Abstract
Analysis of the daily surface solar radiation (SSR) data over the 12 stations spread across India revealed that the data have many missing values and significant variability, which induce large uncertainties in the seasonal and annual means. Since the number of missing values and data variability change over time, ignoring them in the trend analysis, as done in most previous studies, may lead to erroneous results. We propose a method to incorporate the data uncertainty in the trend analysis and compare it with the traditional method, that ignores uncertainty, using the synthetic data with a known trend and having a different percentage of missing values. The proposed method is able to capture the trends in more than 85% of the cases, whereas the traditional method does it for less than 70% of the cases. Analysis of the SSR data by the proposed method revealed a renewed stronger solar dimming of about − 45 W/m^{2} per decade over the last decade (2006–2015) at all the 12 stations. Analysis of an independent satellitederived SSR data from the Breathing Earth System Simulator (generated using the MODIS satellite data) also showed the significantly decreasing trends of about − 20 W/m^{2} per decade in the annual SSR over the entire India. The SSR proxies (sunshine duration and diurnal temperature range) and the wind speed also exhibited the trends consistent with the renewed dimming and suggest that it can be attributed to the increasing aerosol concentration over India in the recent decade.
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Acknowledgments
The authors are thankful to Dr. Martin Wild for his comments/suggestions that helped us to significantly improve this manuscript. The authors thank IMD for providing the SSR, diffuse radiation, sunshine duration, and wind speed data. We also thank MODIS for providing the AOD and cloud fraction data which were obtained from GIOVANI (https://giovanni.gsfc.nasa.gov/giovanni/) website. We are thankful to GHCM for providing the monthly mean maximum and minimum temperature data (https://www.ncdc.noaa.gov/ghcnm/v3.php). We also thank WRDC for providing the radiation data which are available at http://wrdc.mgo.rssi.ru/.
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Appendices
Appendix A: Weighted linear regression (WLR)
The seasonal (or annual) SSR anomalies (μ_{y}) and corresponding standard errors (ε_{y}) were employed in weighted linear regression (WLR) to find the trends. The equation of the trend line fitted by the WLR method is given by the Eq. 13, in which A and B represent the slope and the intercept, respectively.
A and B were found using Eqs. 14 and 15 (Taylor 1997).
where,
In the equations above, t_{y} and μ_{y} represent year and seasonal (or annual) SSR anomaly, respectively, for the y th data point, and N_{years} denotes the number of years for which data are available. The weights (w_{y}) are inversely proportional to the square of the standard error (ε_{y}) of SSR anomaly μ_{y}. The model error (σ_{M}) is given by Eq. 17 and the uncertainties in slope (ΔA) and intercept (ΔB) are given by Eqs. 18 and 19, respectively.
where \(T_{i}^{\prime }s\) are given by the following:
Appendix B: Generation of synthetic data
Following steps were used for generating 30 years of synthetic daily SSR data:

1.
Monthly mean (\({M_{_{y,m}}}\)) and standard deviation (\({S_{_{y,m}}}\)) for the first year (i.e., y = 1) were specified. To make simulations reaslistic, \({M_{_{1,m}}}\) and \({S_{_{1,m}}}\) were estimated using daily SSR data of the Pune station (mean and standard deviations are given in Table S1).

2.
A decreasing trend of − 1.0 W/m^{2} per year was induced for estimating monthly means for the remaining years using the following equations.
$$ {M_{y + 1,m}} = {M_{y,m}}  1 \left\{ {\begin{array}{*{20}{c}} {y = 1,2,\ldots,29}\\ {m = 1,2,\ldots,12} \end{array}} \right. $$(20)The monthly standard deviations were assumed to be constant, i.e.,
$$ {S_{y + 1,m}} = {S_{y,m}} \left\{ {\begin{array}{*{20}{c}} {y = 1,2,\ldots,29}\\ {m = 1,2,\ldots,12} \end{array}} \right. $$(21) 
3.
The daily SSR values for the month (m) and the year (y) were generated as independent random samples from the normal distribution with mean M_{y,m} and standard deviation S_{y,m}. Five hundred random samples were generated to create 500 sets of 30 years of the daily SSR data.

4.
From the daily SSR data, thus generated, some data were randomly (5 to 20 days per month) removed to create a missing data set. The random removal was performed 10 times to create 5000 (10× 500) simulations of the missing dataset. Two groups of missing datasets were simulated. In the first group, data were removed from all the months in all the years and in the second group, data were removed only from the monsoon months of each year. The later group was generated to mimic the observed records where most of the missing observations are present during the monsoon season.

5.
Annual trends in the simulated datasets were estimated using the WLR and LR methods as described in Section 3.
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Soni, P., Srivastava, R. & Tripathi, S. Implication of data uncertainty in the detection of surface radiation trends and observational evidence of renewed solar dimming over India. Theor Appl Climatol 137, 2663–2680 (2019). https://doi.org/10.1007/s0070401827437
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