## 4 Results and Discussion
### Comparison between Observed and Lunar Irradiance Model Predicted Radiances
Both the cosine function and bi-directional reflection correction to the GIRO predicted values in Equation (2) played important roles based on testing by omitting each of the terms individually. Second, similar to the VIIRS DNB observed radiances, the GIRO model predicted radiances can be  
The DNB to GIRO radiance ratio in Table 2 quantifies the agreement between observation and model predictions. A value of 1.0 would indicate perfect agreement, while deviating from 1.0 would give the percent difference. The table shows that during the waxing phase, the agreement between DNB observation and GIRO model predictions is within \(\pm\)5% (0.96 to 1.05) for lunar phase angles from 10 to 50 degrees. Note that according to Figure 8, the radiance at 50 degrees of lunar phase angle is about 3 DNB radiance units (nW/cm\({}^{-2}\)sr) which is at the DNB minimum radiance specification L\({}_{\rm min}\). The large differences at 60 degrees is likely due to the very small radiance value and artifacts of the polynomial fit in that area, while at 70 and 80 degrees the radiance values decreases below L\({}_{\rm min}\) according to Figure 8. On the other hand, for the waning phase, the agreement is within \(\pm\)5% in the lunar phase angle range between 10 and 70 degrees, although the best agreement can be around 1% at low lunar phase angles. As with any statistical analysis, the polynomial fits to both the DNB observations and GIRO model predictions have uncertainties. The standard error in the estimate using the 4 th order polynomial fit is on the order of 1.19 (nW/cm\({}^{-2}\)-sr) radiance unit, which is about 3% at the radiance level of 40 nW/cm\({}^{-2}\)-sr, but can become significant in percentage at low radiances such as L\({}_{\rm min}\). On the other hand, the R-square or goodness of fit for our polynomial curve fitting is typically better than 0.98. It was found that the ADM is very useful in accounting for the DCC BRDF effects and reducing the biases. However, the ADM was developed for day time applications under solar illumination, and its fitness for lunar illumination needs to be further evaluated despite the positive results in this study. It is possible that the ADM can potentially be further refined with long-term VIIRS DNB observations of the reflected lunar radiance over DCC. To ensure the consistency of our study results, we also performed comparison between the Suomi NPP and GIRO predicted time series with the near-nadir-samples (with an 86 second sampling interval), and the ADM correction to account for DCC bi-directional reflectance effects. The major inputs to ADM are the lunar zenith angle from the DNB data, and view zenith angle of 10 degrees (half of the view zenith angle range used in the DCC granule level sampling), which is \(\pm\)8.8 degrees from nadir. In this part of the analysis, an additional criterion for filtering the data with a threshold of \(>\)400 DCC pixels for each granule is applied, which reduced the standard deviation of the biases. The results are presented in Figure 9 below. In this figure, the lower panel (left vertical axis) shows the observed vs. GIRO predicted radiances. The maximum lunar radiance changes month by month, thus leading to different coefficients for the polynomial functions in the curve fitting as discussed earlier. We also found that for some months,  
Figure 9: Time Series Comparing Suomi NPP VIIRS DNB Observations with GIRO Predictions (lower panel); and the DNB/GIRO ratio (top panel, right vertical axis). such as January 2019, a few granules contained data near full moon (lunar phase angle \(<\) 2 degrees) in which case the GIRO failed to generate a result because it is out of the range and capability of the model The upper panel of the figure (right vertical axis) shows the radiance ratio (DNB observed/GIRO predicted). A statistical analysis of the ratio shows that the mean ratio is \(\sim\)0.97, and the standard deviation is 0.05, which suggests that the DNB observations are lower than the model predicted values by \(\sim\)3%, with an uncertainty of \(\pm\)5%.