# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
## 4 Discussion  
An FM plays an important role during GNSS RO data processing and assimilation. The more accurate the FM, the more accurate are the two processes. Although the Abel algorithm and interpolation methods separately are sufficiently discussed, to the best of our knowledge, there has been no systematic discussion on the combination of interpolation and integration methods. In addition, no results on estimating the error of an FM on a fixed model level, for example 137 EC, have been reported. In this study, we first reviewed the principles of four Abel integral methods. Subsequently, we conducted experiment 1 to compare three Abel integral algorithms (i.e., direct, exp, and exp_T) and two interpolation methods (i.e., log-cubic and log-linear) via the mean value analysis of the difference and relative difference between the experimental value  
Figure 8: BA difference (\(\mathrm{E-T}\), in gray) and RD (\(\mathrm{E-T}\)) / \(\mathrm{T}\), in red) for EC4 Dvar, FNL, ERA5, and FY3D based on the exp algorithm and log-cubic interpolation (**a–c**) and log-linear interpolation (**d–f**). (**g–i**) The difference between the RD of log-cubic interpolation and that of log-linear interpolation. (forward-modeled EC4 Dvar) and true value (FY3D BA). The results suggested that (1) the exp algorithm with log-cubic interpolation is the most accurate one in every level because it has better integral accuracy (errors approximately 2%) to inputs, especially when the input is lower than an order of magnitude of 1 \(\times\) 10\({}^{-2}\) (that is, above a height of 60 km). Above this height, the direct algorithm induced a 10% error. (2) When errors exist in inputs, the exp algorithm tends to amplify the error compared with the direct algorithm. (3) The exp algorithm with log-linear interpolation is often feasible for its computation efficiency. Its errors were 0.2%, \(-\)2%, and \(-\)20% larger than those of log-cubic for the heights of 5-50, 55-65, and 70-80 km, respectively. (4) The improvement of exp_T algorithm to the exp algorithm is limited. Our results are consistent with results of [35; 36]. Subsequently, we conducted experiment 2 to determine the exact errors caused by the exp algorithm, which would be affected by the level interval of inputs. Therefore, we selected refractivity profiles on model levels of three widely used analyses (i.e., 137 levels of EC4 Dvar, 31 FNL, and 37 ERA5) as the inputs. Results demonstrated that the forward-modeled BA of denser resolution analysis EC4 Dvar was closer to the true value FY3D. The errors of the exp algorithm with log-cubic and log-linear were less than 1% and 0.5% below 38 km, less than 4% within 38-50 km, and less than 1.8% and 2% above 58 km, respectively. By contrast, the other two analyses had low accuracy. Our results are in good agreement with the results of [28]. This study paves a way to better understand the errors of the FM in RO data assimilation and processing. Simulation errors on model levels of three analyses can be a helpful FM error reference. This experiment only compared direct, exp, and exp_T algorithms, while excluding the fourth Abel integral: the linear algorithm. Such an algorithm is not feasible for data assimilation but can be used in RO data processing. In the future, linear algorithm and other integral algorithms can be addressed to compare the FM. Regarding the interpolation method, we only considered log-linear and log-cubic interpolation methods, which might have inferior representativeness above 60 km, where the interval is too large. Machine learning methods might be resorted to solve this problem. Conceptualization, N.D. and W.B. ; Methodology, N.D.; Software, J.X., X.W., Y.C. and X.M. ; Validation, N.D., W.B., C.L., C.Y., F.H., G.T. and P.H. ; Formal analysis, N.D. and X.L. ; Investigation, N.D.; Resources, Y.S. and Q.D. ; Data curation, N.D.; Writing--original draft preparation, N.D.; Writing--review and editing, W.B. ; Visualization, N.D.; Supervision, W.B. and Y.S. ; Project administration, Y.S. and Q.D. ; Funding acquisition, Q.D.