# New Higher-Order Correction of GNSS RO Bending Angles Accounting for Ionospheric Asymmetry: Evaluation of Performance and Added Value

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}) and geomagnetic (BNe) terms appear to be dominant and comparable in magnitude. The BNe term takes negative or positive values, depending on the angle between the geomagnetic field vector and the direction of RO ray paths, while the Ne

^{2}term is generally negative. We evaluated the new approach against the existing “Kappa approach” and the standard linear dual-frequency correction of bending angles and found it to perform well and in many average conditions similar to the simpler Kappa approach. On top of this, the bi-local approach can provide added value for RO missions with low LEO altitudes and for regional-scale applications, where its capacity to account for the ionospheric inbound-outbound asymmetry as well as for the geomagnetic term plays out.

## 1. Introduction

## 2. Correction Approach and Implementation Setup

^{−2}term of Equation (1) but has the other higher-order terms uncorrected, as well as a dominating error related to the ray path splitting, which is proportional to f

^{−4}[16].

^{−4}, with these terms formulated as:

#### 2.1. Formulation of Bi-Local RIE Model

_{e}(r

_{L}) is the local electron density at the LEO satellite, and other symbols are as defined previously. The functional $\mathrm{F}\left(X\right)$ is a ray path integral function from GNSS to LEO that can be described by inbound, outbound, and another local-LEO component as follows:

#### 2.2. Processing Setup for the Bi-Local Correction in the rOPS

#### 2.2.1. rOPS as the System for RIE Correction Evaluation

^{−4}, and (2) integrating a complete uncertainty propagation chain from RO raw tracking data and high accuracy GNSS orbit data, and relevant side influences, through four processors to atmospheric profiles [35]. Figure 2 shows the main components of the rOPS as well as the main interactions, where the occultation data processing (ODP) subsystem is the core subsystem provided for the retrieval of the level one and level two profiles [36,37,38,39,40].

#### 2.2.2. Implementation of the Bi-Local Correction Approach

#### 2.3. Setup of the Performance Evaluation Simulations

#### 2.3.1. Solar and Ionospheric Conditions

_{e}) profiles used to implement the bi-local correction in this study are normalized N

_{e}values by combining the NeUoG model electron density profiles and IGS vTEC maps. After the normalization, the N

_{e}profiles have inherited the profile shapes of the NeUoG model electron density profiles and at the same time the vTEC value of the IGS vTEC maps (cf. Table 2). Using observation-based vTECs in this way, the model profiles are adjusted to be consistent with observed ionospheric conditions. Figure 3b,c show example IGS vTEC maps of 00:00 UT and 12:00 UT, respectively, on the test day 15 July 2013.

#### 2.3.2. Geomagnetic Conditions

## 3. Results

#### 3.1. Statistical Ensemble of Bending Angle RIE Profiles

^{2}), and NeL terms, wherein both the BNe and Ne

^{2}terms have three subcomponents, i.e., inbound (TxTP), outbound (TPRx) and Rx local (Rxloc) terms. According to our study, for the MetOp-A/-B missions (Figure 5), at 60 km altitude and upwards, the order of magnitude of the mean total bi-local correction RIE is 0.01 μrad, and the BNe and Ne

^{2}terms appear to be both significant terms with the same order of magnitude (see Figure 5).

^{2}terms it is still small and hence a minor term in practical application.

^{2}correction term in Figure 5c,g. One can see that both the bi-local Ne

^{2}term and the Kappa bending angle RIE have a clear negative tendency and similar magnitude values. The noisier behavior in Figure 5d,h is a result of using observed bending angles in the Kappa correction approach (Equation (5)), whereas the bi-local approach is based primarily on model estimations (except for using observed bending angles in the small last term of Equation (7)). Very large outliers of observed bending angle profiles were excluded according to quality control criteria, which led to bad-profile exclusion if one or more sampling points within the impact altitude range from 25 to 80 km showed values of ${\left[{\alpha}_{1}(a)-{\alpha}_{2}(a)\right]}^{2}$ that exceeded 0.1 μrad

^{2}. As a second step, the selected profiles were smoothed by a moving average filter with a window width of 10 km, before being used for Equation (5).

^{2}terms, the BNe terms have negative and positive values, depending on the angle between the geomagnetic field vector and the ray path direction, while all the Ne

^{2}terms are negative. The values of the Ne

^{2}terms being systemically negative results in almost all the total bending angle RIE terms being negative as well. Comparing the statistics of the total bending angle RIE and Ne

^{2}term, the average bias is similar, which means the effect of the BNe term on the global-mean climatology bending angle is small. This is because globally the BNe terms have almost the same amounts of positive-sign and negative-sign occurrences and hence cancel each other out when averaging over the RO events.

^{2}terms are negative. However, in the 2008 CHAMP mission ensemble data set, the average bias of the BNe term (Figure 6f) has a clear negative tendency, which causes the mean of the total bending angle RIE to be larger than that of the Ne

^{2}terms alone. In both Figure 5 and Figure 6, comparing the profiles and statistics of the inbound Ne

^{2}(Figure 5i,m and Figure 6i,m) and outbound Ne

^{2}(Figure 5k,o and Figure 6k,o) terms, one can see that the inbound Ne

^{2}terms are larger than those of the outbound. Whereas, comparing the profiles and statistics of the inbound BNe (Figure 5j,n and Figure 6j,n) and outbound BNe (Figure 5l,p and Figure 6l,p) terms, the inbound BNe terms are not always larger than those of the outbound. Specifically, the outbound BNe terms of the MetOp-A/-B missions (Figure 5l,p) are smaller than those of the inbound, while the outbound BNe terms of the GRACE (Figure 6l) and CHAMP (Figure 6p) missions are slightly larger than those of the inbound.

^{2}and BNe terms is given in the discussion in Section 4.

^{2}term and the Kappa bending angle RIE, we furthermore used Equation (9) to obtain modeled Kappa bending angle RIE correction profiles, based on the IGS vTEC maps and NeUoG ionospheric model. Results comparing these with the bi-local Ne

^{2}term for different categories of ionospheric asymmetry are shown in Figure 7. One can see that the modeled Kappa bending angle RIEs are smoother than when using observed bending angles, and the total mean is similar to the total mean of the bi-local Ne

^{2}terms, as shown in Figure 7a,b. However, in individual profiles, Kappa bending angle RIEs can be larger in size than those of the bi-local Ne

^{2}terms. The reason for this is discussed in more detail below.

^{2}terms for the symmetric ensembles are larger than those of the asymmetric ensembles, because the nearly symmetrical RO events include those cases that have large vTEC at both inbound and outbound sides. For the nearly symmetric and moderately asymmetric ensembles, the sizes of the mean of the bi-local Ne

^{2}terms are smaller than those of the Kappa bending angle RIEs, but in the strongly asymmetrical cases, it is the other way around.

^{2}term and modeled Kappa bending angle RIEs (i.e., mean of bi-local Ne

^{2}minus mean of modeled Kappa) for the three classified ensembles in Figure 7. Obviously, the difference profiles of the nearly symmetric and moderately asymmetric ensembles are positive, while the difference profile of the strongly asymmetrical ensemble is negative. This implies that the assumptions of the traditional spherical symmetry and the bi-local spherical symmetry, as shown in Figure 1, play key roles in causing the different behaviors.

^{2}term and the modeled Kappa term bending angle RIEs, scatter plots of the Kappa factors and RIE terms are shown in Figure 9. Regarding the modeled and computed Kappa factors, the model-Kappa factor was calculated through Equation (6), using the ${c}_{1}$, ${c}_{2}$, ${c}_{3}$, ${c}_{4}$ coefficients as given in [25], whereas the computed Kappa factor was calculated via $\kappa (a)={\delta \alpha (a)/\left[{\alpha}_{1}(a)-{\alpha}_{2}(a)\right]}^{2}$, using for $\delta \alpha (a)$ the bi-local Ne

^{2}RIE term and computing ${\alpha}_{1}(a)-{\alpha}_{2}(a)$ by Equation (9). Note for the following discussion that, given the ${\left[{\alpha}_{1}(a)-{\alpha}_{2}(a)\right]}^{2}$, the modeled Kappa RIE term was calculated by multiplying with the model-Kappa factor, whereas the bi-local Ne

^{2}RIE term was obtained by instead multiplying the same ${\left[{\alpha}_{1}(a)-{\alpha}_{2}(a)\right]}^{2}$ with the computed Kappa factor.

^{2}terms is smaller than that of the Kappa bending angle RIEs for those ensembles. However, for the strongly asymmetrical ensemble, both individually and on average, the computed Kappa values are systematically larger than the model-Kappa values. This is why the size of the mean of the bi-local Ne

^{2}terms is larger than that of the Kappa bending angle RIEs for the strongly asymmetric ensemble.

^{2}terms are not numerically larger than the most negative Kappa bending angle RIEs in Figure 7. The reasons for this are, as shown in Figure 9b,c, that the largest computed Kappa values roughly correspond to the smallest RIEs, while the smallest computed Kappa values roughly correspond to the largest RIEs.

#### 3.2. Statistical Results Inspected by Histograms

^{2}terms among the MetOp-A/-B, GRACE and CHAMP missions, by way of histograms. To also enable a regional cross-comparison in a challenging region, the data sets of equatorial day time zone (EDT) (latitude 10°S−30°N local time 9:00–21:00 h) are used in addition to the global dataset here. Based on the bi-local RIE data sets illustrated in Section 3.1, the BNe and Ne

^{2}term averages over the 50–60 km impact altitude range were calculated profile by profile for the individual RO events, then the mean and median of these average BNe and Ne

^{2}values were computed as well as the data counted into histogram bins.

^{2}term has a clear negative tendency. Compared with the global dataset, the EDT-zone dataset has somewhat more negative mean and median values. These results are consistent with the Kappa-correction term characteristics, which according to Equation (5) always exhibit negative average values by construction.

## 4. Discussions

## 5. Conclusions

^{2}term has a clear negative tendency, an aspect where bi-local and Kappa approaches are closely consistent; (3) apart from some outliers, the bi-local RIE corrections are somewhat larger than those of the Kappa correction, while the mean profiles are often reasonably close. Thus, the bi-local approach and the smart simplified setup of the Kappa approach are, in this sense, in reasonably good agreement.

^{2}term plays a dominant role, and the geomagnetic term (BNe term) can be important regionally; (5) for all the RO missions, the magnitude of the local-LEO (NeL) term is very small and can be neglected; (6) the bi-local approach has more capacity and clear added value, compared to the Kappa correction, to represent the variability of the ionospheric and geomagnetic conditions that affect RO events. This is particularly important for regional-scale averages, where the geomagnetic term is of key relevance, which is not accounted for by the Kappa approach.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Melbourne, W.G.; Davis, E.S.; Duncan, C.B.; Hajj, G.A.; Hardy, K.R.; Kursinski, E.R.; Meehan, T.K.; Young, L.E.; Yunck, T.P. The Application of Spaceborne GPS to Atmospheric Limb Sounding and Global Change Monitoring; JPL Publication 94-18; Jet Propulsion Laboratory: Pasadena, CA, USA, 1994; p. 147. [Google Scholar]
- Kursinski, E.R.; Hajj, G.A.; Schofield, J.T.; Linfield, R.P.; Hardy, K.R. Observing earth’s atmosphere with radio occultation measurements using the global positioning system. J. Geophys. Res.
**1997**, 102, 23429–23465. [Google Scholar] [CrossRef] - Steiner, A.K.; Lackner, B.C.; Ladstädter, F.; Scherllin-Pirscher, B.; Foelsche, U.; Kirchengast, G. GPS radio occultation for climate monitoring and change detection. Radio Sci.
**2011**, 46, RS0D24. [Google Scholar] [CrossRef] [Green Version] - Lackner, B.C.; Steiner, A.K.; Hegerl, G.C.; Kirchengast, G. Atmospheric climate change detection by radio occultation data using a fingerprinting method. J. Clim.
**2011**, 24, 5275–5291. [Google Scholar] [CrossRef] [Green Version] - Anthes, R.A.; Rocken, C.; Kuo, Y.H. Applications of COSMIC to meteorology and climate. Terr. Atmos. Ocean. Sci.
**2000**, 11, 115–156. [Google Scholar] [CrossRef] [Green Version] - Gobiet, A.; Kirchengast, G. Advancements of global navigation satellite system radio occultation retrieval in the upper stratosphere for optimal climate monitoring utility. J. Geophys. Res.
**2004**, 109, D24110. [Google Scholar] [CrossRef] [Green Version] - Sun, Y.Q.; Bai, W.H.; Liu, C.L.; Liu, Y.; Du, Q.F.; Wang, X.Y.; Yang, G.L.; Liao, M.; Yang, Z.D.; Zhang, X.X.; et al. The Fengyun-3c radio occultation sounder GNOS: A review of the mission and its early results and science applications. Atmos. Meas. Tech.
**2018**, 11, 5797–5811. [Google Scholar] [CrossRef] [Green Version] - Buontempo, C.; Jupp, A.; Rennie, M. Operational NWP assimilation of GPS radio occultation data. Atmos. Sci. Lett.
**2008**, 9, 129–133. [Google Scholar] [CrossRef] - Healy, S.B. Forecast impact experiment with GPS radio occultation measurements. Geophys. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] [Green Version] - Vorobev, V.V.; Krasilnikova, T.G. An estimation of accuracy of the atmospheric refractive-index recovery from measurements of doppler shifts at frequencies used in the navstar system. Izv. Fiz Atmos. Okeana
**1994**, 29, 626–633. [Google Scholar] - Hoque, M.M.; Jakowski, N. Ionospheric bending correction for GNSS radio occultation signals. Radio Sci.
**2011**, 46, RS0D06. [Google Scholar] [CrossRef] - Liu, C.L.; Kirchengast, G.; Sun, Y.Q.; Bai, W.H.; Du, Q.F.; Wang, X.Y.; Meng, X.G.; Wang, D.W.; Cai, Y.R.; Wu, D.; et al. Study of bending angle residual ionosphric error in real RO data. Int. Geosci. Remote Sens.
**2016**, 4171–4174. [Google Scholar] - Mannucci, A.J.; Ao, C.O.; Pi, X.; Iijima, B.A. The impact of large scale ionospheric structure on radio occultation retrievals. Atmos. Meas. Tech.
**2011**, 4, 2837–2850. [Google Scholar] [CrossRef] [Green Version] - Liu, C.L.; Kirchengast, G.; Zhang, K.F.; Norman, R.; Li, Y.; Zhang, S.C.; Carter, B.; Fritzer, J.; Schwaerz, M.; Choy, S.L.; et al. Characterisation of residual ionospheric errors in bending angles using GNSS RO end-to-end simulations. Adv. Space Res.
**2013**, 52, 821–836. [Google Scholar] [CrossRef] - Liu, C.L.; Kirchengast, G.; Zhang, K.; Norman, R.; Li, Y.; Zhang, S.C.; Fritzer, J.; Schwaerz, M.; Wu, S.Q.; Tan, Z.X. Quantifying residual ionospheric errors in GNSS radio occultation bending angles based on ensembles of profiles from end-to-end simulations. Atmos. Meas. Tech.
**2015**, 8, 2999–3019. [Google Scholar] [CrossRef] [Green Version] - Syndergaard, S. On the ionosphere calibration in GPS radio occultation measurements. Radio Sci.
**2000**, 35, 865–883. [Google Scholar] [CrossRef] - Healy, S.B.; Culverwell, I.D. A modification to the standard ionospheric correction method used in GPS radio occultation. Atmos. Meas. Tech.
**2015**, 8, 3385–3393. [Google Scholar] [CrossRef] [Green Version] - Danzer, J.; Healy, S.B.; Culverwell, I.D. A simulation study with a new residual ionospheric error model for GPS radio occultation climatologies. Atmos. Meas. Tech.
**2015**, 8, 3395–3404. [Google Scholar] [CrossRef] [Green Version] - Bassiri, S.; Hajj, G.A. Higher-order ionospheric effects on the GPS observables and means of modeling them. Adv. Astronaut. Sci.
**1993**, 82, 1071–1086. [Google Scholar] - Hoque, M.M.; Jakowski, N. Higher order ionospheric propagation effects on GPS radio occultation signals. Adv. Space. Res.
**2010**, 46, 162–173. [Google Scholar] [CrossRef] - Danzer, J.; Scherllin-Pirscher, B.; Foelsche, U. Systematic residual ionospheric errors in radio occultation data and a potential way to minimize them. Atmos. Meas. Tech.
**2013**, 6, 2169–2179. [Google Scholar] [CrossRef] [Green Version] - Qu, X.C.; Li, Z.H.; An, J.C.; Ding, W.W. Characteristics of second-order residual ionospheric error in GNSS radio occultation and its impact on inversion of neutral atmospheric parameters. J. Atmos. Sol. Terr. Phys.
**2015**, 130, 159–171. [Google Scholar] [CrossRef] - Zeng, Z.; Sokolovskiy, S.; Schreiner, W.; Hunt, D.; Lin, J.; Kuo, Y.H. Ionospheric correction of GPS radio occultation data in the troposphere. Atmos. Meas. Tech.
**2016**, 9, 335–346. [Google Scholar] [CrossRef] [Green Version] - Zeng, Z.; Sokolovskiy, S. Effect of sporadic E clouds on GPS radio occultation signals. Geophys. Res. Lett.
**2010**, 37, L18817(1–5). [Google Scholar] [CrossRef] - Angling, M.J.; Elvidge, S.; Healy, S.B. Improved model for correcting the ionospheric impact on bending angle in radio occultation measurements. Atmos. Meas. Tech.
**2018**, 11, 2213–2224. [Google Scholar] [CrossRef] [Green Version] - Liu, C.L.; Kirchengast, G.; Sun, Y.Q.; Zhang, K.F.; Norman, R.; Schwaerz, M.; Bai, W.H.; Du, Q.F.; Li, Y. Analysis of ionospheric structure influences on residual ionospheric errors in GNSS radio occultation bending angles based on ray tracing simulations. Atmos. Meas. Tech.
**2018**, 11, 2427–2440. [Google Scholar] [CrossRef] [Green Version] - Danzer, J.; Schwaerz, M.; Kirchengast, G.; Healy, S.B. Sensitivity analysis and impact of the Kappa-Correction of residual ionospheric biases on radio occultation climatologies. Earth Space Sci.
**2020**, 7. [Google Scholar] [CrossRef] - Syndergaard, S.; Kirchengast, G. A bi-local estimation approach for residual ionospheric correction of radio occultation bending angles. Presented at EUMETSAT ROM SAF-IROWG International Workshop, Elsinore, Denmark, 19–25 September 2019; Fri, 20 Sep; Poster P23. Available online: http://www.romsaf.org/romsaf-irowg-2019/en/content/21/program-agenda-by-day (accessed on 8 September 2020).
- Leitinger, R.; Kirchengast, G. Easy to use global and regional ionospheric models—A report on approaches used in Graz. Acta Geodaet. Geophys. Hung.
**1997**, 32, 329–342. [Google Scholar] - Thébault, E.; Finlay, C.C.; Beggan, C.D.; Alken, P.; Aubert, J.; Barrois, O.; Bertrand, F.; Bondar, T.; Boness, A.; Brocco, L.; et al. International geomagnetic reference field: The 12th generation. Earth Planets Space
**2015**, 67. [Google Scholar] [CrossRef] - Feltens, J. The international GPS service (IGS) ionosphere working group. Adv. Space Res.
**2003**, 31, 635–644. [Google Scholar] [CrossRef] - Petrie, E.J.; Hernández Pajares, M.; Spalla, P.; Moore, P.; King, M.A. A review of higher order ionospheric refraction effects on dual frequency GPS. Surv. Geophys.
**2011**, 32, 197–263. [Google Scholar] [CrossRef] [Green Version] - Kirchengast, G.; Schwärz, M.; Schwarz, J.; Scherllin-Pirscher, B.; Pock, C.; Innerkofler, J.; Proschek, V.; Steiner, A.K.; Danzer, J.; Ladstädter, F.; et al. The reference occultation processing system approach to interpret GNSS radio occultation as SI-traceable planetary system refractometer. Presented at OPAC-IROWG International Workshop, Leibnitz, Austria, 8–14 September 2016; Available online: http://wegcwww.unigraz.at/opacirowg2016/data/public/files/opacirowg2016_Gottfried_Kirchengast_presentation_261.pdf (accessed on 8 September 2020).
- Kirchengast, G.; Schwärz, M.; Angerer, B.; Schwarz, J.; Innerkofler, J.; Proschek, V.; Ramsauer, J.; Fritzer, J.; Scherllin-Pirscher, B.; Rieckh, T.; et al. Reference OPS DAD—Reference Occultation Processing System (rOPS) Detailed Algorithm Description; Technical Report for ESA and FFG No. 1/2018, Doc-Id: WEGC-rOPS-2018-TR01, Issue 2.0; Wegener Center, University of Graz: Graz, Austria, 2018. [Google Scholar]
- Schwarz, J. Benchmark Quality Processing of Radio Occultation Data with Integrated Uncertainty Propagation; Scientific Report No. 77-2018; Wegener Center Verlag: Graz, Austria, 2018. [Google Scholar]
- Schwarz, J.; Kirchengast, G.; Schwärz, M. Integrating uncertainty propagation in GNSS radio occultation retrieval: From excess phase to atmospheric bending angle profiles. Atmos. Meas. Tech.
**2018**, 11, 2601–2631. [Google Scholar] [CrossRef] [Green Version] - Gorbunov, M.E.; Kirchengast, G. Uncertainty propagation through wave optics retrieval of bending angles from GPS radio occultation: Theory and simulation results. Radio Sci.
**2015**, 50, 1086–1096. [Google Scholar] [CrossRef] [Green Version] - Gorbunov, M.E.; Kirchengast, G. Wave-optics uncertainty propagation and regression-based bias model in GNSS radio occultation bending angle retrievals. Atmos. Meas. Tech.
**2018**, 11, 111–125. [Google Scholar] [CrossRef] [Green Version] - Schwarz, J.; Kirchengast, G.; Schwärz, M. Integrating uncertainty propagation in GNSS radio occultation retrieval: From bending angle to dry-air atmospheric profiles. Earth Space Sci.
**2017**, 4, 200–228. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Kirchengast, G.; Scherllin-Pirscher, B.; Schwärz, M.; Nielsen, J.K.; Ho, S.P.; Yuan, Y.B. A new algorithm for the retrieval of atmospheric profiles from GNSS radio occultation data in moist air and comparison to 1DVar retrievals. Remote Sens.
**2019**, 11, 2729. [Google Scholar] [CrossRef] [Green Version] - Innerkofler, J.; Kirchengast, G.; Schwärz, M.; Pock, C.; Jäggi, A.; Andres, Y.; Marquardt, C. Precise orbit determination for climate applications of GNSS radio occultation including uncertainty estimation. Remote Sens.
**2020**, 12, 1180. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Radio occultation geometry between global navigation satellite system (GNSS) transmitter and low Earth orbit (LEO) receiver satellites, schematically illustrating the signal ray path passing through the ionosphere and the atmosphere, as well as the overall local spherical symmetry assumption (

**a**) in contrast to the bi-local spherical symmetry assumption (

**b**).

**Figure 2.**Overview of the reference occultation processing system (rOPS) in the form of a functional block diagram, illustrating the logic of the system modeling and data analysis approach, indicating its main subsystems and the occultation data processing with the processors, highlighting the level 1b (L1b) data processor where the bi-local bending angle residual ionospheric error (RIE) correction is implemented with the red box.

**Figure 3.**Time series of monthly mean F10.7 index for the years from 2000 to 2018 (

**a**) illustrating the three test days with square marks. The International GNSS Service (IGS) vTEC maps of the 00:00 UT (universal time) and 12:00 UT on 15 July 2013 are shown in panels (

**b**,

**c**), respectively. In panels (

**b**,

**c**), the equatorial day time (EDT) zone is highlighted with a red box. (

**d**) vTEC scatter plot illustrating inbound vTEC (x-axis) vs. outbound vTEC (y-axis), classifying the ionospheric conditions into nearly symmetric (green; inbound-vs-outbound vTEC variation against mean vTEC within +/−10%), moderately asymmetric (blue; vTEC variation in between green and red), and strongly asymmetric (red; inbound-vs-outbound vTEC variation against mean vTEC larger than +/−50%).

**Figure 4.**Panel (

**a**) shows the global distribution of mean tangent point locations of 493 European Meteorological Operational satellites mission (MetOp-B) RO events on 15 July 2013, including 255 RO events with negative geomagnetic (BNe) term (red downward-pointing triangles) and 238 RO events with positive BNe term (green upward-pointing triangles), the region highlighted with a red box includes 35 positive-BNe RO events but only 17 negative-BNe RO events. The background color map illustrates the mean geomagnetic field intensity for 15 July 2013. Panel (

**b**) shows the local time (x-axis) vs. latitude (y-axis) distribution of the mean tangent points (black dots), inbound ion.mean-altitude points (magenta triangles), and outbound ion.mean-altitude points (cyan triangles) of 493 MetOp-B RO events on 15 July 2013. Similar to panel (

**a**), the upward-pointing triangles denote positive BNe while downward-pointing triangles denote negative BNe. The background color map illustrates the IGS vTEC maps at 12:00 UT on 15 July 2013, and the equatorial day time (EDT) zone is highlighted with a red box.

**Figure 5.**Bi-local and Kappa bending angle RIE correction profiles and their statistics for the global ensemble high solar activity data set (left columns, MetOp-A/-B missions on 15 May and 15 July 2013) and low solar activity data set (right columns, MetOp-A mission on 15 July 2008). In the upper composite panels, the total bi-local bending angle correction (Total) and its BNe and electron-density-squared (Ne

^{2}) component terms, and the observed Kappa correction term are shown by subpanels (

**a**–

**d**) and (

**e**–

**h**). The lower composite panels show the inbound (TxTP) and outbound (TPRx) components of the Ne

^{2}and BNe terms, in panels (

**i**–

**l**) and (

**m**–

**p**), respectively. In each subpanel, the individual bending angle RIEs (grey) and their mean (heavy red) and standard deviation (heavy green) are shown.

**Figure 6.**Bi-local bending angle RIE correction profiles and their statistics for the global ensemble high solar activity data set (left columns, the Gravity Recovery And Climate Experiment (GRACE) mission on 15 May and 15 July 2013) and low solar activity data set (right columns, Challenging Mini-satellite Payload (CHAMP) mission on 14–16 July 2008). In the upper composite panels (top two rows), the total bi-local bending angle correction (Total) and its BNe, Ne

^{2}, and NeL component terms are shown by subpanels (

**a**–

**d**) and (

**e**–

**h**). The lower composite panels (third and fourth row) show the inbound (TxTP) and outbound (TPRx) components of the Ne

^{2}and BNe terms, in panels (

**i**–

**l**) and (

**m**–

**p**), respectively. The bottom row panels show the Rx local (Rxloc) component of the Ne

^{2}and BNe terms, in panels (

**q**,

**r**) and (

**s**,

**t**), respectively.

**Figure 7.**Bi-local Ne

^{2}term (top) and modeled Kappa (bottom) bending angle RIE correction profiles and their statistics, shown for the global ensemble total data set in (

**a**,

**b**) from the MetOp-A/-B RO events on all three test days as well as for the classified nearly symmetric, moderately asymmetric, and strongly asymmetric partial data sets in (

**c**,

**d**), (

**e**,

**f**), and (

**g**,

**h**), respectively.

**Figure 8.**Difference profiles of the mean bi-local Ne

^{2}term and modeled Kappa bending angle RIEs (i.e., mean bi-local Ne

^{2}minus mean modeled Kappa) for the classified nearly symmetric (

**green line**), moderately asymmetric (

**blue line**), and strongly asymmetric (

**red line**) data sets, as illustrated in Figure 7.

**Figure 9.**Scatter plots of Kappa factor and bending angle RIE term values at 60 km impact altitude level illustrating the computed Kappa factor vs. the model-Kappa factor (according to [25]) in panel (

**a**) as well as vs. the bi-local Ne

^{2}term in panel (

**b**) and vs. the modeled Kappa term in panel (

**c**), respectively. Colors of the circles indicate the three classes of nearly symmetric (green), moderately asymmetric (blue), and strongly asymmetric (red) data sets (as illustrated in Figure 3d).

**Figure 10.**Relationship between the histogram distribution of the number of RO events and the average value of the bi-local BNe term over 50–60 km impact altitude range for high solar activity data (left column, MetOp-A/-B (

**a**) and GRACE (

**c**) on 15 May and 15 July 2013) and low solar activity data (right column, CHAMP (

**b**) and GRACE (

**d**) on 14–16 July 2008). In each panel, the global (blue) and EDT (red) ensembles are presented as well as the global mean, global median, EDT mean, and EDT median as vertical lines in the corresponding colors (see legend in panel (

**a**)).

**Figure 11.**Relationship between the histogram distribution of the number of RO events and the average value of the bi-local Ne

^{2}term over 50–60 km impact altitude range for high solar activity data (left column, MetOp-A/-B (

**a**) and GRACE (

**c**) on 15 May and 15 July 2013) and low solar activity data (right column, CHAMP (

**b**) and GRACE (

**d**) on 14–16 July 2008). In each panel, the global (blue) and EDT (red) ensembles are presented as well as the global mean, global median, EDT mean, and EDT median as vertical lines in the corresponding colors (see legend in panel (

**a**)).

**Figure 12.**Profiles and their statistics of the BNe remainder term (rightmost term of Equation (11)) for the global ensemble data sets of MetOp-A/-B missions on 15 May and 15 July 2013 (

**left panel**), GRACE mission on 15 May and 15 July 2013 (

**middle panel**), and CHAMP mission on 14–16 July 2008 (

**right panel**), respectively. In each subpanel, the individual bending angle RIEs (gray), their mean (heavy red), and their standard deviation (heavy green) are shown.

**Figure 13.**Vertical profiles of geomagnetic field intensity parallel to the RO ray path for the 2013 GRACE (15 May and 15 July 2013; (

**a**,

**c**)) and 2008 GRACE (14–16 July 2008; (

**b**,

**d**)) ensemble data sets. RO events are shown as “bundle plot”, with grey profiles and blue profiles for the global and EDT regions, respectively.

Variable | Unit | Description |
---|---|---|

φ, λ | degree | Location (i.e., latitude, longitude) of the RO event |

UT | s | Reference time of the RO event |

R_{c}, h_{G} | m | Radius of curvature (WGS84 ^{1}) and geoid undulation (EGM2008 ^{2}) |

f_{k} | Hz | (GPS) Transmitter signal carrier frequency, elements f_{k} (L1 and L2 signals) |

z_{a} | m | Impact altitude (profile, joint one for L1 and L2) |

α_{1} | rad | Bending angle of L1 signal (profile) |

α_{2} | rad | Bending angle of L2 signal (profile) |

F10.7 | s.f.u. | Solar flux index |

r_{T} | m | Position vector of Tx satellite for ref.altitude ^{3} (ECEF ^{4}: x y z) |

v_{T} | m/s | Velocity vector of Tx satellite for ref.altitude (ECEF: v_{x} v_{y} v_{z}) |

r_{R} | m | Position vector of Rx satellite for ref.altitude (ECEF: x y z) |

v_{R} | m/s | Velocity vector of Rx satellite for ref.altitude (ECEF: v_{x} v_{y} v_{z}) |

s_{TR} | m | Straight line unit vector from Tx (r_{T}) to Rx (r_{R}) (ECEF: x y z) |

x_{I350T} | m | Position vector of ion.mean-altitude point ^{5} inbound (ECEF and Geodetic) |

x_{I350R} | m | Position vector of ion.mean-altitude point outbound (ECEF and Geodetic) |

${\mathrm{vTEC}\text{}}_{\mathrm{I}350\mathrm{T}}^{\mathrm{IGS}}$ | TECU | vTEC of ion.mean-altitude point inbound (from IGS) |

${\mathrm{vTEC}\text{}}_{\mathrm{I}350\mathrm{R}}^{\mathrm{IGS}}$ | TECU | vTEC of ion.mean-altitude point outbound (from IGS) |

^{1}WGS84: World Geodetic System 1984

^{2}EGM2008: The Earth Gravitational Model 2008

^{3}ref.altitude: mean tangent-point reference altitude (z

_{a}= 60 km) for the radio occultation (RO) event for computing all geometrical transmitter (Tx) and receiver (Rx) variables and the corresponding ray path-related variables

^{4}ECEF: Earth-centered Earth-fixed coordinate system

^{5}ion.mean-altitude point: the point of intersection of the GNSS RO ray path (for the reference altitude) and the 350 km height spherical surface, where the ionospheric mean vertical total electron content (vTEC) values are located.

**Table 2.**Derived inputs for the bi-local correction, from the Electron density (Ne) University of Graz model (NeUoG) and the International Geomagnetic Reference Field (IGRF)-12 model.

Variable | Unit | Description |
---|---|---|

${\mathrm{Ne}}_{\mathrm{I}350\mathrm{T}}^{\mathrm{NeUoG}}(z)$ ${\mathrm{Ne}}_{\mathrm{I}350\mathrm{R}}^{\mathrm{NeUoG}}(z)$ | m^{−3} | Electron density inbound/outbound, calculated using NeUoG model. NeUoG inputs: height (km), latitude (deg), longitude (deg), time (UT) and F10.7 (s.f.u.); we use impact altitude for z. |

${\mathrm{B}}_{//,\text{}\mathrm{I}350\mathrm{T}}^{\mathrm{IGRF}}(z)$ ${\mathrm{B}}_{//,\text{}\mathrm{I}350\mathrm{R}}^{\mathrm{IGRF}}(z)$ | nT | Geomagnetic field strength along the ray path inbound/outbound derived from the total geomagnetic field vector B (X, Y, Z), which is obtained using the IGRF-12 model; then computed by B_{//} = B⋅s_{TR}. |

${\mathrm{vTEC}}_{\mathrm{I}350\mathrm{T}}^{\mathrm{NeUoG}}$ ${\mathrm{vTEC}}_{\mathrm{I}350\mathrm{R}}^{\mathrm{NeUoG}}$ | TECU (10 ^{16}m^{−2}) | Integration of N_{e} from 80 km to 2000 km inbound/outbound |

${\mathrm{Ne}}_{\mathrm{I}350\mathrm{T}}^{}(z)$ ${\mathrm{Ne}}_{\mathrm{I}350\mathrm{R}}^{}(z)$ | m^{−3} | N_{e}(z) = (vTEC^{IGS}/vTEC^{NeUoG})⋅N_{e}^{NeUoG}(z) inbound/outbound, which is the observational TEC-adjusted N_{e}(z) |

Constant | Unit | Description |
---|---|---|

C | m^{3}s^{−1} | 40.308, basic ionization constant in the ion.refractive index equation |

K | m^{3}T^{−1}s^{−3} | 1.1283 × 10^{12}, magnetic term constant in the ion.refractive index equation |

Variable | Unit | Description |
---|---|---|

$\delta {\alpha}^{\mathrm{RIE}}$ | rad | Total residual ionospheric error (RIE) from Equation (7), i.e., ${\alpha}_{c}(a)-\alpha (a)=\delta {\alpha}^{\mathrm{BNe}}(a)+\delta {\alpha}^{{\mathrm{Ne}}^{2}}(a)+\delta {\alpha}^{\mathrm{NeL}}(a)$ |

$\delta {\alpha}^{\mathrm{BNe}}$ $\delta {\alpha}_{\mathrm{TxTP}}^{\mathrm{BNe}}$ $\delta {\alpha}_{\mathrm{TPRx}}^{\mathrm{BNe}}$ $\delta {\alpha}_{\mathrm{Rxloc}}^{\mathrm{BNe}}$ | rad | Total geomagnetic residual term–based on Equation (8) with $X={B}_{//}{N}_{e}$ Tx-to-TP geomagnetic residual term–from 1st integral in Equation (8) TP-to-Rx geomagnetic residual term–from 2nd integral in Equation (8) Rx-local geomagnetic residual term–from last term in Equation (8) |

$\delta {\alpha}^{{\mathrm{Ne}}^{2}}$ $\delta {\alpha}_{\mathrm{TxTP}}^{{\mathrm{Ne}}^{2}}$ $\delta {\alpha}_{\mathrm{TPRx}}^{{\mathrm{Ne}}^{2}}$ $\delta {\alpha}_{\mathrm{Rxloc}}^{{\mathrm{Ne}}^{2}}$ | rad | Total squared electron density residual term–based on Equation (8) with $X={N}_{e}{}^{2}$ Tx-to-TP squared electron density residual term–from 1st integral in Equation (8) TP-to-Rx squared electron density residual term–from 2nd integral in Equation (8) Rx-local squared electron density residual term–from last term in Equation (8) |

$\delta {\alpha}^{\mathrm{NeL}}$ | rad | LEO electron density residual term–last term in Equation (7) |

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## Share and Cite

**MDPI and ACS Style**

Liu, C.; Kirchengast, G.; Syndergaard, S.; Schwaerz, M.; Danzer, J.; Sun, Y.
New Higher-Order Correction of GNSS RO Bending Angles Accounting for Ionospheric Asymmetry: Evaluation of Performance and Added Value. *Remote Sens.* **2020**, *12*, 3637.
https://doi.org/10.3390/rs12213637

**AMA Style**

Liu C, Kirchengast G, Syndergaard S, Schwaerz M, Danzer J, Sun Y.
New Higher-Order Correction of GNSS RO Bending Angles Accounting for Ionospheric Asymmetry: Evaluation of Performance and Added Value. *Remote Sensing*. 2020; 12(21):3637.
https://doi.org/10.3390/rs12213637

**Chicago/Turabian Style**

Liu, Congliang, Gottfried Kirchengast, Stig Syndergaard, Marc Schwaerz, Julia Danzer, and Yueqiang Sun.
2020. "New Higher-Order Correction of GNSS RO Bending Angles Accounting for Ionospheric Asymmetry: Evaluation of Performance and Added Value" *Remote Sensing* 12, no. 21: 3637.
https://doi.org/10.3390/rs12213637