# Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
## Appendix A Abel Integral Algorithms
### Abel Integral
#### a.2.2 Exp Algorithm  
This method was proposed by the EUMETSAT GRAS-SAF and was summarized and applied in numerical assimilation by [23; 40]. The refractivity is assumed to exponentially decrease with height since it has a relationship with temperature according to Equation (1). At a height of over 12 km, where water vapor is sparse, we can assume \(e\) equals zero. Based on the isothermal atmosphere assumption, \(p=\rho R_{d}T\), and Equation (1), the refractivity is exponentially decreased with height [26; 43; 44]. Therefore, the refractivity \(N\) as the \(j\)th level can be expressed as [35]:  
\[\begin{array}{c}N=N_{j}\exp\left(-k_{j}\left(x-x_{j}\right)\right)\\ \frac{dN}{dx}=-k_{j}N_{j}\exp\left(-k_{j}\left(x-x_{j}\right)\right)\end{array} \tag{5}\]  
where \(k_{j}=\ln(N_{j}/N_{j+1})/(x_{j+1}-x_{j})\), and the value is positive with a minimum of \(10^{-6}\). In addition, the term \(\ln n=\ln(10^{-6}N+1)\) in Equation (2) is equivalent to \(10^{-6}N\) based on infinitesimal assumption. Moreover, at the tangent point, we have \(\sqrt{x^{2}-a^{2}}\approx\sqrt{2a(x-a)}\) when we assume the impact parameter \(x\) is equivalent to the impact parameter \(a\) because \(a\) and \(x\) of air in the higher magnitude of order (6000 km) and their difference are less (<80 km).  
Therefore, the differential at the \(j\)th level is expressed as [35]  
\[\Delta\alpha_{j}=10^{-6}k_{j}N_{j}\exp\left(k_{j}\left(x_{j}-a\right)\right) \sqrt{2a}\int_{x_{j}}^{x_{j+1}}\frac{\exp\left(-k_{j}\left(x-a\right)\right)} {\sqrt{\left(x-a\right)}}dx \tag{6}\]To improve the integral accuracy and compute it economically, [45] used the error function (erf) by writing \(t=\sqrt{k_{j}(x-a)}\).  
Notably, the algorithm is not feasible under the condition of super refraction (often in the troposphere, below 12 km) because the gradient of N is large (resulting in a large \(k_{j}\)) such that the BA is overestimated [35]. In such circumstances, the second line of Equation (7) is written as:  
\[\frac{dN}{dx}=\frac{N_{j+1}-N_{j}}{x_{j+1}-x_{j}} \tag{10}\]  
This formula is still under the exponential decay assumption; However, it is acceptable because the interval is relatively small in the troposphere.  
The error of the exp algorithm comes from its advantage. When the input refractivity is smaller in high levels, the exponential decay is faster than linear decay, making it closer to the actual refractivity. However, when an abnormally larger (less) refractivity exist at some level, the results of the exp algorithm will be considerably larger (less) than the results of the direct algorithm, as shown in Equation (11).  
The exp algorithm is widely used in the FM for NWP data assimilation since the interpolation of Equation (10) is a reliable approximation between levels. However, this interpolation can yield oscillations when the interval is too large, specifically between 25 km and 45 km [36]. Furthermore, the desirable interval for avoiding this phenomenon is less than 100 m. Focused on this problem, [36] improved this algorithm with a physical association between refractivity and temperature. Thus, it is called the temperature revised exp algorithm.