# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
### FM Algorithm  
The core of an FM is the Abel integral, whose original form is given by Equation (2), in which two singularities exist: \(x=a\) and \(x=\infty\). To solve such problem, [22] used hyperbolic transformation and an algorithm that closely resembles the Abel integral was obtained, which we call the direct algorithm in this study. To further improve the approximation accuracy, the interpolation and numerical approximation methods were employed. For example, based on the physical assumption of isothermal atmosphere and temperature exponential approximation with height, refractivity was assumed to decrease exponentially with impact parameters. Methods of exponential extrapolation and interpolation of refractivity were summarized by the European Center for Medium-Range Weather Forecasts (ECMWF) [23; 24; 25]. In such a method, numerical approximation, such as the error function (erf), was also employed. This method is called the exp algorithm in this paper. Subsequently, the linear assumption of the BA was considered in the Abel integral inversion, called the linear algorithm in this paper. Both exp and linear algorithms are incorporated in the radio occultation processing package (ROPP) software developed by the Radio Occultation Meteorology Satellite Application Facility [24; 26]. Later, it was found that oscillations occur between levels (vertical grids) of the forward modeled BA using the exp algorithm. In order to correct this oscillation, [24] revised the exp algorithm by adopting more temperature operators in the refractivity operator; the modified algorithm is called the exp_T algorithm in this paper. For more details on these four algorithms, refer to Appendix A: Abel Integral Algorithms. In addition to the four methods, the NOAA National Centers for Environmental Prediction (NCEP) used quadratic Lagrangian polynomials [27], which avoided the exponential decay of refractivity with height. The FM is the integral algorithm as shown in Equation (2), in which the refractivity was firstly interpolated along a denser grid of impact parameter to improve integration accuracy. Therefore, the accuracy of an FM is closely associated with the interpolation method and numerical integral method. In terms of interpolation methods, authors of [28] found that the log-cubic spline (hereafter, log-cubic) is the most accurate one. Log-cubic interpolation projects refractivity into the log space and then interpolates it by the log-cubic method. In addition to this interpolation method, log-linear interpolation is tested in this paper. To the best of our limited knowledge, most studies either focus on the numerical integration method or only on the interpolation methods. Thus far, a systematic discussion on the combination of interpolation and integration methods has not been reported. In addition, to the best of our knowledge, only a few results have been reported on the estimation of the error of the FM occurring on a fixed model level, for example 137 EC. The present study aims to discuss both of these aspects. In this study, first, we compare three Abel integral algorithms, namely direct, exp, and exp_T algorithms, and two interpolation methods, namely log-cubic and log-linear. Second, we evaluate errors of FMs on the model levels of three analyses. To achieve these two aims, we set two experiments. In experiment 1, we compare FMs and interpolation methods. In experiment 2, the most accurate FM is employed to calculate the BA on model levels of three widely used analysis datasets, namely ECMWF 4 Dvar analysis with 137 levels,ECMWF ERA5 analysis with 37 levels, and final operational global analysis data (FNL) with 31 levels. Furthermore, the errors of FMs are simulated, providing an error reference for FMs under the analyzed model levels. The simulation also demonstrates to what extent a low-resolution model level can be used for conducting comparative research of RO products. Our research provides a better understanding of the error of FMs in RO data processing and assimilation.