## 3 Topside Effective Scale Height Calculation Methodology
### Topside Effective Scale Height Vertical Gradient Modeling  
To gain information on the \(\mathrm{d}H/\mathrm{d}z\) median behavior, we applied the procedure developed in [25], as briefly outlined in Section 3.1, to the COSMIC--1 RO dataset described in Section 2.4. Figure 1 depicts an example of topside \(H\) calculation and modeling with data measured by COSMIC--1 on the 11 th of October 2011 at 10:19:45 UT (Universal Time) at latitude = 39.86\({}^{\circ}\) N and longitude = 0.69\({}^{\circ}\) E. Topside \(N_{\mathrm{e}}(h)\) values measured by COSMIC--1--represented by blue points in panel (a)--are used to calculate \(H_{\mathrm{Epstein}}(h)\) through Equation (7)--represented as blue points in panel (b). Calculated \(H_{\mathrm{Epstein}}(h)\) values show a clear linear dependence with height. The fit of Equation (6) to the calculated \(H_{\mathrm{Epstein}}(h)\) values allows for the obtainment of the linear fit parameters, i.e., \(\mathrm{d}H/\mathrm{d}z\) (the slope) and \(H_{0}\) (the intercept). \(H(z)\) values obtained with fitting parameters are represented as the red line in panel (b). By using calculated \(H(z)\) values as the effective scale height in the NeQuick topside representation of Equation (4), we obtain the \(N_{\mathrm{e}}(h)\) topside profile--represented by the red curve in panel (a). The comparison between \(N_{\mathrm{e}}(h)\) values measured by COSMIC--1 and those modeled by using \(H(z)\) assumes that it is possible to accurately reproduce the topside profile with the linear approximation of the effective scale height. This assumption is true in the range of altitudes covered by COSMIC--1 observations [25]. For each COSMIC--1 RO profile, we obtain a couple of linear parameters (\(\mathrm{d}H/\mathrm{d}z\), \(H_{0}\)). We applied this procedure to the selected COSMIC--1 RO dataset, which covers the diurnal and solar activity conditions probed by CSES--01. \(H_{0}\) values from COSMIC--1 have been used as references for comparison (see Section 4), while \(\mathrm{d}H/\mathrm{d}z\) values are used in the topside \(H\) modeling along with CSES--01 and Swarm B observations. Since the topside ionosphere exhibits seasonal variations, we binned calculated \(\mathrm{d}H/\mathrm{d}z\) values as a function of the months of the year in four bins around solstices and equinoxes:  
* NDJ: November, December, January--the months around the December solstice;
* FMA: February, March, April--the months around the March equinox;
* MJJ: May, June, July--the months around the June solstice;
* ASO: August, September, October--the months around the September equinox. Ref. [51] highlighted also how topside parameters are strongly connected with the geomagnetic field lines configuration. Then, magnetic coordinates are the natural basis for a spatial description of the topside parameters. Therefore, we further binned \(\mathrm{d}H/\mathrm{d}z\) values as a function of the Quasi-Dipole (QD) geomagnetic latitude [52], from 90\({}^{\circ}\) S to 90\({}^{\circ}\) N in steps of 2.5\({}^{\circ}\). Figure 2 represents the \(\mathrm{d}H/\mathrm{d}z\) mean values as a function of QD latitude for the four seasons and for the two LT sectors encompassed by CSES--01 orbit. Error bars are the standard deviation values inside each bin. \(\mathrm{d}H/\mathrm{d}z\) ranges between 0.1 and 0.3, with very large differences between daytime and nighttime conditions and large seasonal variations during nighttime. During daytime, \(\mathrm{d}H/\mathrm{d}z\) shows an absolute minimum at the geomagnetic equator and values increasing with latitude. This pattern is consistent across different seasons. Conversely, in the nighttime sector, \(\mathrm{d}H/\mathrm{d}z\) shows large seasonal variations with maxima at mid latitudes. At equinoxes (red and blue curves), \(\mathrm{d}H/\mathrm{d}z\) maximizes at mid latitudes in both hemispheres, with small hemispheric differences. Instead, at solstices, \(\mathrm{d}H/\mathrm{d}z\) maximizes in the winter hemisphere at mid latitudes.