A General Model for Converting All-Wave Net Radiation at Instantaneous to Daily Scales Under Clear Sky

Abstract

Surface all-wave net radiation ($R_n$) is one of the essential parameters to describe surface radiative energy balance, and it is of great significance in scientific research and practical applications. Among various acquisition approaches, the estimation of $R_n$ from satellite data is gaining more and more attention. In order to obtain the daily $R_n$ ($R_{nd}$) from the instantaneous satellite observations, a parameter $C_d$, which is defined as the ratio between the $R_n$ at daily and at instantaneous under clear sky was proposed and has been widely applied. Inspired by the sinusoidal model, a new model for $C_d$ estimation, namely New Model, was proposed based on the comprehensive clear-sky $R_n$ measurements collected from 105 global sites in this study. Compared with existing models, New Model could estimate $C_d$ at any moment during 9:30~14:30 h, only depending on the length of daytime. Against the measurements, New Model was evaluated by validating and comparing it with two popular existing models. The results demonstrated that the $R_{nd}$ obtained by multiplying $C_d$ from New Model had the best accuracy, yielding an overall $R^2$ of 0.95, root mean square error (RMSE) of 14.07 Wm⁻², and Bias of −0.21 Wm⁻². Additionally, New Model performed relatively better over vegetated surfaces than over non- or less-vegetated surfaces with a relative RMSE (rRMSE) of 11.1% and 17.89%, respectively. Afterwards, the New Model $C_d$ estimate was applied with MODIS data to calculate $R_{nd}$. After validation, the $R_{nd}$ computed from $C_d$ was much better than that from the sinusoidal model, especially for the case MODIS transiting only once in a day, with $R_{nd}$-validated $R^2$ of 0.88 and 0.84, RMSEs of 19.60 and 27.70 Wm⁻², and Biases of −0.76 and 8.88 Wm⁻². Finally, more analysis on New Model further pointed out the robustness of this model under various conditions in terms of moments, land cover types, and geolocations, but the model is suggested to be applied at a time scale of 30 min. In summary, although the new $C_d$ model only works for clear-sky, it has the strong potential to be used in estimating $R_{nd}$ from satellite data, especially for those having fine spatial resolution but low temporal resolution.

1. Introduction

Surface all-wave net radiation ($R_n$) is the difference between the incoming and outgoing shortwave and longwave radiation at surface, which represents the balance state of energy absorption, reflection, and emission on the Earth’s surface. $R_n$ can be mathematically expressed as follows:

$$R_n=R_{ns}+R_{nl}$$

(1)

$$R_{ns}=R_s^{\downarrow }-R_s^{\uparrow }=\left(1-\alpha \right)R_s^{\downarrow }$$

(1a)

$$R_{nl}=R_l^{\downarrow }-R_l^{\uparrow }$$

(1b)

where $R_{\mathit{ns}}$ is the net shortwave radiation (Wm⁻²), $R_{\mathit{nl}}$ is the net longwave radiation (Wm⁻²), $R_s^{\downarrow }$ and $R_s^{\uparrow }$ are the incoming and outgoing shortwave radiations (Wm⁻²), $R_l^{\downarrow }$ and $R_l^{\uparrow }$ are the incoming and outgoing longwave radiations (Wm⁻²), and $\alpha$ is the broadband albedo. The downward direction is defined as positive in this study. $R_n$ is one of the most essential parameters in Earth’s energy radiative balance [1,2]. It drives various natural phenomena and processes on Earth and determines its energy redistribution, so it is of great significance in the surface–atmospheric energy exchange [1,2,3]. Hence, $R_n$ has been widely used in various applications, such as agriculture, climate, hydrology, and so on.

In practical use, due to the different limitations in ground measurements, the alternative way to obtain $R_n$ is from model reanalysis or remotely sensed products. Relatively, the satellite-based radiative products are becoming more and more popular because of their superior performance, as pointed out by several studies [4,5,6,7,8]. Additionally, with the development of society, $R_n$ at high spatial resolution is highly required nowadays. Hence, satellite observations, particularly those from polar-orbiting satellites with finer spatial size across kilometer-to-meter scales, have become one of the most popular data sources for $R_n$ estimation [6], such as data from Moderate Resolution Imaging Spectroradiometer (MODIS) [9,10,11], Advanced Very High Resolution Radiometer (AVHRR) [12], and Landsat sensors [13], have been successfully applied in estimating $R_n$. However, the number of available polar-orbiting satellite observations in one day varies around the globe, with the value gradually decreasing when the satellite approaches the equator. In addition, for those polar-orbiting satellites with the finer spatial resolution (e.g., Landsat, Sentinel), the available observations in one day are much less because of their longer revisit period (>1 day). Thus, the accuracy of the estimated $R_n$ for a time period in some regions (i.e., low latitudes), such as the daily mean $R_n$ ($R_{\mathit{nd}}$) at finer resolution, which is required more often in practical use, would be heavily influenced by the very limited availability of satellite observations.

To address this issue by expanding the instantaneous satellite observations into $R_{\mathit{nd}}$ reasonably, several methods have been proposed. Sinusoidal model, which was first proposed by Bisht et al. [14] for estimating daily mean shortwave radiation, was also commonly used to retrieve an accurate $R_{\mathit{nd}}$ from the instantaneous values [2,14]. The assumption of sinusoidal model is that the instantaneous $R_n$ ($R_{\mathit{ni}}$) varies, conforming to a theoretical sinusoidal curve during daytime without clouds, so the daytime average $R_n$ (R_nD) could be estimated through at least one $R_{\mathit{ni}}$ in daytime that could be derived from the corresponding instantaneous satellite observations, after which $R_{\mathit{nd}}$ could be obtained from the R_nD based on their statistical relationship [2,14]. Sinusoidal model is popular because it is simple to understand and be conducted with its clear physical mechanisms. But the model performance is significantly influenced by the number of available $R_{\mathit{ni}}$; the more $R_{\mathit{ni}}$ used, the more accurate the estimated R_nD. Furthermore, the $R_{\mathit{nd}}$ cannot be obtained directly with this method, and its relationship with R_nD was not easy to model [11]. Hence, it is not ideal to apply sinusoidal model with polar-orbiting satellite data within mid-low latitudes where there are few available satellite data [11]. With the rapid development of machine learning methods and its wide application in remote sensing, several algorithms were developed to estimate $R_{\mathit{nd}}$ directly from instantaneous satellite observations at the top of the atmosphere (TOA) with machine learning methods [9,11,15]. Compared to the sinusoidal model, this kind of algorithm is easily to be implemented and its outputs are satisfactory, although they lack mechanisms and are heavily dependent on the quality and quantity of the samples for modeling [9,11].

An empirical parameter referred as $C_d$ is the alternative temporal expansion method, which was first proposed by Seguin and Itier [16], was thought to be a general model to expand $R_{\mathit{ni}}$ into $R_{\mathit{nd}}$ [16,17,18]. $C_d$ is defined as the ratio between $R_{\mathit{nd}}$ and $R_{\mathit{ni}}$:

$$C_d={\displaystyle \frac{R_{nd}}{R_{ni}}}$$

(2)

Hence, if the value of $C_d$ in a certain region at a certain time was determined, then the corresponding $R_{\mathit{nd}}$ could be easily obtained from $R_{\mathit{ni}}$ at that time by multiplying $C_d$. $C_d$ was first proposed based on the analysis results of the ground measurements during the summer midday in the Avignon region of France [16]. The authors found that the ratio between $R_{\mathit{nd}}$ and $R_{\mathit{ni}}$ at a certain time and region under a clear sky was very close to a constant, they thus defined $C_d$ as 0.3 $\pm$ 0.03. Afterwards, $C_d$ was further studied and was found to have different values under different cases. Thereby, $C_d$ was further revised as a function of various parameters (e.g., time, the day of year <doy>, and latitude), such as the linear function [19] and quadratic function [17]. Because of its simple format and being easy to calculate, $C_d$ has been widely used in practical applications [17,18]. Lately, the definition of $C_d$ has been applied with satellite data by Carmona [20]. Being different from the previous models, the calculation of $C_d$ in this study was not from a model with a specific format but by the calculated $R_{\mathit{ni}}$ and $R_{\mathit{nd}}$ from satellite data using parameterization models; hence, this $C_d$ contains some physical meanings of the temporal expansion of $R_n$. Additionally, the authors claimed that this $C_d$ could be the same for one image from any satellites, so it can be used generally. However, this $C_d$ is difficult to be widely applied because of its unspecific format, with its accuracy highly dependent on the estimated $R_{\mathit{ni}}$ and $R_{\mathit{nd}}$. Generally, the previous $C_d$ calculation models were usually developed using the ground measurements within a small region, but their performance in other regions has not been thoroughly evaluated. Therefore, there is still a great development space for $C_d$ estimation for its application with satellite data.

In this study, based on comprehensive clear-sky ground measurements, a new $C_d$ model for clear sky that could be applied at any moment around the globe during 9:30~12:00 hours was proposed. Against the ground measurements, this model was evaluated by validating and inter-comparing it with other two existing models, and then it was applied in $R_{\mathit{nd}}$ estimation with MODIS data. This paper is organized as follows: Section 2 introduces the data and methods, results and analysis are provided in Section 3, and the discussion and conclusions are given in Section 4.

2. Materials and Methods

2.1. Data and Pre-Process

2.1.1. In Situ Measurements

The $R_n$ measurements were collected from 105 globally distributed sites in three radiative measuring networks (

Table 1. Information about the three measuring networks.

Abbr.	Number of Sites	Time Period	Temporal Resolution	Reference
ARM	29	1994~2017	1 min	[21]
La Thuile	70	1996~2015	30 min	[8]
SURFRAD	6	1995~2017	1 or 3 min	[22]

ARM: Atmospheric Radiation Measurement (http://www.archive.arm.gov/), La Thuile: Global Fluxnet (La Thuile dataset) (https://fluxnet.org/), SURFRAD: Surface Radiation Network (http://www.esrl.noaa.gov/gmd/grad/surfrad/).

) from 1994 to 2017. Note that $R_n$ was obtained directly or by adding the four radiative components measurements as Equation (1).

Figure 1a shows the geographic locations of all sites and their land cover types classified according to International Geosphere–Biosphere Program (IGBP) [23]. These sites mainly distributed over North America (Figure 1c) and Europe (Figure 1d), with ten main land cover types, including deciduous broadleaf forest (DBF), evergreen broadleaf forest (EBF), evergreen needleleaf forest (ENF), mixed forest (MF), croplands (CRO), grasslands (GRA), close shrublands (CSH), open shrublands (OSH), woody savannas (WSA), and permanent wetlands (WET). The statistics of the main land cover types of these sites are given in Figure 1b.

All original R_n measurements underwent strict quality control and were converted to local time format first, and then they were aggregated into two time scales instantaneous and daily. As Table 1 shows, the sampling frequency at different sites is different; hence, the “instantaneous” scale was defined as half-hour (30 min) according to the study of Wang et al. [24] to ensure their consistency. Thereby, the $R_n$ measurements at each site sampled at the frequency less than half hour were aggregated into half hour without any missed out, and then all half-hour samples (R_ni) were aggregated into daily means without any missing in one day. In addition, to satisfy the requirements of the R model, the hourly averages of R_n measurements during 10:00~11:00 hours were calculated by averaging the corresponding R_ni. Afterwards, since all sites are on flat surfaces, the R_n measurement at 12:30 hour of each day was determined as the $R_{\mathit{nmax}}$ for all sites, as Bisht et al. [14] suggested. Note that only the R_ni during 9:30~14:30 hours were used. Therefore, a piece of sample for one day contained three variables R_ni, R_nd, and $R_{\mathit{nmax}}$. Finally, all samples under clear sky, which were determined as long as their CI values (see Section 2.1.3) at both instantaneous and daily scales were both larger than 0.7, were screened for the following study.

Finally, all clear-sky samples were randomly divided into training datasets and validation datasets in the ratio of 7:3. Specifically, the number of samples for model training and validation were 49,927 and 21,403 for New Model, 15,828 and 6788 in total for S model at three moments (12:00, 13:00, 14:00), and 7730 and 3314 for R model, respectively. Figure 2 presents the number of samples at each half-hour from 9:30 to 12:30 hours. The sample size gradually increased when the time was getting close to noon, after which they were nearly the same.

2.1.2. Remotely Sensed Data

A.

NDVI from Landsat 5/7/8

In this study, NDVI (normalized difference vegetation index) was used to determine the vegetated and non-vegetated surface, and the Landsat data (http://glovis.usgs.gov/) including Landsat5 Thematic Mapper (TM), Landsat7 Enhanced TM (ETM+), and Landsat8 Operational Land Imager (OLI) were applied for NDVI calculation according to Equation (3). If NDVI ≤ 0.1, then the surface was defined as non-vegetated, and vice versa.

$$NDVI={\displaystyle \frac{B_{IR}-B_R}{B_{IR}+B_R}}$$

(3)

where $B_{IR}$ is the reflectance of the near-infrared band (band4 of TM/ETM+ and band5 of OLI), and $B_R$ is the reflectance of the red band (band3 of TM/ETM+ and band4 of OLI). In order to ensure the consistency between the NDVI calculated from the different Landsat sensors, the NDVI calculated from TM/ETM+ (namely ${\mathrm{NDVI}}_{\mathrm{TM}}$/${\mathrm{NDVI}}_{\mathrm{ETM}+}$) was transferred to match the one calculated from OLI (namely ${\mathit{NDVI}}_{\mathrm{OLI}}$) according to [25]:

$$NDV{I}_{OLI}=1.0778NDV{I}_{TM}-0.0848$$

(4a)

$$NDV{I}_{OLI}=1.0747NDV{I}_{ETM+}-0.0839$$

(4b)

In this study, only the clear-sky Landsat bands, whose qc value from their cloud mask provided by the Landsat Ecosystem Disturbance-Adaptive Processing System (LEDAPS) atmosphere correction tool (version 3.3) [26] was 0/3/4, were used to compute NDVI. If the NDVI was unavailable for some day, then the nearest NDVI that was within an 8-day period (4 days before and after) was replaced. By referring to previous studies, the surface was classified as vegetated if NDVI > 0.1, and vice versa.

B.

MODIS data

In order to examine the effectiveness of C_d in R_nd estimation with satellite data, the model developed by Li et al. [11] (Li’s model hereinafter) to calculate R_ni directly from MODIS top-of-atmosphere (TOA) data was applied in this study(

Table 2. MODIS data used in the study.

MODIS Product	Spatial Resolution	Parameters Used
MOD/MYD02	1 km	1 km_RefSB, 1 km_Emissive
MOD/MYD03	1 km	SolarZenith (SZA), SolarAzimuth (SAA), SensorZenith (VZA), SensorAzimuth (VAA), Height
MOD/MYD35	1 km	Cloud Mask

). Therefore, the instantaneous TOA reflectance and radiance data with a spatial resolution of 1 km from MOD/MYD021KM, the corresponding geolocation and view geometry information including the solar zenith angle (SZA), solar azimuth angle (SAA), sensor zenith angle (VZA), sensor azimuth angle (VAA), and height from MOD/MYD03, as well as the cloud mask from MOD/MYD35 at all 105 sites during 2000–2017 were extracted. For comparison, only the clear-sky MODIS observations, which were determined by MOD/MYD35 with “confident clear” or “probably clear”, were used in this study. After matching, a total of 1022 R_nd samples were obtained for comparison, and all other samples (No. of MODIS TOA samples = 131,563) were applied for training Li’s model. All MODIS data were obtained from the NASA Remote Sensing Imagery Data Download Web site (https://ladsweb.modaps.eosdis.nasa.gov/), and Li’s model was implemented on Python 3.6.

2.1.3. Clearness Index (CI) Calculation

$\mathrm{CI}$ is usually used to represent the atmospheric transmittance and determine the sky condition. It is defined as the ratio of the downward solar radiation and the extraterrestrial radiation ($R_{\mathit{se}})$ [27], as Equation (5) shows:

$$CI={\displaystyle \frac{R_s^{\downarrow }}{R_{se}}}$$

(5)

To obtain the $\mathrm{CI}$ at instantaneous (${\mathit{CI}}_i$) and $\mathrm{daily}$ $\left({\mathit{CI}}_d\right)$ scales, the corresponding instantaneous $R_{\mathit{se}}\left(R_{\mathit{se}\_i}\right)$ and daily $R_{\mathit{se}}$($R_{\mathit{se}\_d}$) need to be calculated first. $R_{se\_i}$ is calculated by Equation (6) [28]:

$$R_{se\_i}=I_0\mathit{cos}z$$

(6)

$$I_0=1353\left\{1+0.034\mathit{cos}\left[{\displaystyle \frac{2\pi \left(doy-1\right)}{365}}\right]\right\}$$

(6a)

$$\mathit{cos}z=\mathit{sin}\gamma \mathit{sin}\delta +\mathit{cos}\gamma \mathit{cos}\delta \mathit{cos}H$$

(6b)

$$H={\displaystyle \frac{\pi }{12}}\left(t_{noon}-t\right)$$

(6c)

where $I_0$ is the effective solar constant, $z$ is the solar zenith angle (unit: rad), $H$ is the hour angle (unit: rad), $t_{\mathit{noon}}$ is the local noon time defined as 12:00 hours.

$R_{\mathit{se}\_d}$ could be calculated by Equation (7):

$$R_{se\_d}={\displaystyle \frac{1440G_{sc}{d}_r}{\pi }}\left({\omega }_s\mathit{sin}\gamma \mathit{sin}\delta +\mathit{cos}\delta \mathit{cos}{\omega }_s\right)$$

(7)

$$d_r=1+0.033\mathit{cos}\left({\displaystyle \frac{2\pi doy}{365}}\right)$$

(7a)

$${\omega }_s=\mathit{arccos}\left(-\mathit{tan}\gamma \mathit{tan}\delta \right)$$

(7b)

where $G_{\mathit{sc}}$ is the solar constant ($0.0820 \mathrm{MJ}{ \mathrm{m}}^{-2}·{\min }^{-1}$), $d_r$ is the inverse relative distance from the Earth to the Sun, and ${\omega }_s$ is the sunset hour angle (unit: rad). If the sky was clear, then the values of ${\mathit{CI}}_i$ and ${\mathit{CI}}_d$ should be both larger than 0.7.

2.2. Methods

In this study, the performance of the new proposed model (namely New Model) was assessed with sinusoidal model and other two existing $C_d$ models, including the model proposed by [17] (named S Model) and the other one developed by Rivas and Carmona [19] (named R Model). The details are given below:

2.2.1. Temporal Expansion Model

A.

Sinusoidal model

According to [14], the R_ni under clear sky during daytime was thought to be theoretically conforming to the sinusoidal model as Equation (8):

$$R_n\left(t\right)=R_{n\_max}\mathit{sin}\left[\left({\displaystyle \frac{t-t_{rise}}{t_{set}-t_{rise}}}\right)\pi \right]$$

(8)

where t_rise and t_set are the sunrise and sunset times, respectively, and R_{n_max} is the maximum R_n in a day. Hence, the R_nD could be obtained by

$$R_{nD}={\displaystyle \frac{{\int }_{t_{rise}}^{t_{set}}{R}_n\left(t\right)}{{\int }_{t_{rise}}^{t_{set}}dt}}={\displaystyle \frac{{\int }_{t_{rise}}^{t_{set}}{R}_n\left(t\right)}{LDt}}={\displaystyle \frac{2R_{n\_max}}{\pi }}$$

(9)

where LDt (unit: hour) is defined as the daytime lasting hours between sunset and sunrise, and it was calculated by

$$LDt=2\times {\displaystyle \frac{180}{15\pi }}\times \mathit{arccos}\left(\mathit{tan}\delta \mathit{tan}\gamma \right)$$

(10)

$$\delta =0.409\mathit{sin}\left({\displaystyle \frac{2\pi doy}{365}}-1.39\right)$$

(10a)

where $\gamma$ is the latitude (unit: rad) and $\delta$ is the solar declination (unit: rad).

B.

Two existing $C_d$ models (S Model and R Model) and new $C_d$ model

Aware of the limitations of setting $C_d$ as a constant, Sobrino et al. [17] proposed S Model to calculate $C_d$ by linking it with the second-degree polynomial equation of $\mathit{doy}$ based on the ground measurements from the east coast of the Iberian Peninsula. S Model was designed for three specific local times including 12:00, 13:00, and 14:00 hours in a uniform format as Equation (11):

$$C_d=a_{1}do{y}^2+a_{2}doy+a_3$$

(11)

where $a_1$, $a_2,$ and $a_3$ are the empirical coefficients and should be defined separately for the three moments(see Section 3.1.1). This model could characterize the variations in clear-sky C_d in a year well, but with a heavy saturation issue.

Afterwards, based on the ground measurements from Tandil Argentina during 2007–2009, Rivas and Carmona [19] found that a linear relationship existed between $R_{\mathit{nd}}$ and $R_{\mathit{ni}}$. Hence, they proposed R Model to calculate $C_d$ as follows:

$$C_d=b_1-{\displaystyle \frac{b_2}{R_{ni}}}$$

(12)

where $b_1$ and $b_2$ are the empirical coefficients and were defined as 0.43 and 54 in the original study. However, being limited by the available ground measurements, R Model only works for the averaged $R_{\mathit{ni}}$ during 10:00~11:00 h. The authors validated the $C_d$ from R Model through the estimate R_nd with a Bias of 2.0 Wm⁻² and a root mean square error (RMSE) of 12.0 Wm⁻², respectively. However, the accuracy of the $C_d$ calculated from this model heavily depends on $R_{\mathit{ni}}$, especially when the value of $R_{\mathit{ni}}$ was small. So far, the S and R model has not been widely validated.

In this study, a new $C_d$ model (namely New Model) that could work under clear sky at any moment during 9:30~14:30 hours has been proposed. In this model, the $R_{\mathit{nd}}$ under clear sky could be regarded as consisting of $R_{\mathit{nD}}$ and nighttime-averaged as R_n ($R_{\mathit{nN}}$). By combining with sinusoidal model (see Equations (8) and (9)), the theoretical R_nd (R_{nd_theoretical}) could be expressed as follows:

$$\begin{array}{cc}\hfill R_{nd\_theoretical}& ={\displaystyle \frac{1}{24}}\left[LDt\cdot R_{nD}+\left(24-LDt\right)\cdot R_{nN}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{24}}\left[LDt\cdot {\displaystyle \frac{{\int }_{t_{rise}}^{t_{set}}{R}_n\left(t\right)dt}{{\int }_{t_{rise}}^{t_{set}}dt}}+\left(24-LDt\right)\cdot R_{nN}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{24}}\left[LDt\cdot {\displaystyle \frac{2R_{nmax}}{\pi }}+(24-LDt)\cdot R_{nN}\right]\hfill \end{array}$$

(13)

Hence, the corresponding theoretical C_d ($C_{d\_\mathit{theoretical}}$) could be obtained as follows:

$$\begin{array}{cc}\hfill C_{d\_theoretical}\left(t\right)& ={\displaystyle \frac{R_{nd\_theoretical }}{R_n\left(t\right)}} \hfill \\ \hfill & ={\displaystyle \frac{\frac{1}{24}\left[LDt\cdot \frac{2R_{n\_max}}{\pi }+(24-LDt)\cdot R_{nN}\right]}{R_{n\_max} sin\left[\left(\frac{t-t_{rise}}{LDt}\right)\pi \right] }}\hfill \\ \hfill & =\left[{\displaystyle \frac{LDt}{12\pi }}+\left(1-{\displaystyle \frac{LDt}{24}}\right)\cdot {\displaystyle \frac{R_{nN}}{R_{n\_max}}}\right]\cdot {\displaystyle \frac{1}{sin\left[\left(\frac{t-t_{rise}}{LDt}\right)\pi \right]}}\hfill \end{array}$$

(14)

Since

$$\begin{array}{cc}\hfill t-t_{rise}& ={\displaystyle \frac{\left[t_{max}-t_{rise}-(t_{max}-t)\right]+[t_{set}-t_{max}-(t_{max}-t\left)\right]}{2}}\hfill \\ \hfill & ={\displaystyle \frac{t_{set}-t_{rise}+2(t-t_{max})}{2}}\hfill \\ \hfill & ={\displaystyle \frac{LDt+2(t-t_{max})}{2}}\hfill \end{array}$$

(15)

Then,

$$C_{d\_theoretical}\left(t\right)=\left[{\displaystyle \frac{LDt}{12\pi }}+\left(1-{\displaystyle \frac{LDt}{24}}\right)\cdot {\displaystyle \frac{R_{nN}}{R_{n\_max}}}\cdot {\displaystyle \frac{1}{\mathit{sin}\left[\left(\frac{1}{2}+\frac{t-t_{max}}{LDt}\right)\pi \right]}}\right]$$

(16)

However, after analysis, it was found that the value of the actual $C_d$ was a little bit different from that of $C_{d\_\mathit{theoretical}}$, and their difference was closely related to t. Besides that, the ratio of R_nN and R_{n_max} was related to LDt in the form of a quadratic function (as shown in Figure 3), so ${\displaystyle \frac{R_{nN}}{R_{n\_max}}}$ could be calculated according to Equation (17a).

Hence, after multiple experiments, the final $C_d$ was expressed as follows:

$$C_d\left(t\right)=c_1\left[{\displaystyle \frac{LDt}{12\pi }}+\left(1-{\displaystyle \frac{LDt}{24}}\right)\cdot {\displaystyle \frac{R_{nN}}{R_{n\_max}}}\right]\cdot {\displaystyle \frac{1}{\mathit{sin}\left(\left(\frac{1}{2}+\frac{t-t_{max}}{LDt}\right)\pi \right)}}+c_2\cdot t+c_3$$

(17)

$${\displaystyle \frac{R_{nN}}{R_{n\_max}}}=d_1\cdot LD{t}^2+d_2\cdot LDt+d_3$$

(17a)

where $c_1$~${ c}_3$ and $d_1$~${ d}_3$ are the empirical coefficients, and t_max is the time of R_{n_max} defined as 12:30 hour by referring to Bisht et al. [14].

From Equation (17), the $C_d$ for any specific location at any time could be obtained theoretically as long as the LDt (Equation (10)) was known. However, by considering the extremely unstable values of C_d around sunrise and sunset due to the very small R_ni, as well as the transit times of most polar-orbiting satellites, the range of t in Equation (17) was defined as 9:30~14:30 hours at local time. Additionally, according to previous study [29], the NDVI that is usually used to characterize the surface vegetation condition is closely related to R_n, and the variations in R_n over the vegetated surface were different from that over the non-vegetated surface. Hence, it is suggested to apply the new $C_d$ model for vegetated and non-vegetated surfaces separately.

For assessment, the New Model, S model, and R model were trained or calibrated by first using the corresponding training samples (

Table 3. The training and validation samples for the three $C_d$.

	Training		Validation
	Cd		Rni	Rnd *
	Time	No. of Samples	No. of Samples	No. of Samples
S Model	12:00	5393	2315	4992
	13:00	5288	2267
	14:00	5147	2206
R Model	10:00~11:00	7730	3314	3314
New Model	9:30~12:30 (30 min an interval)
	Vegetated	31,450	13,624	4668
	Non-vegetated	18,431	7759	2730
Common validation samples: No. = 517

* means the duplicated values have been removed.

), respectively. Afterwards, the accuracy of their $R_{\mathit{nd}}$ estimates, as well as the ones from the two existing models but with their original coefficients, were validated against their corresponding validation samples (Table 3). For better comparison, the three $C_d$ models were inter-compared using the common validation samples (No. of samples = 517). If more than one R_nd estimate was obtained in one day, then the final $R_{\mathit{nd}}$ was the average of all estimates. Table 3 lists the training and validation samples used for each case. Note that the samples were different for the three $C_d$ models because of their different use requirements.

After that, the C_d estimated from New Model was applied to expand the R_ni, which was estimated from the MODIS TOA data (Table 2) by implementing Li’s model into R_nd (namely R_nd-cd-NM). Meanwhile, the R_ni was also used with the sinusoidal method to obtain R_nD, which was then used to calculate $R_{\mathit{nd}}$ (namely R_nd-SIN) according to the empirical relation between R_nD and $R_{\mathit{nd}}$. More details could be referred to Li et al. [11]. Afterwards, the two results were validated and compared.

2.2.2. Evaluation Criteria

The validation accuracy was represented by four common statistic measures: the determination coefficient (${\mathrm{R}}^2$), root mean square error ($\mathrm{RMSE}$), Bias, and relative root mean square error (rRMSE), in which the rRMSE was used to eliminate the influence of different sample size. They were calculated by the following equations.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}}$$

(18a)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(18b)

$$Bias={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left(y_i-{\widehat{y}}_i\right)$$

(18c)

$$rRMSE={\displaystyle \frac{RMSE}{mean\left({\sum }_{i=1}^n{y}_i\right)}}$$

(18d)

where $y_i$ is the $i$-th ground observation, ${\widehat{y}}_i$ is the corresponding estimation, and n is the number of samples.

3. Results

3.1. Evaluation on the Three C_d Models

3.1.1. Validation Accuracy at Site Scale

Based on the corresponding samples shown in Table 3, the accuracy of the three C_d model was calculated separately, and then they were inter-compared with the common samples. Meanwhile, the accuracy of the two existing models before and after calibration was also presented.

A.

S Model

Table 4. The original and calibrated coefficients of S Model.

Time	Original Coefficient			Calibrated Coefficient
	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$
12:00	$-7 \times {10}^{-6}$	0.0026	0.0756	$-7.483 \times {10}^{-6}$	0.0026	0.0383
13:00	$-8 \times {10}^{-6}$	0.0028	0.0820	$-7.486 \times {10}^{-6}$	0.0026	0.0375
14:00	$-7 \times {10}^{-6}$	0.0027	0.1240	$-7.862 \times {10}^{-6}$	0.0027	0.0467

gives the calibrated and original coefficients of S Model at each of the three moments 12:00 hour, 13:00 hour, and 14:00 hour. Relatively, among the three groups of coefficients, the difference in a₃ before and after calibration was the largest, especially at 14:00 hour. Additionally, the training accuracy of S Model at the three moments was very similar to one other with the range of R² being 0.91~0.92, RMSE being18.07~18.91 Wm⁻², and Bias being −1.02~−1.23 Wm⁻², respectively.

Against the corresponding validation samples, the validation results of S Model overall and at specific moments, which was represented by the accuracy of the estimated R_nd (named as R_nd-Cd-SM) calculated by using the C_d derived from S model, were calculated and presented in Figure 4. In Figure 4a, the top chart represents the validated RMSE and rRMSE of the calculated R_nd-Cd-SM before (the bar and dash dot line in red) and after (the bar and dash dot line in blue) calibration, respectively, and the table below represents the corresponding validated Bias. Figure 4b and c display the overall validation accuracy of R_nd-Cd-SM when using the original and calibrated coefficients, respectively.

The validation accuracy of S model at the three moments was also similar. It turns out that the estimation accuracy of S model was significantly improved after calibrating with the more comprehensive training samples at all three moments, with the overall validated RMSE (rRMSE) decreasing by 19.59 Wm⁻² (18%) and the magnitude of Bias decreasing by 30.53 Wm⁻². Among the three specific moments, the model performance improved the most at 14:00 hour, with the RMSE (rRMSE) decreasing by 29.35 Wm⁻² (27%) and the magnitude of Bias decreasing by 41.52 Wm⁻². Moreover, the calibrated S Model has a more robust performance compared to its original version, yielding all R² better than 0.9, Bias of −0.85~−1.5 Wm⁻², and rRMSE of ~17%, respectively. These results indicated the necessity of calibration for the S model.

B.

R model

Table 5. Calibrated coefficients of R Model.

Original Coefficient		Corrected Coefficient
${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_1$	${\mathit{b}}_2$
0.43	54	0.3819	68.27

gives the coefficients of R model before and after calibration, where the change in b₂ was large. The training accuracy of the calculated R_nd from the C_d estimated from R model (named as R_nd-Cd-RM) yielded an R² of 0.89, RMSE of 21.55 Wm⁻², and Bias of 2.05 Wm⁻².

Figure 5 presents the scatter plots between the R_nd-Cd-RM estimates before and after model calibration and the validation samples, respectively. From Figure 5b, the accuracy of the calibrated R model improved a lot, with an RMSE of 21.36 Wm⁻² and Bias of 1.33 Wm⁻². Compared to the results of the original model (Figure 5a), the rRMSE decreased by 19.02%, and the magnitude of the Bias decreased by 38.26 Wm⁻². However, although the overestimation tendency of the original R model was revised after calibration to a certain extent, the tendency of underestimation was still observed at high values (>150 Wm⁻²).

In summary, the two existing models for $C_d$ estimation demonstrated a reasonable relationship between $C_d$ and other parameters, but the two models should be calibrated with more comprehensive samples before application.

C.

New Model

As described above, the New Model was trained separately for vegetated and non-vegetated surfaces using the corresponding training samples (Table 3), and their coefficients were given in

Table 6. Coefficients of the New Model for vegetated (NDVI ≥ 0.1) and non-vegetated (NDVI < 0.1) surfaces, respectively.

Condition	${\mathbf{c}}_1$	${\mathbf{c}}_2$	${\mathbf{c}}_3$	${\mathbf{d}}_1$	${\mathbf{d}}_2$	${\mathbf{d}}_3$
NDVI ≥ 0.1	0.9204	−0.0052	0.0280	−0.0039	0.1146	−0.9468
NDVI < 0.1	0.9041	−0.0070	0.0519	−0.0036	0.0939	−0.7710

.

Afterwards, the R_nd (named as R_nd-Cd-NM) calculated through the $C_d$ derived from New Model were validated against the in situ measurements. Figure 6a presents the overall validation accuracy of the R_nd-Cd-NM estimates. It is shown that the R_nd-Cd-NM estimates were nearly in the 1:1 line with the measurements, and the accuracy was satisfactory with an R² of 0.95, RMSE of 14.07 Wm⁻², and Bias of −0.21 Wm⁻². But relatively speaking, the accuracy of the R_nd-Cd-NM estimates for the vegetated surface (Figure 6b) was a little bit better than that for the non-vegetated surface (Figure 6c), with a larger R² of 0.95 and a smaller rRMSE of 11.17%. Moreover, the non-vegetated surface R_nd-Cd-NM was observed to be underestimated at high values (>150 Wm⁻²) and overestimated at low values (<50 Wm⁻²), which means this tendency also existed in the C_d estimates in this case.

3.1.2. Models Inter-Comparison

In order to evaluate objectively, the three C_d models were inter-compared against the common validation samples (No. = 517), and the results were further divided into vegetated and non-vegetated surfaces as shown in Figure 7.

From Figure 7, the overall validation accuracy of the R_nd-Cd-NM from the New Model (Figure 7c) was the best with a RMSE of 14.20 Wm⁻² and Bias of 0.21 Wm⁻², followed by the R_nd-Cd-SM from S model (Figure 7a). The performance of the R_nd-Cd-RM from R model (Figure 7b) was the worst with its validated RMSE and Bias being larger than that of the R_nd-Cd-NM by 9.65 and 2.33 Wm⁻², respectively. In terms of the performance of the three models on the vegetated and non-vegetated surfaces, the three models worked generally better over the vegetated surface than that over the non-vegetated surface, yielding rRMSEs of 13.96% and 19.51% for S model, 17.23% and 26.80% for R model, and 10.44% and 15.50% for New Model, respectively. However, the tendency of underestimation for the vegetated surface was the most remarkable for S model (Bias = −7.07 Wm⁻²), and tendency of overestimation for the non-vegetated surface for S and R models with the corresponding Biases of 6.78 and 7.77 Wm⁻², respectively. Only the New Model has no tendency, yielding Biases of 0.27 and 0.11 Wm⁻².

Furthermore, by considering New Model could work at any time, the R_nd-Cd-NM was estimated and validated by using the same validation samples of S and R model (Table 3) for further comparison. The results are provided in Figure 8.

Figure 8a presents the validation results of the estimated R_nd-Cd-NM using the validation samples corresponding to S model. Compared with Figure 4, the overall validation RMSE (rRMSE) of New Model witnessed a reduction by 3.44 Wm⁻² (rRMSE = 3.27%), accompanied by a decrease in Bias amplitude by 1.02 Wm⁻². At the three specific moments, the Bias ranged from −0.54 to 0.27 Wm⁻² and rRMSE remained below ~14.45%, which were both superior to that of S model. Regarding Figure 8b, the performance of New Model with the same validation samples of R model (Figure 5) was much better, yielding an R² value of 0.95, a Bias of 2.50 Wm⁻², and an RMSE (rRMSE) of 13.54 Wm⁻² (11.2%).

To sum up, the New Model proposed in this study explored more about C_d and thus outperformed the existing models.

3.2. Further Analysis of New Model

3.2.1. Model Performance Under Different Conditions

By considering the superior performance of New Model, more analysis on this model was conducted in terms of its performance under various conditions including time scales, moments, geolocations, and land cover types. Furthermore, the estimated R_nd-Cd-NM was used to represent the accuracy of New Model.

It is known that the largest difference between New Model and the existing models is that it could estimate $C_d$ at any moment during 9:30~14:30 hours. To illustrate the influence of different time scales on this model, the R_nd-Cd-NM estimates at the interval of 1/10/20/30/60 min were validated against the SURFRAD measurements at the central moments of 10:30 hour, 12:00 hour, and 13:30 hour, respectively, and the results are shown in Figure 9. From the results, the performance of New Model represented by R_nd estimation was relatively stable for different time scales at the three specific moments, with the changes in the validated RMSE all being less than 1.5 Wm⁻², but 30 min was the relatively optimal one. Hence, it is suggested to apply New Model at a time scale within 60 min, especially 30 min.

Additionally, the R_nd-Cd-NM estimation accuracy at different moments during 9:30~14:30 hours (30 min as an interval) was also explored and shown in Figure 10. The results demonstrated the accuracy of the estimated R_nd-Cd-NM was basically stable across all moments with a validated rRMSE of 12.10~15.01% (RMSE of 13.98~16.37 Wm⁻²), which indicated the robustness of New Model. Relatively, the model performed the best at 12:30 hour with the smallest rRMSE of 12.96% (RMSE = 13.98 Wm⁻²). Hence, the further it is from noontime (12:30 hour), the worse the estimated R_nd-Cd-NM accuracy, especially in the late afternoon. Meanwhile, it was also found that the C_d value is the smallest during noontime, which means that the uncertainty of the estimated R_nd-Cd-NM would decrease with decreasing C_d.

Afterwards, the validation results of the R_nd-Cd-NM were examined by their land cover types and shown in Figure 11. It can be seen that the R_nd-Cd-NM accuracy was generally stable for different land covers except for WET, with their validated R² values ranging from 0.90 to 0.96, RMSE of 12.49 to 18.22 Wm⁻², and Bias of −0.67 to 5.81 Wm⁻². Relatively, the rRMSE of the forest types including DBF, EBF, DNF, and MF were smaller, with their values between 10.80 and 12.66%, whereas for other vegetation types having a small NDVI, such as GRA, OSH, and so on, their R_nd-Cd-NM-validated rRMSE was larger, ranging 14.29~15.53%. The results were consistent with our pervious findings shown in Figure 6. But the poor accuracy for WET most likely resulted from the small sample size.

Finally, the spatial distribution of the R_nd-Cd-NM validation accuracy (Figure 12) demonstrated that New Model generally worked well around the globe, with most of the validated R² being larger than 0.90 (Figure 12a) and the RMSE (rRMSE) was within 20 Wm⁻² (15%) (Figure 12c,d), but the overestimation tendency was mainly observed over the Eastern hemisphere (red points in Figure 12b), and vice versa over the Western hemisphere (blue points in Figure 12b).

In summary, the performance of New Model was superior to other existing C_d models, with higher accuracy and more robustness, but it works better over the vegetated surface than over the non-vegetated surface.

3.2.2. Application with the MODIS Data to Estimate R_nd

Since C_d could be easily estimated from New Model at any time during 9:30~14:30 hours, it was thus used to obtain R_nd-cd-NM from the $R_{\mathit{ni}}$ estimated from the MODIS TOA data by using Li’s model. Meanwhile, the R_nd-SIN was also calculated from the same R_ni with the sinusoidal model (see Section 2.2.1) for comparison. Note that the average would be taken as the final R_nd-cd-NM or R_nd-SIN estimate if more than one group of MODIS TOA data were available in one day. Afterwards, against the ground measurements, the two results (Figure 13) were compared and further analyzed by the available MODIS data in one day.

Overall, the R_nd-cd-NM estimates (Figure 13b) almost distributed near to the 1:1 line and were closer to the ground measurements than that of the R_nd-SIN estimates (Figure 13a), with overall validated R² of 0.88 and 0.84, RMSE values of 19.60 and 27.70 Wm⁻², and Biases of −0.76 and 8.88 Wm⁻², respectively. The significant overestimation tendency was observed for the R_nd-SIN when its value was less than 150 Wm⁻², and vice versa for the other values. Combined with the used MODIS TOA data, it can be seen that more R_ni in one day would help in obtaining a more accurate R_nd when either using the C_d or sinusoidal model, particularly for the R_nd-cd-NM when its RMSE decreased by 4.03 Wm⁻². If only one R_ni was available in one day, then the application of C_d could improve the accuracy of the estimated R_nd (red points in Figure 13a) significantly compared with that from the sinusoidal model (red points in Figure 13b), yielding RMSE values of 20.56 and 28.14 Wm⁻² and Biases of 0.30 and 8.63 Wm⁻², respectively. Furthermore, in this case, the estimated R_nd-cd-NM were even more accurate than the R_nd-SIN from multiple R_ni (blue points in Figure 13a) by decreasing the rRMSE and the magnitude of Bias by 5.11% and 9.29 Wm⁻², respectively.

Therefore, the C_d obtained from New Model has strong potential to be widely used in R_nd estimation from the instantaneous satellite data in the future.

4. Conclusions

To address the issue of improving the estimation of R_nd from R_ni or the instantaneous satellite data, several temporal expansion methods have been proposed. In particular, an empirical parameter C_d, which was defined as the ratio between R_nd and R_ni, under clear sky used to be popular because of its simple format and easy implementation. Although C_d could be calculated from the existing models easily, the robustness of these models around the globe has not been well verified. In this study, a C_d estimation model, namely New Model, has been proposed. Being different from the existing models, New Model could obtain C_d at any time during 9:30~14:30 hours under clear sky, only depending on LDt. Against the comprehensive measurements collected from 105 global sites, New Model and two commonly used existing models (S model and R model) were validated and compared. Afterwards, the performance of New Model was further analyzed under various conditions. Finally, the estimated C_d from New Model were applied on the R_ni computed from the MODIS TOA data with Li’s model to obtain R_nd, which were also compared with the ones computed from the sinusoidal model. According to all results, several main conclusions could be drawn as follows:

(1) After comprehensive evaluation, the New Model proposed in this study exhibited superior performance compared to other two existing models, with the R_nd-Cd-NM obtained from the New Model C_d estimates yielding the best accuracy with an overall validated R² of 0.95, RMSE of 14.07 Wm⁻², and Bias of −0.21 Wm⁻², followed by the R_nd calculated from S model and R model with their validation R² values of 0.91 and 0.89, RMSEs of 18.26 Wm⁻² and 21.36 Wm⁻², and Biases of −1.03 Wm⁻² and 1.33 Wm⁻², respectively.

(2) Generally speaking, New Model exhibited robust performance under various cases in terms of time scales, moments, spatial distribution, and land cover types. However, New Model performed relatively better over the vegetated surface (i.e., DBF, EBF, DNF, and MF) than that over the non- or less-vegetated surface (i.e., GRA, OSH, and WET). Furthermore, New Model is suggested to be applied near noontime at a time scale within 60 min.

(3) The two existing models are strongly suggested to be applied after calibrating with the comprehensive samples. After calibration, the estimation accuracy represented by R_nd of the two models could be improved significantly by decreasing the RMSE values by 19.59 and 23.00 wm⁻² and the magnitude of Bias by 30.53 wm⁻² and 38.26 wm⁻², respectively. Regarding the vegetated and non-vegetated surfaces, similar results were also observed for S and R models as that for New Model, but they both tended to be overestimated for non-vegetated surface.

(4) The C_d from New Model has been successfully applied to expand the R_ni estimated from the MODIS TOA data into R_nd, which had a much better accuracy than the ones from the same R_ni but using the sinusoidal model, yielding overall R² values of 0.88 and 0.84, RMSEs of 19.60 and 27.70 Wm⁻², and Bias of −0.76 and 8.88, respectively. In addition, the more R_ni estimates in one day, the more accurate of the final obtained R_nd-Cd-NM.

In summary, the proposed New Model could estimate C_d more accurately and conveniently. Therefore, this model has a strong potential to be used for R_nd estimation from the satellite data, especially the ones with low temporal resolutions, such as the Landsat data. However, so far, the new C_d model could be only applied during 9:30~14:30 hours at flat surfaces under clear sky; thus, the application of the new C_d model is limited. In particular, after multiple experiments, we found that the cloud effects cannot be ignored, but the uncertainty by using the new C_d model under overcast sky was relatively smaller than that under mixed cloudy sky. Thus, for other cases apart from limitations, the application of New Model should be approached with more caution. More efforts to improve New Model are on the way.