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Net radiation is a key component of the energy balance, whose estimation accuracy has an impact on energy flux estimates from satellite data. In typical remote sensing evapotranspiration (ET) algorithms, the outgoing shortwave and longwave components of net radiation are obtained from remote sensing data, while the incoming shortwave (
^{2} ≥ 0.92), (2)
_{n}) estimates caused by the use of
_{n} estimates using recommended incoming radiation estimating equations could improve ET estimates.

Net radiation (_{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 _{n} after accounting for sensible heat flux (_{n} to vegetation index, and _{n}. Llasat and Snyder reported that 65%–85% of the error in _{n} estimation directly propagates to crop-reference ET in the Catalonia region of Spain [_{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 [

In algorithms that estimate energy fluxes from remotely senses data, _{n} is estimated by summing up estimates of its shortwave and longwave components:
_{s} is the surface emissivity, ^{−8} W/m^{2}/K^{4}), and _{s} [K] is the surface temperature. The incoming components (

Several studies have been conducted to evaluate empirical estimating equations of incoming components. For example, Gubler _{n} calculated from

The purpose of this study was to evaluate the accuracy of commonly used empirical equations that estimate
_{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

This paper is organized as follows. Section 2 provides the commonly-used estimating equations for
_{n} estimates against observations are provided and discussed in Section 4. Conclusions are drawn in Section 5.

_{0}, 1,367 W/m^{2}), solar zenith angle (

The Earth-Sun distance can be calculated using the day of year (DOY) [

The solar zenith angle can be calculated using geographical latitude (

For sloping surfaces,

For horizontal surfaces,

The solar hour angle is calculated using local time (

The solar declination is calculated using DOY:

The challenge in

We considered the following estimating equations commonly used in remote sensing ET algorithms for estimating

SW1—the equation of Tasumi

SW2—the equation of Zillman [

SW3—the modified equation of Zillman [

SW4—the equation of Shine [

SW5—the modified equation of Shine [

SW6—the scheme of Allen

SW7—the scheme of Kondo [

_{0}) at the screen level [_{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 [

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

The _{Bo} is calculated as:
_{t} is the empirical turbidity coefficient, _{t} varies between 0 (extremely turbid, dusty or polluted air) and 1 (clean air, typical of agricultural and natural vegetation regions). _{0}) and atmospheric pressure (_{0}

The _{Do} is calculated from _{Bo} as:

_{0} [kPa] is the standard atmospheric pressure at sea level, _{w}_{e} is an intermediate variable for _{d} [°C] is the dew point temperature, and

Shortwave radiation is the sum of a direct beam component and a diffuse component. Both components depend on atmospheric transmissivity (_{Bo} and _{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.

_{a} is the atmospheric emissivity, and _{a} [K] is the screen-level air temperature. _{a} can be obtained from weather station data, but _{a} depends on vertical profiles of temperature and radiatively active constituents that are not available from typical weather station data. _{a} is often estimated from weather station data using empirical equations, and therefore the challenge in
_{a}

We considered the following estimating equations commonly used in remote sensing ET algorithms for estimating

LW1—the equation of Brutsaert [

LW2 (a–d)—the equation of Prata [

LW3—the scheme of Kondo [

Depending on the

LW2a refers to

LW2b refers to _{e} [mm] is obtained from

LW2c refers to _{d} [K] measurement is at the screen level.

LW2d refers to

_{e} and air temperature to estimate clear-sky

Our data came from flux tower stations at five sites (three sites in China, one site in South Africa, and one site in USA). _{a}, relative humidity (RH), _{s},
_{n} [_{a}, RH, _{a}, RH, _{n} [

The instruments were well maintained and calibrated yearly, and so the measurement errors can be safely assumed to arise only from the known instrument manufacturing errors. For example, a Q7 net radiometer (see

Our approach was to evaluate estimates of
_{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}_{i}

Only clear-sky cases were collected. The total number of clear-sky cases was 999, 1,449, 2,948, 897 and 738 cases, at the Taoyuan, Yucheng, Skukuza, Fukang, and OPEC sites, respectively. ^{2} ≥ 0.92) between the estimates and observations. The bias (^{2} to 86.12 W/m^{2}, the variability (^{2} to 89.18 W/m^{2}, and the relative variability (^{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

^{2} ≥ 0.92) between the estimates and observations. The relative variability (^{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

As discussed in Section 1, _{n} is typically estimated as the sum of the incoming and outgoing shortwave and longwave radiation fluxes. Here, we assess the impacts of
_{n} estimates. We compared the _{n} estimates (obtained using
_{n} observations at the three sites in China. _{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
_{n} estimate. This is because the
_{n} estimates.

This study reveals that the
_{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

Although our evaluations on incoming radiation estimation equations span wide ranges of climate and land cover types, it is still hard to affirm that the best estimating equations in this study work best across the lands. For universal empirical estimating equations for
_{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.

We have evaluated the accuracy of seven estimating equations for incoming shortwave radiative flux (

Both
^{2} ≥ 0.92).

The

The

As a whole, the

The MRE in the net radiation (_{n}) estimates caused by the use
_{n} involves (1) the best

This study suggests that incoming radiation estimation equations with less empirical coefficients or well-calibrated equations could be used for better estimating _{n} and then evapotranspiration (ET) in remote sensing ET algorithms. The best _{n} estimates still have at least 10% error, which will be inevitably propagated to ET estimates. Therefore, the accuracy of _{n} estimation should be carefully considered in developing and applying remote sensing ET algorithms in future studies and applications.

This study was supported by the NASA NIP Grant NNX08AR31G to the University of Connecticut and by the Environment Research and Technology Development Fund (E-1203) of the Ministry of the Environment, Japan. Authors thank anonymous reviewers and editors for their constructive comments.

The authors declare no conflict of interest.

_{2}fluxes in terrestrial ecosystems based on the APEIS-flux system

Scatter plot of

Diurnal variation of relative absolute error (|_{i}_{i}_{i}

Scatter plot of

Diurnal variation of relative absolute error (|_{i}_{i}_{i}

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.

Ground validation sites and ground-based measurements.

Taoyuan (China) | 111.469°E |
Humid (16.5 °C, 1,450 mm) | Paddy | _{a}, RH, _{s},
_{n} |
2003 |

Yucheng (China) | 116.571°E |
Semi-arid (13.1 °C, 610 mm) | Irrigated crop | _{a}, RH, _{s},
_{n} |
2007 |

Fukang (China) | 87.937°E |
Arid, (6.6 °C, 164 mm) | Shrub | _{a}, RH, _{s},
_{n} |
2003 |

Skukuza (South Africa) | 31.497°E |
Semi-arid (21.9 °C, 547 mm) | Savanna | _{a}, RH, |
2008 |

OPEC (New Mexico, USA) | 106.756°W |
Arid, (17.8 °C, 280 mm) | Pecan orchard | Ta, RH, P,
_{n} |
2003 |

Note:

_{a} is the air temperature, RH is the relative humidity, _{s} is the surface temperature,
_{n} is the net radiation.

Mean absolute relative error (%) in _{n} estimates resulting from various combinations of

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 |