Global Navigation Satellite System (GNSS) radio occultation (RO; Melbourne et al., 1994; Kursinski et al., 1997; Hajj et al., 2002) is a relatively new atmospheric sounding technique. It can deliver data traceable to the international standard of time (the SI second) and has a demonstrated capacity for monitoring decadal-scale climate change in the free atmosphere (Steiner et al., 2009, 2011, 2013; Anthes, 2011; Foelsche et al., 2011; Lackner et al., 2011; Ho et al., 2012; Angerer et al., 2017). This capacity rests on RO's unique combination of characteristics such as high vertical resolution, high accuracy, long-term stability, and global coverage (Kursinski et al., 1997; Scherllin-Pirscher et al., 2011; Anthes, 2011; Steiner et al., 2011). Figure 1 illustrates the GNSS RO geometry that constitutes the basis of the RO technique. The focus is to schematically show essential aspects relevant to this study on along-ray ionospheric influences on RO bending angles, which deepens insight on top of our recent Liu et al. (2015) study.

Ionospheric error is significant in GNSS RO observations (e.g., Kursinski et al., 1997; Mannucci et al., 2011; Liu et al., 2013), and a dual-frequency linear combination of RO bending angles is usually implemented to correct for the first-order ionospheric effect (Vorob'ev and Krasil'nikova, 1994; Ladreiter and Kirchengast, 1996). However, the higher-order residual ionospheric error (RIE) after this correction is still not negligible for high-accuracy applications such as RO-based climate change monitoring (Steiner et al., 2011, 2013). This applies especially above about 35 km altitude, where the RIE becomes increasingly relevant compared to the exponentially decreasing magnitude of the neutral atmospheric bending angle (Syndergaard, 2000; Mannucci et al., 2011; Danzer et al., 2013, 2015; Liu et al., 2013, 2015; Healy and Culverwell, 2015).

Moreover, the RIE can propagate downwards into the lower-stratospheric retrievals of refractivity and temperature through the Abel integral and the hydrostatic integral (Kursinski et al., 1997; Gobiet and Kirchengast, 2004; Steiner and Kirchengast, 2005; Gobiet et al., 2007). It is therefore essential to better understand and further mitigate the RIE in order to enable benchmark-quality stratospheric RO retrievals.

A wide array of studies related to a better understanding of higher-order ionospheric errors in GNSS RO data have been conducted already by a range of scientists (Bassiri and Hajj, 1993; Vorob'ev and Krasil'nikova, 1994; Ladreiter and Kirchengast, 1996; Syndergaard, 2000; Gorbunov, 2002; Hoque and Jakowski, 2010, 2011; Mannucci et al., 2011; Danzer et al., 2013, 2015; Healy and Culverwell, 2015; Coleman and Forte, 2017). A few of these also suggested ways of correcting higher-order RIEs in RO bending angles (Syndergaard, 2000; Danzer et al., 2013; Healy and Culverwell, 2015), which may be applied on top of the standard dual-frequency correction introduced by Vorob'ev and Krasil'nikova (1994).

The convenient formulation introduced by Healy and Culverwell (2015), which adds a fairly simple higher-order squared-bending-angle difference term to the standard correction, is meanwhile applied in operational processing of the data from the European MetOp (Meteorological Operational Satellites) RO mission (Luntama et al., 2008; Christian Marquardt, EUMETSAT Darmstadt, personal communications, 2017). Recently, Angling et al. (2018) further improved the empirical modeling of the “kappa coefficient” in this formulation, by accounting for solar zenith angle, solar flux (F10.7 index), and altitude dependencies.

In our work over the recent years we have assessed the variation of bending angle RIEs (biases and standard deviations) with solar activity, with latitudinal region, and with or without the assumption of ionospheric spherical symmetry and of co-existing RO observing system errors, using end-to-end simulations for single RO events (Liu et al., 2013) and a full-day ensemble of RO events (Liu et al., 2015). As shown in these explanatory simulation studies, in overall agreement with the empirical study of Danzer et al. (2013), the RIE biases have a clear negative tendency and a bias magnitude increasing with solar activity, as well as being affected by deviations from ionospheric spherical symmetry (Mannucci et al., 2010) where increasing asymmetries also tend to increase the biases.

What remained unexplored in our Liu et al. (2015) study and had also not yet been explored elsewhere – but is critical to be understood for further improvement of the existing RIE corrections that apply spherical symmetry (Syndergaard, 2000; Healy and Culvervell, 2015; Angling et al., 2018) – is the influences of the three-dimensional and asymmetric ionospheric structures along the GNSS-to-LEO (low Earth orbit) signal paths on the RIE, in particular the conditions that may lead to anomalously high RIEs.

A first step in this direction, though not focusing on bending angle RIEs, was the study by Mannucci et al. (2011), which found that under ionospheric storm conditions anomalous effects can be significant. Recently also Coleman and Forte (2017) reported RIE investigations for asymmetry conditions, including on the effect of traveling ionospheric disturbances upon the RIE. Another step was the somewhat puzzling side result in our Liu et al. (2015) study that the end-to-end simulations of an ensemble of about 700 RO events produced about two dozen RIE outlier profiles. The basis was 3-D ray tracing simulations, where the ionospheric model NeUoG (University of Graz electron density model; Leitinger and Kirchengast, 1997) was used as a quasi-realistic model for large-scale 3-D ionospheric structures, together with the atmospheric model MSIS-90 (Mass Spectrometer and Incoherent Scatter neutral atmosphere model 1990; Hedin, 1991) for simple but representative neutral atmosphere reference conditions. More precisely, the RIE standard deviation of 26 profiles from the simulations exceeded a threshold value of 7 µrad within the upper stratosphere and mesosphere. These were therefore rejected from the ensemble statistics results reported by Liu et al. (2015).

In this study we now place focus on these 26 exceptional profiles and, by way of detailed along-ray analyses of ray tracing simulations, aim to shed light on the causes of anomalously high RIEs, with the additional goal of deepening quantitative insight into how RIEs accumulate during signal propagation, along with accumulation of the total (atmospheric) bending angles that are the desired RO observables. In Sect. 2, the exceptional RO events and the simulation setup for exploring their bending angle RIEs are introduced. Section 3 provides the results, which we mainly discuss through detailed inspection of example events. A summary and conclusions are finally given in Sect. 4.

Figures 4 to 7 sequentially illustrate for the three representative RO events (Occ.25, Occ.20, Occ.530) the along-ray behavior of the key variables atmospheric and ionospheric refractivity (Eq. 4), L1 and L2 delta impact parameter (Eq. 3), L1 and L2 accumulated bending angle (Eq. 5), and ionosphere-corrected Lc bending angle and bending angle RIE (Eq. 7). This is done in the form of imaging these variables for the three RO events in the along-ray distance vs. impact height coordinate system (panels a and b of Figs. 4–7; ±3500 km along-ray distance about ray tangent points; impact height range: 20 to 80 km) and in the form of depicting the along-ray behavior of the two exceptional events along representative impact heights (80, 50, 30 km; panels c–f in Fig. 4 and panels c–d in Figs. 5–7).

In order to enable close inspection of the critical role of ionospheric symmetries, each of the Figs. 4–7 directly intercompares the non-spherical and spherical symmetry conditions. In terms of solar activity only the results for the medium solar activity level (F10.7 = 140) are illustrated, since we found that the influence of solar activity (which mainly drives the ionization level in the NeUoG model) is primarily to steer the magnitude of the effects (see Liu et al., 2013, 2015). The typical along-ray characteristics are therefore reasonably well represented by just illustrating the medium-solar-activity case.

Figure 4 shows the atmospheric and ionospheric refractivities and underpins that the along-ray differences of inbound ionosphere (from the GPS) and outbound ionosphere (towards LEO) refractivities can be substantial. For example, in the case of the Occ.20 event these refractivities differ by more than a factor of 2 near the ionospheric F layer maximum, where the refractivities are largest (e.g., L2 inbound refractivity near 10 NU; L2 outbound refractivity reaching more than 20 NU). As expected, the atmospheric refractivity starts to exceed 1 NU only below about 35 km, and of course it exhibits no frequency dependence. It is thus essential to have a reliable first-order and higher-order ionospheric correction to strongly mitigate the ionospheric effects that appear prominent down to the lower stratosphere.

Figure 5 shows the L1 and L2 delta impact parameters, which first of all verifies the reliability of the numerical ray tracing estimates, since the spherically symmetric ionosphere conditions indeed lead to along-ray impact parameter changes of within 1 m only. This confirms that under such spherical symmetry conditions the bending angle retrievals (e.g., Schwarz et al., 2017) will be highly accurate, including for the impact height that is decisive for enabling accurate vertical geolocation (Scherllin-Pirscher et al., 2017). Under non-spherical-symmetry ionosphere conditions, the Occ.20 event with the largest asymmetry of the example events shows that along-ray L1 and L2 impact parameter variations of more than 10 to 20 m are possible and are generally found to be negative (relative to the initial impact parameter at the GPS transmitter). Lc bending angle retrievals with their intrinsic spherical symmetry assumption should thus receive higher-order ionospheric correction to mitigate such possible impacts.

Figure 6 depicts the accumulated L1 and L2 bending angles, which highlight the significant along-ray modulations that the bending angle receives due to the ionospheric influences relative to the atmospheric bending angle, in particular above about 35 km in the upper stratosphere and mesosphere, where the neutral atmospheric bending angle is rather small. Below 35 km the dominating influence of the atmosphere in the vicinity of the tangent point location becomes prominently visible, in line with the exponential increase of atmospheric refractivity (Fig. 4) and hence atmospheric bending down into the lower stratosphere. Nevertheless even at 30 km the ionospheric contribution is still visible, which underscores that an accurate ionospheric correction with minimized residual error will be vital.

Figure 7a shows the ionosphere-corrected Lc bending angle and indicates, compared to Figs. 6a and b, that the standard linear dual-frequency correction of bending angles basically does a very effective job in eliminating the ionospheric bending angle contributions. The Lc bending angle images look visually very clean and are highly dominated by just the atmospheric accumulated bending angle accruing at all heights around the tangent point location. Directly inspecting the bending angle RIE, finally, shows that the along-ray behavior and accumulated magnitude of the higher-order RIE left by the linear correction significantly depend in particular on asymmetry conditions. Technical ray tracer effects are also visible as intermittent spiky behavior, since the RIEs are at the sub-microradian to microradian magnitude level only, which is a challenge for the ionospheric model smoothness as discussed in Sect. 2.1.

The Occ.530 event under non-spherical-symmetry appears to accumulate the highest RIEs of near 2 to 4 µrad at LEO, while the spherical symmetry cases both accumulate RIEs up to around 0.5 to 1 µrad or less only. This is in line with Fig. 2, which shows for the majority of the 26 exceptional events the dominance of asymmetry and 3-D ray tracing effects in driving the RIE magnitude. Also, as shown by Fig. 7c and d (and found for other RO events inspected but not separately shown), the mesospheric RIEs above about 50 km generally appear to be higher than the upper-stratospheric ones from 50 km downwards. This is in line with findings of Syndergaard (2000) and likely driven by the increased closeness of the tangent point height to the ionospheric E layer peaking near 105 km, which makes the Lc bending angle more vulnerable to higher-order RIEs.

Figure 7c and d (and along-ray results for further exceptional events not separately shown) also clearly indicate the mixing-in of technical ray tracer effects in our simulations. These render it more difficult to rigorously quantify the (smooth) physical RIE effects from the large-scale ionospheric model structure since, despite the reasonable smoothing applied, the spiky components may somewhat perturb the smooth accumulated results as well. In the future we will therefore aim to further improve the simulation setup to fully isolate the technical from the physics-based propagation effects. For now we have found clear evidence, nevertheless, that currently both technical effects and cases with physically high RIEs from ionospheric asymmetries play important roles in explaining the anomalous behavior of the exceptional RO events.

Previous theoretical and simulation studies as well as empirical studies that we surveyed in the Introduction have characterized and quantified higher-order residual ionospheric errors (RIEs) in bending angles by analyses of individual events as well as ensembles of events. The statistical results showed that the mean bending angle RIE biases are predominantly negative, typically at the 0.03 to 0.1 µrad level, and these biases may lead to systematic errors in stratospheric climatologies built from retrieved profiles. The RIE standard deviations are typically at the 0.1 to 0.5 µrad level, and they have a clear tendency to increase with increasing solar activity, i.e., with increasing ionization level (electron density) in the ionosphere.

In our previous Liu et al. (2015) study we had contributed to these findings but were left with 26 exceptional RO events with very high RIEs, at the 1 to 10 µrad standard deviation level, in the context of about 700 standard events with low-noise RIEs within 0.5 µrad standard deviation. In this study we therefore placed focus on these 26 exceptional events and, by way of detailed along-ray analyses of ray tracing simulations over the stratosphere and mesosphere, inspected the causes of anomalously high RIEs. The goal at the same time was to deepen quantitative insight into how RIEs accumulate during signal propagation, along with accumulation of the total atmospheric bending angles that are the desired RO observables.

From the results of these analyses we conclude with the following main findings on the causes of the exceptional RO events:

Strengthening previous results by Mannucci et al. (2010, 2011), we find that asymmetric ionospheric conditions play an important role for anomalously high RIEs, more than ionization levels driven by solar activity and possible geographic location dependencies that seemed to be present from salient geographic clustering of the majority of exceptional RO events in two regions (European–Asian region and Indian Ocean region).

In the future we will therefore aim to further improve our along-ray simulation and analysis system to fully isolate the technical from the physics-based propagation effects. For now we have found clear evidence, nevertheless, that currently both technical effects and cases with physically high RIEs from ionospheric asymmetries play major roles in explaining the anomalous behavior of the exceptional RO events. The detailed along-ray modeling system will also be valuable beyond this work for additional GNSS RO signal propagation studies.