# Evaluation of Clear-Sky Incoming Radiation Estimating Equations Typically Used in Remote Sensing Evapotranspiration Algorithms

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## Abstract

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^{2}≥ 0.92), (2) ${R}_{\text{S}}^{\downarrow}$ estimating equations tend to overestimate, especially at higher values, (3) ${R}_{\text{L}}^{\downarrow}$ estimating equations tend to give more biased values in arid and semi-arid regions, (4) a model that parameterizes the diffuse component of radiation using two clearness indices and a simple model that assumes a linear increase of atmospheric transmissivity with elevation give better ${R}_{\text{S}}^{\downarrow}$ estimates, and (5) mean relative absolute errors in the net radiation (R

_{n}) estimates caused by the use of ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimating equations varies from 10% to 22%. This study suggests that R

_{n}estimates using recommended incoming radiation estimating equations could improve ET estimates.

## 1. Introduction

_{n}) is a key component of the energy balance, whose estimation accuracy has an impact on energy flux estimates from remotely sensed data. In typical algorithms that handle remote sensing data, evapotranspiration (ET) is estimated as a residual of R

_{n}after accounting for sensible heat flux (H) and soil heat flux (G) [1–4]; G is estimated from empirical equations that relate G/R

_{n}to vegetation index, and H is estimated such that the maximum value of H over a “hot” surface does not exceed R

_{n}. Llasat and Snyder reported that 65%–85% of the error in R

_{n}estimation directly propagates to crop-reference ET in the Catalonia region of Spain [5]. Sun et al. showed that a 10% error in R

_{n}could result in a 25% error in actual ET when the latter is estimated through the Sim-ReSET algorithm over an irrigated crop field in a semi-arid climate [6].

_{n}is estimated by summing up estimates of its shortwave and longwave components:

_{s}is the surface emissivity, σ is the Stefan-Boltzmann constant (i.e., 5.670373×10

^{−8}W/m

^{2}/K

^{4}), and T

_{s}[K] is the surface temperature. The incoming components ( ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$) might be indirectly estimated from remote sensing data and atmospheric profile observational data through radiative transfer models [7–10], but they are typically estimated from available weather station data using empirical but straightforward equations in remote sensing ET algorithms [1,6,11–13]. The outgoing components ( ${R}_{\text{S}}^{\uparrow}$ and ${R}_{\text{L}}^{\uparrow}$) could be directly estimated from remote sensing optical and thermal information of land surface.

_{n}calculated from ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ was not evaluated.

_{n}from weather station data under clear sky conditions. Our approach was to compare these estimates to ground-based measurements (observations) across continents with contrasting climates and land cover types. We considered 7 equations for estimating ${R}_{\text{S}}^{\downarrow}$ and 6 equations for estimating ${R}_{\text{L}}^{\downarrow}$. The study was limited to clear-sky conditions (given our focus on satellite remote sensing ET algorithms and the fact that satellite remote sensing cannot provide useful visual and thermal data of land surface during cloudy sky conditions).

_{n}estimates against observations are provided and discussed in Section 4. Conclusions are drawn in Section 5.

## 2. Methodology

#### 2.1. Estimating Equations for Incoming Shortwave Radiative Flux ( ${R}_{\text{S}}^{\downarrow}$)

#### 2.1.1. Theoretical Framework

_{0}, 1,367 W/m

^{2}), solar zenith angle (θ, rad), atmospheric transmissivity (τ), and Earth-Sun distance in astronomical unit (d, AU):

- For sloping surfaces,$$\begin{array}{ll}\text{cos}\hspace{0.17em}\theta =\hfill & \text{sin}\hspace{0.17em}\delta (\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}s-\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}s\hspace{0.17em}\text{cos}\hspace{0.17em}\gamma )\hfill \\ \hfill & +\text{cos}\hspace{0.17em}\delta \hspace{0.17em}\text{cos}\omega (\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}s+\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}s\hspace{0.17em}\text{cos}\hspace{0.17em}\gamma )\hfill \\ \hfill & +\text{cos}\hspace{0.17em}\delta \hspace{0.17em}\text{sin}\hspace{0.17em}\gamma \hspace{0.17em}\text{sin}\hspace{0.17em}s\hspace{0.17em}\text{sin}\hspace{0.17em}\omega \hfill \end{array}$$
- For horizontal surfaces,$$\text{cos}\hspace{0.17em}\theta =\text{sin}\hspace{0.17em}\delta \hspace{0.17em}\text{sin}\hspace{0.17em}\phi +\text{cos}\hspace{0.17em}\delta \hspace{0.17em}\text{cos}\hspace{0.17em}\phi \hspace{0.17em}\text{cos}\hspace{0.17em}\omega $$

#### 2.1.2. Equations for Estimating Clear-Sky ${R}_{\text{S}}^{\downarrow}$

_{0}) at the screen level [21]:

_{0}is in kPa. The seasonal variation of the Earth-Sun distance is not considered in this equation. This equation was adopted to estimate net radiation from Moderate Resolution Imaging Spectroradiometer (MODIS) data [25], and also was used in existing remote sensing ET algorithms [6,26,27].

_{Bo}is the clearness index for direct beam radiation and K

_{Do}is the clearness index for diffuse beam radiation.

_{Bo}is calculated as:

_{t}is the empirical turbidity coefficient, P [kPa] is the surface atmospheric pressure, β [rad] is the angle of the sun above the horizon, and W [mm] is the equivalent depth of precipitable water in the atmosphere. K

_{t}varies between 0 (extremely turbid, dusty or polluted air) and 1 (clean air, typical of agricultural and natural vegetation regions). β is calculated from solar declination (δ) geographical latitude (φ), and solar hour angle (ω) as sinβ = sinδ sinφ +cosδ cosφ cosω. W is calculated from the water vapor pressure at the screen level (e

_{0}) and atmospheric pressure (P) as W = 0.14e

_{0}P+2.1.

_{Do}is calculated from K

_{Bo}as:

_{0}[kPa] is the standard atmospheric pressure at sea level, C and F are two aerosol optical parameters related to the turbidity coefficient (η, 0.03 for clear sky conditions), J is the parameter that accounts for the effect of surface albedo (as a constant of 0.2) on scattered light, I is the vapor optical parameter, F

_{w}is an intermediate variable used to calculate I, W [mm] is the atmospheric precipitable water, W

_{e}is an intermediate variable for W estimation, t

_{d}[°C] is the dew point temperature, and b is the parameter that accounts for the effect of air pressure on atmospheric precipitable water. The SW7 scheme was used in the remote sensing ET algorithm of Nishida et al. [12], among others.

#### 2.1.3. Summary of Clear-Sky ${R}_{\text{S}}^{\downarrow}$ Estimating Equations

_{Bo}and K

_{Do}, with the clearness index for the diffuse component that depends on the clearness index for the direct component and the latter is computed from an empirical turbidity coefficient which is assigned a constant value (equal to 1.0) for clear sky conditions. SW7 is more complex, and it is based on a number of relations that rely on two aerosol optical parameters related to the turbidity coefficient which is assigned a constant value for clear sky conditions. Molecular absorption is parameterized either from elevation (SW1) or from water vapor pressure (SW2 to SW5) or from atmospheric precipitable water (SW6 and SW7). The only difference between SW2 and SW3, and between SW4 and SW5, is the accounting for (or lack thereof) seasonal variation of the Earth-Sun distance.

#### 2.2. Estimating Equations for Incoming Longwave Radiative Flux ( ${R}_{\text{L}}^{\downarrow}$)

#### 2.2.1. Theoretical Framework

_{a}is the atmospheric emissivity, and T

_{a}[K] is the screen-level air temperature. T

_{a}can be obtained from weather station data, but ε

_{a}depends on vertical profiles of temperature and radiatively active constituents that are not available from typical weather station data. ε

_{a}is often estimated from weather station data using empirical equations, and therefore the challenge in ${R}_{\text{L}}^{\downarrow}$ estimation lies in obtaining accurate estimates of ε

_{a}

#### 2.2.2. Equations for Estimating Clear Sky ${R}_{\text{L}}^{\downarrow}$

- LW2a refers to Equations (19a) and (19b), where air temperature [K] and vapor pressure [kPa] measurements are at the screen level.
- LW2d refers to Equations (19a) and (19e), where air temperature [K] and vapor pressure [kPa] measurements are at the screen level.

_{e}and air temperature to estimate clear-sky ${R}_{\text{L}}^{\downarrow}$ [24]:

#### 2.2.3. Summary of Clear-Sky ${R}_{\text{L}}^{\downarrow}$ Estimating Equations

## 3. Data and Approach

#### 3.1. Data

_{a}, relative humidity (RH), P, T

_{s}, ${R}_{\text{S}}^{\downarrow}$, ${R}_{\text{S}}^{\uparrow}$, ${R}_{\text{L}}^{\downarrow}$, ${R}_{\text{L}}^{\uparrow}$, and R

_{n}[32]. The site in South Africa has 30-min measurements of T

_{a}, RH, P, ${R}_{\text{S}}^{\downarrow}$, and ${R}_{\text{S}}^{\uparrow}$ [33], and the site in USA has 60-min measurements of T

_{a}, RH, P, ${R}_{\text{S}}^{\downarrow}$, and R

_{n}[34].

#### 3.2. Approach

_{n}through comparison of the estimates with flux tower observations. The estimates were obtained from empirical equations using input data (e.g., air temperature) provided by the flux tower observations. We used the following statistics to measure the performance of the estimates:

_{i}is the estimated value, O

_{i}is the flux tower observed value, ME is the mean error, MAE is the mean absolute error, MRE is the mean relative absolute error, and n is the number of pairs of estimated and observed values.

## 4. Results and Discussions

#### 4.1. Evaluation of Clear-Sky Incoming Shortwave Radiation ( ${R}_{\text{S}}^{\downarrow}$) Estimating Equations

^{2}≥ 0.92) between the estimates and observations. The bias (i.e., ME) varies from 3.35 W/m

^{2}to 86.12 W/m

^{2}, the variability (i.e., MAE) varies from 30.92 W/m

^{2}to 89.18 W/m

^{2}, and the relative variability (i.e., MRE) varies from 4.66% to 13.09%. According to ME, SW6 performs better than the other methods at all sites (ME: 3.35–19.76 W/m

^{2}). According to MRE, SW1 performs better than the other methods at four of the five sites (MRE: 4.66%–8.71%). Mainly because of location-specific empirical coefficients, the simple equations (SW2–SW5) that estimate transmissivity based on vapor pressure and the complex SW7 that involves several equations give worse performance (ME: 18.59–86.12 W/m

^{2}; MRE: 6.30%–13.09%). The ${R}_{\text{S}}^{\downarrow}$ values from SW1 to SW7 tend to be overestimated over the higher ranges, which indicates that atmospheric transmissivity tends be overestimated when ${R}_{\text{S}}^{\downarrow}$ values are relatively high. The SW1 and SW6 perform better because both estimate atmospheric transmissivity better.

#### 4.2. Evaluation of Clear-sky Incoming Longwave Radiation ( ${R}_{\text{L}}^{\downarrow}$) Estimating Equations

^{2}≥ 0.92) between the estimates and observations. The relative variability (i.e., MRE) varies from 2% to 8%, which is better than that of ${R}_{\text{S}}^{\downarrow}$ estimates. All equations give negatively-biased estimates (ME from −8.97 W/m

^{2}to −3.88 W/m

^{2}) at the Fukang site (arid), positively-biased estimates (ME from 10.25 W/m

^{2}to 23.61 W/m

^{2}) at the Yucheng site (semi-arid), and relatively less-biased estimates (ME from −9.01 W/m

^{2}to 7.78 W/m

^{2}) at the Taoyuan site (humid). This suggests that all the equations considered were not calibrated well for semi-arid and arid regions. The equation that gives the best ${R}_{\text{L}}^{\downarrow}$ estimate depends on the climate/site: LW3 at the humid site, LW2d and LW3 at the semi-arid site, and LW2c at the arid site. This indicates that the empirical ${R}_{\text{L}}^{\downarrow}$ estimating equations perform better while local climatic conditions are similar to that under which they were developed.

#### 4.3. Evaluation of Clear-Sky Net Radiation (R_{n}) Estimating Equations

_{n}is typically estimated as the sum of the incoming and outgoing shortwave and longwave radiation fluxes. Here, we assess the impacts of ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimating equations on the accuracy of R

_{n}estimates. We compared the R

_{n}estimates (obtained using ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimates and ${R}_{\text{S}}^{\uparrow}$ and ${R}_{\text{L}}^{\uparrow}$ observations) to R

_{n}observations at the three sites in China. Table 2 presents comparison results for all combinations of estimating equations. The MRE varies from 10.53% to 21.57%, depending on the site and estimating equation. The MRE is smaller at the humid site (11.14% to 15.29%) and semi-arid site (10.53% to 16.73%) compared to the arid site (13.65% to 21.57%). The ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimating equations that give the best R

_{n}estimate are: SW1 and LW2d in the humid site, SW1 and LW2d in the semi-arid site, and SW1 and LW3 in the arid site. This is consistent with our earlier finding that SW1 performs better for estimating ${R}_{\text{S}}^{\downarrow}$ (see Figure 1). However, the best estimating equations for ${R}_{\text{L}}^{\downarrow}$ (LW3 at the humid and semi-arid sites, and LW2c at the arid site; See Figure 3) are not involved in the set of equations that give the best R

_{n}estimate. This is because the ${R}_{\text{L}}^{\downarrow}$ estimates that have large negative biases (therefore not the best ${R}_{\text{L}}^{\downarrow}$ estimates) tend to counter the large positive biases in the SW1 ${R}_{\text{S}}^{\downarrow}$ estimates, leading to the best R

_{n}estimates.

#### 4.4. Suggestions on Incoming Radiation Estimation Equations for Remote Sensing ET Algorithms and Further Studies

_{n}and then ET. We conducted a case study at an arid site (Fukang) to demonstrate the improvement of remote sensing-based ET estimates using recommended incoming radiation estimating equations. The SW2 and LW2a equations originally embedded in the Sim-ReSET model were replaced by the SW6 and LW2c equations recommended in this study, respectively. By comparing with eddy covariance flux measurements, the Sim-ReSET model using the SW6 and LW2c equations could better estimate actual ET than that using the original SW2 and LW2a equations, with the MRE decreasing from 30% to 21% (see Figure 5). Refer to Sun et al. [6,35] for details about the Sim-ReSET model and ground measurements.

_{n}, robust radiative transfer models could be used to simulate full ranges of climates and land cover types to obtain universal empirical coefficients or make a look-up table of empirical coefficients in further studies. Meanwhile, more ground observations representing all kinds of climates land cover types are in the request to validate the results of simulations.

## 5. Conclusions

- Both ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimates from all evaluated equations well correlate with observations (R
^{2}≥ 0.92). - The ${R}_{\text{S}}^{\downarrow}$ estimating equations tend to overestimate, especially at higher values. The equations give large errors in the morning and late afternoon hours, where diffuse radiation is substantial. Of all the estimating equations, the equation treats the diffusive radiation component using two clearness indices and the equation assumes a linear increase of atmospheric transmissivity with elevation give the best estimates, and the mean relative absolute errors (MRE) are less than 10%. The equations that estimate atmospheric transmissivity from vapor pressure data or involve several complex relations produce worse results, and their MREs tend to be more than 10%.
- The ${R}_{\text{L}}^{\downarrow}$ estimating equations produce biased estimates at the arid and semi-arid sites (MRE: >4%) and less-biased estimates at the humid site (MRE: <3%).
- As a whole, the ${R}_{\text{L}}^{\downarrow}$ estimating equations tend to perform better than the ${R}_{\text{S}}^{\downarrow}$ estimating equations.
- The MRE in the net radiation (R
_{n}) estimates caused by the use ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimating equations varies from 10% to 22%. The equation that gives the best estimate of R_{n}involves (1) the best ${R}_{\text{S}}^{\downarrow}$ estimating equation for ${R}_{\text{S}}^{\downarrow}$ estimation, and (2) the ${R}_{\text{L}}^{\downarrow}$ estimating equation that gives the largest negative bias or the smallest positive bias for ${R}_{\text{L}}^{\downarrow}$ estimation to compensate for the large positive bias in the ${R}_{\text{S}}^{\downarrow}$ estimates.

_{n}and then evapotranspiration (ET) in remote sensing ET algorithms. The best R

_{n}estimates still have at least 10% error, which will be inevitably propagated to ET estimates. Therefore, the accuracy of R

_{n}estimation should be carefully considered in developing and applying remote sensing ET algorithms in future studies and applications.

## Acknowledgments

## Conflicts of Interest

## References

- Ruhoff, A.L.; Paz, A.R.; Collischonn, W.; Aragao, L.E.O.C.; Rocha, H.R.; Malhi, Y.S. A MODIS-based energy balance to estimate evapotranspiration for clear-sky days in Brazilian tropical savannas. Remote Sens
**2012**, 4, 703–725. [Google Scholar] - Mariotto, I.; Gutschick, V.P. Non-lambertian corrected albedo and vegetation index for estimating land evapotranspiration in a heterogeneous semi-arid landscape. Remote Sens
**2010**, 2, 926–938. [Google Scholar] - Cuenca, R.; Ciotti, S.; Hagimoto, Y. Application of Landsat to evaluate effects of irrigation forbearance. Remote Sens
**2013**, 5, 3776–3802. [Google Scholar] - Hankerson, B.; Kjaersgaard, J.; Hay, C. Estimation of evapotranspiration from fields with and without cover crops using remote sensing and in situ methods. Remote Sens
**2012**, 4, 3796–3812. [Google Scholar] - Llasat, M.C.; Snyder, R.L. Data error effects on net radiation and evapotranspiration estimation. Agric. For. Meteorol
**1998**, 91, 209–221. [Google Scholar] - Sun, Z.G.; Wang, Q.X.; Matsushita, B.; Fukushima, T.; Ouyang, Z.; Watanabe, M. Development of a simple remote sensing evapotranspiration model (Sim-ReSET): Algorithm and model test. J. Hydrol
**2009**, 376, 476–485. [Google Scholar] - Gueymard, C.A. Clear-sky irradiance predictions for solar resource mapping and large-scale applications: Improved validation methodology and detailed performance analysis of 18 broadband radiative models. Solar Energy
**2012**, 86, 2145–2169. [Google Scholar] - Liang, S.L.; Wang, K.C.; Zhang, X.T.; Wild, M. Review on estimation of land surface radiation and energy budgets from ground measurement, remote sensing and model simulations. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens
**2010**, 3, 225–240. [Google Scholar] - Wu, H.R.; Zhang, X.T.; Liang, S.L.; Yang, H.; Zhou, G.Q. Estimation of clear-sky land surface longwave radiation from MODIS data products by merging multiple models. J. Geophys. Res.-Atmos.
**2012**, 117. [Google Scholar] [CrossRef] - Chen, L.; Yan, G.J.; Wang, T.X.; Ren, H.Z.; Calbo, J.; Zhao, J.; McKenzie, R. Estimation of surface shortwave radiation components under all sky conditions: Modeling and sensitivity analysis. Remote Sens. Environ
**2012**, 123, 457–469. [Google Scholar] - Bastiaanssen, W.G.M.; Menenti, M.; Feddes, R.A.; Holtslag, A.A.M. A remote sensing surface energy balance algorithm for land (SEBAL)-1. Formulation. J. Hydrol
**1998**, 212, 198–212. [Google Scholar] - Nishida, K.; Nemani, R.R.; Running, S.W.; Glassy, J.M. An operational remote sensing algorithm of land surface evaporation. J. Geophys. Res.-Atmos.
**2003**. [Google Scholar] [CrossRef] - Gao, Y.C.; Long, D.; Li, Z.L. Estimation of daily actual evapotranspiration from remotely sensed data under complex terrain over the upper Chao River Basin in North China. Int. J. Remote Sens
**2008**, 29, 3295–3315. [Google Scholar] - Gubler, S.; Gruber, S.; Purves, R.S. Uncertainties of parameterized surface downward clear-sky shortwave and all-sky longwave radiation. Atmos. Chem. Phys
**2012**, 12, 5077–5098. [Google Scholar] - Trnka, M.; Žalud, Z.; Eitzinger, J.; Dubrovský, M. Global solar radiation in central European lowlands estimated by various empirical formulae. Agric. For. Meteorol
**2005**, 131, 54–76. [Google Scholar] - Marthews, T.R.; Malhi, Y.; Iwata, H. Calculating downward longwave radiation under clear and cloudy conditions over a tropical lowland forest site: An evaluation of model schemes for hourly data. Theor. Appl. Climatol
**2012**, 107, 461–477. [Google Scholar] - Carmona, F.; Rivas, R.; Caselles, V. Estimation of daytime downward longwave radiation under clear and cloudy skies conditions over a sub-humid region. Theor. Appl. Climatol.
**2013**. [Google Scholar] [CrossRef] - Duffie, J.A.; Beckman, W.A. Solar Engineering of Thermal Process, 1st ed.; John Wiley and Sons: New York, NY, USA, 1980. [Google Scholar]
- Garner, B.J.; Ohmura, A. A method for calculating direct shortwave radiation income of slopes. J. Appl. Meteorol
**1968**, 7, 796–800. [Google Scholar] - Tasumi, M.; Allen, R.G.; Bastiaanssen, W.G.M. The Theoretical Basis of Sebal; University of Idaho: Moscow, ID, USA, 2000; pp. 46–69. [Google Scholar]
- Zillman, J.W. A Study of Some Aspects of the Radiation and Heat Budgets of the Southern Hemisphere Oceans; Bureau of Meteorology, Department of the Interior: Canberra, ACT, Australia, 1972. [Google Scholar]
- Shine, K.P. Parameterization of the shortwave flux over high albedo surfaces as a function of cloud thickness and surface albedo. Q. J. Roy. Meteor. Soc
**1984**, 110, 747–764. [Google Scholar] - Allen, R.G.; Trezza, R.; Tasumi, M. Analytical integrated functions for daily solar radiation on slopes. Agric. For. Meteorol
**2006**, 139, 55–73. [Google Scholar] - Kondo, J. Atmospheric Science near the Ground Surface; University of Tokyo Press: Tokyo, Japan, 2000. [Google Scholar]
- Bisht, G.; Venturini, V.; Islam, S.; Jiang, L. Estimation of the net radiation using MODIS (moderate resolution imaging spectroradiometer) data for clear sky days. Remote Sens. Environ
**2005**, 97, 52–67. [Google Scholar] - Venturini, V.; Islam, S.; RodrigueZ, L. Estimation of evaporative fraction and evapotranspiration from MODIS products using a complementary based model. Remote Sens. Environ
**2008**, 112, 132–141. [Google Scholar] - Jiang, L.; Islam, S.; Guo, W.; Jutla, A.S.; Senarath, S.U.S.; Ramsay, B.H.; Eltahir, E.A.B. A satellite-based daily actual evapotranspiration estimation algorithm over south florida. Glob. Planet. Change
**2009**, 67, 62–77. [Google Scholar] - Brutsaert, W. On a derivable formula for longwave radiation from clear skies. Water Resour. Res
**1975**, 11, 742–744. [Google Scholar] - Prata, A.J. A new long-wave formula for estimating downward clear-sky radiation at the surface. Q. J. Roy. Meteor. Soc
**1996**, 122, 1127–1151. [Google Scholar] - Reitan, C.H. Surface dew point and water vapor aloft. J. Appl. Meteorol
**1963**, 2, 776–779. [Google Scholar] - Venäläinen, A. The Spatial Variation of Mean Monthly Global Radiation in Finland; University of Helsinki: Helsinki, Finland, 1994. [Google Scholar]
- Watanabe, M.; Wang, Q.X.; Hayashi, S. Monitoring and simulation of water, heat, and CO
_{2}fluxes in terrestrial ecosystems based on the APEIS-flux system. J. Geogr. Sci**2005**, 15, 131–141. [Google Scholar] - Kutsch, W.L.; Hanan, N.; Scholes, B.; McHugh, I.; Kubheka, W.; Eckhardt, H.; Williams, C. Response of carbon fluxes to water relations in a savanna ecosystem in South Africa. Biogeosciences
**2008**, 5, 1797–1808. [Google Scholar] - Wang, J.M.; Miller, D.R.; Sammis, T.W.; Gutschick, V.P.; Simmons, L.J.; Andales, A.A. Energy balance measurements and a simple model for estimating pecan water use efficiency. Agric. Water Manage
**2007**, 91, 92–101. [Google Scholar] - Sun, Z.G.; Wang, Q.X.; Matsushita, B.; Fukushima, T.; Ouyang, Z.; Watanabe, M.; Gebremichael, M. Further evaluation of the Sim-ReSET model for et estimation driven by only satellite inputs. Hydrol. Sci. J
**2013**, 58, 994–1012. [Google Scholar]

**Figure 1.**Scatter plot of ${R}_{\text{S}}^{\downarrow}$ estimates versus measurements for different estimating equations and sites.

**Figure 2.**Diurnal variation of relative absolute error (|E

_{i}− O

_{i}|/O

_{i}) in ${R}_{\text{S}}^{\downarrow}$ estimates on four typical days in four seasons at five sites.

**Figure 3.**Scatter plot of ${R}_{\text{L}}^{\downarrow}$ estimates versus measurements for different estimating equations and sites.

**Figure 4.**Diurnal variation of relative absolute error (|E

_{i}− O

_{i}|/O

_{i}) in ${R}_{\text{L}}^{\downarrow}$ estimates on four typical days in four seasons at three sites.

**Figure 5.**Comparison of actual ET estimates from the Sim-ReSET model involving original (ET_V1) and recommended (ET_V2) incoming radiation estimating equations against eddy covariance flux measurements (ET_EC) at the Fukang site, respectively.

Site | Location | Climate (Annual Average Temperature, Annual Precipitation) | Land Cover | Measurements * | Date Available over the Whole Year of |
---|---|---|---|---|---|

Taoyuan (China) | 111.469°E 28.944°N 108 m a.s.l. | Humid (16.5 °C, 1,450 mm) | Paddy | T_{a}, RH, P, T_{s},
${R}_{\text{S}}^{\downarrow}$,
${R}_{\text{S}}^{\uparrow}$,
${R}_{\text{L}}^{\downarrow}$,
${R}_{\text{L}}^{\uparrow}$, R_{n} | 2003 |

Yucheng (China) | 116.571°E 36.829°N 28 m a.s.l. | Semi-arid (13.1 °C, 610 mm) | Irrigated crop | T_{a}, RH, P, T_{s},
${R}_{\text{S}}^{\downarrow}$,
${R}_{\text{S}}^{\uparrow}$,
${R}_{\text{L}}^{\downarrow}$,
${R}_{\text{L}}^{\uparrow}$, R_{n} | 2007 |

Fukang (China) | 87.937°E 44.292°N 470 m a.s.l. | Arid, (6.6 °C, 164 mm) | Shrub | T_{a}, RH, P, T_{s},
${R}_{\text{S}}^{\downarrow}$,
${R}_{\text{S}}^{\uparrow}$,
${R}_{\text{L}}^{\downarrow}$,
${R}_{\text{L}}^{\uparrow}$, R_{n} | 2003 |

Skukuza (South Africa) | 31.497°E 25.020°S 365 m a.s.l. | Semi-arid (21.9 °C, 547 mm) | Savanna | T_{a}, RH, P,
${R}_{\text{S}}^{\downarrow}$,
${R}_{\text{S}}^{\uparrow}$ | 2008 |

OPEC (New Mexico, USA) | 106.756°W 32.225°N 1177 m a.s.l. | Arid, (17.8 °C, 280 mm) | Pecan orchard | Ta, RH, P,
${R}_{\text{S}}^{\downarrow}$, R_{n} | 2003 |

^{*:}T

_{a}is the air temperature, RH is the relative humidity, P is the air pressure, T

_{s}is the surface temperature, ${R}_{\text{S}}^{\downarrow}$ is the downwelling shortwave radiation, ${R}_{\text{S}}^{\uparrow}$ is the upwelling shortwave radiation, ${R}_{\text{L}}^{\downarrow}$ is the downwelling longwave radiation, ${R}_{\text{L}}^{\uparrow}$ is the upwelling longwave radiation, and R

_{n}is the net radiation.

**Table 2.**Mean absolute relative error (%) in R

_{n}estimates resulting from various combinations of ${R}_{\text{S}}^{\downarrow}$ and ${R}_{\text{L}}^{\downarrow}$ estimating equations.

SW1 | SW2 | SW3 | SW4 | SW5 | SW6 | SW7 | ||
---|---|---|---|---|---|---|---|---|

LW1 | Taoyuan (paddy, humid) | 12.84 | 15.05 | 14.44 | 13.83 | 13.17 | 12.25 | 13.06 |

Yucheng (irrigated crop, semi-arid) | 11.07 | 16.69 | 15.21 | 14.36 | 12.88 | 14.80 | 14.43 | |

Fukang (shrub, arid) | 13.84 | 20.85 | 18.17 | 17.78 | 15.50 | 15.75 | 17.88 | |

LW2a | Taoyuan (paddy, humid) | 12.86 | 15.01 | 14.51 | 13.91 | 13.36 | 12.18 | 13.15 |

Yucheng (irrigated crop, semi-arid) | 10.90 | 16.56 | 15.36 | 14.18 | 13.05 | 14.50 | 14.72 | |

Fukang (shrub, arid) | 13.81 | 21.38 | 18.53 | 18.10 | 15.78 | 15.61 | 18.22 | |

LW2b | Taoyuan (paddy, humid) | 12.92 | 15.00 | 14.45 | 13.95 | 13.34 | 12.11 | 13.06 |

Yucheng (irrigated crop, semi-arid) | 10.88 | 16.63 | 15.39 | 14.21 | 13.04 | 14.56 | 14.74 | |

Fukang (shrub, arid) | 13.81 | 21.28 | 18.45 | 18.03 | 15.70 | 15.63 | 18.13 | |

LW2c | Taoyuan (paddy, humid) | 13.21 | 15.29 | 14.76 | 14.35 | 13.72 | 12.11 | 13.30 |

Yucheng (irrigated crop, semi-arid) | 11.05 | 16.73 | 15.55 | 14.39 | 13.28 | 14.45 | 14.88 | |

Fukang (shrub, arid) | 13.76 | 21.57 | 18.69 | 18.24 | 15.93 | 15.48 | 18.35 | |

LW2d | Taoyuan (paddy, humid) | 11.14 | 14.18 | 13.78 | 12.53 | 12.21 | 12.40 | 12.67 |

Yucheng (irrigated crop, semi-arid) | 10.53 | 15.99 | 14.88 | 13.51 | 12.53 | 14.74 | 14.43 | |

Fukang (shrub, arid) | 13.74 | 20.44 | 17.73 | 17.31 | 15.22 | 15.63 | 17.49 | |

LW3 | Taoyuan (paddy, humid) | 12.03 | 14.68 | 14.12 | 13.20 | 12.60 | 12.30 | 12.83 |

Yucheng (irrigated crop, semi-arid) | 10.81 | 16.43 | 15.01 | 14.02 | 12.66 | 14.78 | 14.37 | |

Fukang (shrub, arid) | 13.65 | 20.39 | 17.82 | 17.36 | 15.23 | 15.51 | 17.56 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Gebremichael, M.; Wang, Q.; Wang, J.; Sammis, T.W.; Nickless, A. Evaluation of Clear-Sky Incoming Radiation Estimating Equations Typically Used in Remote Sensing Evapotranspiration Algorithms. *Remote Sens.* **2013**, *5*, 4735-4752.
https://doi.org/10.3390/rs5104735

**AMA Style**

Sun Z, Gebremichael M, Wang Q, Wang J, Sammis TW, Nickless A. Evaluation of Clear-Sky Incoming Radiation Estimating Equations Typically Used in Remote Sensing Evapotranspiration Algorithms. *Remote Sensing*. 2013; 5(10):4735-4752.
https://doi.org/10.3390/rs5104735

**Chicago/Turabian Style**

Sun, Zhigang, Mekonnen Gebremichael, Qinxue Wang, Junming Wang, Ted W. Sammis, and Alecia Nickless. 2013. "Evaluation of Clear-Sky Incoming Radiation Estimating Equations Typically Used in Remote Sensing Evapotranspiration Algorithms" *Remote Sensing* 5, no. 10: 4735-4752.
https://doi.org/10.3390/rs5104735