## 2 Materials and Methods
### Methodology
#### 2.2.1 Heliosat-2  
Below is a detailed explanation of the methodology and implementation of the Heliosat-2 algorithm for estimating solar irradiance, presenting a systematic approach to analyzing satellite imagery and ground observations to derive accurate solar irradiance values. Figure 2 shows the flowchart of the Heliosat-2 algorithm followed in this study. * Step 1: Generate Clear-Sky Global Horizontal Irradiance (GHI\({}_{\rm cs}\)). Figure 2: Flowchart showing the algorithms used for the all-sky GHI estimation using Heliosat-2. Here, the method utilizes the most recent ground GHI observations from the past to generate the clear-sky GHI for a specific timestamp (t):  
\[\mathrm{GHI_{cs}(t)}=\mathrm{max}\{\mathrm{GHI(t-d)}:1\leq d\leq d_{max}\} \tag{1}\]  
where, \(d_{max}\) is the number of days considered in the calculation of clear-sky conditions. Alternatively, a Bird clear-sky model [39] can be used to calculate \(\mathrm{GHI_{cs}(t)}\), depending on the availability of on-site clear-sky solar radiation dataset to estimate the coefficients in a solar radiation model [40]. A Bird clear-sky model takes into account factors such as the position of the sun, solar zenith angle, atmospheric conditions (such as water vapor and aerosols), and surface albedo (reflectivity). * Step 2: Calculate Satellite Reflectance Value. Determine the satellite reflectance value for the given timestamp (t) by analyzing the pixel value of the satellite image. \[\rho_{\mathrm{sat}}(t)=\frac{\mathrm{Pixel(t)}}{1.361\times\cos\theta} \tag{2}\]  
Here, \(\rho_{\mathrm{sat}}(t)\) is the reflectance of the satellite, \(\mathrm{Pixel(t)}\) represents the pixel value, and the solar constant is taken as 1.361 kW\(\cdot\)m\({}^{-2}\), while \(\theta\) is the solar zenith angle. * Step 3: Calculate Minimum and Maximum Satellite Reflectance Values. \[\rho_{\mathrm{max}}(t) =\mathrm{max}\{\rho(t-d):1\leq d\leq d_{max}\} \tag{3}\] \[\rho_{\mathrm{min}}(t) =\mathrm{max}\{\rho(t-d):1\leq d\leq d_{max}\} \tag{4}\]  
This helps in establishing the minimum and maximum satellite reflectance values to delineate the range of potential cloud cover. In determining the albedo parameters, we take cues from both [PERSON] et al. [27] and [PERSON] et al. [41]. The models developed by [PERSON] et al. and [PERSON] et al. both represent significant advancements in the field of solar irradiance estimation, though they focus on different aspects of the issue. The model involved in [PERSON] et al. is primarily concerned with quantifying cloud cover through a cloudiness index labeled 'n', which is integral to understanding and predicting the solar irradiation that reaches the Earth's surface within a given area. The model operates on the principle that cloud cover significantly influences solar irradiance; by assessing the clarity of the sky through digital counts, it estimates the GHI. This methodology provides a focused approach for dealing with variability in cloud cover and determining its direct impact on solar energy reception. Conversely, the method in [PERSON] et al. broadens the scope by not only addressing cloud cover but also integrating a comprehensive analysis of other atmospheric conditions, including aerosols, ozone, and water vapor, through RT modeling. Their technique retrieves a long time-series of spectrally resolved all-sky radiance, enhancing the accuracy of irradiance data under both clear and cloudy-sky conditions. For \(\rho_{\mathrm{max}}(t)\), representing the maximum reflectance from cloudy pixels, we follow [PERSON] et al. 's approach of selecting the 95 th percentile of reflectance values at local noon, ensuring robustness against outliers and changes in sensors. Local noon is when the sun is highest in the sky, marking its maximum altitude angle, which is known as the solar zenith angle. This varies based on location and time of year due to factors like Earth's tilt and orbit. This is why the solar zenith angle is always at its minimum during local noon, making it distinct from the surrounding times.