## 3 Topside Effective Scale Height Calculation Methodology
### NeQuick Model Topside Representation and Effective Scale Height Linear Approximation  
The NeQuick model describes the topside ionosphere through a semi-Epstein layer anchored at the F2-layer peak [11; 19; 25]:  
\[N_{\rm e}(h)=4 Nm\mathrm{F2}\frac{\exp\left(\frac{h-hm\mathrm{F2}}{H}\right)}{ \left[1+\exp\left(\frac{h-hm\mathrm{F2}}{H}\right)\right]^{2}}. \tag{4}\]  
The \(N_{\rm e}\) decrease with height is driven by the empirical effective scale height \(H\), which the NeQuick model describes with the following formulation:  
\[H(h)=H_{0}\bigg{[}1+\frac{rg(h-hm\mathrm{F2})}{rH_{0}+g(h-hm\mathrm{F2})} \bigg{]}. \tag{5}\]  
According to (5), \(H\) is a function of the three empirical topside parameters: \(H_{0}\), \(g\), and \(r\). \(H_{0}\) is the scale height value at the F2-layer peak; \(g\) represents the scale height vertical gradient near the F2-layer peak; \(r\) is the parameter controlling the \(H\) behavior very distant from the F2-layer peak. The NeQuick model calculates \(H_{0}\) from F2-layer and bottomside parameters [11], while \(g\) = 0.125 and \(r\) = 100.  
The formulation (5) was designed to allow a smooth transition from the linear behavior near the F2-layer peak to the non-linear behavior in the overlying plasmasphere. The authors of [6; 23; 25] investigated the altitudinal range in which the \(H\) linear approximation holds. Using topside sounder data, the authors of [23] found that the \(H\) linear approximation is valid up to 1300 km above the F2-layer peak. The authors of [6; 25] found similar results using COSMIC--1 RO data up to the maximum altitude sounded by COSMIC--1 satellites, i.e., 800 km from ground. As a consequence, as demonstrated by the authors of [23] and [6; 25], the \(H\) linear approximation is well-verified at the altitudes of CSES--01 and Swarm B orbits (\(\sim\)500 km from ground), and there is no added value in introducing the non-linear terms in the \(H\) modeling. Then, we did not apply the NeQuick non-linear formulation of Equation (5) and relied on the linear approximation given by:  
\[\left\{\begin{array}{l}H(z)=\frac{\mathrm{d}H}{\mathrm{d}z}z+H_{0},\\ z\equiv h-hm\mathrm{F2},\end{array}\right. \tag{6}\]  
where the slope \(\mathrm{d}H/\mathrm{d}z\) represents the vertical gradient of the modeled topside linear scale height, while the intercept \(H_{0}\) represents the value at the F2-layer peak (\(h\) = \(hm\mathrm{F2}\) at \(z\) = 0).  
An exact application of Equation (6) requires the knowledge of the \(H\) topside vertical profile. The authors of [25] developed a mathematical procedure to obtain \(H\) values from F2-layer peak and topside \(N_{\rm e}\) values through the NeQuick topside formulation of Equation (4):  
\[H_{\rm Epstein}(h)=\frac{h-hm\mathrm{F2}}{\ln\bigg{\{}\frac{1}{N_{\rm e}(h)} \bigg{[}(2 Nm\mathrm{F2}-N_{\rm e}(h))+2\sqrt{Nm\mathrm{F2}^{2}-N_{\rm e}(h) \cdot Nm\mathrm{F2}}\bigg{]}\bigg{\}}}. \tag{7}\]  
\(H_{\rm Epstein}\) is the effective scale height obtained by mathematically inverting the semi-Epstein topside formulation of Equation (4).  
According to (7), when the F2-layer peak characteristics (\(Nm\mathrm{F2}\) and \(hm\mathrm{F2}\)) and the \(N_{\rm e}(h)\) topside profile are both available from observations, an \(H_{\rm Epstein}(h)\) profile can be obtained for the entire topside profile. This is the case with COSMIC--1 RO profiles, which allow for the calculation of \(H_{\rm Epstein}(h)\) through Equation (7), as well as allow for the obtainment of \(\mathrm{d}H/\mathrm{d}z\) and \(H_{0}\) values by fitting Equation (6) to the calculated \(H_{\rm Epstein}(h)\) values [25]. The same approach used by the authors of [25] cannot be applied to CSES--01 and Swarm B satellite data because only the in situ \(N_{\rm e}\) value is provided by satellites. Toapply Equation (6) to in situ LEO satellite observations, we need to constrain \(\mathrm{d}H/\mathrm{d}z\) in Equation (6) by using additional information provided by COSMIC--1 RO observations.