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Journal topic
**Atmospheric Measurement Techniques**
An interactive open-access journal of the European Geosciences Union

Journal topic

- About
- Editorial board
- Articles
- Special issues
- Highlight articles
- Manuscript tracking
- Subscribe to alerts
- Peer review
- For authors
- For reviewers
- EGU publications

- Abstract
- Introduction
- Methods and data
- Algorithm sequence and example results
- Algorithm validation
- Performance demonstration
- Conclusions
- Data availability
- Appendix A: Algorithm description
- Appendix B: Variance propagation for comparison
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- References

AMT | Articles | Volume 11, issue 5

Atmos. Meas. Tech., 11, 2601–2631, 2018

https://doi.org/10.5194/amt-11-2601-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/amt-11-2601-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 3.0 License.

Special issue: Observing Atmosphere and Climate with Occultation Techniques...

**Research article**
04 May 2018

**Research article** | 04 May 2018

Integrating uncertainty propagation in GNSS radio occultation retrieval: from excess phase to atmospheric bending angle profiles

^{1}Wegener Center for Climate and Global Change (WEGC), University of Graz, Graz, Austria^{2}Institute for Geophysics, Astrophysics and Meteorology/Institute of Physics, University of Graz, Graz, Austria

^{1}Wegener Center for Climate and Global Change (WEGC), University of Graz, Graz, Austria^{2}Institute for Geophysics, Astrophysics and Meteorology/Institute of Physics, University of Graz, Graz, Austria

**Correspondence**: Jakob Schwarz (jakob.schwarz@uni-graz.at)

**Correspondence**: Jakob Schwarz (jakob.schwarz@uni-graz.at)

Abstract

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Global Navigation Satellite System (GNSS) radio occultation (RO) observations are highly accurate, long-term stable data sets and are globally available as a continuous record from 2001. Essential climate variables for the thermodynamic state of the free atmosphere – such as pressure, temperature, and tropospheric water vapor profiles (involving background information) – can be derived from these records, which therefore have the potential to serve as climate benchmark data. However, to exploit this potential, atmospheric profile retrievals need to be very accurate and the remaining uncertainties quantified and traced throughout the retrieval chain from raw observations to essential climate variables. The new Reference Occultation Processing System (rOPS) at the Wegener Center aims to deliver such an accurate RO retrieval chain with integrated uncertainty propagation. Here we introduce and demonstrate the algorithms implemented in the rOPS for uncertainty propagation from excess phase to atmospheric bending angle profiles, for estimated systematic and random uncertainties, including vertical error correlations and resolution estimates. We estimated systematic uncertainty profiles with the same operators as used for the basic state profiles retrieval. The random uncertainty is traced through covariance propagation and validated using Monte Carlo ensemble methods. The algorithm performance is demonstrated using test day ensembles of simulated data as well as real RO event data from the satellite missions CHAllenging Minisatellite Payload (CHAMP); Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC); and Meteorological Operational Satellite A (MetOp). The results of the Monte Carlo validation show that our covariance propagation delivers correct uncertainty quantification from excess phase to bending angle profiles. The results from the real RO event ensembles demonstrate that the new uncertainty estimation chain performs robustly. Together with the other parts of the rOPS processing chain this part is thus ready to provide integrated uncertainty propagation through the whole RO retrieval chain for the benefit of climate monitoring and other applications.

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Schwarz, J., Kirchengast, G., and Schwaerz, M.: Integrating uncertainty propagation in GNSS radio occultation retrieval: from excess phase to atmospheric bending angle profiles, Atmos. Meas. Tech., 11, 2601–2631, https://doi.org/10.5194/amt-11-2601-2018, 2018.

1 Introduction

Back to toptop
Observation systems of the free atmosphere, focusing on the range from the
top of the atmospheric boundary layer upwards, were historically designed for
weather research and forecasting purposes. They have considerable
shortcomings from a climate monitoring perspective
(Karl et al., 1995), and so the related global climate monitoring infrastructure
remains fragile and incomplete even today (Bojinski et al., 2014).
The Global Climate Observing System (GCOS) aims to
improve the observational foundation for the climate sciences
(GCOS, 2015). For this purpose the establishment of *climate benchmark data records* is essential. To qualify as climate benchmark,
records need to be (1) of global coverage, (2) of
high accuracy, (3) long-term stable, (4) tested for systematic errors
on orbit, (5) and tied to irrefutable standards, and they need to (6) measure
Essential Climate Variables (ECVs) (NRC, 2007; GCOS, 2015).

Based on the quality and abundance of Global Navigation Satellite System (GNSS) signal sources, in particular from the Global
Positioning System (GPS) so far, the GNSS radio occultation (RO)
observation record is globally available (continuously since 2001),
long-term stable (due to the so-called self-calibration and high
signal stability during the event), and highly accurate (accuracy
traceable to the SI second). Due to the self-calibrating property, the
accuracy is also ensured on orbit; i.e., there is no need for
calibration or bias correction in post-processing on ground
(Leroy et al., 2006). The basic RO excess phase data can therefore serve
as a fundamental climate data record (FCDR) as defined by
GCOS (2010). From this FCDR with its unique properties,
ECVs – in particular the thermodynamic ECVs pressure, temperature, and
humidity in the free atmosphere – can be derived using an RO
*retrieval* chain.

In order to reliably serve as climate benchmark data record, however, the retrieved ECV profiles and their claimed accuracy – expressed by the uncertainties provided – need to be traceable back to the (small) uncertainties of the FCDR and in turn to the raw data. This requires that (1) the RO retrieval is highly accurate and avoids any undue amplification of uncertainties associated with the quantities in the FCDR and that (2) the uncertainties are propagated through the entire retrieval chain, from the raw data to the ECV profiles, duly accounting for relevant side influences such as from background information. Developed at the Wegener Center of the University of Graz (WEGC), together with international partners, the Reference Occultation Processing System (rOPS) (Kirchengast et al., 2015) aims to establish such a fully traceable RO processing for the first time (Kirchengast et al., 2016a, b).

In Fig. 1 the basic steps of the RO retrieval chain in the rOPS, i.e., the precise orbit determination (POD) and excess phase processing (labeled “L1a” in Fig. 1), the subsequent atmospheric bending angle retrieval (“L1b”), the refractivity and dry-air retrieval (“L2a”), and the moist-air retrieval (“L2b”) are sketched.

Kursinski et al. (1997) – and more recently Hajj et al. (2002), Anthes (2011), and Steiner et al. (2011) – provided detailed introductions and reviews of the RO technique and its applications in meteorology and climate. Ho et al. (2012) and Steiner et al. (2013) included comparative current RO retrieval chain descriptions of leading international RO processing centers, none of which yet include uncertainty propagation. Empirical error (uncertainty) estimates computed statistically from retrieved RO atmospheric profiles and climatologies have been derived by Kuo et al. (2004), Steiner and Kirchengast (2005), and Scherllin-Pirscher et al. (2011a, b, 2017), the last of which with a focus on climate uses also providing simple analytical error models. These studies and many others have described the RO retrieval chain in detail and have shown the high accuracy and quality of RO data, particularly in the upper-troposphere and lower-stratosphere region.

The aim of the integrated uncertainty propagation in the rOPS is to
eventually propagate uncertainties along this *entire* retrieval chain from
the raw measurement data to the ECVs (Kirchengast et al., 2016a, b), whereby the implementation of the rOPS uncertainty
propagation occurs in the sequential blocks illustrated in
Fig. 1. The L2a processing and uncertainty propagation
from atmospheric bending angle to dry-air profiles has already been
introduced by Schwarz et al. (2017, SKS2017 hereafter).

This study is a direct complement to the work in SKS2017. Using the same
propagation and validation methods as applied in SKS2017,
it focuses on the uncertainty propagation from excess phase to
atmospheric bending angle profiles, i.e., the L1b processing. As in
SKS2017, *random uncertainties* are propagated using covariance
propagation (CP) and validated using Monte Carlo (MC) ensemble methods.
As in the L2a processor, we also propagate (conservative bound) estimates for
*systematic uncertainties* along the retrieval chain of the L1b processor.
Additionally, *correlation length* profiles and *resolution* profiles are
provided.

Uncertainty propagation as covariance propagation from excess phase to bending angle profiles has been outlined and demonstrated in a basic form, by Syndergaard (1999) and Rieder and Kirchengast (2001), but not been implemented yet in processing center retrieval chains or applied to real RO data. As visible in Fig. 1, the L1b processor consists of three major retrieval parts, which are expanded into detailed substructure in Fig. 2. We propagate estimated random uncertainties from excess phase profiles to Doppler shift profiles (Sect. (1) in Fig. 2); further to geometric-optics (GO) bending angle profiles, merged with wave-optics (WO) bending angle profiles (2); and finally to atmospheric bending angle profiles (3), using a full-CP approach. In combination with the definitions of the main operators and variables in Table 1, and of the vertical grid and coordinate variables in Table 2, Fig. 2 provides a concise overview on the detailed workflow of the L1b processor.

Uncertainty propagation for the WO bending angle retrieval has been implemented and demonstrated for simulated events by Gorbunov and Kirchengast (2015); estimation of random and systematic uncertainties for real events including boundary layer bias correction is introduced by Gorbunov and Kirchengast (2018).

Other ongoing rOPS retrieval advancements relevant to this study are the inclusion of the high-altitude initialization algorithm, introduced by Li et al. (2013, 2015), in the L2a processor and the reduction of remaining higher-order ionospheric effects in the retrieved bending angle profiles of the L1b processor (based on work by Syndergaard, 2000; Liu et al., 2015; Healy and Culverwell, 2015; and Danzer et al., 2013, 2015). Furthermore, the precise orbit determination (POD) of the RO receiver satellite and the excess phase processing, also including the associated uncertainty propagation, are part of ongoing work (Innerkofler et al., 2017).

Finally, related work and manuscript preparation on a new moist-air retrieval algorithm (L2b) and corresponding L2b uncertainty propagation are ongoing (Li et al., 2018; Kirchengast et al., 2017a).

The paper is structured as follows. In Sect. 2 we introduce the uncertainty estimation, propagation, and validation methods and the data sources and preparation. In Sect. 3, with the help of an example RO event, the uncertainty propagation sequence is introduced. In Sect. 4 we present the results from the MC validation of the CP uncertainty estimates. In Sect. 5 the performance of the algorithm is then evaluated using test day ensembles with real data from the RO missions CHAllenging Minisatellite Payload (CHAMP) (Wickert et al., 2001); FORMOSAT-3 Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) (Anthes et al., 2008); and Meteorological Operational Satellite A (MetOp) (Luntama et al., 2008); as well as with simulated data approximating characteristics of the Meteorological Operational Satellite A (Luntama et al., 2008; simMetOp data hereafter). We close with conclusions and outlook in Sect. 6. A detailed description of the implemented uncertainty propagation algorithms can be found in Appendix A.

2 Methods and data

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We follow the Guide to the Expression of Uncertainty in Measurement (JCGM, 2008a, b, 2011; GUM hereafter) and aim to follow terminology as provided by the International Vocabulary of Metrology (JCGM, 2012), a terminology also adopted by the GUM. SKS2017 provides a more thorough introduction, including the motivation for using the respective uncertainty estimation, propagation, and validation methods; we refer the particularly interested reader to this companion (open-access) work and provide the essential methods needed more in a summarized form below.

We categorize uncertainties into *estimated random uncertainties*
and *estimated systematic uncertainties*. Effects of
unpredictable or stochastic temporal and spatial variations in
repeated observations, like effects from fluctuations in the atmosphere
or the thermal noise of the receiver system, could in principle be
estimated by ensemble statistics from multiple RO events. However,
since such effects are essentially stationary in a statistical sense,
we can estimate their statistics also from individual RO event data,
given their high noise-resolving sampling rate. These effects are
included in the estimated random uncertainties.

Systematic effects (biases), which can not be quantified using statistical data
analysis based on just one individual RO profile, are estimated and corrected for
when known, as recommended by the GUM. The remaining residual
biases are assumed to stay within a (conservative) bound estimate, which we
refer to as *estimated systematic uncertainty* and by which we aim to provide at
least 90 % likelihood coverage (confidence) that residual biases stay within
the plus/minus envelope range of this uncertainty.

Depending on their nature, components of the systematic uncertainty that we
need to estimate can be fundamentally systematic across different RO events,
a subtype we term *estimated basic systematic uncertainties*, or
appear systematic just for individual RO events, a second subtype that
we term *estimated apparent systematic uncertainties*. It is important
to distinguish these two subtypes, since the apparent systematic uncertainties
will essentially behave as random uncertainties in ensemble averaging over
many RO events, such as when generating climatologies, while the basic
systematic ones will not average out and therefore fundamentally limit
the (absolute) accuracy of ensemble averages such as climatologies.

Since the noise-type effects giving rise to short-range-correlated random uncertainties can be considered uncorrelated to the bias-type effects inducing long-range-correlated apparent systematic uncertainties, and since both are uncorrelated to basic systematic uncertainties, it is insightful and possible with due care to estimate and propagate each of these uncertainties independently.

As for the L2a processor (SKS2017), the operators of
the L1b processor (i.e., the boldfaced items 1.2, 1.4, 2.1, 2.7, 2.9, 3.1,
and 3.5 in Fig. 2) qualify as *explicit*, *multivariate*,
and *linear* measurement models, as defined in the GUM, with
*correlated input* quantities. They can therefore be formulated as

$$\begin{array}{}\text{(1)}& Y={\mathbf{A}}^{XY}\cdot X,\end{array}$$

where the input quantity *X* and output quantity *Y* are rank 1 vectors
(profiles) of random variables, which we call *state* profiles.
According to the GUM, their *random uncertainties* can be propagated
using

$$\begin{array}{ll}{\displaystyle}{\mathbf{C}}_{Y}& {\displaystyle}=E\left[Y{Y}^{T}\right]=E\left[\left({\mathbf{A}}^{XY}X\right){\left({\mathbf{A}}^{XY}X\right)}^{T}\right]\\ \text{(2)}& {\displaystyle}& {\displaystyle}={\mathbf{A}}^{XY}E\left[X{X}^{T}\right]{\left({\mathbf{A}}^{XY}\right)}^{T}={\mathbf{A}}^{XY}{\mathbf{C}}_{X}{\left({\mathbf{A}}^{XY}\right)}^{T}\end{array}$$

when the uncertainties are normally distributed. This assumption is reasonably justified, since the receiving system noise (i.e., thermal noise and residual clock estimation noise) and the ionospheric noise (from scintillations induced by ionospheric irregularities) are essentially normally distributed overall (Kursinski et al., 1997; Syndergaard, 1999; Gorbunov, 2002b; Sokolovskiy et al., 2009). These noise sources are the main contribution to the random uncertainties in the excess phase profiles feeding into the L1b processor.

**C**_{X} and **C**_{Y} are the covariance
matrices of the input and output variables, respectively, and
**A**^{XY} is the linear (or linearized) operator connecting *X* and
*Y*. Equation (3) formulates how the
covariance matrix **C**_{X} is calculated from random uncertainty
estimates ${u}_{X}^{r}$ and the correlation matrix
**R**_{X}:

$$\begin{array}{}\text{(3)}& {C}_{X,ij}={u}_{X,i}^{r}\cdot {u}_{X,j}^{r}\cdot {R}_{X,ij}.\end{array}$$

As a key variable characterizing **R**_{X}, *correlation length*
profiles *l*_{X} are estimated from the correlation functions assembled in
**R**_{X}. The algorithm used estimates *l*_{X} by searching for the
distances downward and upward of the correlation functions' main peak
at which the correlation function has dropped to a value of 1∕*e*
(≈0.378). The adopted correlation length estimate is the
arithmetic mean of these two upward and downward estimates (as the peak may
be somewhat asymmetric). Additionally
the correlation length is constrained by the data domain; i.e., the
correlation length can never be larger than the profiles' vertical range.

Since the covariance propagation of random uncertainties requires extensive matrix multiplications for each measurement model along the entire retrieval chain, we also tested simpler variance propagation (VP), for which correlations are ignored; Appendix B summarizes the relevant algorithms. However, as shown in Sect. 4, variance propagation unduly overestimates random uncertainties, so that covariance propagation is required.

When the operator is linear, as is the case for the applicable L1b
operators, estimated *systematic uncertainties*
can be propagated by application of the
state retrieval operator on the estimated systematic input uncertainty:

$$\begin{array}{ll}{\displaystyle}{u}_{Y}^{s}& {\displaystyle}={\mathbf{A}}^{XY}\cdot \left(X+{u}_{X}^{s}\right)-Y={\mathbf{A}}^{XY}\cdot X+{\mathbf{A}}^{XY}\cdot {u}_{X}^{s}-Y\\ \text{(4)}& {\displaystyle}& {\displaystyle}=Y+{\mathbf{A}}^{XY}\cdot {u}_{X}^{s}-Y={\mathbf{A}}^{XY}\cdot {u}_{X}^{s},\end{array}$$

where ${u}_{X}^{s}$ and ${u}_{Y}^{s}$ are the rank 1 systematic uncertainty profiles of the input and output variables.

In addition to random uncertainties, systematic uncertainties, and the correlation
length, we also estimate *resolution* profiles *w*_{X} as
context information along with the provided random uncertainty profiles
(necessary, e.g., because smoothing can decrease random uncertainties,
while making resolution coarser). This is enabled by careful selection and
formulation of low-pass filter operations, in particular explicit filter
cutoff frequency specification as the main driver of the resolution
remaining after low-pass filtering.

We note that the (half-)Fresnel scale *physical* resolution often
ascribed to RO bending angle profiles retrieved by geometric-optics
methods (e.g., Kursinski et al., 1997; Gorbunov et al., 2004) will generally be
somewhat coarser than the *filter-limited* resolution estimated
here. This is intentional to maximize available information in the
bending angle profiles provided by the L1b processor. In the rOPS, on
input to the L2a processor and before high-altitude initialization by
statistical optimization, the resolution of all profiles is brought to
a common altitude-dependent resolution, which reflects the
half-Fresnel scale (SKS2017).

The input variables needed for the L1b uncertainty propagation, visible in
Fig. 2 and defined in Table 1, are
the retrieved excess phase profiles *L*_{r,k}(*t*) and the associated
systematic uncertainty profiles ${u}_{{L}_{\mathrm{r},k}}^{s}\left(t\right)$,
random uncertainty profiles ${u}_{{L}_{\mathrm{r},k}}^{r}\left(t\right)$, and correlation matrices
${\mathbf{R}}_{{L}_{\mathrm{r},k}}$, as well as the orbit positions and velocities of
receiver and transmitter satellite – *r*_{R}(*t*),
*v*_{R}(*t*), *r*_{T}(*t*), and
*v*_{T}(*t*) – and their (systematic) uncertainties,
${u}_{\mathit{r}\mathrm{R}}^{s}\left(t\right)$, ${u}_{\mathit{v}\mathrm{R}}^{s}\left(t\right)$,
${u}_{\mathit{r}\mathrm{T}}^{s}\left(t\right)$, and ${u}_{\mathit{v}\mathrm{T}}^{s}\left(t\right)$. For due limitation
of depth of workflow detail in Fig. 2 we do not separately show
the propagation of the basic and apparent systematic uncertainties as they are both
identically propagated through the operator chain shown for ${u}_{{L}_{\mathrm{r},k}}^{s}\left(t\right)$.
All variables are provided on the time grid *t* with elements *t*_{i}, at *f*_{s}=50 Hz
sampling rate, and for the two GPS carrier frequencies: *f*_{Tk},
with $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}$, *f*_{T1}=1.57542 GHz, and
*f*_{T2}=1.22760 GHz.

We used excess phase state profiles *L*_{r,k}(*t*) and the orbit state
profiles *r*_{R}(*t*), *v*_{R}(*t*),
*r*_{T}(*t*), and *v*_{T}(*t*) from 15 July
2008 as a test day ensemble. For CHAMP, COSMIC, and MetOp, orbit state and excess phase profiles
were provided by the COSMIC Data Analysis and Archiving Center (CDAAC) of the University
Corporation for Atmospheric Research (UCAR), Boulder, Colorado.
The End-to-End GNSS Occultation Performance Simulation and Processing
System (EGOPS) (Fritzer et al., 2009) was used for generating the simulated
MetOp orbit state and excess phase profiles with realistic receiver noise (simMetOp).
Figure 3 shows *L*_{r,k}(*t*) in panel (a),
${u}_{{L}_{\mathrm{r},k}}^{s}\left(t\right)$ in panel (b), ${u}_{{L}_{\mathrm{r},k}}^{r}\left(t\right)$ in panel (c), and
${\mathbf{R}}_{{L}_{\mathrm{r},k}}$ in terms of representative correlation functions in panels (d) and
(e), for a typical COSMIC RO event of the test day ensemble from
15 July 2008 (example case).

Exploiting the linearity of the (linearized) retrieval operators, the
so-called *baseband approach* (Kirchengast et al., 2016a) is applied
throughout the rOPS. Hereby a zero-order model profile is subtracted
from the input state profile, and only the remaining delta profile is
processed through the operator. After application of the operator, the
zero-order model profile of the output state profile is added back to
the resulting delta profile. This approach effectively avoids biases
from numerical operations on (near-)exponentially varying RO profiles,
since the model profiles that we derive from short-range (24 h)
forecasts of the European Centre for Medium-Range Weather Forecasts
(ECMWF) skillfully subtract the (near-)exponential variation. The
remaining increment profiles that we need to treat numerically then
appear to be very linear and with low dynamical range, which leads to
very low residual numerical errors of operators such as filters and
derivatives.

The model profiles used as zero-order states in the retrieval – i.e.,
*L*_{m}, *D*_{m}, and *α*_{m} (cf. Table 1) – were created
from ECMWF short-range (24 h)
forecast refractivity fields, accurately forward-modeled to bending angle (*α*_{m}),
Doppler shift (*D*_{m}), and excess phase (*L*_{m}) profiles,
collocated to the latitude, longitude, and
time of the respective RO event processed in the rOPS. The ECMWF fields used
have a horizontal resolution of about 300 km (triangular truncation
T42) – which corresponds to the approximate horizontal resolution of RO
profiles (e.g., Kursinski et al., 1997) – and are available at 91 vertical levels (L91).

ECMWF fields were chosen for their proven leading quality (Untch et al., 2006; Bauer et al., 2015)
and thus high suitability for serving as zero-order state profiles; any other
reasonable model profiles could be used as well since the retrieval results
negligibly depend on the exactly chosen zero-order model profiles.
For comparison we plotted *L*_{m}(*t*) for the COSMIC
example case into Fig. 3a, which demonstrates that the ECMWF
short-range forecast lies very close to *L*_{r1}(*t*) and
*L*_{r2}(*t*) and thus is well suited as the model profile.

While in the future the excess phase random and systematic uncertainty
profiles will be more rigorously estimated by the rOPS L1a processor
(Innerkofler et al., 2017) and provided as input to the L1b processor,
they had to be estimated for this study from existing excess phase
profiles with realistic noise and
simplified modeling. To this end, each *estimated random uncertainty* profile ${u}_{{L}_{\mathrm{r},k}}^{r}\left(t\right)$ was estimated based on
the noise of the respective retrieved excess phase profile
*L*_{r,k}(*t*). The noise was determined following the
estimation scheme for bending angle observation errors described by
Li et al. (2015, Sect. 2.2 therein), so we just briefly summarize how
we used it here.

First, for both, the retrieved profile *L*_{r,k} and for the model profile
*L*_{m}, the mean over all grid points between 60 and 70 km was
determined. Then *L*_{m} was offset-corrected towards *L*_{r,k} by subtracting the
difference of these two means from *L*_{m}, giving the offset-corrected
model profile ${L}_{\stackrel{\mathrm{\u0303}}{\mathrm{m}}}$. Next, the delta profile
$\mathit{\delta}{L}_{\mathrm{r}\stackrel{\mathrm{\u0303}}{\mathrm{m}},k}={L}_{\mathrm{r},k}-{L}_{\stackrel{\mathrm{\u0303}}{\mathrm{m}}}$ was
calculated. After smoothing $\mathit{\delta}{L}_{\mathrm{r}\stackrel{\mathrm{\u0303}}{\mathrm{m}},k}$ with a 10 km
moving-average boxcar (BC) filter, the smoothed profile was subtracted from $\mathit{\delta}{L}_{\mathrm{r}\stackrel{\mathrm{\u0303}}{\mathrm{m}},k}$ again, to get $\mathit{\delta}\mathit{\delta}{L}_{\mathrm{r}\stackrel{\mathrm{\u0303}}{\mathrm{m}},k}$, the
random noise profile component of *L*_{r,k} isolated in this way.
Finally, the estimated random uncertainty was determined as

$$\begin{array}{}\text{(5)}& {u}_{L\mathrm{r},ik}^{r}=\sqrt{\sum _{j=i-M/\mathrm{2}}^{i+M/\mathrm{2}}\mathit{\delta}\mathit{\delta}{L}_{\mathrm{r}\stackrel{\mathrm{\u0303}}{\mathrm{m}},jk}^{\mathrm{2}}},\end{array}$$

where *M* is the number of grid points equivalent to a window width of
10 km. To avoid boundary effects of the filter, ${u}_{{L}_{\mathrm{r},k}}^{r}$ was only
determined up to *z*_{aTop}−5 km and down to
*z*_{aGradr} at 30 km. It was constantly extended at the upper end and extended by a
linear gradient below *z*_{aGradr}, using (in units [m])

$$\begin{array}{}\text{(6)}& {u}_{L\mathrm{r},ik}^{r}={u}_{{L}_{\mathrm{r},k}}^{r}\left({z}_{\mathrm{aGradr}}\right)+{\displaystyle \frac{{z}_{\mathrm{aGradr}}-{z}_{\mathrm{a},i}}{\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}}},\end{array}$$

for all elements of ${u}_{{L}_{\mathrm{r},k}}^{r}\left(t\right)$ below *z*_{aGradr},
roughly following estimates of ESA/EUMETSAT (1998) and the
overall behavior of estimates from real excess phase profiles (the latter became too
vulnerable to biases and fluctuations to continue using them below 30 km).

Since the noise components responsible for the random uncertainty at
excess phase level are essentially uncorrelated at a sampling rate of
50 Hz (Syndergaard, 1999; Hajj et al., 2002), the correlation matrix
**R**_{Lr} is set to unity in the diagonal and to zero
outside (i.e., a Kronecker-*δ* assignment) for both channels (Fig. 3d–f). In
case the future excess phase data from the rOPS L1a processor exhibit
non-negligible correlations for some data from some of the RO
missions, we will account for these correlations in
**R**_{Lr}, since our L1b algorithm
(Sect. 3) is prepared for full
covariance propagation. The elements of the covariance matrix
**C**_{Lr} are hence (item 1.1 in Fig. 2)

$$\begin{array}{}\text{(7)}& {C}_{L\mathrm{r},ijk}={u}_{L\mathrm{r},ik}^{r}\cdot {u}_{L\mathrm{r},jk}^{r}\cdot {R}_{L\mathrm{r},ijk}={u}_{L\mathrm{r},ik}^{r}\cdot {u}_{L\mathrm{r},jk}^{r}\cdot {\mathit{\delta}}_{ij}.\end{array}$$

For the MC validation of the CP, error profile realizations ${\mathit{\u03f5}}_{L\mathrm{r}}^{r}$ were superimposed onto simulated “true” excess phase profiles ${L}_{\mathrm{r},k}^{\mathrm{T}}\left(t\right)$. As a source for ${L}_{\mathrm{r},k}^{\mathrm{T}}$, we used an EGOPS-simulated “error-free” CHAMP event from 8 August 2008 (i.e., no receiver system errors superimposed). Using an “error-free” profile as the basis, with the particular simulated profile just serving as a representative RO profile to illustrate the MC validation, allows us to strictly ensure the consistency of the random uncertainty of the input profile with the ensemble of superimposed error profile realizations.

To create the error profiles, a representative ${u}_{L\mathrm{r}}^{r,\mathrm{Std}}$ uncertainty profile was selected from a COSMIC ensemble of uncertainty profiles, created according to Eqs. (5) and (6). The error profile realizations are random draws from a distribution characterized by these uncertainties, again assuming that ${R}_{L\mathrm{r},ij}={\mathit{\delta}}_{ij}$, i.e., that there are no correlations between the individual grid levels (item (a) in Fig. 2; Fig. 3f). The same standard profile ${u}_{L\mathrm{r}}^{r,\mathrm{Std}}$ was used as input for the CP to which the MC results are then compared. This MC validation method applied to test the rOPS L1b uncertainty propagation steps is essentially the same as in SKS2017, and it is described therein in more detail.

The *estimated systematic uncertainty* ${u}_{L\mathrm{r},k}^{s}$ was
determined based on a simple model roughly following error estimates
from ESA/EUMETSAT (1998), with constant uncertainty from 80 km down to
*z*_{aGrads} at 8 km and a linear uncertainty gradient in
the troposphere; as noted above, this simplified modeling will be
replaced in the future by realistic uncertainty estimates received as L1b
retrieval input from the L1a processor (Innerkofler et al., 2017).

The constant ${u}_{L\mathrm{r},k}^{s}$ above
*z*_{aGrads} is 0.1 mm for *k*=1 and 0.2 mm
for *k*=2 for MetOp and simMetOp. This uncertainty is interpreted as an estimated basic
systematic uncertainty, i.e., as a lower-bound estimate of available accuracy.

For CHAMP and COSMIC we set ${u}_{L\mathrm{r},\mathrm{1}}^{s}=\mathrm{0.2}$ mm and
${u}_{L\mathrm{r},\mathrm{2}}^{s}=\mathrm{0.4}$ mm, to roughly reflect the fact that
these RO receivers are lower-cost instruments with lower gain, and
thus somewhat lower tracking performance, than the RO receiver
on MetOp (e.g., Luntama et al., 2008; Angerer et al., 2017). From
*z*_{aGrads} downwards, ${u}_{L\mathrm{r},k}^{s}$ (in units [m])
increases by

$$\begin{array}{}\text{(8)}& {u}_{L\mathrm{r},ik}^{s}={u}_{{L}_{\mathrm{r},k}}^{s}\left({z}_{\mathrm{aGrads}}\right)+{\displaystyle \frac{{z}_{\mathrm{aGrads}}-{z}_{\mathrm{a},i}}{\mathrm{3}\times {\mathrm{10}}^{\mathrm{7}}}}.\end{array}$$

In order to avoid a sharp kink in the ${u}_{{L}_{\mathrm{r},k}}^{r}$ profiles
at *z*_{aGradr}, and in the ${u}_{{L}_{\mathrm{r},k}}^{s}$ profiles at
*z*_{aGrads}, a 2 km width moving-average boxcar filter
was applied to smooth these simple uncertainty models around these
transition altitudes (for the example profile ${u}_{L\mathrm{r}}^{s}$ is
visible in Fig. 3b).

The orbit position and velocity uncertainties of the transmitter and the receiver satellites show little variation within the short duration of an individual RO event of about 45 s to 2 min (Innerkofler et al., 2017) and can be assumed to be constant biases. They are thus counted to the systematic uncertainties, more precisely the apparent systematic uncertainties, since the actual values of the orbit-borne biases will generally change in a pseudo-random manner from event to event.

We set the transmitter position and velocity
uncertainties to ${u}_{\mathit{r}\mathrm{T}}^{s}=\mathrm{3}$ cm and ${u}_{\mathit{v}\mathrm{T}}^{s}=\mathrm{0.01}$ mm s^{−1}, consistent with accuracies for GPS orbits available from GNSS
orbit providers like the International GNSS Service (IGS).
The receiver position and velocity uncertainties, ${u}_{\mathit{r}\mathrm{R}}^{s}=\mathrm{5}$ cm and
${u}_{\mathit{v}\mathrm{R}}^{s}=\mathrm{0.05}$ mm s^{−1} for CHAMP and MetOp, are adopted
4 times smaller than those for COSMIC with ${u}_{\mathit{r}\mathrm{R}}^{s}=\mathrm{20}$ cm and
${u}_{\mathit{v}\mathrm{R}}^{s}=\mathrm{0.2}$ mm s^{−1}, as found by ongoing
rOPS-related POD studies (Innerkofler et al., 2017), consistent with previous
literature (e.g., Montenbruck et al., 2009; Schreiner et al., 2010).

3 Algorithm sequence and example results

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In this section the L1b uncertainty propagation algorithm sequence is introduced. We illustrate the effects of the algorithm on the main uncertainty variables by way of the COSMIC example case already used for Fig. 3.

For each L1b retrieval step – i.e., segments (1), (2), and (3) in
Fig. 2 – the results for the principal variables are shown
in Figs. 4 to 8. These variables are the state
profiles *X*_{r} (with ${X}_{\mathrm{r}}\in \mathit{\left\{}{L}_{\mathrm{F},k}\right(t)$,
*D*_{r,k}(*t*), *α*_{G,k}(*t*),
*α*_{G,k}(*z*_{a}), *α*_{M,k}(*z*_{a}),
*α*_{F,k}(*z*_{a}), and
*α*_{r}(*z*_{a})*}*), the estimated systematic uncertainty
profiles ${u}_{X\mathrm{r}}^{s}$, the estimated random uncertainty profiles
${u}_{X\mathrm{r}}^{r}$, representative correlation functions
*R*_{Xr,i} (with *i* such that ${z}_{\mathrm{a},i}\in \mathit{\{}\mathrm{10},\mathrm{30},\mathrm{50},\mathrm{70}\phantom{\rule{0.125em}{0ex}}\mathrm{km}\mathit{\}}$), and the correlation length profiles
*l*_{Xr} and resolution profiles *w*_{Xr}. Along with
the dual-frequency state profiles, we also show the collocated
forward-modeled short-range forecast profiles, i.e., model profiles
*X*_{m} with ${X}_{\mathrm{m}}\in \mathit{\{}{L}_{\mathrm{m}},{D}_{\mathrm{m}},{\mathit{\alpha}}_{\mathrm{m}}\mathit{\}}$ for comparison.

A concise definition of the variables involved is provided in Table 1, as introduced above. The summary description in this section is complemented by a complete step-by-step description of the algorithm along the entire L1b retrieval chain in Appendix A, which is organized for convenience into the same sequence of subsections.

To simplify the notation in the description, we suppress index *k* whenever steps are
applied in an identical way to the data of both GNSS L-band channels with frequencies *f*_{T1} and
*f*_{T2}. Only if the two channels are treated differently, such as in
Sect. 3.3, is the index considered again. For conciseness
we also do not illustrate both the estimated basic and estimated apparent
systematic uncertainty but rather the total estimated systematic uncertainty as the
overall result.

A Blackman windowed sinc (BWS) low-pass filter with a filter cutoff
frequency *f*_{c}=2.5 Hz (boxcar-equivalent filter width of 0.2 s) (item 1.2 in
Fig. 2) is applied onto the excess phase profile
*L*_{r}(*t*), before the Doppler differentiation (item 1.4 in
Fig. 2), to avoid an amplification of high-frequency
noise in the phase profile by the derivative operation. This filter suppresses the noise;
consequentially the filtered excess phase profile *L*_{F}(*t*),
shown in Fig. 4a,
is expected to have random uncertainties ${u}_{L\mathrm{F}}^{r}$ of smaller
magnitude, but correlated over the length of the filter window. The
uncertainties obtained through the implemented algorithm confirm these
expectations, i.e., random uncertainty profiles in Fig. 4c are
less than a third in magnitude of those in Fig. 3c, and
Fig. 4d–e show how the correlation functions widened and
the correlation length and vertical resolution reached ∼0.5 and ∼0.6 km, respectively, above about 30 km impact altitude (Fig. 4f).

The random uncertainty propagation algorithm, i.e., the covariance
propagation from **C**_{Lr} to **C**_{LF}, is
described by Eq. (A6) and item 1.3 in
Fig. 2 and is justified by Eq. (2). To obtain
${u}_{L\mathrm{F}}^{r}$ and **R**_{LF}, we use
Eqs. (A7) and (A8).

To propagate the estimated systematic uncertainty ${u}_{L\mathrm{r}}^{s}$,
which characterizes long-range-correlated offsets or biases, we use the
same BWS filter as for the state profile, i.e., making use of
Eq. (4). Because the input uncertainty profile
${u}_{L\mathrm{r}}^{s}$ is chosen to be constant down to *z*_{aGrads},
the filter has little effect, and ${u}_{L\mathrm{F}}^{s}$, shown in
Fig. 4b, is essentially equal to ${u}_{L\mathrm{r}}^{s}$,
shown in Fig. 3b.

The resolution profile *w*_{Lr} is determined by the filter width
according to Eqs. (A11) and
(A13). After the BWS filtering, the resolution is roughly equal to
the correlation length *l*_{LF}, amounting to ∼0.6 km above about
30 km impact altitude and becoming finer downwards due to
the increasing refraction (Fig. 4f).

The next step is a five-point differentiation operation (item 1.4 in
Fig. 2) used to
calculate the Doppler shift profile *D*_{r}(*t*) from the filtered excess phase
profile *L*_{F}(*t*). The resulting dual-frequency Doppler shift profiles are plotted
along with the model profile *D*_{m}(*t*) in
Fig. 5a for the example case.

As for the filtered excess phase, we apply CP (Eq. A18,
item 1.5 in Fig. 2) to first
calculate the covariance matrix **C**_{Dr} and then extract
${u}_{D\mathrm{r}}^{r}$ (shown in Fig. 5c) and **R**_{Dr}
(Fig. 5d and 5e) from it. The choice of the x-axis
range shows the random uncertainties increased, but the differentiation actually
does increase relative random uncertainties (relative to the state profile).
It also causes anti-correlation with neighboring elements, as visualized by the
negative side peaks of the correlation functions in
Fig. 5d and 5e. The correlation length
*l*_{Dr} (of the main correlation function peak)
decreases accordingly (now smaller than 0.3 km throughout), because the
correlation functions fall off more steeply on both sides of the main peak (Fig. 5f).

For calculating the estimated systematic uncertainty, we use the state operator; i.e.,
we just differentiate ${u}_{L\mathrm{F}}^{s}$ and get ${u}_{D\mathrm{r}}^{s}$
(shown in Fig. 5b). With the current illustrative choice of input
uncertainties the systematic uncertainty of the Doppler shift
profile is zero above the transition to the troposphere, where the
estimated systematic uncertainty of the excess phase is assumed constant;
in the troposphere a Doppler shift offset of ∼ 0.02 mm s^{−1}
occurs.

The resolution profile *w*_{Dr} shows that the vertical resolution
stays unaffected by this operator (cf. Figs. 5f and
4f), because the BWS filter width of the preceding low-pass
filtering (intentionally) stretched beyond the five neighboring points involved
in the differentiation.

The next operator is the GO bending angle retrieval in
which retrieved GO bending angle profiles *α*_{G}(*t*) are calculated from Doppler shift profiles
*D*_{r}(*t*) and the orbit position and velocity vectors
*r*_{T}(*t*), *r*_{R}(*t*),
*v*_{T}(*t*), and *v*_{R}(*t*) (item 2.1 in
Fig. 2) and then
interpolated to the (common monotonic) impact altitude grid *z*_{a} (item 2.6
in Fig. 2).

Figure 6a shows retrieved
*α*_{G} profiles together with the model profile *α*_{m}.
The mildly nonlinear implicit-type bending angle retrieval operator needs to be
solved iteratively, and it requires linearization for both random and
systematic uncertainty propagation, as described in detail in Appendix A
(Sect. A2). Because this retrieval step is performed level
by level, keeping levels independent, the GO bending angle retrieval leaves
correlation functions and resolution unchanged
(cf. Figs. 6d–f and
5d–f).

The estimated random uncertainties ${u}_{\mathit{\alpha}\mathrm{G}}^{r}$, as shown in
Fig. 6c, now increase more strongly in the lower stratosphere
and troposphere (to about 40 to 50 µrad near 10 km), because they are
dependent on the vertical gradient of the impact parameter *a*_{t}, which is increasingly
larger towards lower altitudes from the increasing refraction.

The main contributions to the estimated systematic uncertainty ${u}_{\mathit{\alpha}\mathrm{G}}^{s}$ are induced by systematic uncertainties in orbit velocity and position of the transmitter and the receiver satellite (details in Sect. A2), which in total amount to about 0.05 µrad (Fig. 6b). Compared to this magnitude, the systematic uncertainty contributed by the Doppler shift uncertainty is very small.

Due to strong refractivity gradients and multipath effects, the GO bending angle retrieval can be inadequate in the troposphere, and therefore WO algorithms are applied to reconstruct the geometric optical ray structure of the wave field (e.g., Gorbunov, 2002a; Gorbunov and Lauritsen, 2004).

In the rOPS, along with the WO bending angle profile
*α*_{W}(*z*_{a}), the systematic uncertainty profile
${u}_{\mathit{\alpha}\mathrm{W}}^{s}$, the random uncertainty profile ${u}_{\mathit{\alpha}\mathrm{W}}^{r}$, the correlation matrix **R**_{αW}, and the resolution
profile *w*_{αW} are retrieved (item 2.7 in
Fig. 2).

The WO bending angle retrieval algorithm used is a canonical transform (CT2) algorithm (Gorbunov et al., 2004), and the associated uncertainty propagation algorithm is not described here, but separately by Gorbunov and Kirchengast (2015, 2018). The WO retrieval and uncertainty propagation results are supplied up to 20 km impact altitude by the WO algorithms.

In the rOPS bending angle retrieval the results from the WO retrieval,
*α*_{W}, are merged with GO retrieval results,
*α*_{G}, at around a transition altitude ${z}_{\mathrm{a}}^{\mathrm{GW}}$ in a
transition range ${z}_{\mathrm{a}}^{\mathrm{GW}}\pm \mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$, to get merged
profiles *α*_{M} (item 2.9 in Fig. 2). The
determination of the transition altitude
and the merging algorithm are described in Appendix A2.3. We use a specialized covariance propagation to
propagate the GO and WO uncertainties, expressed by the covariance matrices
**C**_{αG} and **C**_{αW},
to properly obtain the covariance matrix of the merged bending angle
**C**_{αM} (Eqs. A37 and
A38, item 2.10 in Fig. 2).

Because the rOPS implementation of the WO uncertainty propagation
(Gorbunov and Kirchengast, 2018) was still in test phase and not yet available for
integration into the simulations here, all examples in this study are
GO-only; i.e., only the GO retrieval is performed. Results for
*α*_{M} are thus unchanged from those shown in
Fig. 6 and not separately illustrated.

In order to nevertheless test and validate the uncertainty propagation of the merging algorithm, WO retrieval results were artificially substituted by the GO results for the MC validation (Sect. 4); i.e., GO was used as a proxy for WO since reasonably capturing expected WO variability as indicated by tests of Gorbunov and Kirchengast (2018).

To prepare the merged bending angle profiles *α*_{M,k} for the
ionospheric correction, they are first filtered by another BWS filter
operation (item 3.1 in Fig. 2) in order to ensure adequately
smoothed bending angle profiles *α*_{F,k}, with $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}$.

The chosen filter cutoff frequency for *k*=1 is *f*_{c1}=2.5 Hz, same as the basic filtering (Sect. 3.1.1), just to
ensure clearness of any higher-frequency effects from operators after
the initial excess phase filtering (e.g., from Doppler shift
derivation that induces short-range anti-correlation effects). For
*k*=2, the cutoff frequency *f*_{c2} is set noise-dependent,
between 2.5 and 0.5 Hz (boxcar-equivalent width of 0.2 to
1.0 s). In events in which the *α*_{F2} profile does
not reach down as far as *α*_{F1}, it is extrapolated down
to the bottom of *α*_{F1}, *z*_{aBot}. The
results for the filtered bending angle state profiles
*α*_{F,k} are displayed in Fig. 7a,
together with the associated model bending angle profile
*α*_{m}. The filter has considerably reduced the noise of
the profile, particularly for *α*_{F2}, where the algorithm selected a cutoff
frequency ${f}_{\mathrm{c}\mathrm{2}}=\mathrm{10}/\mathrm{7}$ Hz in this example case.

The relevant covariance-propagated random uncertainties ${u}_{\mathit{\alpha}F,k}^{r}$ are shown in
Fig. 7c (blue and red), illustrating the reduced noise,
especially for *α*_{F2}. In return, the peaks of the correlation functions
broaden (cf. Figs. 7d–e and 6d–e), with correlation lengths
*l*_{αF,k} at near 0.4 km for *α*_{F1} and
above 0.5 km for *α*_{F2} (Fig. 7f).

The estimated systematic uncertainty remains largely unchanged (Fig. 7b) due to its smooth character.

The resolution of the filtered bending angle profiles (according to Eqs. A11 and
A13) is determined by the cutoff frequencies *f*_{c,k} of
the BWS filters. In the example case it is therefore essentially unchanged
for *α*_{F1}, while it is significantly decreased for
*α*_{F2} (cf. Figs. 7f and 6f)
since ${f}_{\mathrm{c}\mathrm{2}}=\mathrm{10}/\mathrm{7}$ Hz. That is, the resolution
*w*_{αF2} in the upper stratosphere for example,
where the vertical scanning velocity of this RO event is about
3.2 km s^{−1}, is near 1.1 km (Fig. 7f).

The final step of the L1b processor is the ionospheric correction (item 3.5
in Fig. 2). The atmospheric bending angle
*α*_{r} is obtained by applying a linear dual-frequency combination of
*α*_{F1} and *α*_{F2}, such that ionospheric effects
are largely removed (details are described in Sect. A3). The
final retrieved atmospheric bending angle *α*_{r} of the
example case is shown in Fig. 8a.
The propagation results for the estimated random uncertainty are shown in
Fig. 8c. The linear combination of the ionospheric
correction amplifies noise and ${u}_{\mathit{\alpha}\mathrm{r}}^{r}$ is therefore
considerably larger than ${u}_{\mathit{\alpha}\mathrm{F}\mathrm{1}}^{r}$, and
${u}_{\mathit{\alpha}\mathrm{F}\mathrm{2}}^{r}$ (cf. Figs. 8c and 7c).

Figure 8d shows how the correlation functions – as obtained
through covariance propagation – combine the characteristics
of the correlation functions from the two matrices **R**_{αF1}
and **R**_{αF2}, while essentially inheriting the
*α*_{F1} behavior, since the *α*_{F2}
influence on the ionospheric correction is comparatively minor (see
Sect. A3).

The residual higher-order ionospheric effects are accounted for by a “conservative best-guess” value (0.05 µrad, reflecting results of Liu et al., 2015, and Danzer et al., 2013, 2015) and added (in root-mean-square form) to the systematic uncertainty profile ${u}_{\mathit{\alpha}\mathrm{r}}^{s}$, leading to a total estimated systematic uncertainty in this example case of ∼ 0.07 µrad (Fig. 8b). Within this uncertainty, the one dominating component from orbit uncertainties (∼ 0.05 µrad; cf. Fig. 6b) can be considered an apparent systematic uncertainty that will essentially average out in ensemble averaging (e.g., climatologies), while the other dominating component from residual higher-order ionospheric biases (also estimated ∼ 0.05 µrad as noted above) can be considered a basic systematic uncertainty. For the latter it is therefore useful and prepared for in the rOPS – in line with GUM recommendations and as discussed in the introductory Sect. 1 – to correct for the quantifiable part of it in the future so that the total basic systematic uncertainty may be mitigated down to the ∼ 0.01 µrad level.

The resolution profile *w*_{αr} of the retrieved bending
angle (Fig. 8f) is dominated by the contribution of
*α*_{F1} that strongly dominates (intentionally by
construction) the ionospheric correction results in terms of the
small-scale bending angle variability. Similar to the correlation length
profile *l*_{αr}, it is therefore very close to
*w*_{αF1} and only slightly larger.

4 Algorithm validation

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The GUM advises to use a MC method for uncertainty propagation if the retrieval operators do not fulfill the criteria for a GUM-type CP. In our case the MC method is put to another beneficial use, to validate the results of the CP, as recommended by JCGM (2011).

For the validation of the covariance propagation by the MC method, we sampled the input excess phase profile random error distribution, statistically described by

$$\begin{array}{}\text{(9)}& {C}_{L\mathrm{r},ij}^{\mathrm{MC}}={u}_{L\mathrm{r},i}^{r,\mathrm{Std}}\cdot {u}_{L\mathrm{r},j}^{r,\mathrm{Std}}\cdot {\mathit{\delta}}_{ij},\end{array}$$

by a large number *M* of draws ${L}_{\mathrm{r}}^{\mathrm{T}}+{\mathit{\u03f5}}_{L\mathrm{r},j}^{r}$ (with $j\in \mathit{\{}\mathrm{1},\mathrm{\dots},M\mathit{\}}$ and *M*=1000). For
each of these *M* profile realizations, the state retrieval is run through
the L1b retrieval chain, to give *M* realizations of the output variable
*X*_{j} (with ${X}_{j}\in \mathit{\left\{}{L}_{\mathrm{F},kj}\right(t),{D}_{\mathrm{r},kj}(t),{\mathit{\alpha}}_{\mathrm{G},kj}({z}_{\mathrm{a}}),{\mathit{\alpha}}_{\mathrm{F},kj}({z}_{\mathrm{a}}),{\mathit{\alpha}}_{\mathrm{r},j}({z}_{\mathrm{a}}\left)\mathit{\right\}}$ and $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}$). From these
individual realizations the mean profiles
*X*^{MC} and the covariance matrices
${\mathbf{C}}_{X}^{\mathrm{MC}}$,

$$\begin{array}{ll}{\displaystyle}{\mathbf{C}}_{X}^{\mathrm{MC}}=& {\displaystyle \frac{\mathrm{1}}{M-\mathrm{1}}}\left[\left({X}_{\mathrm{1}}-{X}^{\mathrm{MC}}\right){\left({X}_{\mathrm{1}}-{X}^{\mathrm{MC}}\right)}^{T}+\mathrm{\dots}\right.\\ \text{(10)}& {\displaystyle}& {\displaystyle}\left.+\left({X}_{M}-{X}^{\mathrm{MC}}\right){\left({X}_{M}-{X}^{\mathrm{MC}}\right)}^{T}\right],\end{array}$$

are calculated (items b–g in Fig. 2). Using the same input
profile and uncertainty information as used to specify the MC runs (described
in Sect. 2.2), the retrieval is then also run with
covariance-based uncertainty propagation, and the resulting CP-propagated
covariance matrices ${\mathbf{C}}_{X}^{\mathrm{CP}}$ are compared to the
MC-derived matrices ${\mathbf{C}}_{X}^{\mathrm{MC}}$. In order to be able to
attribute potential changes between CP and MC covariance matrices better, we
decompose **C**_{X} into ${u}_{X}^{r}$ and **R**_{X}
(Eqs. A7 and A8), and compare them separately.

Figure 9 shows the different steps along the retrieval
chain from *L*_{F,k}(*t*) to *D*_{r,k}(*t*),
*α*_{G,k}(*z*_{a}), *α*_{F,k}(*z*_{a}), and
*α*_{r}(*z*_{a}) in the rows, for *k*=1 (GPS *f*_{T1}
frequency) in the left column and
for *k*=2 (GPS *f*_{T2} frequency) in the middle column. The right column shows multiple
representative correlation functions, from near 10 to near 70 km. Due to the limited number of MC
draws, the MC results (black lines) show some jitter both in the estimated
random uncertainty and in the correlation functions. Since the purpose of
the MC results is only to demonstrate the correctness of the CP result, we
can disregard this behavior.

Figure 9a (light blue) and b (orange) show the random uncertainties ${u}_{L\mathrm{r},\mathrm{1}}^{r}$ and ${u}_{L\mathrm{r},\mathrm{2}}^{r}$, respectively, which characterize the input distribution and from which the random error profiles ${\mathit{\u03f5}}_{L\mathrm{r},j}^{r}$ are drawn. They also show the CP results for the random uncertainty ${u}_{L\mathrm{F}\mathrm{1}}^{r}$ (dark blue in Fig. 9a) and ${u}_{L\mathrm{F}\mathrm{2}}^{r}$ (red in Fig. 9b), compared to the MC propagated random uncertainties (black).

The CP and MC lines match very well and show that the implemented CP
algorithm delivers correct results for the basic filtering step. For
*f*_{T2}, the MC uncertainties do not reach down as far as
the CP uncertainties, because the shortest of all draws of the large
ensemble of size *M* determines how far down the recombined MC
covariance matrix (Eq. 10) reaches.
Figure 9c compares CP correlation functions
*R*_{LF,i1} (blue) and *R*_{LF,i2} (red) to the
corresponding MC correlation functions (black dashed).

The CP and MC correlation functions also agree well. Both capture the
narrow peak, broadened by the BWS filter. Again the MC correlation
functions fluctuate around zero left and right of the peak, from the
finite ensemble size, but it is obvious that the CP delivers the
correct off-peak results (i.e., zero; the off-peak elements outside
the BWS filter window must nominally be zero).
The MC validation (black) of ${u}_{D\mathrm{r}\mathrm{1}}^{r}$
(Fig. 9d), ${u}_{D\mathrm{r}\mathrm{2}}^{r}$
(Fig. 9e), and **R**_{Dr,i} (Fig. 9f, blue and red) demonstrates
that the CP through the Doppler shift derivation performs correctly as well.

The next row, Fig. 9g to i, shows the results for the GO
bending angle *α*_{G}(*z*_{a}), i.e., after the
interpolation of all quantities to the (common monotonic) impact altitude
grid *z*_{a}. For comparison, in Fig. 9a to f, all
quantities have been computed on the common time grid (“setting time”
relative to time zero at 80 km altitude) with 50 Hz sampling rate;
the corresponding impact altitude of the “true” profile
${L}_{\mathrm{r}}^{\mathrm{T}}$ is shown for additional convenience on the
right-hand-side (RHS) axis. In Fig. 9g to o, these bending angle quantities have
been computed on the impact altitude grid; in these cases therefore the
corresponding setting time of the “true” profile is shown for additional
convenience on the RHS axis.

The results for the filtered bending angle *α*_{F} follow in
Fig. 9j to l. Also here the MC results match the CP result
well. Due to the lower BWS cutoff frequency for *α*_{F2}, now
${u}_{\mathit{\alpha}\mathrm{F}\mathrm{2}}^{r}$ is smaller than ${u}_{\mathit{\alpha}\mathrm{F}\mathrm{1}}^{r}$, even
though ${u}_{\mathit{\alpha}\mathrm{G}\mathrm{2}}^{r}$ was larger than
${u}_{\mathit{\alpha}\mathrm{G}\mathrm{1}}^{r}$. Correspondingly the peak of the correlation
functions *R*_{αF,i2} widened more than those of
*R*_{αF,i1} (cf. Fig. 9l and i).

Finally, Fig. 9m to o show the CP results for retrieved
atmospheric bending angle *α*_{r}, where Fig. 9n
is included as a special cross-comparison in the case of only variance propagation
being used instead of CP. Figure 9m and o confirm that CP
results are also correct for this final L1b variable, in terms both of random
uncertainty and correlation functions.

In order to demonstrate that a full CP is necessary to propagate
random uncertainties correctly, we also calculated random uncertainties
${u}_{\mathit{\alpha}\mathrm{r}}^{r}$ based on mere VP from
*α*_{G} to *α*_{r} for comparison. A description
of this VP algorithm (i.e., only diagonal elements of the
covariance matrices are considered) is provided in
Appendix B. Figure 9n clearly shows that
VP would overestimate random uncertainties in *α*_{r}
considerably, pointing to the importance of the complete CP implementation
in the L1b retrieval chain, even though the correlation lengths involved
in the processing steps are rather small.

5 Performance demonstration

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To statistically evaluate the performance of the new L1b uncertainty propagation
algorithm, we also processed a complete test day of real (CHAMP, COSMIC, MetOp)
and simulated (simMetOp) data from GNSS RO satellite missions.
Figure 10 shows the results for estimated systematic and random
uncertainty profiles, as well as correlation
length and resolution profiles for filtered excess phase profiles
*L*_{F,1}. Figure 11 subsequently illustrates the
ensemble mean of the same variables for
*L*_{F1}, *D*_{r1}, *α*_{G1}, and *α*_{r}
for the test day ensemble. In Fig. 10 we also co-illustrate the number
of events processed for each of the RO missions (middle column).

About 5 % of the total number of processed profiles for each mission have been discarded, because they were detected as outliers based on the magnitude of their random uncertainty profiles (these outliers are not included in the number of profiles shown). All profiles are shown as function of impact altitude, because each of the profiles in the ensembles needed to be interpolated to the same (standard) impact altitude grid, to orderly calculate their mean profiles.

Figure 10a shows ${u}_{L\mathrm{F}\mathrm{1}}^{r}$ and ${u}_{L\mathrm{F}\mathrm{1}}^{s}$ for all ∼100 CHAMP events. It is visible (also in Fig. 10d and g) that the random uncertainty is estimated based on excess phase noise between 30 and 75 km and synthetically extended above and below, as described in Sect. 2.2. For the large majority of events, ${u}_{L\mathrm{F}\mathrm{1}}^{r}$ lies between about 0.5 and 3 mm in the range between 30 and 75 km. Note that these results show the random uncertainties after the application of the basic BWS filter (Sect. 3.1.1), but the input uncertainties ${u}_{L\mathrm{r}\mathrm{1}}^{r}$ are of similar shape (though larger in magnitude).

Figure 10b shows that the correlation length profiles of the
CHAMP ensemble (gray) and its ensemble mean (yellow) are of relatively constant
magnitude from 35 to 80 km but then get smaller downward, because the
RO event's scan velocity decreases (see
Eq. A13). Since the BWS filter determines the vertical
resolution and the correlation length at the same time, the resolution profiles
*w*_{LF1} (Fig. 10c) are quite similar to the
correlation length profiles *l*_{LF1} (Fig. 10b).

The number-of-events profile shows that most CHAMP events end between 5 and 12 km (Fig. 10b, black). This is because the GO profiles illustrated here are cut off right at the lower end of the GO–WO transition range at ${z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$ (cf. Table 2).

Compared to CHAMP, the mean random uncertainty ${u}_{L\mathrm{F}\mathrm{1}}^{r}$
(Fig. 10d) for the ∼1500 events of the COSMIC
ensemble is smaller, particularly above 30 km, indicating the improved
data quality of this later mission. The mean of the correlation length
profiles *l*_{LF1} (Fig. 10e) is higher than for
CHAMP (Fig. 10b), and correspondingly the resolution of the
COSMIC profiles is also somewhat coarser (Fig. 10f and c). The
cutoff frequency and sampling rate – and thus the resolution in time – are
set to be the same in the rOPS, irrespective of the missions; these
differences hence are due to the different vertical scan velocities of the
missions induced by the differences in orbit altitudes (CHAMP ∼400 km,
COSMIC ∼700 km).

For the real MetOp data (available here as a data set from UCAR/CDAAC, as for CHAMP and COSMIC), ${u}_{L\mathrm{F}\mathrm{1}}^{r}$ appears similar to COSMIC (cf. Fig. 10d, g), while for simMetOp (with best possible simulated MetOp-type receiver noise) it is clearly smaller than for COSMIC. From 35 to 80 km the mean random uncertainty profile for simMetOp stays below 1 mm (Fig. 10j). Three individual profiles exhibit comparatively high uncertainties of larger than 2 mm within about 40 to 55 km, however, reflecting that the simMetOp error simulations are capable of partly generating higher-noise profiles of the type more frequently seen in the real MetOp data (Fig. 10g).

On the other hand, the average correlation length/resolution profile of the ∼500 real MetOp and ∼700 simMetOp ensemble members is very similar, driven by the orbit being essentially the same for the real data and the simulations (Fig. 10h, i, k, l). Compared to COSMIC (Fig. 10e, f), the correlation length and resolution are again somewhat larger/coarser, due to an even somewhat higher scan velocity of the MetOp satellite (∼820 km orbit altitude). The systematic uncertainty ${u}_{L\mathrm{F}\mathrm{1}}^{s}$ – just co-illustrated for completeness in Fig. 10a, d, g, and j – is almost left unchanged by the BWS filter and is essentially equal to the preset input uncertainty for all three missions (Sect. 2.2).

Figure 11 shows how the ${u}_{L\mathrm{F}\mathrm{1}}^{s}$,
${u}_{L\mathrm{F}\mathrm{1}}^{r}$, *l*_{LF1}, and *w*_{LF1} profiles
are on average affected
by the uncertainty propagation. The color code for the different satellite missions
is the same as in Fig. 10.
The propagation effects visible are similar to those already seen in
Figs. 3 to 8. The Doppler shift derivation
increases the relative uncertainties and reduces correlation length (of the
main peak), while the
resolution stays the same (Fig. 11d–f). The GO bending angle retrieval leaves
correlation length and resolution unchanged, while random uncertainties
increase strongly in the lower stratosphere and troposphere due to the
increasing refractive effects (Fig. 11g–i).

Finally, the BWS filtering before the ionospheric correction decreases random
uncertainties and increases correlation length, and resolution somewhat.
However, the linear combination of the two bending angle profiles
*α*_{F1} and *α*_{F2} then increases the random
uncertainty again (cf. Fig. 11j and g). The adaptive
minor-channel cutoff frequency *f*_{c2} for the relatively noisy CHAMP
profiles is generally lower than for the other two missions, and the filter
effect is therefore stronger for CHAMP (indicated by the larger *l*_{αr} in Fig. 11k)

The estimated systematic uncertainty of the atmospheric bending angle ${u}_{\mathit{\alpha}\mathrm{r}}^{s}$, indicated for completeness in Fig. 11 (left column, inflated by a factor of 10 in panel a and 100 in panels d, g, and j for somewhat better visibility), stays below 0.1 µrad for all three missions.

6 Conclusions

Back to toptop
In order to deliver climate benchmark data sets, it is essential to integrate uncertainty propagation in RO retrievals. In this study we presented the uncertainty propagation algorithm chain from excess phase profiles to atmospheric bending angle profiles (L1b processing), as newly implemented in the rOPS at the WEGC. Along with the basic profile retrieval, we provide estimates for systematic and random uncertainties, error correlation matrices, and vertical resolution profiles, which is unique amongst all existing RO processing systems so far (Ho et al., 2012; Steiner et al., 2013).

We validated the implemented algorithm via comparison to Monte Carlo sample propagation results and demonstrated the performance of the algorithm using real-data ensembles. We find close agreement between the implemented covariance propagation of random uncertainties and the Monte Carlo validation runs, verifying the correctness of the implemented algorithm. The test day ensembles for three different missions (CHAMP, COSMIC, MetOp) show reliable, robust, and consistent results that provide valuable insight and understanding of retrieval chain details.

Together with the integration of the uncertainty propagation algorithm from atmospheric bending angle profiles to dry-air profiles (L2a processing) presented by Schwarz et al. (2017), the rOPS can now provide estimates of systematic and random uncertainty profiles, of error correlation matrices and resolution, and of observation-to-background weighting ratio profiles from excess phase to dry-air profiles.

The next step towards the final atmospheric profiles, currently ongoing, is the introduction of integrated uncertainty propagation for the moist-air retrieval (L2b processing). Implementation of uncertainty propagation for the wave-optics bending angle retrieval and for the orbit determination and excess phase processing (L1a processing) is ongoing as well.

Once completed, the full rOPS retrieval chain will run with integrated uncertainty estimation, a major step towards climate benchmark data provision, and beneficial for the wide diversity of uses in atmospheric and climate science and applications.

Data availability

Back to toptop
Data availability.

The RO excess phase and orbit data used in the study are available from UCAR/CDAAC Boulder, CO, USA, at http://cdaac-www.cosmic.ucar.edu/ (last access: 26 March 2018). The ECMWF analysis and forecast data are accessible from ECMWF Reading, UK, at http://www.ecmwf.int/en/forecasts/datasets (last access: 26 March 2018). To access the relevant result files of the uncertainty propagation please contact the corresponding author. The developed algorithm is provided in the Appendix.

Appendix A: Algorithm description

Back to toptop
In this appendix the rOPS L1b uncertainty propagation algorithm is
introduced, following the L1b retrieval chain (Fig. 2;
Sect. 3) step by step,
starting with excess phase profile *L*_{r} as input and
proceeding to *L*_{F}, *D*_{r}, *α*_{G},
*α*_{M}, *α*_{F} and finally
*α*_{r}. The relevant variable definitions and symbol
explanations are summarized in Tables 1 and
2. A fully detailed algorithmic description is provided by
Kirchengast et al. (2017b).

If not stated otherwise, elements of the vector-type vertical profiles are addressed
using subscript *i* (with $i\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},N\mathit{\}}$), and optionally *j* (with $j\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},N\mathit{\}}$), running from top downward towards the
bottom of the profile, where *N* is the number of vertical grid
levels. Until the interpolation of all quantities to the common monotonic
impact altitude grid *z*_{a}, all quantities are provided on an equidistant
50 Hz time grid *t* with grid points *t*_{i}.

All steps in Sects. A1 and A2 are applied to each
of the GNSS transmitter channels' carrier frequencies *f*_{Tk}, as also indicated by the index *k* in
Fig. 2. In the notation of these sections we therefore
suppress the index *k* for the convenience of simplified readability. Also
for conciseness we write the estimated systematic uncertainty equations only
for the total systematic uncertainties *u*^{s} and briefly address the type
of the relevant components (whether basic systematic uncertainty
*u*^{b} or apparent systematic uncertainty *u*^{a}) in the
surrounding text.

The Doppler differentiation (item 1.4 in Fig. 2) would potentially amplify high-frequency noise in the excess phase profiles. To avoid this amplification, a BWS low-pass filter (e.g., Smith, 1999) is applied onto the excess phase profiles first (item 1.2 in Fig. 2).

For this basic filtering the relative cutoff frequency
*f*_{c}∕*f*_{s} is set to 0.05,
equivalent to *f*_{c}=2.5 Hz, 21 grid points, or a cutoff period
${\mathit{\tau}}_{\mathrm{c}}=\mathrm{1}/{f}_{\mathrm{c}}=\mathrm{0.4}$ s, for the standard
sampling rate *f*_{s} of 50 Hz used for all RO profiles in the
L1b processor of the rOPS. The corresponding sample width
of the Blackman window $\stackrel{\mathrm{\u0303}}{M}$ (with samples $m\in \mathit{\{}\mathrm{0},\mathrm{\dots},M\mathit{\}}$) is
set to $\stackrel{\mathrm{\u0303}}{M}=\mathrm{2}\cdot {f}_{\mathrm{s}}/{f}_{\mathrm{c}}$, yielding 41 grid points. This
ensures a reliable filter performance, also allowing the vertical resolution
of the filtered data to be robustly quantified.

With such a design, the BWS low-pass filter combines efficient removal of
high-frequency noise with a narrow smoothing window. The BWS filter thus
achieves a better smoothing effect, while keeping a *w*_{LF} of
higher resolution than a simple moving-average BC filter. Based on
a time segment of a few seconds of the excess phase
delta profile of the COSMIC example event (also used for
Figs. 3 to 8), Fig. A1 illustrates
how the BWS filter compares to boxcar
filters of 11 and 21 grid points. The corresponding filter functions are
displayed in Fig. A1a, while Fig. A1b
compares the filter results.

It is clearly seen that the smoothing window width of the BWS filter best corresponds to an 11-point boxcar filter (confirmed numerically by minimization of the sum of squared differences between boxcar and BWS filter result), while giving considerably better filtering results (as for example visible between 31.5 and 32.0 s, where the 11-point boxcar filter zigzags around the BWS result). The effective filter width of the BWS filter, which we also term “boxcar-equivalent width”, is therefore its full width at half maximum (see Fig. A1a), corresponding to $\stackrel{\mathrm{\u0303}}{M}/\mathrm{4}+\mathrm{1}$ samples with our design.

The actually used sample width *M*
of the BWS filter is equal to $\stackrel{\mathrm{\u0303}}{M}$, except that it decreases at the top and
bottom of the profile such that it does not reach beyond the first/last
element of the vector to be filtered. At the *i*th grid point
(with $i\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},N\mathit{\}}$, and *N* being the profile length in grid
points), the filter width *M* is thus

$$\begin{array}{}\text{(A1)}& {\displaystyle}M=\left\{\begin{array}{ll}\stackrel{\mathrm{\u0303}}{M}& \text{for}\stackrel{\mathrm{\u0303}}{M}/\mathrm{2}iN-\stackrel{\mathrm{\u0303}}{M}/\mathrm{2}\\ \mathrm{2}i-\mathrm{1}& \text{for}\mathrm{1}\le i\le \stackrel{\mathrm{\u0303}}{M}/\mathrm{2}\\ \mathrm{2}(N-i)+\mathrm{1}& \text{for}N-\stackrel{\mathrm{\u0303}}{M}/\mathrm{2}iN\end{array}\right..\end{array}$$

The *state* profile of the filtered phase
*L*_{F} is obtained using the “baseband approach”
(Kirchengast et al., 2016a), i.e., by first subtracting a zero-order model
profile *L*_{m} and applying the filter only to the delta profile
$\mathit{\delta}{L}_{\mathrm{rm}}={L}_{\mathrm{r}}-{L}_{\mathrm{m}}$ (with the model profile
being adequately smooth over the scale of the filter window width). This
approach efficiently mitigates residual numerical biases. After the
application of the BWS filter, the model profile is added back again. We
express the BWS filter as a linear matrix operator
**A**^{BWS} and get (item 1.2 in Fig. 2)

$$\begin{array}{}\text{(A2)}& {L}_{\mathrm{F}i}={L}_{\mathrm{m}i}+\sum _{j=\mathrm{0}}^{N}{A}_{ij}^{\mathrm{BWS}}\cdot \mathit{\delta}{L}_{\mathrm{rm}j}\end{array}$$

for the filtered excess phase, where $j\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},N\mathit{\}}$. The band matrix
operator **A**^{BWS} has elements

$$\begin{array}{}\text{(A3)}& {\displaystyle}{A}_{ij}^{\mathrm{BWS}}=\left\{\begin{array}{ll}\mathrm{0}& \text{for}ji-M/\mathrm{2}\text{and}\\ & \text{for}ji+M/\mathrm{2}\\ {w}_{j-i+M/\mathrm{2}}& \text{for}i-M/\mathrm{2}ji+M/\mathrm{2}\end{array}\right..\end{array}$$

The central filter weight ${w}_{\mathrm{0}+M/\mathrm{2}}$ at *j*=*i* is the (*M*∕2)th
filter element (according to the definition of the BWS weights below); therefore its index is *M*∕2.
With $m=j-i+M/\mathrm{2}$ (and therefore $\mathrm{0}\le m\le M$), each single BWS weight is calculated using

$$\begin{array}{}\text{(A4)}& {w}_{m}={\displaystyle \frac{{w}_{\mathrm{raw},m}}{{\sum}_{m=\mathrm{0}}^{M}{w}_{\mathrm{raw},m}}}\end{array}$$

and

$$\begin{array}{}\text{(A5)}& {\displaystyle}{w}_{\mathrm{raw},m}=\left\{\begin{array}{l}\frac{\mathrm{sin}\left(\mathrm{2}\mathit{\pi}{f}_{\mathrm{c}}/{f}_{\mathrm{s}}\left(m-M/\mathrm{2}\right)\right)}{m-M/\mathrm{2}}[\mathrm{0.42}-\mathrm{0.5}\mathrm{cos}\\ \phantom{\rule{1em}{0ex}}\left(\mathrm{2}\mathit{\pi}\frac{m}{M}\right)+\mathrm{0.08}\mathrm{cos}\left(\mathrm{4}\mathit{\pi}\frac{m}{M}\right)]\\ \phantom{\rule{1em}{0ex}}\text{for}m\ne M/\mathrm{2}\\ \mathrm{2}\mathit{\pi}{f}_{\mathrm{c}}/{f}_{\mathrm{s}}\\ \phantom{\rule{1em}{0ex}}\text{for}m=M/\mathrm{2}\end{array}\right..\end{array}$$

The *estimated random uncertainty* is then propagated by covariance
propagation (item 1.3 in Fig. 2),

$$\begin{array}{}\text{(A6)}& {\mathbf{C}}_{L\mathrm{F}}={\mathbf{A}}^{\mathrm{BWS}}\cdot {\mathbf{C}}_{L\mathrm{r}}\cdot {\left({\mathbf{A}}^{\mathrm{BWS}}\right)}^{\mathrm{T}}.\end{array}$$

The random uncertainty profile ${u}_{L\mathrm{F}}^{r}$ and the error correlation
matrix **R**_{LF} are not needed for the subsequent random
uncertainty propagation but are calculated from
**C**_{LF} for being available for the L1b output, using

$$\begin{array}{}\text{(A7)}& {u}_{L\mathrm{F},i}^{r}=\sqrt{{C}_{L\mathrm{F},ii}}\end{array}$$

and

$$\begin{array}{}\text{(A8)}& {R}_{L\mathrm{F},ij}={\displaystyle \frac{{C}_{L\mathrm{F},ij}}{{u}_{L\mathrm{F},i}^{r}{u}_{L\mathrm{F},j}^{r}}}.\end{array}$$

The *correlation length* profile *l*_{Lr} has elements

$$\begin{array}{}\text{(A9)}& {l}_{L\mathrm{r},i}={\left.{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}t}}\right|}_{i}\cdot |{t}_{i}-t({R}_{L\mathrm{F},ij}=\mathrm{1}/e\left)\right|\end{array}$$

computed upward and downward from the main peak of the correlation function
and then averaged. Here d*z*∕d*t* is the scan velocity
profile, obtained from using the MSL altitude grid *z*_{t} calculated as part
of the forward modeling towards *L*_{m} at the corresponding time grid
*t* (cf. Table 2).

We note that after the L2a refractivity retrieval also
the MSL altitude grid consistent with the retrieved refractivity profile could be used
(as described by SKS2017, Appendix A therein), from a repeated forward modeling.
The difference for the scan velocity estimate is found to be very small, however, since
the forward-modeled *z*_{t} based on collocated refractivity profiles from
ECMWF short-range forecast fields is already sufficiently reliable, and this also
keeps the L1b processor as a decoupled predecessor of the L2a processor.

For the *estimated systematic uncertainty*, interpreted as a basic systematic
uncertainty (Sect. 2.2), we apply the same low-pass filter
as used for the *state* profile (item 1.2 in
Fig. 2), but with no zero-order profile subtracted, i.e.,

$$\begin{array}{}\text{(A10)}& {u}_{L\mathrm{F}i}^{s}=\sum _{j=\mathrm{0}}^{N}{A}_{ij}^{\mathrm{BWS}}\cdot {u}_{L\mathrm{r}j}^{s}.\end{array}$$

The *resolution* in time of *L*_{F} and its uncertainties, *τ*_{BW},
is the boxcar-equivalent width (cf. Fig. A1a) determined
by the cutoff frequency *f*_{c} of the BWS filter,

$$\begin{array}{}\text{(A11)}& {\mathit{\tau}}_{L\mathrm{F}}\approx {\displaystyle \frac{\mathrm{1}}{{f}_{\mathrm{c}}+\mathrm{\Delta}{f}_{\mathrm{c}}/\mathrm{2}}}\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}{f}_{\mathrm{c}}}},\end{array}$$

with our design choice $\stackrel{\mathrm{\u0303}}{M}=\mathrm{2}({f}_{\mathrm{s}}/{f}_{\mathrm{c}})$ and with the BWS filter stopband-to-passband transition width being (Smith, 1999)

$$\begin{array}{}\text{(A12)}& \mathrm{\Delta}{f}_{\mathrm{c}}\approx {\displaystyle \frac{\mathrm{4}{f}_{\mathrm{s}}}{\stackrel{\mathrm{\u0303}}{M}}}.\end{array}$$

Given *f*_{c}=2.5 Hz, this results in an effective resolution
*τ*_{LF}=0.2 s and corresponds to the resolution obtained
when applying a 11-point boxcar filter as explained at the beginning of
this section above. The filter window intercomparison in
Fig. A1a also illustrates this, because the full
width at half maximum of the 2.5 Hz 41-point BWS filter is 11 points.

This resolution in time can finally be converted to the vertical (MSL altitude) resolution in space:

$$\begin{array}{}\text{(A13)}& {w}_{L\mathrm{F},i}={\left.{\displaystyle \frac{\mathrm{d}z}{\mathrm{d}t}}\right|}_{i}\cdot {\mathit{\tau}}_{L\mathrm{F}},\end{array}$$

where, as for the correlation length estimation (Eq. A9), the scan velocity profile is employed to convert from the time domain to MSL altitude domain.

After the application of the BWS filter to the excess phase profiles
*L*_{r} (for both carrier frequencies of the given GNSS
system), the *state* profile of the Doppler is derived from the
filtered phase profiles *L*_{F} (item 1.4 in
Fig. 2). To minimize systematic errors from the
numerical differentiation to negligible magnitude, the model profile
*L*_{m} is again subtracted from the filtered phase profile,

$$\begin{array}{}\text{(A14)}& \mathit{\delta}{L}_{\mathrm{Fm}}={L}_{\mathrm{F}}-{L}_{\mathrm{m}},\end{array}$$

and the resulting delta profile *δ**L*_{Fm} is then
differentiated. After the derivative, the zero-order Doppler shift
model profile *D*_{m} is added (the latter also available from
the forward modeling, in a form strictly consistent with the excess
phase model profile *L*_{m}).

Based on careful tests of different formulations, we use a five-point
derivative scheme. The discretization of this five-point derivative
*δ**D*_{rm,i} is given by

$$\begin{array}{ll}\text{(A15)}& {\displaystyle}& {\displaystyle}\mathit{\delta}{D}_{\mathrm{rm},i}={\left.{\displaystyle \frac{\mathrm{d}\mathit{\delta}{L}_{\mathrm{Fm}}\left(t\right)}{\mathrm{d}t}}\right|}_{i}{\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}={\displaystyle \frac{-\mathit{\delta}{L}_{\mathrm{Fm},i-\mathrm{2}}+\mathrm{8}\mathit{\delta}{L}_{\mathrm{Fm},i-\mathrm{1}}-\mathrm{8}\mathit{\delta}{L}_{\mathrm{Fm},i+\mathrm{1}}+\mathit{\delta}{L}_{\mathrm{Fm},i+\mathrm{2}}}{-{t}_{i-\mathrm{2}}+\mathrm{8}{t}_{i-\mathrm{1}}-\mathrm{8}{t}_{i+\mathrm{1}}+{t}_{i+\mathrm{2}}}}\end{array}$$

for each of the frequencies (e.g., Syndergaard, 1999). This can be expressed in matrix form as

$$\begin{array}{}\text{(A16)}& {D}_{\mathrm{r},i}={D}_{\mathrm{m},i}+\mathit{\delta}{D}_{\mathrm{rm},i}={D}_{\mathrm{m},i}+\sum _{j=\mathrm{1}}^{N}{A}_{ij}^{\mathrm{L}\mathrm{2}\mathrm{D}}\cdot \mathit{\delta}{L}_{\mathrm{Fm},j},\end{array}$$

using matrix operator **A**^{L2D} with

$$\begin{array}{ll}\text{(A17)}& {\displaystyle}& {\displaystyle}{\mathbf{A}}^{\mathrm{L}\mathrm{2}\mathrm{D}}={\displaystyle}& {\displaystyle}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\mathrm{1}}{\mathrm{12}\mathrm{\Delta}t}}\left[\begin{array}{rrrrrrrrr}-\mathrm{18}& \mathrm{24}& -\mathrm{6}& \mathrm{0}& \mathrm{0}& \mathrm{0}& & \mathrm{0}& \mathrm{0}\\ -\mathrm{6}& \mathrm{0}& -\mathrm{6}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{\ddots}& \mathrm{0}& \mathrm{0}\\ -\mathrm{1}& \mathrm{8}& \mathrm{0}& -\mathrm{8}& \mathrm{1}& \mathrm{0}& & \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -\mathrm{1}& \mathrm{8}& \mathrm{0}& -\mathrm{8}& \mathrm{1}& \mathrm{\ddots}& \mathrm{0}& \mathrm{0}\\ & \mathrm{\ddots}& & \mathrm{\ddots}& & \mathrm{\ddots}& & \mathrm{\ddots}& \\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{\ddots}& -\mathrm{8}& \mathrm{1}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& & \mathrm{0}& -\mathrm{6}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& & \mathrm{24}& -\mathrm{18}\end{array}\right],\end{array}$$

where $\mathrm{\Delta}t={t}_{i+\mathrm{1}}-{t}_{i}$, being 0.02 s in our case of *f*_{s}=50 Hz.

The *estimated random uncertainty* can then be propagated (item 1.5 in Fig. 2) using

$$\begin{array}{}\text{(A18)}& {\mathbf{C}}_{D}={\mathbf{A}}^{\mathrm{L}\mathrm{2}\mathrm{D}}\cdot {\mathbf{C}}_{L\mathrm{F}}\cdot {\left({\mathbf{A}}^{\mathrm{L}\mathrm{2}\mathrm{D}}\right)}^{\mathrm{T}}.\end{array}$$

The covariance matrix is again (cf. Eqs. A7 and A8) decomposed into estimated random uncertainties and error correlation matrix (item 2.2 in Fig. 2) using

$$\begin{array}{}\text{(A19)}& {u}_{D\mathrm{r},i}^{r}=\sqrt{{C}_{D\mathrm{r},ii}}\end{array}$$

and

$$\begin{array}{}\text{(A20)}& {R}_{D\mathrm{r},ij}={\displaystyle \frac{{C}_{D\mathrm{r},ij}}{{u}_{D\mathrm{r},i}^{r}{u}_{D\mathrm{r},j}^{r}}}.\end{array}$$

For the *estimated systematic uncertainty*, further on interpreted as
basic systematic uncertainty (cf. Eq.A10), we apply the derivative
operator (item 1.4 in Fig. 2) to the systematic uncertainties,
with no zero-order profile subtracted, i.e.,

$$\begin{array}{}\text{(A21)}& {u}_{D\mathrm{r},i}^{s}=\sum _{j=\mathrm{1}}^{N}{A}_{ij}^{\mathrm{L}\mathrm{2}\mathrm{D}}\cdot {u}_{L\mathrm{F},j}^{s}.\end{array}$$

The *resolution* remains unaffected by the Doppler shift derivation,
since the five-point sample width of the derivative operator is fully
within the 11-point effective filter width (stopband) of the BWS filter
applied before, so that *τ*_{Dr}=*τ*_{LF} and
*w*_{Dr}=*w*_{LF}.

From the Doppler shift *state* profile *D*_{r} (again for
both frequencies of the given GNSS system) we can derive the
impact parameter profile *a*_{t} and GO bending angle profile
*α*_{G} (item 2.1 in Fig. 2) using first the
geometric relation

$$\begin{array}{}\text{(A22)}& {D}_{\mathrm{r},i}=\left[{v}_{\mathrm{R},i}\mathrm{cos}\left({\mathit{\varphi}}_{\mathrm{R},i}\right)-{v}_{\mathrm{T},i}\mathrm{cos}\left({\mathit{\varphi}}_{\mathrm{T},i}\right)\right]-{\dot{r}}_{\mathrm{RT},i},\end{array}$$

where

$$\begin{array}{}\text{(A23)}& {\mathit{\varphi}}_{\mathrm{R},i}={\mathit{\eta}}_{\mathrm{R},i}-\mathrm{arcsin}\left({\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{R},i}}}\right)\end{array}$$

and

$$\begin{array}{}\text{(A24)}& {\mathit{\varphi}}_{\mathrm{T},i}=\left(\mathit{\pi}-{\mathit{\eta}}_{\mathrm{T},i}\right)-\mathrm{arcsin}\left({\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{T},i}}}\right)\end{array}$$

for each individual level of the time grid *t*_{i}, in order to determine
*a*_{t} from sequential application to all levels (Kursinski et al., 1997; Syndergaard, 1999).
Here ${v}_{\mathrm{R},i}=\left|{\mathit{v}}_{\mathrm{R},i}\right|$ is the receiver velocity;
${r}_{\mathrm{R},i}=\left|{\mathit{r}}_{\mathrm{R},i}\right|$ the receiver radial position;
*η*_{R,i} the angle between the receiver velocity and position
vectors; *ϕ*_{R,i} then the angle between the receiver velocity and ray path vectors
(and all these equivalently for the transmitter); and ${\dot{r}}_{\mathrm{RT},i}={\left|\frac{d({\mathit{r}}_{\mathrm{T}}-{\mathit{r}}_{\mathrm{R}})}{dt}\right|}_{i}$ is
the time derivative of the distance between the transmitter and the
receiver at time *t*_{i}, i.e., the “kinematic straight-line Doppler
shift” to be subtracted in Eq. (A22) to match the
(excess) Doppler shift *D*_{r,i} induced by the atmosphere (and
ionosphere).

Based on *a*_{t}, the elements of the GO bending angle profile
*α*_{G} are subsequently calculated using another geometrical
relation:

$$\begin{array}{}\text{(A25)}& {\mathit{\alpha}}_{\mathrm{G},i}={\mathit{\theta}}_{\mathrm{RT},i}-\mathrm{arccos}\left({\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{R},i}}}\right)-\mathrm{arccos}\left({\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{T},i}}}\right),\end{array}$$

where *θ*_{RT,i} is the opening angle between the transmitter and
receiver position vectors. Syndergaard (1999, Fig. 1.5 therein) provides an
illustration of the relevant geometry.

All the variables in Eqs. (A22)–(A25) are defined in the occultation plane spanned by the receiver and transmitter position vectors after oblateness correction (Syndergaard, 1998), i.e., after they have been transformed to originate in the Earth ellipsoid's center of local curvature in the occultation plane at the mean tangent point (MTP) location of the RO event.

The MTP location is defined as the geodetic (geographic) location on the WGS84 ellipsoid, where the straight-line path between transmitter and receiver touches this ellipsoid, i.e., where the straight-line tangent height is zero. This can be computed with very high accuracy at the sub-meter level (see Scherllin-Pirscher et al., 2017, for more details on the geolocation accuracy of RO). Using the MTP location's center of local curvature rather than the Earth's center of mass as the origin is essential to ensure that the assumption of spherical symmetry, implicit in Eqs. (A22) to (A25), is accurately valid geometrically.

The impact parameter retrieval is solved iteratively, because it is
impossible to rearrange Eqs. (A22) to
(A24) into an explicit expression for the retrieval of
the impact parameter, but it is mildly nonlinear and converges fast,
in particular if the initial guess for *a*_{t,i} is estimated from the
previous level (starting at the top level with the straight-line
impact parameter).

After the GO bending angle retrieval, the bending angles of all GNSS
frequencies are interpolated to a common monotonic impact altitude
grid *z*_{a} (item 2.6 in Fig. 2), based on the monotonically
sorted impact parameter grid of the leading channel, *a*_{t1} (i.e., *k*=1).

For each element of *z*_{a} we get (item 2.3 in Fig. 2)

$$\begin{array}{}\text{(A26)}& {z}_{\mathrm{a},i}={a}_{t,j\mathrm{1}}-{h}_{\mathrm{G}}-{R}_{\mathrm{C}},\end{array}$$

where *j* is the index of the elements of the sorted impact parameter grid
*a*_{t1}. *h*_{G} is the geoid undulation (see
Scherllin-Pirscher et al., 2017, for a detailed discussion of its use in RO
analysis), and *R*_{C} is the local radius of curvature of the RO
event.

Because the impact parameter is only implicitly expressed in Eqs. (A22)–(A24), but GUM-type uncertainty
propagation along Eqs. (2) and (4) requires an
explicit measurement model, we make use of a linearization of the bending
angle retrieval. We use the approach described by Melbourne et al. (1994), and applied to
uncertainty propagation by Syndergaard (1999), for the propagation
of the *estimated random uncertainty* from Doppler shift *D*_{r} to
GO bending angle *α*_{G} (item 2.5 in Fig. 2).

This linearization establishes a direct relation between random uncertainties of the Doppler shift ${u}_{D\mathrm{r}}^{r}$ and the uncertainties of the bending angle ${u}_{\mathit{\alpha}\mathrm{G}}^{r}$, using

$$\begin{array}{}\text{(A27)}& {u}_{\mathit{\alpha}\mathrm{G}\left(t\right),i}^{r}\approx {\left.-{\left({\displaystyle \frac{\mathrm{d}{a}_{\mathrm{SL}}}{\mathrm{d}t}}\right)}^{-\mathrm{1}}\right|}_{i}{u}_{D\mathrm{r}\left(t\right),i}^{r},\end{array}$$

where *a*_{SL} is the straight-line impact
parameter. These bending angle uncertainties ${u}_{\mathit{\alpha}\mathrm{G}}^{r}$ are
relative to the time grid as independent coordinate. To get the desired
uncertainties with respect to the impact altitude grid *z*_{a} (introduced in
Eq. A26), the uncertainties of the impact altitude *z*_{a}
need to be transferred to the bending angle, so that the *z*_{a} grid can
subsequently be considered free of error. Syndergaard (1999) showed
that this can be done by replacing Eq. (A27) with

$$\begin{array}{}\text{(A28)}& {u}_{\mathit{\alpha}\mathrm{G}\left({z}_{\mathrm{a}}\right),i}^{r}\approx {\left.-{\left({\displaystyle \frac{\mathrm{d}{a}_{t}^{T}}{\mathrm{d}t}}\right)}^{-\mathrm{1}}\right|}_{i}{u}_{D\mathrm{r}\left(t\right),i}^{r},\end{array}$$

where ${a}_{t}^{T}$ is the “true” impact parameter. We use
the forward-modeled impact parameter *a*_{tm} instead (i.e.,
adopt ${a}_{t\mathrm{m}}={a}_{t}^{T}$) and accept the additional
error thus incurred, assuming it is smaller than the 2 % relative
error due to the linearization estimated by
Melbourne et al. (1994). This is a reasonable assumption given the high
quality of our forward-modeled profiles derived from ECMWF short-range
forecast refractivity fields.

As a consequence we have to accept that the overall inaccuracy of our random
uncertainty estimate cannot be brought below 2 %. Therefore, to ensure that
our simplified estimate does not underestimate the real uncertainty, we
account for the linearization error by multiplying a factor *f*_{uαlin}=1.02 to the uncertainty of the
retrieved GO bending angle

$$\begin{array}{}\text{(A29)}& {u}_{\mathit{\alpha}\mathrm{G},i}^{r}={f}_{u\mathit{\alpha}\mathrm{lin}}\cdot {u}_{\mathit{\alpha}\mathrm{G}\left({z}_{\mathrm{a}}\right),i}^{r}.\end{array}$$

In this way we acknowledge that although the calculation of the
*state* of the bending angle does not make use of the
linearization, and therefore the linearization does not increase the
uncertainty of the *state* profile, it may increase the error in
the uncertainty estimate itself.

Finally, the ${u}_{\mathit{\alpha}\mathrm{G}}^{r}$ profile is also interpolated to the
common monotonic impact altitude grid *z*_{a}.

In the GO approximation, the bending angle values at each grid point only
depend on the Doppler shift values of the same grid points; i.e., the existing
correlations between the errors at different levels are left
unchanged: i.e., **R**_{αG}=**R**_{Dr}.
The covariance matrix can hence be calculated by recombining the
Doppler shift correlation matrix with the propagated uncertainties (item 2.8 in Fig. 2),

$$\begin{array}{}\text{(A30)}& {C}_{\mathit{\alpha}\mathrm{G},ij}={u}_{\mathit{\alpha}\mathrm{G},i}^{r}\cdot {u}_{\mathit{\alpha}\mathrm{G},j}^{r}\cdot {R}_{D\mathrm{r},ij}.\end{array}$$

For the propagation of the *estimated systematic uncertainty* (item 2.4 in Fig. 2) three types of potential
systematic errors adding to the impact parameter uncertainty ${u}_{at}^{s}$, and
consequentially the bending angle uncertainty ${u}_{\mathit{\alpha}\mathrm{G}}^{s}$, are taken into
account: systematic errors in the Doppler shift, i.e., ${u}_{D\mathrm{r}}^{s}$;
systematic errors in the velocities of the satellites, i.e., ${u}_{\mathit{v}R}^{s}$
and ${u}_{\mathit{v}T}^{s}$; and systematic errors in the positions of the
satellites, i.e., ${u}_{\mathit{r}R}^{s}$ and ${u}_{\mathit{r}T}^{s}$. The latter two
orbit-borne types are interpreted as apparent systematic uncertainties
(Sect. 2.2), while the excess phase-borne uncertainty
${u}_{D\mathrm{r}}^{s}$ is a basic systematic uncertainty.

For the propagation of these estimated systematic uncertainties to
${u}_{\mathit{\alpha}\mathrm{G}}^{s}$,
Eqs. (A22)–(A24) are linearized around
the retrieved state quantities (serving as zero-order state), and no terms
higher than first-order are kept. Then *ϕ*_{R} and *ϕ*_{T} in
Eq. (A22) are substituted by the linearized versions of
Eqs. (A23) and (A24), and the resulting
equation is solved (level by level) for the impact parameter ${a}_{t,i}=f({D}_{\mathrm{r},i},{r}_{\mathrm{R},i},{r}_{\mathrm{T},i},{v}_{\mathrm{R},i},{v}_{\mathrm{T},i})$,
with $i=\mathrm{1},\mathrm{2},\mathrm{\dots},N$. Adopting the first-order deviations to represent the estimated systematic
uncertainties, we obtain

$$\begin{array}{ll}\text{(A31)}& {\displaystyle}& {\displaystyle}{u}_{at,i}^{s}={\displaystyle \frac{\mathrm{1}}{{k}_{at,i}}}{\displaystyle}& {\displaystyle}\phantom{\rule{0.125em}{0ex}}\sqrt{{\left({u}_{D\mathrm{r},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{v\mathrm{R},i}\cdot {u}_{v\mathrm{R},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{r\mathrm{R},i}\cdot {u}_{r\mathrm{R},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{v\mathrm{T},i}\cdot {u}_{v\mathrm{T},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{r\mathrm{T},i}\cdot {u}_{r\mathrm{T},i}^{s}\right)}^{\mathrm{2}}},\end{array}$$

where

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{k}_{at,i}={\left.{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {\mathit{\varphi}}_{\mathrm{R}}}}\right|}_{i}\cdot {\left.{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{R}}}{\partial {a}_{t}}}\right|}_{i}+{\left.{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {\mathit{\varphi}}_{\mathrm{T}}}}\right|}_{i}\cdot {\left.{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{T}}}{\partial {a}_{t}}}\right|}_{i}=-{v}_{\mathrm{R},i}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\cdot \mathrm{sin}{\mathit{\varphi}}_{\mathrm{R}i}\cdot {\displaystyle \frac{\mathrm{1}}{\sqrt{{r}_{\mathrm{R},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}}-{v}_{\mathrm{T},i}\cdot \mathrm{sin}{\mathit{\varphi}}_{\mathrm{T}i}{\displaystyle \frac{\mathrm{1}}{\sqrt{{r}_{\mathrm{T},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}},\\ {\displaystyle}& {\displaystyle}{k}_{v\mathrm{R},i}={\left.-{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {v}_{\mathrm{R}}}}\right|}_{i}=-\mathrm{cos}{\mathit{\varphi}}_{\mathrm{R}i},\\ {\displaystyle}& {\displaystyle}{k}_{v\mathrm{T},i}={\left.-{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {v}_{\mathrm{T}}}}\right|}_{i}=-\mathrm{cos}{\mathit{\varphi}}_{\mathrm{T}i},\\ {\displaystyle}& {\displaystyle}{k}_{r\mathrm{R},i}={\left.{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {\mathit{\varphi}}_{\mathrm{R}}}}\right|}_{i}\cdot {\left.{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{R}}}{\partial {r}_{\mathrm{R}}}}\right|}_{i}={\displaystyle \frac{{v}_{\mathrm{R},i}\cdot \mathrm{sin}{\mathit{\varphi}}_{\mathrm{R}i}\cdot {a}_{t,i}}{{r}_{\mathrm{R},i}\sqrt{{r}_{\mathrm{R},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}},\\ \text{(A32)}& {\displaystyle}& {\displaystyle}{k}_{r\mathrm{T},i}={\left.{\displaystyle \frac{\partial {D}_{\mathrm{r}}}{\partial {\mathit{\varphi}}_{\mathrm{T}}}}\right|}_{i}\cdot {\left.{\displaystyle \frac{\partial {\mathit{\varphi}}_{\mathrm{T}}}{\partial {r}_{\mathrm{T}}}}\right|}_{i}={\displaystyle \frac{{v}_{\mathrm{T},i}\cdot \mathrm{sin}{\mathit{\varphi}}_{\mathrm{T}i}\cdot {a}_{t,i}}{{r}_{\mathrm{T},i}\sqrt{{r}_{\mathrm{T},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}}.\end{array}$$

A number of simplifications have been made to arrive at this result. First, the last term in Eq. A22 is disregarded since errors in the positions are assumed to be constant with respect to the short time duration of an RO event; remaining errors $\mathrm{\Delta}{\dot{r}}_{\mathrm{RT}}$ after taking the derivative are therefore of higher order. Next, orbit position and velocity uncertainties are both assumed to be constant within the short duration of an event, and the velocity uncertainties obtained are interpreted as uncertainties along the direction of the velocity vector. Consequentially, the uncertainty is also projected along with the vector into the ray path direction. A more conservative estimation (which we consider overly conservative in context) would interpret the uncertainties as ellipsoids at the velocity vectors' heads and would hence take the full magnitude of the uncertainties along the ray path direction (without projection).

Furthermore, since all error sources (the processing of the occultation tracking data and
the POD for transmitter and receiver) are essentially independent from each other,
the different input uncertainties are assumed to be uncorrelated.
Finally, we reasonably assumed the errors of the angle between the position and
velocity vectors (*η*) to be negligible (${u}_{\mathit{\eta}}^{s}\approx \mathrm{0}$) for the purpose here,
for both the transmitter and receiver.

In order to finally derive the systematic uncertainty of the bending angle from the impact parameter's uncertainty, we continue with a linearization of Eq. (A25) and arrive at

$$\begin{array}{ll}\text{(A33)}& {\displaystyle}& {\displaystyle}{u}_{\mathit{\alpha}\mathrm{G},i}^{s}={\displaystyle}& {\displaystyle}\phantom{\rule{0.125em}{0ex}}\sqrt{{\left({u}_{\mathit{\theta}\mathrm{RT},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{at,i}\cdot {u}_{at,i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{r\mathrm{R},i}\cdot {u}_{r\mathrm{R},i}^{s}\right)}^{\mathrm{2}}+{\left({k}_{r\mathrm{T},i}\cdot {u}_{r\mathrm{T},i}^{s}\right)}^{\mathrm{2}}},\end{array}$$

where

$$\begin{array}{ll}\text{(A34)}& {\displaystyle}& {\displaystyle}{k}_{at,i}={\left.{\displaystyle \frac{\partial \mathit{\alpha}}{\partial {a}_{t}}}\right|}_{i}={\displaystyle \frac{\mathrm{1}}{\sqrt{{r}_{\mathrm{R},i}^{\mathrm{2}}+{a}_{t,i}^{\mathrm{2}}}}}+{\displaystyle \frac{\mathrm{1}}{\sqrt{{r}_{\mathrm{T},i}^{\mathrm{2}}+{a}_{t,i}^{\mathrm{2}}}}},{\displaystyle}& {\displaystyle}{k}_{r\mathrm{R},i}={\left.{\displaystyle \frac{\partial \mathit{\alpha}}{\partial {r}_{\mathrm{R}}}}\right|}_{i}=-{\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{R},i}\sqrt{{r}_{\mathrm{R},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}},\\ {\displaystyle}& {\displaystyle}{k}_{r\mathrm{T},i}={\left.{\displaystyle \frac{\partial \mathit{\alpha}}{\partial {r}_{\mathrm{T}}}}\right|}_{i}=-{\displaystyle \frac{{a}_{t,i}}{{r}_{\mathrm{T},i}\sqrt{{r}_{\mathrm{T},i}^{\mathrm{2}}-{a}_{t,i}^{\mathrm{2}}}}}.\end{array}$$

In practice we separately calculate the basic and apparent systematic uncertainty estimates (${u}_{\mathit{\alpha}\mathrm{G}}^{\mathrm{b}}$ from the first RHS terms in Eqs. A31 and A33, ${u}_{\mathit{\alpha}\mathrm{G}}^{\mathrm{a}}$ from the orbit-borne terms) and afterwards obtain ${u}_{\mathit{\alpha}\mathrm{G}}^{s}$ as a combined result, in order to enable separate propagation in subsequent processing steps.

The *resolution* profile remains unaffected by the bending angle retrieval,
since the level-by-level approach of the algorithm does not create extra
correlation and further vertical smoothing, so that
*τ*_{αG}=*τ*_{Dr} and *w*_{αG}=*w*_{Dr}.

After the GO bending angle, the WO bending angle *state* profile
*α*_{W}(*z*_{a}) is retrieved (item 2.7 in Fig. 2) from excess phase profile *L*_{r}(*t*) (and its uncertainties) and the amplitude profile *A*_{r}(*t*)
(and uncertainties) in a WO retrieval following
Gorbunov and Kirchengast (2015, 2018). Along with the state
profile, the *systematic uncertainty* profile ${u}_{\mathit{\alpha}\mathrm{W}}^{s}$,
the covariance matrix **C**_{αW}, and the
*resolution* profile *w*_{αW} are derived.

The covariance matrix **C**_{αW} is then decomposed to
*random uncertainty* profile ${u}_{\mathit{\alpha}\mathrm{W}}^{r}$ and
correlation matrix **R**_{αW} in the same form as done
above for **C**_{Dr} (Eqs. A19 and A20)
and **C**_{LF} (Eqs. A7 and A8). The
estimated systematic uncertainty ${u}_{\mathit{\alpha}\mathrm{W}}^{s}$ is
composed of a basic systematic uncertainty ${u}_{\mathit{\alpha}\mathrm{W}}^{\mathrm{b}}$,
propagated through the wave-optical retrieval from the excess phase
uncertainty ${u}_{L\mathrm{r}}^{s}$, and an apparent systematic
uncertainty ${u}_{\mathit{\alpha}\mathrm{W}}^{\mathrm{a}}$, estimated in the lower
troposphere as residual bias uncertainty of a regression-based boundary layer
bias correction (Gorbunov and Kirchengast, 2018).

The WO bending angle retrieval algorithm and the associated uncertainty propagation algorithm are not explicitly described here; the reader is referred to Gorbunov and Kirchengast (2015) and Gorbunov and Kirchengast (2018). However, we have prepared the merging with the WO bending angle variables (they will be actually merged in when the WO tests within the rOPS are complete), which is described next.

The *α*_{W} profile, prepared on the common
grid *z*_{a}, and the *α*_{G} profile are
merged over an upper-tropospheric transition range (item 2.9 in Fig. 2). The
gradual transition, weighted by a symmetric half-sine function, has a defined impact
altitude transition of half-width $\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}=\mathrm{2}$ km
around transition altitude ${z}_{\mathrm{a}}^{\mathrm{GW}}$, allowed within 9 km to
14 km, estimated from *α*_{G} data quality. The resulting
merged bending angle profile *α*_{M} is

$$\begin{array}{}\text{(A35)}& {\mathit{\alpha}}_{\mathrm{M},i}={\mathit{\gamma}}_{i}\cdot {\mathit{\alpha}}_{\mathrm{G},i}+(\mathrm{1}-{\mathit{\gamma}}_{i})\cdot {\mathit{\alpha}}_{\mathrm{W},i},\end{array}$$

where the weighting profile *γ* is formulated as

$$\begin{array}{}\text{(A36)}& {\mathit{\gamma}}_{i}=\left\{\begin{array}{l}\mathrm{1}\phantom{\rule{1em}{0ex}}\text{for}{z}_{\mathrm{a},i}\ge {z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\\ \mathrm{0.5}\cdot \left[\mathrm{sin}\left(\frac{\mathit{\pi}}{\mathrm{2}}\cdot \frac{{z}_{\mathrm{a},i}-{z}_{\mathrm{a}}^{\mathrm{GW}}}{\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}}\right)+\mathrm{1}\right]\\ \phantom{\rule{1em}{0ex}}\text{for}|{z}_{\mathrm{a},i}-{z}_{\mathrm{a}}^{\mathrm{GW}}|\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\\ \mathrm{0}\phantom{\rule{1em}{0ex}}\text{for}{z}_{\mathrm{a},i}\le {z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}.\end{array}\right.\end{array}$$

To determine the random uncertainties for the merged GO–WO input bending angle, we need to merge the covariance matrices of both bending angles.

We can assume both incoming covariance matrices **C**_{αG} and **C**_{αW} are provided on
the common monotonic target grid *z*_{a} (i.e., also the WO uncertainties and
correlations are interpolated to this common grid before the
merger). We further can reasonably assume that there are no
cross-correlations between GO and WO errors, given the very different
retrieval schemes. Based on this we can compose the covariance matrix of
the merged bending angle profile, **C**_{αM} (item 2.10 in Fig. 2) as follows. Outside the merging zone
(i.e., outside of ${z}_{\mathrm{a}}^{\mathrm{GW}}\pm \mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$) we can assign

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{C}_{\mathit{\alpha}\mathrm{M},ij}=\\ \text{(A37)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\left\{\begin{array}{ll}{C}_{\mathit{\alpha}\mathrm{G},ij}& \text{for}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},i}{z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\\ & \text{and}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},j}{z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\\ {C}_{\mathit{\alpha}\mathrm{W},ij}& \text{for}{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}}\\ & \text{and}{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}}\\ \mathrm{0}& \text{for}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},i}{z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\\ & \text{and}{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}}\\ \mathrm{0}& \text{for}{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}}\\ & \text{and}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},j}{z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}\end{array}\right.,\end{array}$$

while within the merging zone we can assign

$$\begin{array}{}\text{(A38)}& {C}_{\mathit{\alpha}\mathrm{M},ij}={\mathit{\gamma}}_{i}{\mathit{\gamma}}_{j}{C}_{\mathit{\alpha}\mathrm{G},ij}+(\mathrm{1}-{\mathit{\gamma}}_{i})(\mathrm{1}-{\mathit{\gamma}}_{j}){C}_{\mathit{\alpha}\mathrm{W},ij},\end{array}$$

wherein *i* is understood such that ${z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}>{z}_{\mathrm{a},i}>{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$ and *j* such that ${z}_{\mathrm{aTop}}>{z}_{\mathrm{a},j}>{z}_{\mathrm{aBot}}$.

Because of the symmetry of the covariance matrix, the covariance elements in the merging zone orthogonal to the one above, i.e., for ${z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}>{z}_{\mathrm{a},j}>{z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$ and ${z}_{\mathrm{aTop}}>{z}_{\mathrm{a},i}>{z}_{\mathrm{aBot}}$, are calculated according to the same formula.

Due to the linear relation between *α*_{M}, *α*_{G},
and *α*_{W}, expressed by Eq. (A35), a bias
${u}_{\mathit{\alpha}\mathrm{G}}^{s}$ in the GO bending angle and a
bias ${u}_{\mathit{\alpha}\mathrm{W}}^{s}$ in the WO bending angle can be
combined linearily as well, and we can compute the *estimated systematic uncertainty* of the merged bending angle ${u}_{\mathit{\alpha}\mathrm{M}}^{s}$
according to (item 2.9 in Fig. 2)

$$\begin{array}{}\text{(A39)}& {u}_{\mathit{\alpha}\mathrm{M},i}^{s}={\mathit{\gamma}}_{i}\cdot {u}_{\mathit{\alpha}\mathrm{G},i}^{s}+(\mathrm{1}-{\mathit{\gamma}}_{i})\cdot {u}_{\mathit{\alpha}\mathrm{W},i}^{s}.\end{array}$$

In practice this formulation is again applied separately for the basic and apparent systematic uncertainty estimates, afterwards obtaining the ${u}_{\mathit{\alpha}\mathrm{M}}^{s}$ profile as a combined result, in order to allow separate propagation in subsequent processing steps.

The *resolution* profile of the bending angle, *w*_{αM},
is equal to the GO resolution *w*_{αG} above ${z}_{\mathrm{a}}^{\mathrm{GW}}+\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$, equal to the WO resolution *w*_{αW}
below ${z}_{\mathrm{a}}^{\mathrm{GW}}-\mathrm{\Delta}{z}_{\mathrm{a}}^{\mathrm{GW}}$ and has a transition
with transition weight *γ*_{i} in between, again following the linear
formulation such as in Eqs. (A35) and (A39).

Because the integration and testing of the uncertainty propagation through the rOPS WO bending angle retrieval are currently still ongoing, as noted in Sect. A2.2 above, the examples shown in this study are all GO-only; i.e., only the GO retrieval is performed. The merging algorithm as described is ready to include the WO bending angles, however.

In order to retrieve the atmospheric bending angle profile *α*_{r},
ionospheric effects need to be corrected for, using the retrieved bending angles
from each transmitter frequency channel.
Since the only GNSS constellation currently used for RO is the GPS – except for recent
initial data from the Chinese GNOS instrument using BeiDou signals
(Liao et al., 2016; Bai et al., 2018) – the data characteristics of the GPS case (with $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}$,
*f*_{T1}=1.57542 GHz, and *f*_{T2}=1.22760 GHz)
are in the prime focus of this section.

This concerns in particular special provisions for the minor (L2) channel noise
filtering and its tropospheric extrapolation. In general the algorithms
are applicable for any of the available GNSS systems, however; if the minor channel
(*f*_{T2}) delivers similar data quality to the major one (*f*_{T1}),
the special provisions for the former will practically have no effect.

Before applying the dual-frequency ionospheric correction, the merged
bending angle *state* profiles *α*_{M,k}(*z*_{a})
at the common *z*_{a} grid are filtered with further BWS low-pass filter
operations, and the minor channel is extrapolated.

For *α*_{M1} the filter is set to the same cutoff frequency as the basic
BWS filter preceding the Doppler derivation (i.e., *f*_{c1}=2.5 Hz), ensuring
a reliable reference resolution and basic smoothness of the whole merged profile. For
filtering of *α*_{M2} a (GPS L2) noise-minimization algorithm is used,
following the approach of Sokolovskiy et al. (2009) for optimal filtering for ionospheric correction.
We search for minimized noise employing a flexible cutoff frequency *f*_{c2}∈*{*2.5, 2, 10∕7, 1, 5∕7, 0.5 Hz*}*, corresponding to using cutoff periods *τ*_{c2}
from 0.4 to 2 s and sample widths of *M*=40 to *M*=200 (for BWS filter
design details see Sect. A1.1).

We adopt the cutoff frequency *f*_{c2} for *α*_{M2}
filtering that minimizes the noise fluctuations of the ionosphere-corrected
atmospheric bending angle delta profile
$\mathit{\delta}{\mathit{\alpha}}_{\mathrm{rm}}^{fc\mathrm{2}}\left({z}_{\mathrm{a}}\right)={\mathit{\alpha}}_{\mathrm{r}}^{fc\mathrm{2}}\left({z}_{\mathrm{a}}\right)-{\mathit{\alpha}}_{\mathrm{m}}\left({z}_{\mathrm{a}}\right)$ when
evaluated over the mesospheric altitude range between 50 and 70 km
(similar to the functional minimization of Sokolovskiy et al., 2009; Eq. 4
therein). At these high altitudes the residual atmospheric mean signal after
subtraction of the forward-modeled signal *α*_{m}(*z*_{a})
is very small (< 0.03–0.3 µrad), and therefore the noise level
representative for the given RO event is well quantifiable.

The weight matrix of the BWS filter, ${\mathbf{A}}_{k}^{\mathrm{BWS}}$, is
determined for both frequencies analogously to Eqs. (A3)
to (A5). When using the baseband approach with model profile
*α*_{m} to create the delta profile
*δ**α*_{Mm} with elements

$$\begin{array}{}\text{(A40)}& \mathit{\delta}{\mathit{\alpha}}_{\mathrm{Mm}i,k}={\mathit{\alpha}}_{\mathrm{M}i,k}-{\mathit{\alpha}}_{\mathrm{m}i},\end{array}$$

the filtered bending angle is then (item 3.1 in Fig. 2)

$$\begin{array}{}\text{(A41)}& {\mathit{\alpha}}_{\mathrm{F}i,k}={\mathit{\alpha}}_{\mathrm{m}i}+\sum _{j=\mathrm{0}}^{N}{A}_{ij,k}^{\mathrm{BWS}}\cdot \mathit{\delta}{\mathit{\alpha}}_{\mathrm{Mm}j,k},\end{array}$$

where $i,j\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{\dots},N\mathit{\}}$ and $k\in \mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}$.

Due to the stronger power of the L1 signal for (most of) the GPS satellites, the GPS signals of
both frequencies are not of the same quality, and the L2 data (for those satellites where encrypted
and hence power-degraded L2 signals are transmitted) do not reach down as far as the L1 data
(i.e., *z*_{aBot2}>*z*_{aBot1}).
If due to this reason *α*_{F2} does not reach down as far as
*α*_{F1} and *z*_{aBot2}≤*z*_{aBot2Max} (with
*z*_{aBot2Max} currently set to 15 km), a *tropospheric bending angle extrapolation* (TBAE) is applied in order to artificially extend *α*_{M2} to
also reach down to *z*_{aBot1} (item 3.3 in Fig. 2).

Briefly summarized, this TBAE is currently implemented as follows.
A linear gradient profile for the difference profile between the two bending angles,
${\mathit{\alpha}}_{\mathrm{F}\mathrm{12}}=({\mathit{\alpha}}_{\mathrm{F}\mathrm{1}}-{\mathit{\alpha}}_{\mathrm{F}\mathrm{2}})$,
is estimated by a least squares fit over a sufficiently wide impact altitude range from
*z*_{aBot2} upward (as wide as the extrapolation range, at least 10 km).
This gradient profile is then linearly extended down to *z*_{aBot1} and
subtracted from *α*_{F1}, to obtain the extrapolated part of
*α*_{F2} from *z*_{aBot2} to *z*_{aBot1}.
If *z*_{aBot2}>*z*_{aBot2Max}, then no TBAE is
performed since the extrapolation range is considered too large.
Details are provided by Kirchengast et al. (2017b), where the most recent version
of the atmospheric bending angle derivation is described that includes this
*α*_{F12} extrapolation in a further advanced form.

For the propagation of the *estimated random uncertainty* we get (item 3.2 in Fig. 2)

$$\begin{array}{}\text{(A42)}& {\mathbf{C}}_{\mathit{\alpha}\mathrm{F},k}={\mathbf{A}}_{k}^{\mathrm{BWS}}\cdot {\mathbf{C}}_{\mathit{\alpha}\mathrm{M},k}\cdot {\left({\mathbf{A}}_{k}^{\mathrm{BWS}}\right)}^{\mathrm{T}}\end{array}$$

for the bending angle error covariance matrices of the leading (*k*=1) and minor (*k*=2) channel.

If a TBAE is applied to *α*_{F2}, the random uncertainty
of *α*_{F2} below *z*_{aBot2} is equal to that of
*α*_{F1}, because the noise is “copied” from *α*_{F1}
since the linear gradient profile from fitting *α*_{F12} is noise-free.
As a consequence, in these cases, we set the matrix elements of
**C**_{αF2} to (item 3.4 in Fig. 2)

$$\begin{array}{}\text{(A43)}& {C}_{\mathit{\alpha}\mathrm{F}\mathrm{2},ij}=\left\{\begin{array}{ll}{\mathbf{C}}_{\mathit{\alpha}\mathrm{F}\mathrm{2},ij}& \text{for}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}\mathrm{2}}\\ & \text{and}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}\mathrm{2}}\\ {\mathbf{C}}_{\mathit{\alpha}\mathrm{F}\mathrm{1},ij}& \text{for}{z}_{\mathrm{aBot}\mathrm{2}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}\mathrm{1}}\\ & \text{and}{z}_{\mathrm{aBot}\mathrm{2}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}\mathrm{1}}\\ \mathrm{0}& \text{for}{z}_{\mathrm{aBot}\mathrm{2}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}\mathrm{1}}\\ & \text{and}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}\mathrm{2}}\\ \mathrm{0}& \text{and}{z}_{\mathrm{aTop}}{z}_{\mathrm{a},i}{z}_{\mathrm{aBot}\mathrm{2}}\\ & \text{for}{z}_{\mathrm{aBot}\mathrm{2}}{z}_{\mathrm{a},j}{z}_{\mathrm{aBot}\mathrm{1}}\end{array}\right..\end{array}$$

**C**_{αF1} and
**C**_{αF2} can then be decomposed as needed into
${u}_{\mathit{\alpha}\mathrm{F}\mathrm{1}}^{r}$, **R**_{αF1}, and
${u}_{\mathit{\alpha}\mathrm{F}\mathrm{2}}^{r}$, **R**_{αF2}, respectively.
Kirchengast et al. (2017b) describe the most recent version consistent with a further
advanced form of the TBAE, where the separate assignments according to
Eq. (A43) are no longer needed.

The *estimated systematic uncertainties* ${u}_{\mathit{\alpha}\mathrm{M},k}^{s}$
(in practice the basic and apparent systematic uncertainty estimates separately)
are filtered with the same filter settings as for the state profiles
(item 3.1 in Fig. 2) and are thus obtained in the form

$$\begin{array}{}\text{(A44)}& {u}_{\mathit{\alpha}\mathrm{F}i,k}^{s}=\sum _{j=\mathrm{0}}^{N}{A}_{ij,k}^{\mathrm{BWS}}\cdot {u}_{\mathit{\alpha}\mathrm{M}j,k}^{s}.\end{array}$$

Since these are smooth profiles, they are marginally changed by this low-pass filtering. The systematic uncertainty component contributed by the TBAE to the estimated systematic uncertainty is added after the ionospheric correction (see next subsection).

As for the basic low-pass filtering of excess phases (Sect. A1.1),
the *resolution* profiles of the filtered bending angles *w*_{αF1}
and *w*_{αF2} are determined by the cutoff frequencies
*f*_{c1} and *f*_{c2} of the BWS filters, following
Eqs. (A11) and (A13).

Based on the filtered and sometimes extrapolated *state* profiles
*α*_{F1} and *α*_{F2}, the ionospheric refractive
effects are corrected for by the standard dual-frequency correction of bending angles
(Vorob'ev and Krasil'nikova, 1994) used in the *f*_{T1}–*f*_{T2}
difference profile form (Sokolovskiy et al., 2009) (item 3.5 in Fig. 2).
For the elements of the retrieved atmospheric bending angle profile
*α*_{r} we thus get

$$\begin{array}{}\text{(A45)}& {\mathit{\alpha}}_{\mathrm{r},i}={\mathit{\alpha}}_{\mathrm{F}\mathrm{1},i}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}\cdot \mathit{\delta}{\mathit{\alpha}}_{\mathrm{F}\mathrm{12},i},\end{array}$$

where

$$\begin{array}{}\text{(A46)}& \mathit{\delta}{\mathit{\alpha}}_{\mathrm{F}\mathrm{12},i}={\mathit{\alpha}}_{\mathrm{F}\mathrm{1},i}-{\mathit{\alpha}}_{\mathrm{F}\mathrm{2},i}\end{array}$$

and

$$\begin{array}{}\text{(A47)}& {\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}={\displaystyle \frac{{f}_{\mathrm{T}\mathrm{2}}^{\mathrm{2}}}{{f}_{\mathrm{T}\mathrm{1}}^{\mathrm{2}}-{f}_{\mathrm{T}\mathrm{2}}^{\mathrm{2}}}}.\end{array}$$

Propagated through the operator of the ionospheric correction (Eq. A45,
currently used here in the classical form with *f*_{T1} and *f*_{T2}
terms), the *estimated random uncertainty* of the resulting atmospheric bending angle,
expressed by the error covariance matrix **C**_{αr}
(item 3.6 in Fig. 2), is obtained as

$$\begin{array}{}\text{(A48)}& {\mathbf{C}}_{\mathit{\alpha}\mathrm{r}}={\left(\mathrm{1}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}\right)}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathbf{C}}_{\mathit{\alpha}\mathrm{F}\mathrm{1}}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathbf{C}}_{\mathit{\alpha}\mathrm{F}\mathrm{2}}.\end{array}$$

**C**_{αr} can then also be decomposed into
${u}_{\mathit{\alpha}r}^{r}$ and **R**_{αr} with the usual
equations (cf., e.g., Eqs. A19 and A20).

Equation (A45) is as well applied to propagate the *estimated systematic uncertainty*
(in practice the basic and apparent systematic uncertainty estimates separately)
through the ionospheric correction using (item 3.5 in Fig. 2)

$$\begin{array}{}\text{(A49)}& {u}_{\mathit{\alpha}\mathrm{r},i}^{s}={u}_{\mathit{\alpha}\mathrm{F}\mathrm{1},i}^{s}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}\cdot \left({u}_{\mathit{\alpha}\mathrm{F}\mathrm{1},i}^{s}-{u}_{\mathit{\alpha}\mathrm{F}\mathrm{2},i}^{s}\right),\end{array}$$

where it is assumed that the systematic errors in
*α*_{F1} and *α*_{F2} are positively
correlated, i.e., have the same sign, and the associated uncertainty
estimates are hence subtracted from one another (as the bending angles
are in Eq. A46). This assumption is reasonable,
since the same sources of non-ionospheric systematic effects apply to
both frequency channels (Doppler shift, orbit velocity, and orbit
position uncertainties).

In the case of TBAE, Eq. (A49) needs to be supplemented below
*z*_{aBot2}, since additional
uncertainties ${u}_{\mathit{\alpha}\mathrm{2}\mathrm{TE}}^{s}$ arise from the errors made in
the fitting parameters and in the extrapolation model (linear
extrapolation) of the TBAE. Hence, for the range ${z}_{\mathrm{aBot}\mathrm{2}}>{z}_{\mathrm{a},i}\ge {z}_{\mathrm{aBot}\mathrm{1}}$,

$$\begin{array}{}\text{(A50)}& {u}_{\mathit{\alpha}\mathrm{r},i}^{s}={u}_{\mathit{\alpha}\mathrm{r}}^{s}\left({z}_{\mathrm{aBot}\mathrm{2}}\right)+{u}_{\mathit{\alpha}\mathrm{2}\mathrm{TE},i}^{s},\end{array}$$

with ${u}_{\mathit{\alpha}\mathrm{2}\mathrm{TE}}^{s}$ being the conservative estimate for
additional (apparent) systematic uncertainty within the extrapolated
impact altitude range. We set ${u}_{\mathit{\alpha}\mathrm{2}\mathrm{TE}}^{s}$ to
zero at *z*_{aBot2} and linearly increase it from
there downwards with a gradient of 1 µrad per 10 km (an
experience-based best guess; cf. Scherllin-Pirscher et al., 2011b, a,
who also address aspects of such tropospheric extrapolation in their discussions
of error sources). It is interpreted as an apparent systematic uncertainty estimate,
since due to the linear fit-based TBAE construction its event-to-event bias character
will be essentially random (Scherllin-Pirscher et al., 2011b).

Also, the ionospheric correction currently applied in
the rOPS is just a first-order correction, which will leave higher-order
residual ionospheric errors in *α*_{r} (e.g., Syndergaard, 2000; Danzer et al., 2013; Liu et al., 2013, 2015; Healy and Culverwell, 2015). The uncertainty from higher-order *residual ionospheric biases* (RIBs), ${u}_{\mathrm{RIB}}^{s}$, is therefore added to the
propagated (basic) systematic uncertainty. ${u}_{\mathrm{RIB}}^{s}$ is interpreted
as basic systematic uncertainty, since the higher-order ionospheric residuals may
not vanish in ensemble-of-events averaging.
The other non-ionospheric sources of systematic errors and the RIBs can be
reasonably assumed to be uncorrelated. The total estimated systematic uncertainty
of the retrieved atmospheric bending angle *α*_{r} hence is

$$\begin{array}{}\text{(A51)}& {u}_{\mathit{\alpha}\mathrm{r},i}^{s}=\sqrt{{\left({u}_{\mathit{\alpha}\mathrm{r},i}^{s}\right)}^{\mathrm{2}}+{\left({u}_{\mathrm{RIB}}^{s}\right)}^{\mathrm{2}}}.\end{array}$$

Based on previous studies (e.g., Liu et al., 2013, 2015; Danzer et al., 2013, 2015), ${u}_{\mathrm{RIB}}^{s}$ is taken to be constant along the entire profile and is estimated to amount to 0.05 µrad. These last two components, ${u}_{\mathit{\alpha}\mathrm{2}\mathrm{TE}}^{s}$ and ${u}_{\mathrm{RIB}}^{s}$, are indicated as item 3.7 in Fig. 2. It is clear that this initial systematic uncertainty estimation can be significantly improved by future dedicated work on better quantifying and (if suitable) correcting for the systematic uncertainty components.

The *resolution* of the retrieved bending angle, *w*_{αr}, essentially
corresponds to the higher resolution of the two bending angle profiles *α*_{F1} and
*α*_{F2}, and thus generally closely matches *w*_{αF1} in most
cases. As a simple but robust and suitable estimate, assuming that the
resolutions of *α*_{r} and *α*_{F1} scale in the
same way as the correlation lengths *l*_{αr} and *l*_{αF1}
(derived from **R**_{αr} and **R**_{αF1} as
described for **R**_{LF} in Eq. A9),
we compute *w*_{αr} as

$$\begin{array}{}\text{(A52)}& {w}_{\mathit{\alpha}\mathrm{r},i}={\displaystyle \frac{{l}_{\mathit{\alpha}\mathrm{r},i}}{{l}_{\mathit{\alpha}\mathrm{F}\mathrm{1},i}}}\cdot {w}_{\mathit{\alpha}\mathrm{F}\mathrm{1},i}.\end{array}$$

In concluding, we note that the atmospheric bending angle derivation algorithms
used in this study – i.e., the adaptive filtering, TBAE, and ionospheric correction parts
as described in this section – have recently received further advancement towards
a form fully based on the combination of *α*_{F1} and the
difference profile *α*_{F12} (rather than of *α*_{F1}
and *α*_{F2}), more aligned with the concept of Sokolovskiy et al. (2009).
A detailed description of this most recent version is found in Kirchengast et al. (2017b).

Appendix B: Variance propagation for comparison

Back to toptop
The full covariance propagation applied to propagate random uncertainties
requires numerically “expensive” matrix operations, and therefore considerable
efforts were made to seize opportunities for reducing the number of numerical
operations (e.g., by only calculating with those elements of the band matrix
**A**^{BWS} which lie within the width of the filter window).

However, as demonstrated in Sect. 4, simplification to a mere variance propagation (i.e., only considering the diagonal elements of the covariance matrices) is not reasonably possible because it leads to an unacceptable overestimation of random uncertainties. This overestimation occurs since the influence of the covariance elements – and thus for example the partially compensating impact of the negative side peaks in the correlation functions – is disregarded.

Here we state the two equations used to obtain the variances-only propagation results shown for comparison purposes in Fig. 9: the estimated random uncertainty was propagated through the BWS filter using

$$\begin{array}{}\text{(B1)}& {\left({u}_{\mathit{\alpha}\mathrm{F}i,k}^{r}\right)}^{\mathrm{2}}=\sum _{j=\mathrm{0}}^{N}{\left({A}_{ij,k}^{\mathrm{BWS}}\right)}^{\mathrm{2}}\cdot {\left({u}_{\mathit{\alpha}\mathrm{M}j,k}^{r}\right)}^{\mathrm{2}},\end{array}$$

and subsequently through the ionospheric correction using

$$\begin{array}{}\text{(B2)}& {\left({u}_{\mathit{\alpha}\mathrm{r},i}^{r}\right)}^{\mathrm{2}}={\left(\mathrm{1}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}\right)}^{\mathrm{2}}\cdot {\left({u}_{\mathit{\alpha}\mathrm{F}\mathrm{1},i}^{r}\right)}^{\mathrm{2}}+{\mathit{\gamma}}_{f\mathrm{T}\mathrm{12}}^{\mathrm{2}}\cdot {\left({u}_{\mathit{\alpha}\mathrm{F}\mathrm{2},i}^{r}\right)}^{\mathrm{2}}.\end{array}$$

Author contributions

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Author contributions.

JS and GK designed the study, including comments by MS, based on the concept and framework of uncertainty propagation into the Reference Occultation Processing system (rOPS), formulated by GK. JS elaborated the detailed algorithms, performed the computational implementation and the analysis, prepared the figures, and wrote the first draft of the paper; he was advised and supported in this work by GK and MS. As part of this support GK provided detailed design input and feedback and substantially contributed to the writing of the paper, and MS substantially contributed to the computational work. Based on a primary role of the first two authors, all authors contributed to consolidating the paper for submission and towards publication.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

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.

Acknowledgements

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Acknowledgements.

We thank UCAR/CDAAC Boulder for access to their RO excess phase and
orbit data as well as ECMWF Reading for access to their analysis and forecast
data. The work was funded by the Austrian Aeronautics and Space Agency of the
Austrian Research Promotion Agency (FFG-ALR) under projects
OPSCLIMPROP and OPSCLIMTRACE and by the European Space Agency
(ESA) under project MMValRO-E.

Edited by: Sean Healy

Reviewed by: two anonymous referees

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Short summary

We process global navigation satellite system radio occultation (RO) observations in a new way with integrated uncertainty propagation; in this study we focus on retrieving atmospheric bending angles from RO excess phase profiles. We find that this new approach within our novel Reference Occultation Processing System (rOPS) exploits the strengths of RO such as its high accuracy and long-term stability in a reliable manner for global climate monitoring and other weather and climate uses.

We process global navigation satellite system radio occultation (RO) observations in a new way...

Atmospheric Measurement Techniques

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