# Global GNSS-RO Electron Density in the Lower Ionosphere
## 2 Data and Methods
### RO Excess Phase (\(\phi_{ex}\)) Measurements  
As a key variable for the \(N_{e}\) retrieval, _hTEC_(\(h_{t}\)) is the integral of \(N_{e}\) along the RO path at each \(h_{t}\). It is related to excess phase \(\phi_{ex}\) profile by  
\[\phi_{ex}=\frac{40.3\times 10^{16}}{f^{2}}hTEC \tag{1}\]  
Errors in _hTEC_ can come from higher-order dependence (e.g., \(f^{-3}\)) and multi-path propagation. In the _E_-region \(h_{t}\), the _hTEC_ error contains contributions from both the _F_- and _E_-region, of which the former is much larger because its residual effects cannot be fully removed. Thus, any small error from the _F_-region \(N_{e}\) retrieval would cause a large error in the _E_-region \(N_{e}\) retrieval. The _hTEC_ profile can be derived from the L1 (\(f_{1}\) = 1.57542 GHz) and L2 (\(f_{2}\) = 1.22760 GHz) excess phase (\(\phi_{exL1}\) and \(\phi_{exL2}\) in meters) by  
\[hTEC=(\phi_{exL1}-\phi_{exL2})\frac{f_{1}^{2}f_{2}^{2}}{40.3\times 10^{16}(f_{2}^ {2}-f_{1}^{2})} \tag{2}\]  
In the previous algorithm, [PERSON] [31] used only the \(\phi_{exL1}\) measurements to derive _hTEC_. Therefore, the linear fit for _F_-region effect removal had to be carried out at \(h_{t}\) = 50-80 km to avoid the atmospheric bending (\(\phi_{atm}\)) at \(h_{t}\) < 60 km. Because the \(\phi_{atm}\) contribution is same in the L1 (\(\phi_{exL1}\)) and L2 (\(\phi_{exL2}\)) profiles, Equation (2) uses both L1 and L2 measurements to cancel the \(\phi_{atm}\) contribution, such that the linear fit can be carried out from lower \(h_{t}\). In this study, we fit the _hTEC_ data at \(h_{t}\) = 30-60 km to a linear function and extrapolate it up and subtract it from _hTEC_ to yield \(\Delta hTEC\) for the _D_/_E_-region \(N_{e}\) retrieval. Hence, the use of _hTEC_ data at \(h_{t}\) = 30-60 km allows the \(N_{e}\) retrieval extended to the _D_-region. To better understand and evaluate error sources and their contributions to _hTEC_, we broke down \(\phi_{ex}\) into the ionospheric and atmospheric components. First, the neutral atmospheric component (\(\phi_{iono\_free}\)) in the \(\phi_{ex}\) profile is defined by  
\[\phi_{iono\_free}=\left(f_{1}^{2}\cdot\phi_{exL1}-f_{2}^{2}\cdot\phi_{exL2} \right)/\left(f_{1}^{2}-f_{2}^{2}\right) \tag{3}\]  
Here, \(\phi_{iono\_free}\) includes the atmospheric bending \(\phi_{atm}\) and the residual ionospheric effect (RIE), or \(\phi_{RIE}\), that cannot be accounted for by the assumption from Equation (1):  
\[\phi_{iono\_free}=\phi_{atm}+\phi_{RIE} \tag{4}\]  
Because \(\phi_{RIE}\) is not from the neutral atmosphere and is not negligible at \(h_{t}\) > 80 km, it should be considered as a measurement error of _hTEC_ in Equation (2). With the dual-frequency approach, one would hope to correct the \(f^{-2}\)-dependent ionospheric effects in radio propagation. However, multi-path propagation and higher-order frequency dependence may induce the RIEs at these altitudes [40; 41]. In Figure 1a, the \(\phi_{atm}\) contribution is obtained from fitting an exponential function to the \(\phi_{iono\_free}\) data at \(h_{t}\) = 20-50 km and extrapolated to the heights above. Furthermore, it is important to understand the contributions to \(\phi_{ex\_iono}\) induced by ionospheric bending (\(\phi_{bend\_iono}\)) and phase advance (\(\phi_{p\_iono}\)) effects as radio wave propagates in the ionosphere. The contributions from these processes are significant in the \(\phi_{ex\_iono}\) profile shown in Figure 1b, and these components have the opposite sign to each other. \[\phi_{ex\_iono}=\phi_{bend\_iono}-\phi_{p\_iono} \tag{5}\]This is because bending delay (due to a longer pathlength) and phase advance (due to faster phase speed) in the ionosphere produce the opposite effects in the measured excess phase \(\phi_{ex}\). These components are defined by  
\[\phi_{bend\_iono} =\int_{RO}ds-\int_{SL}ds \tag{6}\] \[\phi_{p\_iono} =\int_{RO}(n-1)ds \tag{7}\]  
where RO and SL in the integral represent the RO and straight-line (SL) path of radio propagation, and \(n\) is refractive index of the ionosphere.