# Global GNSS-RO Electron Density in the Lower Ionosphere
## 2 Data and Methods  
The algorithm developed in this study is an extension of the earlier attempt to retrieve the \(N_{e}\) profile in the lower ionosphere from the GNSS-RO excess phase (\(\phi_{ex}\)) profile [31]. The RO excess phase is the difference between the measured and the free-space straight-line distance of GNSS (transmitter) and LEO (receiver) satellites. In the so-called \"bottom-up\" technique, a linear function is fitted to the \(\phi_{ex}\) profile over a narrow \(h_{t}\) range below the \(D\)-region to remove the \(F\)-region effects before the data are input for the \(D\)/\(E\)-region \(N_{e}\) retrieval. The \(F\)-region effects need to be removed first as cleanly as possible on a profile-by-profile basis, because a small \(\phi_{ex}\) residual from the \(F\)-region variability can induce large errors in the \(D\)/\(E\)-region retrieval. The bottom-up method, self-sufficient in removing \(F\)-region effects, is specially designed for retrieving low \(N_{e}\) concentrations in the lower ionosphere (Appendix A). It has several important differences from the conventional onion-peeling or top-down approach. First, the top-down method assumes spherical homogeneity for all \(N_{e}\) layers along the RO path and requires absolute calibration of the excess phase (\(\phi_{ex}\)) measurements, whereas the bottom-up method assumes only homogeneity in the \(D\)/\(E\)-region. The homogeneity assumption on the \(F\)-region \(N_{e}\) is not required, because it can become problematic for retrieving the \(N_{e}\) below it, as a small error in the \(F\)-region \(N_{e}\) retrieval can induce a large effect on the \(D\)/\(E\)-region \(N_{e}\). This error propagation is caused by a long tail of the Abel weighting function, which does not vanish even at the RO tangent height (\(h_{t}\)) far below the \(F\)-region [31]. As a result, small residual errors of the \(F\)-region \(N_{e}\) retrieval can propagate down to the \(D\)/\(E\)-region \(N_{e}\) in the data inversion process and generate a large error there [33; 34]. A manifestation of this error propagation is clearly seen in the \(N_{e}\) profiles retrieved from the onion-peeling approach, which often exhibit large oscillatory, sometimes negative, values in the lower ionosphere [35; 36; 37]. On the other hand, the bottom-up method shows little contribution from the \(F\)-region in its weighting function in the \(D\)/\(E\)-region, and therefore its \(N_{e}\) retrieval is less impacted by the \(F\)-region error. The differences between the Abel and bottom-up weighting functions are detailed in Section 2.3. Second, the bottom-up method is resilient to complex \(F\)-region variability with capability to cope with most of the \(F\)-region effects (e.g., inhomogeneity, structural variations) on a profile-by-profile basis. It is a self-sufficient approach, without relying on any auxiliary data. In other words, the algorithm contains a standalone data process using the same \(\phi_{ex}\) measurement profile for both \(E\)-region \(N_{e}\) retrieval and \(F\)-region effect removal. Previous studies found that the \(E\)-region \(N_{e}\) retrieval using an auxiliary model or constrained to an assimilated 3D \(N_{e}\) distribution had limited improvements for mitigating unrealistic \(F\)-region \(N_{e}\) features [38] and gradient effects on the derived \(E\)-region \(N_{e}\)[39]. These model-aided approaches are prone to ionospheric variability and tend to bias towards the quiet-time condition, not working reliably for a disturbed ionosphere. Third, the bottom-up algorithm does not require absolute \(\phi_{ex}\) calibration that is needed to derive a line-of-sight (LOS) or horizontal \(TEC\) (\(hTEC\)) profile prior to the \(N_{e}\) inversion. It uses a differential \(\phi_{ex}\) profile, or the \(hTEC\) difference (\(\Delta hTEC\)), with respect to a linear function fitted to the \(\phi_{ex}\) or \(hTEC\) data below the \(D\)/\(E\)-region.