# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
## 3 Results
### Experiment 1: Algorithm Comparison
#### 3.1.2 Relative Difference Analysis of Direct and Exp Algorithms  
Next, we analyzed the RD, (E \(-\) T)/T \(\times\) 100%, of the FY3D BA (T) and the forward modeled EC4 Dvar BA (E) in different FMs and show the results in Figure 4a-c. The orders of magnitude of the differences in the ranges of 5-55 km, 55-70 km, and 70-80 km are within 1-1.5%, 4--10%, and 10-80%, respectively. This result is consistent with those of the previous studies [28; 35; 36]. The results shown in Figure 4a-c are divided into three aspects for clarity. (i) The RD of the exp is near to zero than that of the direct, indicating the BA of the exp algorithm is closer to the true value. In other words, the exp algorithm is more accurate than the direct algorithm. The result of the direct algorithm is larger than the true value below 60 km (approximately 0.3% larger than exp) and considerably less than the true value above 60 km. (ii) Since these two algorithms have the same input and same vertical-level setting, this gap is probably due to the integral accuracy of the algorithm. The integral accuracy depends on the order of magnitude of refractivity (input), which is shown in Figure A5 in Appendix B. Intriguingly, when the order of magnitude is 1 \(\times\) 10\({}^{-2}\) above (below) a height of 60 km, the direct algorithm's result is less (larger) than the true value. (iii) Note that at heights of 18, 48, and 58 km, larger errors are incurred by refractivity rather than the errors of the FM. Unexpectedly, the errors of the exp algorithm at these heights are larger than those of the direct algorithm. This difference can be attributed to the fact that refractivity of the exp algorithm exponentially decays faster than that of the direct algorithm.  
Figure 3: Difference between the BA of FY3D (T) and EC4 Dvar BA (E) using the direct and exp Abel integral algorithms, together with log-linear and log-cubic interpolation methods. (**a,b**) (\(\text{E}_{\text{exp}\text{/dir}}-\text{T}\)) using the log-cubic interpolation; (**c,d**) difference of the Abel integral algorithms (\(\text{E}_{\text{dir}}-\text{E}_{\text{exp}}\)) based on the log-linear and log-cubic interpolation, respectively; (**e,f**) difference of interpolation methods (\(\text{E}_{\text{in}}-\text{E}\)) based on exp and direct Abel integral algorithms, respectively.  
The green and blue lines in Figure 4d-f illustrate the RD between the direct and exp algorithms. Below 60 km, their difference is less than \(\pm\)2%, whereas above 60 km, it is considerably large. The RDs between the two algorithms based on the two interpolation methods are indicated by the orange and pink lines; the RD is even larger when the algorithms are based on log-cubic interpolation. In general, the errors due to log-linear interpolation of the FM with the exp algorithm are larger than those with the direct algorithm. The RD of two interpolations with the exp algorithm is less than 0.2%, \(-\)2%, and \(-\)20% for heights of 5-50, 55-65, and 70-80 km, respectively. Specifically, because these errors are almost equal to the monthly errors of FY3D to ECMWF recorded in the ROM website (see Figure 2) the log-linear interpolation can be considered to be not sufficiently accurate. However, if such an error is acceptable, the exp algorithm with log-linear interpolation will be a timely and effective FM, since the log_linear interpolation is more cost-efficient than the log-cubic interpolation.