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.

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

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

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)

In Fig.

Schematic view of the main processors of the retrieval chain in the rOPS (L1a, L1b highlighted, L2a, L2b) and the main operators of the L1b processor (1, 2, 3), which are in the focus of this study.

The aim of the integrated uncertainty propagation in the rOPS is to
eventually propagate uncertainties along this

Detailed workflow for state retrieval and uncertainty propagation of
the main L1b operators from excess phase to atmospheric bending angle
profiles (1)–(3) and of the subroutines used in the MC testing framework
(a)–(g). The mathematical notation, including all symbols, is introduced in
Tables

Principal variables for the rOPS L1b uncertainty propagation.

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,

Vertical grids, coordinate variables, and specific settings for the rOPS L1b processing system.

Uncertainty propagation as covariance propagation from excess phase to
bending angle profiles has been outlined and demonstrated in a basic form,
by

Uncertainty propagation for the WO bending angle retrieval has
been implemented and demonstrated for simulated events by

Other ongoing rOPS retrieval advancements relevant to this study are the inclusion
of the high-altitude initialization algorithm, introduced by

Finally, related work and manuscript preparation on a new moist-air retrieval
algorithm (L2b) and corresponding L2b uncertainty propagation are ongoing

The paper is structured as follows. In Sect.

We follow the
Guide to the Expression of Uncertainty in Measurement
(

We categorize uncertainties into

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

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

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.

As a key variable characterizing

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

When the operator is linear, as is the case for the applicable L1b
operators, estimated

In addition to random uncertainties, systematic uncertainties, and the correlation
length, we also estimate

We note that the (half-)Fresnel scale

The input variables needed for the L1b uncertainty propagation, visible in
Fig.

Input profiles of retrieved excess phase

We used excess phase state profiles

Exploiting the linearity of the (linearized) retrieval operators, the
so-called

The model profiles used as zero-order states in the retrieval – i.e.,

ECMWF fields were chosen for their proven leading quality

While in the future the excess phase random and systematic uncertainty
profiles will be more rigorously estimated by the rOPS L1a processor

First, for both, the retrieved profile

Since the noise components responsible for the random uncertainty at
excess phase level are essentially uncorrelated at a sampling rate of

For the MC validation of the CP, error profile realizations

To create the error profiles, a representative

The

The constant

For CHAMP and COSMIC we set

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

We set the transmitter position and velocity
uncertainties to

Results for filtered excess phase profiles

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.

For each L1b retrieval step – i.e., segments (1), (2), and (3) in
Fig.

A concise definition of the variables involved is provided
in Table

To simplify the notation in the description, we suppress index

A Blackman windowed sinc (BWS) low-pass filter with a filter cutoff
frequency

The random uncertainty propagation algorithm, i.e., the covariance
propagation from

To propagate the estimated systematic uncertainty

Results for retrieved Doppler shift profiles

The resolution profile

The next step is a five-point differentiation operation (item 1.4 in
Fig.

As for the filtered excess phase, we apply CP (Eq.

For calculating the estimated systematic uncertainty, we use the state operator; i.e.,
we just differentiate

Results for geometric optics bending angle profiles

The resolution profile

The next operator is the GO bending angle retrieval in
which retrieved GO bending angle profiles

Figure

The estimated random uncertainties

The main contributions to the estimated systematic uncertainty

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

In the rOPS, along with the WO bending angle profile

The WO bending angle retrieval algorithm used is a canonical transform
(CT2) algorithm

In the rOPS bending angle retrieval the results from the WO retrieval,

Because the rOPS implementation of the WO uncertainty propagation

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.

Results for filtered bending angle profiles

To prepare the merged bending angle profiles

The chosen filter cutoff frequency for

The relevant covariance-propagated random uncertainties

Results for atmospheric bending angle profile

The estimated systematic uncertainty remains largely unchanged
(Fig.

The resolution of the filtered bending angle profiles (according to Eqs.

The final step of the L1b processor is the ionospheric correction (item 3.5
in Fig.

Figure

The residual higher-order ionospheric effects are accounted for by a
“conservative best-guess” value (

The resolution profile

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

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

Results from the validation of CP covariance matrices

Figure

Figure

The CP and MC lines match very well and show that the implemented CP
algorithm delivers correct results for the basic filtering step. For

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

The next row, Fig.

The results for the filtered bending angle

Finally, Fig.

Uncertainty propagation results for real-data ensembles from 15 July
2008, for the filtered excess phase profile

In order to demonstrate that a full CP is necessary to propagate
random uncertainties correctly, we also calculated random uncertainties

Uncertainty propagation results for real-data ensembles from 15 July
2008 for output profiles of the leading channel (

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

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

Figure

The number-of-events profile shows that most CHAMP events end between 5 and

Compared to CHAMP, the mean random uncertainty

For the real MetOp data (available here as a data set from UCAR/CDAAC, as
for CHAMP and COSMIC),

On the other hand, the average correlation length/resolution profile
of the

Figure

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

The estimated systematic uncertainty of the atmospheric bending angle

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

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

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.

The RO excess phase and orbit data used in the study are
available from UCAR/CDAAC Boulder, CO, USA, at

In this appendix the rOPS L1b uncertainty propagation algorithm is
introduced, following the L1b retrieval chain (Fig.

If not stated otherwise, elements of the vector-type vertical profiles are addressed
using subscript

All steps in Sects.

The Doppler differentiation (item 1.4 in Fig.

For this basic filtering the relative cutoff frequency

Comparison of the Blackman windowed sinc (BWS) low-pass filter and
boxcar (BC) filters based on a representative segment (between 30.3 and

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

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

The actually used sample width

The

The

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

For the

The

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

After the application of the BWS filter to the excess phase profiles

Based on careful tests of different formulations, we use a five-point
derivative scheme. The discretization of this five-point derivative

The

The covariance matrix is again (cf. Eqs.

For the

The

From the Doppler shift

Based on

All the variables in
Eqs. (

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

The impact parameter retrieval is solved iteratively, because it is
impossible to rearrange Eqs. (

After the GO bending angle retrieval, the bending angles of all GNSS
frequencies are interpolated to a common monotonic impact altitude
grid

For each element of

Because the impact parameter is only implicitly expressed in Eqs. (

This linearization establishes a direct relation between random uncertainties
of the Doppler shift

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

Finally, the

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.,

For the propagation of the

For the propagation of these estimated systematic uncertainties to

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 (

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. (

The

After the GO bending angle, the WO bending angle

The covariance matrix

The WO bending angle retrieval algorithm and the associated
uncertainty propagation algorithm are not explicitly described here;
the reader is referred to

The

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

Because of the symmetry of the covariance matrix, the covariance elements in the merging zone
orthogonal to the one above, i.e., for

Due to the linear relation between

The

Because the integration and testing of the uncertainty propagation through the
rOPS WO bending angle retrieval are currently still ongoing, as noted in
Sect.

In order to retrieve the atmospheric bending angle profile

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
(

Before applying the dual-frequency ionospheric correction, the merged
bending angle

For

We adopt the cutoff frequency

The weight matrix of the BWS filter,

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.,

Briefly summarized, this TBAE is currently implemented as follows.
A linear gradient profile for the difference profile between the two bending angles,

For the propagation of the

If a TBAE is applied to

The

As for the basic low-pass filtering of excess phases (Sect.

Based on the filtered and sometimes extrapolated

Propagated through the operator of the ionospheric correction (Eq.

Equation (

In the case of TBAE, Eq. (

Also, the ionospheric correction currently applied in
the rOPS is just a first-order correction, which will leave higher-order
residual ionospheric errors in

Based on previous studies

The

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

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

However, as demonstrated in Sect.

Here we state the two equations used to obtain the variances-only propagation
results shown for comparison purposes in Fig.

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.

The authors declare that they have no conflict of interest.

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

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