## 4 Results and Validation
### Validation Approach and Performance Indicators  
The retrieved daily mean CLARA-A3 RSF (\(F_{CLARA}\)) was validated against the daily mean TOA Shortwave Flux from the CERES synoptic gridded product SYN1 deg [44] edition 4.1, denoted by \(F_{CERES}\). Note that CERES' Energy Balanced and Filled (EBAF) product [32] was not used since it is a monthly product optimized regarding absolute accuracy (calibration), stability, and long term (decadal) trend detection, whereas this validation was done on daily and sub-daily scale.  
The bias of daily mean RSF for every grid box with indices \((i,j)\) was calculated as \(F_{CLARA,i,j}-F_{CERES,i,j}\). Prior to the validation, the spatial resolution of \(F_{CLARA}\) was first downgraded to match the nested grid of CERES. Maps of this bias were created (daily \"bias maps\"), from which the global Mean Bias (\(MB\)) and corresponding bias-corrected RMS of the bias (\(RMSB\)) were calculated over all grid boxes' biases, as follows:  
\[MB=\frac{1}{m\cdot n}\cdot\sum_{i=1}^{m}\sum_{j=1}^{n}(F_{CLARA,i,j}-F_{CERES,i, j}) \tag{9}\]  
\[RMSB=\sqrt{\frac{1}{m\cdot n}\cdot\sum_{i=1}^{m}\sum_{j=1}^{n}(F_{CLARA,i,j}- F_{CERES,i,j}-MB)^{2}}, \tag{10}\]  
where \(m\) and \(n\) are the number of grid boxes in longitude (360) and latitude (180) dimension, respectively. The \(RMSB\) is a commonly used measure of spatial validation between data records, or with respect to a reference record (considered truth), also called regional uncertainty (precision). The \(MB\) is a measure for the absolute calibration of the data record with respect to the reference data (accuracy). Note that these global statistics were calculated using a latitudinal area weighting to prevent an over-representation of the polar areas, which is not shown in the simplified equations to preserve readability.  
Besides the \(MB\) and \(RMSB\), the absolute bias of daily mean RSF for every grid box \((i,j)\) was calculated as \(|F_{CLARA,i,j}-F_{CERES,i,j}|\). Similar to the mean bias (Equation (9)), the Mean Absolute Bias (\(MAB\)) was then calculated as the global spatial mean over all grid boxes, as shown in Equation (11). The daily \(MAB\) is another measure for regional uncertainty.  
\[MAB=\frac{1}{m\cdot n}\cdot\sum_{i=1}^{m}\sum_{j=1}^{n}|F_{CLARA,i,j}-F_{CERES,i,j}|. \tag{11}\]  
The above mentioned performance indicators (\(MB\), \(RMSB\) and \(MAB\)) are based on biases between _daily mean_ quantities (daily mean RSF from CLARA-A3 and daily mean RSF from CERES-SYN1 deg). Therefore, these indicators may be hiding (temporal) intra-day compensating errors, for instance an overestimation in the morning followed by an underestimation in the afternoon which cancel each other, resulting in an apparently low bias for a given grid box. This was investigated by calculating the \(MAB\) on hourly time scale (\(MABH\)) between the retrieved dataset (CLARA-A3) and the reference dataset (CERES SYN1 deg-Hour Ed.4.1). The daily absolute bias on an hourly time scale for a single grid box \((i,j)\) was thus calculated from 24 h, denoted \(h\), and hence 24 biases (Equation (12)). From this, the spatial global mean was calculated to obtain the \(MABH\):  
\[MABH=\frac{1}{m\cdot n}\cdot\sum_{i=1}^{m}\sum_{j=1}^{n}\left[\frac{1}{24} \cdot\sum_{h=1}^{24}\left|F_{CLARA,i,j,h}-F_{CERES,i,j,h}\right|\right]. \tag{12}\]  
All the mentioned performance indicators are so-called \"indicators of dispersion (Class A)\" in the terminology of [PERSON] [45]. The use of more sophisticated indicators is beyond the scope of this article, in which the validation primarily serves as a first confirmation of the retrieval algorithm's capabilities following the preparatory test phase of CLARA-A3 (with only one year of data). However, once the full data record is generated and released, a more extensive in-depth validation will be performed.