in hydrologic cycle and water resources planning. But the estimation issue still remains until nowadays. This research attempts to make clear this problem by the following way. In a humid region, by applying the Bowen ratio concept and optimum procedure on the soil surface, sensible and latent heat fluxes are estimated using net radiation (
*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.

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 [

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).

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.

A proposed model is somewhat similar to previous research [

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,

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

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 [

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

The governing equation to be solved is obtained by heat balance relationship [_{sat} (Ts) are assumed initially; thus, the heat balance relationship has not closed as Equation (3):

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, _{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:

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

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).

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:

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:

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)):

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.

The heat imbalance is observed in actual data, which is well known as a “closure issue” [

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.

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

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

The constraint for optimization process set as follows:

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.

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 [

To examine the proposed method, six sites were chosen in humid regions having annual precipitation over 1000 mm (

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

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.

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 [

Observed data do not achieve the heat balance relationship, as shown in

To confirm the validity, _{obs} with lE_{est} and

Site name/FLUXNET ID: | Tsukuba/Site2-Jap | Brooks Field Site 11/US-Br3 | Konza Prairie/US-Kon | Goodwin Creek/US-Goo | Dripsey/IE-Dri | Vall dAlinya/ES-VDA |
---|---|---|---|---|---|---|

Country: | Japan | USA | USA | USA | Ireland | Spain |

State/province: | Tsukuba University/Ibaraki Pref. | Iowa | Kansas | Mississippi | Corcaigh | Cataluna |

Latitude (+N/−S): | 36.1135 | 41.9747 | 39.0824 | 34.2547 | 51.9867 | 42.1522 |

Longitude (+E/−W): | 140.0948 | −93.6936 | −96.5603 | −89.8735 | −8.7518 | 1.4485 |

Elevation: | 29.0 m | 314 m | 443 m | 87 m | 186 m | 1787 |

Vegetation (IGBP): | Grasslands | Croplands | Grasslands | Grasslands | Grasslands | Grasslands |

Tower height: | 30.5 m | 5 m | - | 4 m | - | - |

Canopy height: | 0.1 - 1.0 m | - | 0.4 m | 0.20 - 0.40 m | - | - |

Data available | 1999 | 2010 | 2009 | 2006 | 2008 | 2008 |

1/1-12/31 | 1/1-12/31 | 1/1-12/31 | 1/1-12-31 | 1/1-12-31 | 1/1-12-31 |

Site name/FLUXNET ID: | Tsukuba/Site2-Jap | Brooks Field Site 11/US-Br3 | Konza Prairie/US-Kon | Goodwin Creek/US-Goo | Dripsey/IE-Dri | Vall dAlinya/ES-VDA | ||
---|---|---|---|---|---|---|---|---|

Variable | Units | Description | Model | Model | Model | Model | Model | Model |

FG | W∙m^{−2} | Soil heat flux | Soil Heat Flux Plate (CPR-PHF-01, Cmimatec) | Soil Heat Flux Plate (HFT, REBS) | Soil Heat Flux Plate (HFT-3, REBS) | Soil Heat Flux Plate (HFP01SC, REBS) | Soil Heat flux plate (HFP01) | Soil Heat flux plate (HFP10SC, Hukseflux) |

H | W∙m^{−2} | Sensible heat flux | Sonic Anemometer (DA-650, TR-61, AKAIJO SONIC Co.) | Sonic Anemometer (CSAT3, Campbell Scientific) | Sonic Anemometer (CSAT3, Campbell Scientific) | Sonic Anemometer (81,000 V, R. M. Young) | Sonic anemometer (CSAT, Campbell Scientific) | Sonic anemometer (R3A, Gill) |

LE | W∙m^{−2} | Latent heat flux | - | Sonic Anemometer (CSAT3, Campbell Scientific) Open Path CO_{2}/H_{2}O Gas Analyzer (LI-7500, LI-COR) | Sonic Anemometer (CSAT3, Campbell Scientific) Infrared CO_{2}/H_{2}O Gas Analyzer (LI-6262, LI-COR) | Sonic Anemometer (81,001V, R. M. Young) Infrared CO_{2}/H_{2}O Gas Analyzer (Open-Path, ATDD/NOAA) | Sonic Anemometer (CSAT, Campbell Scientific) Open Path CO_{2}/H_{2}O Gas Analyzer (LI-7500, LI-COR) | Sonic Anemometer (R3A, Gill) Open Path CO_{2}/H_{2}O Gas Analyzer (LI-6262, LI-COR) |

PREC | mm | Precipitation | Rain Gauge (WB0013-05, Yokogawa Denshikiki Co.) | - | - | Tipping Bucket Rain Gauge (TB3, Hydrological Services) | Rain gauge (arg100) | Precipitation Sensor (ARG 100, Environmental measurements Ltd) |

PRESS | kPa | Barometric pressure | Barometric Pressure Sensor (PTB210, Vaisala) | - | - | Barometric Pressure Sensor (PTB101, Vaisala) | Barometric Pressure Sensor (PTB1001, Vaisala) | Sensor technics (model 144SC0811BARO) |

RH | % | Relative humidity of air | Temperature/ Humidity Probe (CVS-HMP45D, Climatec) | Temperature/ Humidity Probe (HMP35, Vaisala) | Temperature/ Humidity Probe (HMP45C, Vaisala) | Temperature/ Humidity Probe (HMP50Y, Vaisala) | Temperature/ Humidity Probe (HMP45C, Vaisala) | Temperature & humidity transmitter (MP100, Rotronic) |

Rn | W∙m^{−2} | Net radiation | Net Radiometer (CN-11, EKO Instruments) | Net Radiometer (Q*7.1, REBS) | Net Radiometer (Q*7.1, REBS) | Net Radiometer (Kipp-zonen, CNR1) | Net Radiometer (CNR1, Kipp-zonen) | Net Radiometer (CNR1, Kipp-zonen) |

TA | deg C | Air temperature | Temperature/ Humidity Probe (CVS-HMP45D, Climatec) | Temperature/ Humidity Probe (HMP35, Vaisala) | Temperature/ Humidity Probe (HMP45C, Vaisala) | Temperature/ Humidity Probe (HMP50Y, Vaisala) | Temperature/ Humidity Probe (HMP45C, Vaisala) | Temperature & humidity transmitter (MP100, Rotronic) |

T_{0} | deg C | Soil temperature | Soil temperature Probe (C-PTG-10, Climatec) | Thermocouple (Type T) | - | - | - | - |

D_{To} | cm | Depth of measurement | 2 | 2 | 2 | 2 | 2.5 | 5 |

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

Site name | Year | Period | Unit | Heat balance components (W∙m^{−2}) | Ra_{imb} | Data gap | Precipitation | Remarks | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Rn | G | H | lE | Imbalance | (%) | (%) | (mm∙year^{−1}) | for lE | ||||

Site2-Jap | 1999 | 1/1-12/31 | W∙m^{−2} | 28,584 | −305 | 5229 | 23,659 | 0 | 0 | 8 | 1336 | Heat balance |

mm∙year^{−1} | 1009 | −11 | 185 | 835 | 0 | |||||||

US-Br3 | 2010 | 1/1-12/31 | W∙m^{−2} | 27,783 | 330 | 6286 | 19,385 | 1783 | 6 | 9 | 1392 | Observed |

mm∙year^{−1} | 981 | 12 | 222 | 684 | 63 | |||||||

US-Kon | 2009 | 1/1-12/31 | W∙m^{−2} | 30,344 | −684 | 9220 | 16,603 | 5205 | 18 | 20 | 1054 | Observed |

mm∙year^{−1} | 1071 | −24 | 325 | 586 | 184 | |||||||

US-Goo | 2006 | 1/1-12/31 | W∙m^{−2} | 32,948 | 1060 | 9662 | 19,402 | 2824 | 8 | 29 | 1369 | Observed |

mm∙year^{−1} | 1163 | 37 | 341 | 685 | 100 | |||||||

IE-Dri | 2008 | 1/1-12-31 | W∙m^{−2} | 28,241 | −341 | 3039 | 14,893 | 8941 | 31 | 8 | 1308 | Observed |

mm∙year^{−1} | 616 | −12 | 107 | 526 | 316 | |||||||

ES-VDA | 2008 | 1/1-12-31 | W∙m^{−2} | 21,922 | 330 | 5991 | 11,434 | 4168 | 19 | 2 | 1227 | Observed |

mm∙year^{−1} | 774 | 12 | 211 | 404 | 147 |

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} [

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.

_{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).

_{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 (

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

Site Name | A | B | R^{2} |
---|---|---|---|

Site2-Jap | 1.000 | 1.000 | 0.997 |

US-Br3 | 1.164 | 1.030 | 0.927 |

US-Kon | 1.186 | 1.123 | 0.835 |

US-Goo | 1.082 | 1.252 | 0.970 |

IE-Dri | 1.135 | 1.496 | 0.916 |

ES-VDA | 1.435 | 1.472 | 0.976 |

Average | 1.167 | 1.229 | 0.937 |

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

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.

_{0} with rehs versus rehz. The feature is site specific. For corrected against estimated lE and H, the relationship is already described by

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

Site Name | item | lE_{cor} | H_{cor} | lE_{obs} | H_{obs} | Ts ~ T_{0} | rehs ~ rehz |
---|---|---|---|---|---|---|---|

Site2-Jap | Slope | *1.033 | 0.397 | *1.033 | 0.397 | 1.079 | |

R^{2} | 0.832 | 0.018 | 0.832 | 0.018 | −1.500 | ||

US-Br3 | Slope | *0.944 | 0.777 | *0.966 | 0.848 | 1.121 | 1.070 |

R^{2} | 0.703 | 0.152 | 0.703 | 0.153 | 0.970 | 0.656 | |

US-Kon | Slope | *0.869 | 0.771 | *0.976 | *0.914 | 0.997 | 0.917 |

R^{2} | 0.310 | 0.203 | 0.310 | 0.203 | 0.997 | 0.605 | |

US-Goo | Slope | *0.890 | *1.038 | *1.114 | 1.222 | 1.013 | 0.957 |

R^{2} | 0.811 | 0.559 | 0.811 | 0.559 | 0.951 | 0.953 | |

IE-Dri | Slope | *0.944 | 0.715 | 1.407 | 0.823 | 1.028 | 1.045 |

R^{2} | 0.677 | 0.078 | 0.677 | 0.078 | 0.961 | 0.768 | |

ES-VDA | Slope | *1.037 | 0.838 | 1.383 | 1.231 | 1.135 | 1.075 |

R^{2} | 0.765 | 0.641 | 0.765 | 0.641 | 0.982 | 0.914 | |

Average | Slope | 0.953 | 0.756 | 1.146 | 0.906 | 0.882 | |

R^{２} | 0.683 | 0.275 | 0.683 | 0.275 | 0.810 |

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 ^{−2} equivalents for 3.53 mm∙day^{−1} [

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 _{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) [

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.

Site name | Hest | lEest | Hcor | lEcor | Hobs | lEobs | lE_{est}/lE_{cor} | H_{est}/H_{cor} | Imbalance |
---|---|---|---|---|---|---|---|---|---|

Site2-Jap | 126 | 901 | 188 | 839 | 188 | 839 | *1.07 | 0.67 | 0 |

US-Br3 | 287 | 681 | 241 | 702 | 221 | 686 | *0.97 | 1.19 | 26 |

US-Kon | 589 | 789 | 558 | 680 | 470 | 606 | 1.16 | *1.06 | 140 |

US-Goo | 472 | 810 | 410 | 925 | 379 | 739 | *0.88 | *1.15 | −52 |

IE-Dri | 241 | 706 | 110 | 770 | 96 | 517 | *0.92 | 2.19 | 67 |

ES-VDA | 384 | 540 | 377 | 559 | 257 | 419 | *0.97 | *1.02 | −11 |

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) [_{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 _{H} = C_{E}, that is the same of Penman method’s assumption [_{H} = C_{E} and Ts mathematically because one equation determine two variables.

To verify the reasonability of estimated Ts,

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 _{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 [

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

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

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.

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.

Maruyama, T. and Segawa, M. (2016) Application of the Reciprocal Analysis for Sensible and Latent Heat Fluxes with Evapotranspiration at a Humid Region. Open Journal of Modern Hydrology, 6, 230-252. http://dx.doi.org/10.4236/ojmh.2016.64019

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.

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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