Systematic Ionospheric Residual Errors in GNSS Radio Occultation: Theory for Spherically Stratified Media

Abstract

The standard ionospheric correction of bending angles in Global Navigation Satellite System (GNSS) radio occultation (RO) measurements removes most of the influence from the ionosphere, but leaves small systematic residual errors in the corrected data. The main reasons for residual errors at stratospheric and mesospheric altitudes are the neglect of higher‐order terms in the expansion of the ionospheric refractive index, and the fact that the two GNSS signals follow slightly different paths through the ionosphere. Another reason is the neglect of the local electron density at the receiver in orbit. These residual errors depend on the geomagnetic field and the spatial distribution of the ionospheric electron density and its gradients. In this work we derive this dependency to high accuracy for a spherically stratified ionosphere, including the “bi‐local” situation where the ionosphere is different (though still spherically stratified) on the inbound and outbound side of the RO event tangent points. As part of the derivations, we find a small residual error term not previously noted, which can become appreciable for elliptical satellite orbits. The results are verified by ray tracing through simple ionospheric and geomagnetic field models. The accuracy of a higher‐order bi‐local ionospheric residual error correction based on these results would be limited by the uncertainty in knowledge of the electron density and by horizontal electron density gradients along the ray paths.

Understanding Ionospheric and Geomagnetic Effects on Residual Biases in Radio Occultation Data for Stratospheric Climate Monitoring

“…We also see a similar increase under nearly symmetric conditions (NSY) for the kappa RIBs in Figure 5 (red labeling). This also shows that the kappa‐correction partially already comprises not only the respective Ne 2 term, but due to its construction as a bending‐angle‐difference‐squared term indirectly also incorporates some effects of the geomagnetic conditions, as well as inbound/outbound asymmetry effects toward electron density change (Syndergaard & Kirchengast, 2022). However, when the bi‐local RIBs show a positive geomagnetic term, this cannot be reproduced by the kappa RIBs, since it cannot change its sign by construction.…”

“…The bi‐local correction approach was proposed by Syndergaard and Kirchengast (2019, 2022) and validated using GNSS RO observational data sets (Liu et al., 2020). The key equation known as the bi‐local RIB correction term can be expressed in the form $$\begin{align*}\delta {\alpha }_{\text{Biloc}}=-\frac{KF\left({B}_{//}{N}_{e}\right)}{{f}_{1}{f}_{2}\left({f}_{1}+{f}_{2}\right)}-\frac{1}{2}\frac{{C}^{2}}{{f}_{1}^{2}{f}_{2}^{2}}\frac{1}{a}\frac{d\left[{a}^{2}F\left({N}_{e}^{2}\right)\right]}{da}\\ -\frac{C}{\left({f}_{1}^{2}-{f}_{2}^{2}\right)}\frac{{r}_{\mathrm{L}}{H}_{\mathrm{L}}{N}_{e}\left({r}_{\mathrm{L}}\right)}{\sqrt{{r}_{\mathrm{L}}^{2}-{a}^{2}}}\frac{d\left({\alpha }_{1}-{\alpha }_{2}\right)}{da}\end{align*}$$ where δα Biloc is the total bi‐local higher‐order RIB correction term (see Equation ) composed of (in order of appearance) a geomagnetic F ( B // N e ) term, an electron density‐squared F ( N e 2 ) term, and a local‐LEO component term (the last term, being only appreciable for non‐circular orbits and large electron densities at the LEO).…”

“…Another more complete higher-order ionospheric correction method was introduced by Syndergaard and Kirchengast (2019) and subsequently described in detail as part of Syndergaard and Kirchengast (2022) work that introduced a rigorous theory on the residual ionospheric biases (RIBs) in GNSS RO data for spherically stratified media. This so-called bi-local correction method is a more refined approach, which not only models (a) different ionization levels but also accounts (b) for the impact of the geomagnetic field in the ionospheric refractive index, (c) for ionospheric inbound (GNSS to tangent point) versus outbound (tangent point to LEO) asymmetry, as well as (d) the influence of the local electron density at the receiver LEO satellite (Syndergaard & Kirchengast, 2022; Section 4 therein). The bi-local correction has a high potential to model the RIBs.…”

GNSS-RO Residual Ionospheric Error (RIE): A New Method and Assessment

“…An Es layer can produce large RIEs near the layer heights with a long tail extended to lower tangent heights in the GNSS-RO profile [Syndergaard, 2000;Syndergaard and Kirchengast 2022]. If the RO measurements stop at or below the Es layer, the 𝜙 !"…”

“…Higher-order contributions not removed by the linear combination of L1 and L2 measurements may depend on several factors. Most important among them are ionospheric structure [Ladreiter and Kirchengast 1996;Syndergaard 2000;Mannucci et al, 2011], magnetic field and electron density (Ne) [Hartmann and Leitinger, 1984;Brunner and Gu, 1991;Morton et al, 2009;Hogan and Jakowski, 2011], radio wave propagation path [Coleman and Forte, 2017], and horizontal inhomogeneity [Syndergaard and Kirchengast, 2022].…”