The Global Navigation Satellite System (GNSS) radio occultation (RO)
technique is widely used to observe the atmosphere for applications such as
numerical weather prediction and global climate monitoring. The ionosphere is
a major error source to RO at upper stratospheric altitudes, and a linear
dual-frequency bending angle correction is commonly used to remove the
first-order ionospheric effect. However, the higher-order residual
ionospheric error (RIE) can still be significant, so it needs to be
further mitigated for high-accuracy applications, especially from 35 km
altitude upward, where the RIE is most relevant compared to the decreasing
magnitude of the atmospheric bending angle. In a previous study we quantified
RIEs using an ensemble of about 700 quasi-realistic end-to-end simulated RO
events, finding typical RIEs at the 0.1 to 0.5

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).

Radio occultation geometry between GNSS transmitter and
low Earth orbit (LEO) receiver satellites, schematically illustrating the
separate L1 and L2 signal ray paths and the ionosphere-corrected (Lc) ray
path through the atmosphere–ionosphere system. Key quantities additionally
indicated are the (total accumulated) bending angle

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

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.

The ensemble of RO events used by Liu et al. (2015) was simulated for 15 July 2008, adopting the European MetOp RO mission as an example low Earth orbiter (Edwards and Pawlak, 2000), specifically thinking of MetOp-A, which was launched as the first of the MetOp series in late 2006 (Luntama et al., 2008). Each MetOp satellite is a sun-synchronous LEO satellite at about 820 km with the Global Positioning System (GPS) Receiver for Atmospheric Sounding (GRAS) on board (Loiselet et al., 2000), which acquires about 700 RO events per day (Luntama et al., 2008).

Using, as summarized above, simple spherically symmetric neutral atmospheric
modeling (by MSIS-90) combined with 3-D ionospheric modeling (by NeUoG), we
simulated in that study the ensemble of daily RO events for 14 different
end-to-end simulation cases. These included without-ionosphere (wi) cases as
well as spherical symmetry (ss) and non-spherical-symmetry (ns) ionospheric
cases for low, medium, and high solar activity levels, under the assumption
of either perfect observing system (op) with no errors or realistic
observing system (or) with MetOp-type errors; for details see Liu et al. (2015), Table 2 and Sect. 2.3 therein. The total number of the simulated RO
events found for the day was 723, of which 26 exceptionally noisy ones were
classified as outliers (estimated bending angle RIE exceeding 7

Distribution of the mean tangent point locations of the
723 RO events simulated by Liu et al. (2015) for 15 July 2008

Figure 2a shows the global distribution of mean tangent point (TP) locations of
all 723 events (as small triangles) and highlights the locations of the 26 exceptional events (as red triangles).
The majority of the latter (18 of the 26) appear to cluster over the European–Asian and Indian Ocean regions
(EAC and IOC, highlighted as boxes); the remaining eight events are distributed
more diversely in other extratropical locations, mainly in the Northern
Hemisphere. Figure 2b and c depict the RIE bias and standard deviation,
defined in the same way as by Liu et al. (2013), which are estimated for the
upper stratosphere and mesosphere (30–80 km) for the 26 events, for the
non-spherical-symmetry (“opns”) and spherical symmetry (“opss”)
ionospheric conditions, respectively. Intercomparing Fig. 2b and c shows
that the main driver of anomalously high RIEs is asymmetric ionospheric
conditions and possibly residual error effects from ray tracing through the
3-D ionosphere, since only few events (6 of the 26) exhibit large RIE
standard deviations (exceeding 1

Related to the clusters, one can see that, in the opss case, almost all
noisy exceptional events occurred in the IOC, while in the opns case the
noisiest ones occurred in both the EAC and IOC. Related to solar activity, one
can see that in both the opss and opns cases higher ionization
(F10.7) levels generally lead to increased RIEs, compared to lowest
ionization (F10.7

These overall characteristics revealed by Fig. 2 point, in particular, to two facts that shall guide our detailed investigation for better understanding of anomalous RIEs: (1) asymmetric ionospheric conditions play a key role, more than ionization levels and possible geographic location dependencies (e.g., via solar or geomagnetic influences), and so inspection of the along-ray signal dynamics is essential; (2) the several exceptions from the overall characteristics, and some geographic clustering that has no obvious physics-based explanation, indicate that there is no single clear cause for the anomalous RIEs and that some perturbations also come in from the technical challenge of smooth ray tracing at millimetric excess-phase accuracy through 3-D ionospheric models like NeUoG.

We inspected the bending angle RIE profiles of the 26 events over the 20 to 80 km height range, including also their underlying excess-phase RIE profiles, and chose three representative events that we will explore in detail below for improving RIE insight: an extremely noisy event (Occ.530 from the EAC) and a medium noisy event (Occ.20 from the IOC) from the 26 exceptional events, both used at medium solar activity, and a reference event from the 697 standard events, with low-noise RIE (Occ.25). Table 1 summarizes the main parameters for these three events, and Fig. 3 illustrates them in terms of excess phases, bending angles, and the associated RIEs.

Illustration of the three example events chosen for
detailed inspection (Occ.530, red; Occ.20, green; Occ.25, blue), showing
their excess-phase profiles

Parameters of the three representative RO events used for detailed inspection. Azimuth of the RO event plane is defined relative to north, counting over west.

Figure 3a shows the behavior of the excess phases of the three events. The
L1 and L2 excess phases are around

As Leitinger and Kirchengast (1997) describe, substantial empirical modeling effort went into strict smoothness of the NeUoG electron density field and its 3-D derivatives that are key for high-accuracy ray tracing; nevertheless some slight discontinuities have likely remained in some rare locations of the modeling space spanned by altitude, latitude, longitude, (universal) time, month, and solar activity. It will therefore be important to separate such technical modeling effects from the physical effects on the propagating signals that cause high RIEs.

Figure 3c shows that, for all three events, the difference between

Intercomparing Fig. 3d with b suggests that these waveform-like
perturbations in the bending angle RIE are mainly induced by propagating the
spiky excess-phase perturbations through the bending angle retrieval, which
involves filtering and a derivative operation from excess phase to Doppler
shift (Schwarz et al., 2017). One main cause that has driven many of the
exceptional events into the outlier range (i.e., into exceeding 7

The ray tracing technique is commonly used for calculating propagation paths of an electromagnetic signal in a medium specified by a position-dependent refractive index field. It has become a significant tool for investigating signal propagation in RO technology. For example, Ladreiter and Kirchengast (1996), Syndergaard (2000), Gobiet and Kirchengast (2004), Steiner and Kirchengast (2005), Hoque and Jakowski (2010), Mannucci et al. (2011), Danzer et al. (2013, 2015), and Liu et al. (2013, 2015) have employed this method inter alia or with a main topical focus to investigate the ionospheric effects on GNSS RO signals. Danzer et al. (2015) noted that their analysis was somewhat limited by high noise of the simulated bending angle profiles at mid- to high latitudes, which partly reflected the degrading impact of technical ray tracer effects that we also encounter and more explicitly address in this study.

We employ the 3-D numerical ray tracing technique integrated in the End-to-end GNSS Occultation Performance Simulation and Processing System version 5.6 (EGOPS 5.6; Fritzer et al., 2013) in the same way as used by Liu et al. (2013, 2015) for simulating the GPS-to-LEO signal propagation through the atmosphere–ionosphere system; for a detailed description of the end-to-end simulation setup the reader is therefore referred to these recent studies. Here we specifically refined and enhanced this setup in the 3-D ray tracing part by adding the co-computation and result extraction for a range of key variables along the propagation paths, instead of only providing the final observational variables of an RO event at the LEO receiver position.

We implemented detailed along-ray diagnostic capabilities into the 3-D ray
tracer of the EGOPS 5.6 software (Fritzer et al., 2013), which is an
extensively proven high-accuracy ray tracer originally developed in the
1990s (Syndergaard, 1998, 1999). In particular, we computed the following
key diagnostic variables for all individual numerical steps along the ray
paths simulated for the GPS L1 and L2 frequencies as well as for a reference
case without ionosphere (Lr), with each ray path starting at the GPS
transmitter position and ending at the LEO receiver position:
3-D position in the WGS84-based Earth-centered, Earth-fixed (ECEF) system,
storing both the cartesian (

These along-ray variables are computed for all available ray paths from 80 to
20 km impact height, which are produced at 50 Hz sampling rate for any
RO event investigated. This leads to a dense sampling by roughly 1500
ray paths in this altitude range (i.e., typical average scan velocities of
RO events are near 2 km s

We will inspect the results for the three representative RO events chosen (Occ.530, Occ.20, Occ.25; see Sect. 2.1 above) in this along-ray distance vs. impact height coordinate system. Before turning to this, we briefly summarize here the equations for the along-ray computation of those key variables that we will inspect closely. This aims to facilitate an appropriate understanding and interpretation of the results.

On the basis of Snell's law, when the Earth's atmosphere and ionosphere are
assumed spherically symmetric, Bouguer's rule can be used to describe the
refraction of a ray path in terms of a constant impact parameter (e.g.,
Budden, 1985),

Equation (1) implies that, at each point along the ray path, the impact parameter

In reality non-spherical-symmetry conditions of appreciable size will
frequently occur, in particular between the ionospheric signal propagation
inbound from the GPS and (after propagating through the atmosphere at
tangent heights below 80 km) the one outbound to LEO (cf. Fig. 1); see,
for example, the RO events discussed by Liu et al. (2013). In order to inspect the
impact parameter changes along the ray path in these cases where

In addition, the accumulated bending angle

Images of the atmospheric and ionospheric refractivity
(Eq. 4) in the along-ray distance (relative to tangent point) vs. impact
height coordinate system for non-spherical-symmetry

Images of the delta impact parameter (Eq. 3) in the
along-ray distance (relative to tangent point) vs. impact height
coordinate system for non-spherical-symmetry

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;

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

Images of the accumulated bending angle (Eq. 5) in the
along-ray distance vs. impact height coordinate system for
non-spherical-symmetry

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

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.

Images of the accumulated ionosphere-corrected Lc bending
angle (

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

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

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

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).

The fact that no obvious physics-based explanation was found for the geographic clustering and the intermittent spiky behavior found in simulated RIEs indicates that a portion of the anomalous RIEs of the exceptional RO events were caused by the technical challenge of ray tracing at millimetric excess-phase accuracy through the 3-D ionospheric model NeUoG, which is not perfectly smooth everywhere in its electron density field derivatives.

The detailed along-ray analyses of atmospheric and ionospheric refractivities, impact parameter changes, bending angles, and RIEs also revealed that along-ray L1 and L2 impact parameter variations of more than 10 to 20 m are possible due to ionospheric asymmetries and are generally found to be negative (relative to the initial impact parameter at the GPS transmitter). Standard bending angle retrievals with their intrinsic spherical symmetry assumption should thus receive higher-order ionospheric correction to mitigate such impacts.

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

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.

The ECMWF
(Reading, UK) is thanked for access to their archived analysis and forecast
data (more information available at

The authors declare that they have no conflict of interest.

This article is part of the special issue “Observing Atmosphere and Climate with Occultation Techniques – Results from the OPAC-IROWG 2016 Workshop”. It is a result of the International Workshop on Occultations for Probing Atmosphere and Climate, Leibnitz, Austria, 8–14 September 2016.

This research was partially supported by the National Natural Science Foundation of China (grant nos. 41405039, 41775034, 41405040, 41505030, 41606206, and 41730109) and the FengYun-3 (FY-3) Global Navigation Satellite System Occultation Sounder (GNOS) development and manufacture project led by NSSC, CAS. The research at WEGC/University of Graz was supported by the European Space Agency (ESA) projects OPSGRAS and MMValRO and the Austrian Research Promotion Agency (FFG) project OPSCLIMPROP (ASAP-9 project no. 840070). We acknowledge Johannes Fritzer (WEGC) for his support in EGOPS software developments valuable to this study. The research at SPACE/RMIT University was supported by the Australian Research Council (ARC) (LP0883288), the Australian Antarctic Division (project no. 4159), and the CAS/SAFEA International Partnership Program for Creative Research Teams (grant no. KZZD-EW-TZ-05). The support from the Jiangsu dual creative talents and Jiangsu dual creative team program projects awarded to CUMT in 2017 is acknowledged. Edited by: Sean Healy Reviewed by: two anonymous referees