# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
## 3 Results
### Experiment 2: Evaluation of the Error of the FM on the Fixed Model Level  
Experiment 1 identified the exp algorithm with the log-cubic interpolation as the ideal FM owing to its high accuracy. Therefore, when computation efficiency is the primary focus, the exp algorithm with the log-linear interpolation is also considered. This experiment tested the extent of errors of the FMs. Another factor that introduced error to the FM is the model level (resolution of vertical levels). Thus, in this experiment, we chose three widely used analyses with different levels: 137 levels EC4 Dvar, 31 levels FNL, and 37 levels of ERA5. The refractivity profiles of these analyses were all simulated by FY3D level 2 refractivity products. During this simulation, FY3D data were first interpolated into a resolution of 20 m and then extracted according to the model level. Subsequently, each simulated analysis was transformed to the BA via the FMs. Next, the results were compared with the FM true value, providing FY3D level 2 product refractivity profiles. Their RDs and differences are listed at the end, which can provide an error reference for the FM under the model levels considered here. Figure 6 shows how the level interval, indicated by the rainbow colors, affects the RD \(((\text{E}_{\text{BA}}-\text{T}_{\text{BA}})/\text{T}_{\text{BA}}\times 100\%)\). In general, the less the interval, the less is the RD. EC4 Dvar's RD is less than the RDs for others because of the higher resolution of its levels. For example, EC4 Dvar's resolution is less than 1 km below 35 km, and its RD is less than \(\pm\)0.5%. By contrast, the errors for FNL and ERA5 are less than \(\pm\)1% for heights of 0-30 km owing to their low resolution. Interestingly, ERA5 and FNL show similar vertical resolution. However, the RD for ERA5 below 10 km is mostly less than that for FNL. This is because the Abel integral integrates values from lower levels to higher levels, and ERA5 owns the other six levels (i.e., 875, 825, 775, 225, 175, and 125 hPa), which ensures more accurate interpolation results for ERA5. Figure 6 also shows the comparison of the log-cubic interpolation (a-c) with log-linear interpolation (d-f). The errors of the latter are 1-2% larger than those of the former owing to the lower interpolation accuracy of log-linear. Figure 7 compares the results of the RD (a-c) and difference (d-f) of the exp algorithm with log-cubic interpolation on the model levels of the three analyses. Below 35 km, the RD and difference for EC4 Dvar are less than \(\pm\)0.5% and \(\pm\)0.5 \(\times\) 10\({}^{-5}\), respectively, while those for FNL and ERA5 frequently oscillate in the range of \(\pm\)2% to \(\pm\)4%. Above 35 km, the RD and difference for EC4 Dvar are similar to those for FNL and ERA5. The large bias for FNL and ERA5 resulted from two aspects: (i) the low resolution of the level interval; (ii) the rapid change in the atmospheric condition below 10 km. In such conditions, if the vertical grid density of refractivity profiles is insufficient, a larger oscillation of BA profiles will occur. The same comparison as Figure 7 but for log-linear interpolation is shown in Figure A6 in Appendix B. Figure 8a-f concludes the detailed errors, the difference (grey), and RD (red) for forward-modeled three analyses, which are listed in Table 2. Figure 8g-i concludes the difference in the RD between two interpolation methods based on each analysis. Below  
30 km, the RD of log-cubic interpolation is 0.5-1% lower than that of log-linear interpolation, 2% lower for 30-50 km, and back to 1% lower for a height above 50 km. For FNL, the log-cubic results are 2% lower than the results of log-linear. For ERA5, the log-cubic results are 1.8% lower than the results of log-linear. Figure 6: Effects of various model level intervals (km) on the FM error. The x-axis represents the RD, (\(\rm(E-T)/T\times 100\%\)), of the BA between true value (T) and three simulated analyses (E): (**a,d**) 137 levels of EC4 Dvar; (**b,e**) 31 levels of FNL; (**c,f**) 37 levels of ERA5.