# Calibration of the Ångström–Prescott Model for Accurately Estimating Solar Radiation Spatial Distribution in Areas with Few Global Solar Radiation Stations: A Case Study of the China Tropical Zone

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## Abstract

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## 1. Introduction

^{2}. In response to the characteristics of large area, sparse meteorological stations, and uneven distribution in the tropical regions of China, and aiming to obtain optimal parameters for global solar radiation calculation models, this study proposes a monthly global solar radiation model based on a single-station method and between-groups linkage of the Ångström–Prescott (A–P) model. Utilizing monthly meteorological data from 80 meteorological stations in the tropical regions of China from 1996 to 2016, the study considers the similarity in inter-station monthly sunshine percentage variations. The applicability and accuracy of the adjusted parameters (a and b coefficients) were tested and evaluated, and the corrected parameters were extended to conventional meteorological stations through Thiessen polygons. Finally, the spatial distribution of solar radiation in the tropical regions of China was simulated using Kriging, inverse distance weighting (IDW), and Spline interpolation techniques. This research not only enriches the calibration cases of the Ångström–Prescott formula coefficients but also enhances the accuracy of solar radiation simulation, providing valuable references for exploring the spatial distribution characteristics during the entire year and dry–wet seasons.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Dataset

#### 2.3. Methods

#### 2.3.1. Estimation of the Global Solar Radiation under the A–P Model

^{−2}; ${\mathrm{H}}_{\mathrm{o}}$ is the astronomical radiation, MJ·m

^{−2}; $\mathrm{S}$/${\mathrm{S}}_{0}$ is the fraction of sunlight received, dimensionless, with $\mathrm{S}$ being the number of hours of daylight and ${\mathrm{S}}_{0}$ being the total number of hours; and a and b are empirical coefficients, dimensionless.

^{−2}·d

^{−1}); n indicates the number of days in the current month (d); T indicates the time of a day, the value of which is 1440 (min·d

^{−1}); ${\mathrm{I}}_{\mathrm{o}}$ is the solar constant, the value of which is 0.082 (MJ·m

^{−2}·min

^{−1}); 1/${\mathsf{\rho}}_{\mathrm{o}}$ is the mean distance between the earth and the sun, dimensionless; ${\mathsf{\omega}}_{\mathrm{o}}$ is the solar hour angle (rad); $\mathsf{\phi}$ is the geographical latitude (rad); $\mathsf{\delta}$ is the declination of the sun (rad); $\mathrm{x}$ indicates the calculation parameter, dimensionless; $\mathrm{N}$ is in day order, the value of which is 365 or 366, dimensionless; ${\mathrm{N}}_{\mathrm{O}}$ indicates the calculation parameter, dimensionless; and $\mathrm{y}$ represents the calculation year, dimensionless.

#### 2.3.2. The Between-Groups Linkage

#### 2.3.3. Thiessen Polygons

#### 2.3.4. Spatial Interpolation

#### 2.3.5. Statistical Evaluation

^{−2}; $\overline{{\mathrm{O}}_{\mathrm{i}}}$ is the mean of observed values, MJ·m

^{−2}; ${\mathrm{E}}_{\mathrm{i}}$ is the estimated value of solar radiation, MJ·m

^{−2}; and $\overline{{\mathrm{E}}_{\mathrm{i}}}$ is the mean of estimated value, MJ·m

^{−2}. n is the corresponding number of observations.

## 3. Results and Discussion

#### 3.1. Result of the Between-Groups Linkage

#### 3.2. Error Analysis and Coefficient a and b Optimization of the A–P Model

#### 3.2.1. The Whole Year (January–December)

#### 3.2.2. The Dry Season (November–April)

#### 3.2.3. The Wet Season (May–October)

^{2}was 0.94, MAPE was 5.42%, RMSE was 33.20 MJ·m

^{−2}, MAE was 24.33 MJ·m

^{−2}, and MBE was 14.15 MJ·m

^{−2}. In the comprehensive agricultural area of China, R

^{2}was 0.71, MAPE was 8.64%, RMSE was 79.99 MJ·m

^{−2}, MAE was 38.12 MJ·m

^{−2}, and MBE was −10.67 MJ·m

^{−2}. However, the absolute difference in MBE values between the two regions was small. Based on the single-station model, Xia et al. [32] calculated the average values, and the results met the regional consistency, at the expense of the accuracy of some stations (ignoring the differences within the zone). Therefore, considering the comprehensive metrics including MAE, MAPE, MBE, and R

^{2}, the values of “a” and “b” in this study were determined through multi-station regression within the region, increasing the sample size for regression and simultaneously optimizing the values for each station within the same region.

#### 3.3. Result of Global Solar Radiation Zoning by the Thiessen Polygons

#### 3.4. Verification of Spatial Interpolation Accuracy

#### 3.4.1. The Average Annual Global Solar Radiation during the Whole Year (January–December)

^{−2}, and the mean value was 5045.3 MJ·m

^{−2}. The spatial distribution results showed that the average annual solar global radiation during in the whole year in the tropical zone of China decreased from west to east in Yunnan, from southwest to northeast in Hainan Island, from the coast to inland in Guangdong and Fujian. The maximum value was located in the western part of Hainan Island, and the minimum value was located in Guizhou. According to QX/T 89-2018 standard [34], the areas with highly abundant solar radiation were mainly distributed in most parts of Yunnan and Hainan Island, followed by some coastal areas of Guangdong and Fujian. The average annual global solar radiation in these areas were more than 5000 MJ·m

^{−2}, accounting for 45.2% of the total area of the tropical zone in China. The remaining areas were all classified as abundant areas, with the majority of Guangxi having relatively lower average annual global solar radiation. The range of the average annual global solar radiation in Guangdong and Fujian was mainly between 4500 and 5000 MJ·m

^{−2}.

#### 3.4.2. The Average Annual Global Solar Radiation during the Dry Season (November–April)

^{−2}and an average value of 2093.9 MJ·m

^{−2}. The average annual global solar radiation in most parts of Yunnan and the southwest of Hainan Island was above 2300 MJ·m

^{−2}, with the highest value appearing in Yunnan. The average annual global solar radiation in the central and northern parts of Guangxi was relatively lower, with most values below 1600 MJ·m

^{−2}(Figure 10).

#### 3.4.3. The Average Annual Global Solar Radiation during the Wet Season (May–October)

^{−2}and an average value of 2941.5 MJ·m

^{−2}. The spatial distribution of the average annual solar radiation was different from the whole year and the dry season. The high-value areas of the average annual solar radiation in the wet season were mainly distributed on Hainan Island, the coastal areas of Guangxi, Guangdong, Fujian, and the border area of Yunnan and Sichuan, with an average annual solar radiation above 3000 MJ·m

^{−2}. The distribution pattern was gradually decreasing from coastal areas to inland areas. In addition, the average annual solar radiation in most areas was mainly distributed in the range of 2800–3000 MJ·m

^{−2}, accounting for 56.5% of the tropical zone (Figure 11).

## 4. Conclusions

- (1)
- Based on the between-groups linkage of sunshine percentage, this study divided the meteorological stations into zones. Stations within the same zone were used for the regression coefficient calculation, which effectively increased the amount of regression sample data. This method could effectively compensate for the simulation accuracy of the regression coefficients in most months when the simulation accuracy of a single station was poor. After parameter optimization, the accuracy of the average annual global solar radiation simulation for each station during the dry and wet seasons and the whole year could be improved by 8.1%, 4.4%, and 5.3%, respectively. In addition, due to the increase in the sample number at specific stations, the multi-station simulation accuracy was lower than that of the single station.
- (2)
- To effectively apply the regression coefficients to non-solar radiation meteorological stations, this study used the property of the Thiessen polygons in which the distance between any point inside the polygon and the control point is the shortest. Based on this, the tropical zone of China was divided into 11 zones, and the stations in the same zone used the same a and b of the A–P model. Through validating the spatial interpolation results of solar radiation for the whole year, the dry season, and the wet season, the optimal methods for the spatial interpolation of solar radiation for the whole year were IDW, and those for the dry and wet seasons were Kriging and Spline, respectively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Province | Station | Latitude (°N) | Longitude (°E) | Altitude (m) |
---|---|---|---|---|

Fujian | Shanghang | 25.05 | 116.42 | 198.00 |

Fujian | Longyan | 25.05 | 117.02 | 376.00 |

Fujian | Pingtan | 25.52 | 119.78 | 32.40 |

Fujian | Zhangzhou | 24.50 | 117.65 | 28.90 |

Fujian | Dongshan | 23.78 | 117.50 | 53.30 |

Fujian | Xiamen | 24.48 | 118.07 | 139.40 |

Fujian | Chongwu | 24.90 | 118.92 | 21.80 |

Fujian | Fuzhou | 26.08 | 119.28 | 84.00 |

Guangdong | Xuwen | 20.33 | 110.18 | 56.20 |

Guangdong | Shaoguan | 24.67 | 113.60 | 121.30 |

Guangdong | Fogang | 23.88 | 113.52 | 97.20 |

Guangdong | Lianping | 24.37 | 114.48 | 215.20 |

Guangdong | Meixian | 24.28 | 116.07 | 116.00 |

Guangdong | Guangning | 23.63 | 112.42 | 92.70 |

Guangdong | Gaoyao | 22.98 | 112.48 | 60.00 |

Guangdong | Heyuan | 23.80 | 114.73 | 71.10 |

Guangdong | Zengcheng | 23.33 | 113.83 | 30.80 |

Guangdong | Huiyang | 23.07 | 114.37 | 108.50 |

Guangdong | Wuhua | 23.92 | 115.75 | 135.90 |

Guangdong | Huilai | 22.98 | 116.30 | 42.00 |

Guangdong | Nanao | 23.43 | 117.03 | 8.00 |

Guangdong | Xinyi | 22.35 | 110.93 | 141.40 |

Guangdong | Luoding | 22.72 | 111.60 | 60.00 |

Guangdong | Taishan | 22.25 | 112.78 | 33.10 |

Guangdong | Shenzhen | 22.53 | 114.00 | 63.00 |

Guangdong | Shanwei | 22.80 | 115.37 | 17.30 |

Guangdong | Zhanjiang | 21.15 | 110.30 | 53.40 |

Guangdong | Yangjiang | 21.85 | 111.98 | 90.30 |

Guangdong | Dianbai | 21.55 | 110.98 | 31.80 |

Guangdong | Shangchuan Island | 21.73 | 112.77 | 21.90 |

Guangdong | Shantou | 23.38 | 116.68 | 2.30 |

Guangdong | Guangzhou | 23.22 | 113.48 | 70.70 |

Guangxi | Fengshan | 24.55 | 107.03 | 509.40 |

Guangxi | Hechi | 24.70 | 108.03 | 260.20 |

Guangxi | Duan | 23.93 | 108.10 | 170.80 |

Guangxi | Liuzhou | 24.35 | 109.40 | 96.80 |

Guangxi | Napo | 23.42 | 105.83 | 794.10 |

Guangxi | Baise | 23.90 | 106.60 | 174.70 |

Guangxi | Jingxi | 23.13 | 106.42 | 739.90 |

Guangxi | Pingguo | 23.32 | 107.58 | 108.80 |

Guangxi | Laibin | 23.45 | 109.08 | 96.70 |

Guangxi | Guiping | 23.40 | 110.08 | 42.50 |

Guangxi | Wuzhou | 23.48 | 111.30 | 114.80 |

Guangxi | Longzhou | 22.33 | 106.85 | 128.80 |

Guangxi | Lingshan | 22.42 | 109.30 | 66.60 |

Guangxi | Yulin | 22.67 | 110.12 | 121.60 |

Guangxi | Fangcheng | 21.78 | 108.35 | 32.40 |

Guangxi | Qinzhou | 21.98 | 108.60 | 49.20 |

Guangxi | Dongxing | 21.57 | 107.95 | 56.80 |

Guangxi | Beihai | 21.45 | 109.13 | 12.80 |

Guangxi | Nanning | 22.63 | 108.22 | 121.60 |

Guizhou | Wangmo | 25.18 | 106.08 | 566.80 |

Guizhou | Luodian | 25.43 | 106.77 | 440.30 |

Hainan | Dongfang | 19.10 | 108.62 | 7.60 |

Hainan | Danzhou | 19.52 | 109.58 | 169.00 |

Hainan | Qiongzhong | 19.03 | 109.83 | 250.90 |

Hainan | Qionghai | 19.23 | 110.47 | 24.00 |

Hainan | Lingshui | 18.55 | 110.03 | 35.20 |

Hainan | Sanya | 18.22 | 109.58 | 419.40 |

Hainan | Haikou | 20.00 | 110.25 | 63.50 |

Sichuan | Panzhihua | 26.57 | 101.72 | 1224.80 |

Yunnan | Huaping | 26.63 | 101.27 | 1230.80 |

Yunnan | Baoshan | 25.12 | 99.18 | 1652.20 |

Yunnan | Yuanmou | 25.73 | 101.87 | 1120.60 |

Yunnan | Chuxiong | 25.03 | 101.55 | 1824.10 |

Yunnan | Ruili | 24.00 | 97.85 | 762.90 |

Yunnan | Jingdong | 24.47 | 100.87 | 1162.30 |

Yunnan | Yuxi | 24.33 | 102.55 | 1716.90 |

Yunnan | Gengma | 23.55 | 99.40 | 1104.90 |

Yunnan | Lincang | 23.88 | 100.08 | 1502.40 |

Yunnan | Lancang | 22.57 | 99.93 | 1054.80 |

Yunnan | Simao | 22.78 | 100.97 | 1302.10 |

Yunnan | Yuanjiang | 23.60 | 101.98 | 400.90 |

Yunnan | Mengla | 21.47 | 101.57 | 633.40 |

Yunnan | Jiangcheng | 22.58 | 101.85 | 1120.50 |

Yunnan | Yanshan | 23.62 | 104.33 | 1561.10 |

Yunnan | Pingbian | 22.98 | 103.68 | 1414.10 |

Yunnan | Mengzi | 23.45 | 103.33 | 1313.60 |

Yunnan | Jinghong | 22.00 | 100.78 | 582.00 |

Yunnan | Tengchong | 24.98 | 98.50 | 1695.90 |

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**Figure 2.**Flowchart representing the steps taken for modeling the global solar irradiation of the tropical zone of China.

**Figure 5.**Comparison with the estimated values based on the ${\mathrm{T}}_{1}$, ${\mathrm{T}}_{2}$, and ${\mathrm{T}}_{3}$ models’ measured values.

**Figure 8.**Distribution of the Thiessen polygons. Note: Polygons of the same color represent the same zone.

Province | Station | Abbrev | Latitude (°N) | Longitude (°E) | Altitude (m) |
---|---|---|---|---|---|

Yunnan | Tengchong | Tch | 24.98 | 98.50 | 1695.90 |

Jinghong | Jh | 22.00 | 100.78 | 582.00 | |

Mengzi | Mz | 23.45 | 103.33 | 1313.60 | |

Sichuan | Panzhihua | Pzh | 26.57 | 101.72 | 1224.80 |

Guangxi | Nanning | Nn | 22.63 | 108.22 | 121.60 |

Beihai | Bh | 21.45 | 109.13 | 12.80 | |

Guangdong | Guangzhou | Gzh | 23.22 | 113.48 | 70.70 |

Shantou | Sht | 23.38 | 116.68 | 2.30 | |

Fujian | Fuzhou | Fzh | 26.08 | 119.28 | 84.00 |

Hainan | Haikou | Hk | 20.00 | 110.25 | 63.50 |

Sanya | Sy | 18.22 | 109.58 | 419.40 |

Station Abbrev | Pzh | Tch | Jh | Mz | Fzh | Gzh | Sht | Nn | Bh | Hk | Sy | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | a | b | a | b | a | b | a | b | a | b | a | b | a | b | a | b | a | b | a | b | |

Jan | 0.282 | 0.363 | −0.033 | 0.841 | 0.214 | 0.495 | 0.164 | 0.618 | 0.15 | 0.628 | 0.15 | 0.628 | 0.15 | 0.628 | 0.15 | 0.628 | 0.15 | 0.628 | 0.15 | 0.628 | 0.224 | 0.52 |

Feb | 0.195 | 0.506 | 0.206 | 0.513 | 0.195 | 0.506 | 0.074 | 0.736 | 0.15 | 0.613 | 0.15 | 0.613 | 0.162 | 0.567 | 0.145 | 0.63 | 0.175 | 0.563 | 0.15 | 0.613 | 0.24 | 0.467 |

Mar | −0.008 | 0.738 | 0.227 | 0.43 | 0.25 | 0.39 | 0.306 | 0.332 | 0.143 | 0.626 | 0.143 | 0.626 | 0.163 | 0.562 | 0.143 | 0.626 | 0.188 | 0.476 | 0.19 | 0.512 | 0.372 | 0.156 |

Apr | 0.188 | 0.482 | 0.282 | 0.32 | 0.276 | 0.352 | 0.287 | 0.37 | 0.212 | 0.423 | 0.156 | 0.612 | 0.171 | 0.562 | 0.171 | 0.562 | 0.186 | 0.54 | 0.225 | 0.456 | 0.316 | 0.304 |

May | 0.231 | 0.451 | 0.231 | 0.451 | 0.248 | 0.424 | 0.225 | 0.513 | 0.194 | 0.529 | 0.18 | 0.564 | 0.21 | 0.479 | 0.194 | 0.529 | 0.131 | 0.671 | 0.317 | 0.303 | 0.424 | 0.136 |

Jun | 0.285 | 0.327 | 0.3 | 0.19 | 0.324 | 0.257 | 0.285 | 0.327 | 0.226 | 0.457 | 0.197 | 0.505 | 0.204 | 0.496 | 0.204 | 0.496 | 0.213 | 0.493 | 0.204 | 0.496 | 0.319 | 0.293 |

Jul | 0.261 | 0.389 | 0.261 | 0.389 | 0.261 | 0.389 | 0.252 | 0.5 | 0.263 | 0.408 | 0.194 | 0.507 | 0.195 | 0.509 | 0.234 | 0.442 | 0.181 | 0.553 | 0.195 | 0.509 | 0.271 | 0.373 |

Aug | 0.2 | 0.5 | 0.309 | 0.253 | 0.284 | 0.357 | 0.31 | 0.302 | 0.261 | 0.412 | 0.201 | 0.499 | 0.199 | 0.505 | 0.242 | 0.448 | 0.146 | 0.624 | 0.199 | 0.505 | 0.199 | 0.505 |

Sep | 0.187 | 0.532 | 0.278 | 0.376 | 0.326 | 0.283 | 0.276 | 0.416 | 0.236 | 0.43 | 0.21 | 0.486 | 0.21 | 0.486 | 0.21 | 0.517 | 0.172 | 0.579 | 0.243 | 0.401 | 0.263 | 0.356 |

Oct | 0.253 | 0.432 | 0.261 | 0.435 | 0.259 | 0.432 | 0.274 | 0.419 | 0.213 | 0.49 | 0.213 | 0.49 | 0.213 | 0.49 | 0.229 | 0.487 | 0.213 | 0.49 | 0.214 | 0.474 | 0.213 | 0.487 |

Nov | 0.154 | 0.544 | 0.239 | 0.48 | 0.213 | 0.505 | 0.08 | 0.793 | 0.202 | 0.511 | 0.202 | 0.511 | 0.202 | 0.511 | 0.202 | 0.511 | 0.214 | 0.508 | 0.202 | 0.511 | 0.287 | 0.389 |

Dec | 0.000441 | 0.759 | 0.129 | 0.642 | 0.224 | 0.488 | 0.224 | 0.488 | 0.191 | 0.526 | 0.191 | 0.526 | 0.191 | 0.526 | 0.191 | 0.526 | 0.191 | 0.526 | 0.191 | 0.526 | 0.209 | 0.538 |

R² | 0.953 | 0.864 | 0.897 | 0.857 | 0.951 | 0.971 | 0.98 | 0.981 | 0.938 | 0.934 | 0.839 |

Error Analysis | Agricultural Comprehensive Area of China [32] | The Tropical Zone of China |
---|---|---|

R² | 0.71 | 0.94 |

MAPE (%) | 8.64 | 5.42 |

RMSE (MJ·m^{−2}) | 79.99 | 33.20 |

MAE (MJ·m^{−2}) | 38.12 | 24.33 |

MBE (MJ·m^{−2}) | −10.67 | 14.15 |

Interpolation Method | Error Analysis | The Whole Year | The Dry Season | The Wet Season |
---|---|---|---|---|

Kriging | RMSE (MJ·m^{−2}) | 407.90 | 187.11 | 202.94 |

MAE (MJ·m^{−2}) | 309.61 | 132.61 | 150.97 | |

MBE (MJ·m^{−2}) | 203.52 | 87.89 | 93.45 | |

R² | 0.72 | 0.92 | 0.44 | |

IDW | RMSE (MJ·m^{−2}) | 377.71 | 234.62 | 196.29 |

MAE (MJ·m^{−2}) | 293.42 | 169.51 | 143.98 | |

MBE (MJ·m^{−2}) | 189.13 | 111.77 | 77.36 | |

R² | 0.77 | 0.87 | 0.43 | |

Spline | RMSE (MJ·m^{−2}) | 413.57 | 189.50 | 173.09 |

MAE (MJ·m^{−2}) | 334.77 | 142.17 | 133.80 | |

MBE (MJ·m^{−2}) | 61.60 | 19.27 | 42.35 | |

R² | 0.63 | 0.89 | 0.53 |

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## Share and Cite

**MDPI and ACS Style**

Yu, X.; Yi, X.; Li, M.-F.; Dai, S.; Li, H.; Luo, H.; Zheng, Q.; Hu, Y.
Calibration of the Ångström–Prescott Model for Accurately Estimating Solar Radiation Spatial Distribution in Areas with Few Global Solar Radiation Stations: A Case Study of the China Tropical Zone. *Atmosphere* **2023**, *14*, 1825.
https://doi.org/10.3390/atmos14121825

**AMA Style**

Yu X, Yi X, Li M-F, Dai S, Li H, Luo H, Zheng Q, Hu Y.
Calibration of the Ångström–Prescott Model for Accurately Estimating Solar Radiation Spatial Distribution in Areas with Few Global Solar Radiation Stations: A Case Study of the China Tropical Zone. *Atmosphere*. 2023; 14(12):1825.
https://doi.org/10.3390/atmos14121825

**Chicago/Turabian Style**

Yu, Xuan, Xia Yi, Mao-Fen Li, Shengpei Dai, Hailiang Li, Hongxia Luo, Qian Zheng, and Yingying Hu.
2023. "Calibration of the Ångström–Prescott Model for Accurately Estimating Solar Radiation Spatial Distribution in Areas with Few Global Solar Radiation Stations: A Case Study of the China Tropical Zone" *Atmosphere* 14, no. 12: 1825.
https://doi.org/10.3390/atmos14121825