Solar radiation is the most important energy source in biological, physical, and chemical processes on the surface of Earth (Liu et al., 2014), as well as one of the sources of green energy that support the sustainable development of human society (Paulescu et al., 2016). Accurate measurement or estimation of solar radiation (R_s) with high spatial and temporal resolutions is of great significance for the study of changes in the surface environment, food safety production, and the implementation of solar energy projects (Qin et al., 2011; Pan et al., 2013; Paulescu et al., 2016). Numerous countries have also established R_s observation systems for this purpose (Paulescu et al., 2013). However, due to high investment and maintenance costs, the density of stations that can continuously perform R_s observation is too low (Paulescu et al., 2013). In China, for example, there are more than 2400 national meteorological stations, but only data from 130 stations belong to the radiation dataset released by the China Meteorological Data Service Center (CMDC). Therefore, numerous studies have developed models to estimate R_s, which are more cost-effective than observations of R_s (Paulescu et al., 2016). At present, the developed models include the remote sensing inversion model (Olseth and Skartveit, 2001), stochastic simulation model (Richardson, 1981), empirical model (Ångström, 1924; Prescott, 1940; Li et al., 2010), and machine learning model (Chen et al., 2011). Among them, empirical models based on sunshine duration and temperature are widely popular because of their simplicity and ease of use (Li et al., 2012a), where the results of the Ångström-Prescott equation (AP) yield the best performance (Iziomon and Mayer, 2002; Almorox and Hontoria, 2004; Trnka et al., 2005; Zhang et al., 2018). In addition, in the Crop Evapotranspiration-Guidelines for Computing Crop Water Requirements (FAO 56) (Allen et al., 1998), published by the Food and Agriculture Organization (FAO) of the United Nations in 1998, the Penman-Monteith equation (PM) is recommended as a scheme to calculate the true value for reference crop evapotranspiration (ET₀). In this scheme, the AP was also suggested to calculate missing R_s values to support the PM in a specific study or application area. Therefore, the a_s and b_s coefficients are important to accurately estimate R_s in the AP.

Existing studies use two main methods to determine coefficients a_s and b_s. In areas without measured solar radiation data, previous studies have directly used the recommended values, i.e., a_s = 0.25 and b_s = 0.50 for the Ångström and FAO 56 (Ångström, 1924; Allen et al., 1998; Du et al., 2003; Zhao et al., 2008; Mai et al., 2012; Ma et al., 2013; Cui, 2014; Lu et al., 2016). However, the spatial and temporal differences in atmospheric components exist objectively due to long-term climate and short-term seasonal changes at the surface, such that the disadvantages of globally fixed a_s and b_s coefficients are evident. In areas where portions of measured solar radiation data are available, the measured data are often used to correct the a_s and b_s coefficients; the corrected coefficients can better reflect the influence that atmospheric conditions have on R_s (Wen, 1964; Ju et al., 2005; Yang et al., 2009). Therefore, the coefficient correction of the AP has attracted extensive attention.

Some studies have introduced terrain (Liu et al., 2017), air temperature (Ojosu and Komolafe, 1987), relative humidity (Ododo et al., 1995), water vapor pressure (Liu et al., 2017), and other geographical factors (Glover and McCulloch, 1958; Gopinathan, 1988; Zhou et al., 2005) or key meteorological factors (Halouani et al., 1993) for AP coefficient correction based on the analysis of the factors that influence R_s. These studies have proven that the AP coefficient is characterized by large differences between regions owing to the influence of geographical (such as altitude) and climate/weather (such as cloud cover) factors, recommending corrected coefficients based on their hypothesis and the AP correction scheme (Liu et al., 2019). However, other studies (Ögelman et al., 1984; Akinoğlu and Ecevit, 1990; Ertekin and Yaldiz, 2000; Iziomon and Mayer, 2002; Almorox and Hontoria, 2004; Yorukoglu and Celik, 2006) have pointed out that, due to the correlation between different variables, a complex model with other factors does not show evident advantages compared with the original AP (Liu et al., 2009a; Liu et al., 2010). Therefore, numerous studies continue to adopt simple and feasible traditional regression correction methods (Liu et al., 2010). For example, in China, Yin et al. (2008) used data from 81 meteorological stations from 1971 to 2000 to analyze and determine that when a_s and b_s were 0.20 and 0.79, respectively, nationwide, the R_s error for the 30 remaining verification stations was the smallest. Hu et al. (2010) divided China into seven regions, analyzing and discussing the values of a_s and b_s in different regions based on 20 years of observation data. At a small regional scale, Peng et al. (2006), Liu et al. (2009b), Li et al. (2012b), Cao et al. (2014) and Yuan et al. (2018) discussed the a_s and b_s correction values in the Yellow River Basin, Anhui, Yunnan, Jiangsu, and other regions. Although these studies show that the R_s estimated by the a_s and b_s values suggested by the FAO is larger than the observed R_s, a_s and b_s values, corrected using the observation data, can effectively improve the precision of the estimated R_s. This improvement, however, is mostly concentrated on the time scale of a year, ignoring the fact that the a_s and b_s values may change over time within a year (Paulescu et al., 2016). Data analysis by Liu et al. (2010) at 20 observation stations in northern China showed that fixed a_s and b_s coefficients for the AP can improve the estimation accuracy of R_s, reduce unnecessary complexity in the model, and are convenient for wide application. However, China is located in a typical monsoon climate zone, such that the solar radiation reaching the surface changes significantly throughout a year. In certain regions, unified a_s and b_s values may lead to seasonal error in the R_s estimations (Xia et al., 2020a). For example, Li et al. (2012a), based on observational data from 15 stations in the Yangtze River Basin, showed that the a_s and b_s coefficients change with time over a year, where the coefficients obtained on a shorter time scale can be used to estimate the R_s at a longer time scale. In addition, if the R_s estimated by the fixed a_s and b_s coefficients in a specific year is directly applied to the calculation of ET₀, error transfer will also affect the calculation results for the ET₀ (Xia et al., 2020a). Therefore, the spatio-temporal changes in the a_s and b_s values throughout the country require further discussion.

Based on the above analysis, the purpose of this study is to obtain the corrected coefficients of the AP at a specific spatio-temporal scale in China. Considering the natural geographical background of agricultural production, such as the type of agricultural region and planting system based on the diverse characteristics of water and heat combinations in China, agricultural production management, and comprehensive agricultural regionalization, the comprehensive agricultural divisions were selected as the framework, including nine primary agricultural regions (Regions A-I) and 38 subregions as division benchmarks. We used the least-squares regression method to calculate the correction coefficients for the AP for every month in every subregion. Based on this, we discuss the characteristics of the a_s and b_s coefficients in each subregion. Finally, by comparing the application errors of our values and the FAO recommended values, we provide the optimal correction coefficients a_s and b_s for every month to obtain a parameter basis for the calculation of high precision R_s values.

According to the AP (1), the data used in this study include the following: 1) the total monthly effective R_s data from 1957 to 2016 from the Dataset of Monthly Values of Radiation Data from Chinese Surface Stations (a total of 130 stations), in which data before 2011 were used to correct the a_s and b_s coefficients while data from 2011 to 2016 were used to verify the accuracy of the estimated R_s calculated using the corrected a_s and b_s coefficients; 2) the average daily relative sunshine duration of each month from 1957 to 2016 from the Dataset of Monthly Values of Climate Data from Chinese Surface Stations (a total of 756 stations), i.e., the input data for the AP; 3) longitude and latitude data for each station for the calculation of the R_a. The above three datasets are from the CMDC (http://data.cma.cn/) and 4) The Comprehensive Agricultural Regionalization Map of China (Figure 1), issued by the National Agriculture Commission, which served as the basis for agricultural divisions in this study. We can calculate the R_s as follows:

where R_s is the solar radiation (MJ m^-2 day^-1); n is the actual duration of sunshine (h); N is the maximum possible duration of sunshine or daylight (h); n/N is the relative sunshine duration; R_a is the extraterrestrial radiation (MJ m^-2 day^-1); a_s is a regression constant expressing the fraction of extraterrestrial radiation that reaches the Earth’s surface on overcast days (n = 0); and a_s + b_s is the fraction of extraterrestrial radiation that reaches the Earth’s surface on clear days (n = N).

Data preprocessing included the following steps. 1) The total monthly R_s (MJ m^-2 month^-1) was converted to a daily average (MJ m^-2 day^-1) and associated with the relative sunshine duration by station number. 2) The average daily R_a of each month was obtained according to the procedure for R_a for daily periods recommended by FAO 56 (Allen et al., 1998). 3) Theoretically, due to the existence of the atmosphere, the solar radiation observed on the ground must be lower than the R_a. However, in actual observations, the solar radiation observed on the ground at certain stations was higher than the R_a due to loss of and damage to the instruments and equipment or errors associated with observation operation, which were eliminated in this study. Finally, the number of stations with valid data throughout the land extent of China, obtained via station association and data screening, was 121 (Figure 1). 4) For agricultural regionalization, based on the latitude and longitude in Figure 1 and the vector data at the 1:250,000 administrative regionalization in China, the regional vector data required for this study were obtained after spatial correction under the Albers projection.

The coefficient correction process for the AP mainly includes four parts: data preprocessing, regression calculation of the a_s and b_s coefficients, verification of the R_s estimation results, and determination of the monthly a_s and b_s coefficients in each subregion. The details are as follows.

1) Data preprocessing. This process has been explained in the preceding section and is not repeated here.

2) Regression calculation of the a_s and b_s coefficients. Based on the pre-processed observed R_s and relative sunshine duration data before 2011, the least-squares regression method was adopted to estimate the a_s and b_s coefficients on a station-by-station basis. Then, the average values of the a_s and b_s coefficients at the stations in each subregion were obtained.

3) Verification of the estimated R_s. According to the AP, as well as based on the relative sunshine duration observations and R_a from 2011-2016, using the subregions’ a_s and b_s coefficients from step 2 and FAO recommended a_s and b_s coefficient values, two groups of R_s values were calculated. The relative error of the two estimated R_s values was compared to verify the reliability and applicability of the regression a_s and b_s coefficients taking the R_s observation data from 2011-2016 as the true values.

4) Determination of the monthly a_s and b_s coefficients in each subregion. The R_s errors estimated by the a_s and b_s coefficients obtained using the least-squares regression and those suggested by the FAO were compared month-by-month. The values of the a_s and b_s coefficients with smaller R_s estimation errors were selected as the final optimal coefficients.

The factors that affect R_s mainly include the latitude, season, altitude, slope direction (shelter), climate, and weather. Among these factors, the latitude and seasonal changes determine the position of direct sunlight on the surface and the maximum sunshine duration. Under specific spatio-temporal conditions, these two variables can be regarded as fixed values, thus determining the initial value of the solar radiation that reaches the surface. Under the assumption that the surface morphology is uniform and there is no atmosphere, the initial value is R_a. However, owing to the uneven surface morphology and existence of the atmosphere, the solar radiation that reaches the surface has spatio-temporal changes, leading to the changes in the a_s and b_s coefficients in different agricultural regions.

The shape and gravitation characteristics of the Earth objectively produce regional differences in the surface atmospheric composition and thickness, which first reflects the defects of the fixed a_s and b_s coefficients. In addition, the composition and thickness of the atmosphere also change under the influence of the climate, changes in the seasons, and human activity. However, in the existing regression correction schemes for the a_s and b_s coefficients (Peng et al., 2006; Yin et al., 2008; Liu et al., 2009a; Liu et al., 2009b; Hu et al., 2010; Liu et al., 2010; Li et al., 2012a; Li et al., 2012b; Cao et al., 2014; Paulescu et al., 2016; Yuan et al., 2018; Huang et al., 2019; Xia et al., 2020a; Xia et al., 2020b), fixed a_s and b_s coefficients are mostly obtained based on the time scale of the acquired data, rather than dynamic a_s and b_s coefficients that change with time. Therefore, we conducted a regression analysis of multi-year data month by month, obtaining the a_s and b_s values in different agricultural regions month by month to reflect the seasonal changes in the a_s and b_s coefficients in different agricultural regions throughout a year. Compared with the correction results for the a_s and b_s coefficients of various agricultural regions in China based on an annual scale reported in Xia et al. (2020b), we observe that the root mean square error (RMSE) and mean relative error (MRE) of the R_s calculated using the a_s and b_s coefficients from the monthly regressions are superior to the a_s and b_s coefficients fixed within the year (Figure 7). The Nash-Sutcliffe efficiency coefficient (NSE) results corresponding to the monthly coefficients are larger than the NSE results corresponding to the fixed coefficients (Figure 7). This indicates that seasonal changes in China have a certain influence on the values of the a_s and b_s coefficients. For accurate estimation of R_s, we cannot ignore the fact that the a_s and b_s coefficients change with the seasons.

Xia et al.’s study (2019) is the only one that discusses the influence that the length change in the observed time series has on the a_s and b_s coefficients based on the obtained fixed a_s and b_s coefficients within a year. We are not aware of any other relevant study. In this study, we selected Subregion C3, which is characterized by relatively flat terrain. Discontinuities in the station data were avoided by taking the average value of the valid observation data from stations in the region. Using five years as the starting point and a step size of the data volume, we obtained the a_s and b_s coefficients under different data series lengths of four months via regression (Figure 8). Based on Figure 8, we observe that the a_s coefficient essentially remains unchanged in the winter (January). In the spring (April), its value decreases with an increase in the data series length. In summer (July), the coefficient values for a data series length of five years is the smallest, the values remain essentially unchanged for a data series length of 10 to 40 years, and there is a sudden increase in the values for a data series length of 45 to 50 years. In autumn (October), the values fluctuate, but the range is not large. The b_s coefficient is characterized by stable fluctuations in autumn and winter. The spring exhibits a trend of increased volatility while in summer there is a decrease in the fluctuations. Thus, we can observe that the time series length of the observed data has more or less of an influence on the a_s and b_s coefficients. In other words, the sensitivity of the a_s and b_s coefficients to the length of the regression data series varies in different months, which has not been taken into account in existing studies, including this study. In addition, by observing the change in the average estimated R_s relative error for the application from 2011 to 2015 of the a_s and b_s coefficients for the monthly regression of different time-series data (Figure 9), we observe that the average estimated R_s error corresponds to the 10-year time series data, which is generally lower than that of the other time series. This indicates that the regression of the a_s and b_s coefficients has an optimal data series length, where the influence of the estimated R_s accuracy may be sensitive to a change in the data series length of less than 15 years. The influence of the R_s accuracy may not be sensitive to a change in the data series length of more than 15 years. Therefore, we must discuss the inter-annual changes in the a_s and b_s coefficients month by month across the country in future studies.

The existing regression correction schemes for the a_s and b_s coefficients may also smooth out the random weather changes in different seasons and the changes in the atmospheric characteristics caused by the discharge of large amounts of pollutants into the atmosphere by human activity within a short period in local regions. Higher altitudes have shorter solar radiation transmission paths and elevated a_s and b_s coefficients. A sunny slope can receive direct solar energy, such that the a_s and b_s coefficients of a sunny slope are greater than those of a shaded slope. In existing studies and here, coefficient correction of the AP is based on observation data from meteorological stations, which are typically located on open, sunny ground. Therefore, the influence that the slope orientation (shielding objects) has on the a_s and b_s coefficients is not considered under the existing data observation conditions. This study mainly aims to lend support for the calculation of ET₀. Therefore, the average values of the a_s and b_s coefficients at the regional stations were obtained based on China’s Comprehensive Agricultural Regionalization. However, this division does not actually correspond to the above factors that affect the spatial and temporal characteristics of surface solar radiation, such that a value for the entire region based on the average results of the stations within the agricultural divisions also leads to errors. For example, in studies based on independent stations, altitude is also a fixed value, which does not need to be considered. However, regionalization studies of these coefficients predominantly use the average value of a region (Hu et al., 2010; Huang et al., 2019), which is equivalent to the value based on the average regional altitude. This study is no exception, where the average values of a_s and b_s were used to represent the values of the entire agricultural region based on the regional statistics of the comprehensive agricultural regionalization. In Figure 1, we observe that the spatial distribution of meteorological stations in China is relatively dense in the southeast and sparse in the northwest, such that the density of stations in each agricultural subregion is different. The average value in each subregion may be different with respect to the accuracy due to differences in sample size. Therefore, if these are used to support other research or applications, such as solar energy engineering research and applications, they would require further analysis of the correlations between the station a_s and b_s coefficients and certain elements, such as topography, to improve the spatial scalability of the a_s and b_s coefficients and the realization of grid a_s and b_s coefficients to extend their application fields.

Using the least-squares regression method, this study, based on meteorological observation data from 121 stations in China, calculated the coefficients of the AP month by month in 36 agricultural subregions of China. By comparing the R_s errors estimated using the values of the a_s and b_s coefficients obtained by a monthly regression and the values of the a_s and b_s coefficients suggested by the FAO, we selected the values of the a_s and b_s coefficients corresponding to the R_s values with the smallest estimation errors as the final applied values of the coefficients for the AP. The main conclusions of our study are as follows.

(1) The values of the coefficients for the AP calculated in this study showed no uniform variation trends with respect to the spatial and temporal distribution of the nine agricultural regions, which is consistent with the spatio-temporal difference characteristics of the unstable atmospheric environment. Moreover, the regression values in most agricultural subregions were different from the global fixed values recommended by the FAO (a_s = 0.25, b_s = 0.50). Therefore, we must discuss the differences in the accuracy between the localized values of the coefficients for the AP and the global fixed values to optimize the applied values of the a_s and b_s coefficients.

(2) Overall, the relative accuracy of the R_s calculated using the regression a_s and b_s coefficients is better than that calculated using the values recommended by the FAO. However, for specific applications in each agricultural region, the R_s accuracy based on the a_s and b_s coefficients from the two sources is not equal. Therefore, by integrating the different scales of the R_s estimation accuracy, from an entire application perspective, we suggest that Regions A, B, C, D, and H continue to use the a_s and b_s coefficients recommended by the FAO for the AP, whereas the other four regions (E, F, G and I) should adopt the a_s and b_s coefficients based on the least-squares regression presented in this study. In specific agricultural subregions with higher accuracy requirements or smaller-scale applications, we recommend the use of the optimized coefficients for the monthly calculation of R_s.

This study enriches the overall investigation of the AP coefficient correction and has certain reference value for improving the calculation of ET₀. Although we provide the AP coefficients supporting ET₀ calculations in each agricultural region based on existing data conditions and by monthly regression, verification, and optimization, there are still some deficiencies that require further exploration. 1) This study only discussed the localization of the a_s and b_s coefficients based on monthly scale data. A more sophisticated ten-day scale or daily coefficients may be more pertinent in the fields of Earth science and solar energy engineering, such that we must collect finer time scale data for in-depth research. 2) Discussing the influence that the length of the time series data has on the monthly a_s and b_s coefficients is necessary. Based on the determination of the optimal historical data series length, the factors that affect the a_s and b_s coefficients can be further analyzed to improve the accuracy of the applied values of the a_s and b_s coefficients. 3) The a_s and b_s coefficients themselves are reflections of the solar radiation atmospheric transmittance, where, at its present stage, remote sensing technology has achieved fruitful results in the inversion of atmospheric indicators. Moreover, based on the principle of energy balance, previous studies have proposed numerous expressions for solar radiation atmospheric transmittance. Therefore, to explore the relationship between the a_s and b_s coefficients and atmospheric parameters obtained via remote sensing inversion (such as the aerosol optical thickness or density), as well as based on the existing expression of the atmospheric transmissibility of solar radiation, building values for grid a_s and b_s coefficients are also a potential breakthrough for the refined application of the AP.