A new reference occultation processing system (rOPS) will include a Global Navigation Satellite System (GNSS) radio occultation (RO) retrieval chain with integrated uncertainty propagation. In this paper, we focus on wave-optics bending angle (BA) retrieval in the lower troposphere and introduce (1) an empirically estimated boundary layer bias (BLB) model then employed to reduce the systematic uncertainty of excess phases and bending angles in about the lowest 2 km of the troposphere and (2) the estimation of (residual) systematic uncertainties and their propagation together with random uncertainties from excess phase to bending angle profiles. Our BLB model describes the estimated bias of the excess phase transferred from the estimated bias of the bending angle, for which the model is built, informed by analyzing refractivity fluctuation statistics shown to induce such biases. The model is derived from regression analysis using a large ensemble of Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) RO observations and concurrent European Centre for Medium-Range Weather Forecasts (ECMWF) analysis fields. It is formulated in terms of predictors and adaptive functions (powers and cross products of predictors), where we use six main predictors derived from observations: impact altitude, latitude, bending angle and its standard deviation, canonical transform (CT) amplitude, and its fluctuation index. Based on an ensemble of test days, independent of the days of data used for the regression analysis to establish the BLB model, we find the model very effective for bias reduction and capable of reducing bending angle and corresponding refractivity biases by about a factor of 5. The estimated residual systematic uncertainty, after the BLB profile subtraction, is lower bounded by the uncertainty from the (indirect) use of ECMWF analysis fields but is significantly lower than the systematic uncertainty without BLB correction. The systematic and random uncertainties are propagated from excess phase to bending angle profiles, using a perturbation approach and the wave-optical method recently introduced by Gorbunov and Kirchengast (2015), starting with estimated excess phase uncertainties. The results are encouraging and this uncertainty propagation approach combined with BLB correction enables a robust reduction and quantification of the uncertainties of excess phases and bending angles in the lower troposphere.

The bending angle (BA) and atmospheric profiles retrieval chain for Global
Navigation Satellite System (GNSS) radio occultation (RO) data includes many
steps involving linear and (moderately) nonlinear transformations, starting
from excess phase and amplitude measurements

The uncertainty propagation through the wave-optical bending angle retrieval
block was investigated recently for large-scale (systematic) and small-scale
(random) uncertainties by

A thorough treatment of systematic uncertainty and its propagation from
excess phase to bending angle in the lower troposphere, including the aim to
correct for the known boundary layer bias (BLB) in standard lower troposphere
RO retrievals, often termed the “negative refractivity bias”

Our starting points for the BLB model construction are the approach based on
refractivity fluctuations introduced by

This approach results in the BLB and (residual) systematic uncertainty model
formulated in terms of tropospheric bending angles. In order to incorporate
this uncertainty modeling into the RO retrieval chain with integrated
uncertainty propagation, it needs to be transferred into the equivalent
excess phase BLB and (residual) systematic uncertainty estimate. For its
propagation, a perturbation approach or the approximation derived by

In order to now transform the bending angle uncertainty into the equivalent
excess phase uncertainty, we use the inverse FIO, which was recently employed
by

The paper is organized as follows. In Sect.

The BLB model is formulated to be capable of providing bending angle BLB
profiles over the lower troposphere up to 5 km impact altitude,
corresponding to about 4 km (mean-sea-level) altitude, with the primary bias
effects occurring within the atmospheric boundary layer below about 2 km
altitude. Here we describe its setup by first introducing the underlying
refractivity fluctuations model (Sect.

Along with the description we illustrate the performance of the BLB model to
quantify the boundary layer biases based on the predictors, underpinning that
the BLB profiles obtained for individual RO events can be effectively used
for BLB correction and lead to a significant reduction of systematic
uncertainty. A simple model for the estimated residual systematic uncertainty
after the BLB profile subtraction, which accounts for the residual bias
and the uncertainty (indirectly) incurred from the use of ECMWF analysis
profiles as regression reference, is described in
Sect.

Deviation statistics induced by simulated refractivity fluctuations:
refractivity structure constant

Deviation statistics obtained for real RO data: difference
statistics of COSMIC profiles including real fluctuations relative to ECMWF
profiles without fluctuations for bending angle as function of impact
altitude

In order to formulate our approach to the bending angle BLB in terms of
the negative refractivity bias

It is visible in Fig.

To put these simulation results into direct context with real data, Fig.

Figure

Deviation statistics obtained for real RO data: latitude–longitude
map (10

Our further strategy of the bias correction consists in the following. We perform the numerical simulation of occultation events with superimposed fluctuations and analyze different objective characteristics of RO signals in order to find those that correlate with the simulated bias. These characteristics will be referred to as predictors. Using this set of predictors, we also compare the simulation results with the processing of real COSMIC observations. We assume that this will allow the formulation of a model for BLB correction, which will also effectively mitigate biases in the retrieved refractivity profiles and further-derived atmospheric profiles. We have to formulate the BLB model with a flexible functional behavior in order to reliably serve its purpose.

We model the BLB by a predictor-based empirical model that is flexible enough to capture the BLB behavior by suitable predictors under widely variable predictor value ranges for individual RO events. Because the dependence of the BLB model profiles from predictors is unknown a priori, we solve for this dependence in the form of the linear combination of a set of linear and nonlinear functions of the predictors. We refer to these functions as adaptive functions. The model estimate of the regression coefficients of the linear combination is based on the comparison of a large set of bending angle observations with a reference data set.

In this study, introducing a first reliable BLB model version, the
observations are from the COSMIC mission and the reference data set consists
of gridded fields of meteorological variables from ECMWF. The ECMWF data have
their own systematic uncertainty, which is taken into account by letting
these uncertainties flow into the estimated residual systematic uncertainty
of bending angle profiles after BLB correction
(Sect.

The BLB model is formulated as follows. We used a set of COSMIC bending angle
observations, including 24 representative days from year 2008. We adopted the
15th and 16th day of every month, amounting in total to about 54 000 RO
events. We used the corresponding ECMWF fields as basis for obtaining the
“true” reference bending angles. To this end, we employed the wave-optics
propagator (

Scatter plot of fluctuation-affected bending angle profiles (

Scatter plot of the difference of fluctuation-effected and
reference bending angle profiles (

Because we need to derive the regression model for widely diverse BLB
behavior, we start with very general regression relations. Consider two
series of random variables, vector

After having solved for the regression coefficient vector

Here we consider the predictors that we may reasonably choose. Besides
predictors depending on RO event altitude and latitude (discussed separately
below), we adopt the following four predictors that are derived from
observational RO data – all as function of impact parameter

Figure

Figure

Comparing the behavior of these predictors, their correlation with the bending angle difference is clearly more salient in the simulations but some smaller asymmetry can also be noticed for the COSMIC observation differences. We therefore kept all four predictors in this study and left possible further reduction of these predictors (and associated adaptive functions) to future fine-tuning of the BLB model regression. An important conclusion from these comparisons is that the fluctuation model alone does not explain the patterns observed in the real observations. However, the role of this model is to help in finding reasonable predictors. The further bias correction procedure is only based on the predictors that can be readily derived from observations, rather than on the fluctuation model.

In addition to these four predictors we utilize the RO event
coordinates (impact altitude

General adaptive functions as we use here are constructed in the form of
different degrees of the predictors and their cross products from degree
zero, which produces unity, up to some maximum degree

For our choice of

As described in Sect.

Given this basis, we define a simple initial systematic uncertainty model for
the BLB-corrected bending angle profiles of the lower troposphere,

From experience with estimated biases of ECMWF analysis fields in other
studies

The estimated residual bias uncertainty profile after BLB correction is
formulated from experience with other bias corrections, such as sampling bias
correction

For the estimated residual systematic uncertainty finally attributed to the
BLB-corrected lower-tropospheric bending angle at any impact altitude, we then
simply adopt the larger one of the two uncertainties,

The propagation of systematic and random uncertainties
through the wave-optical retrieval chain was investigated by

Practically the application of this approximation was shown by

The reason and underlying problem is that the perturbation of the excess phase due to the superimposing of the systematic uncertainty of the bending angle is not smooth. The variation in the bending angle profile in each realization results in the different phase perturbation corresponding to a different ray manifold with a different caustic structure. Therefore, the excess phase perturbation has a complicated nonlinear relation with the phase (eikonal) uncertainty in impact parameter space, and this perturbation corresponds to a complicated coherent signal being a superposition of multiple signals corresponding to different rays.

To overcome this difficulty, we do apply the linearized approximation only
for the propagation of random uncertainty, i.e., the covariance propagation
according to

First, the BLB profile and its estimated systematic uncertainty profile after
BLB subtraction are computed according to Sect.

Second, the BLB-corrected L1 bending angle profile and this profile
perturbed by the estimated systematic uncertainty profile are each projected
back to excess phase by applying the inverse FIO approach recently introduced
by

Deviation statistics based on original BLB-corrected bending angles:
difference statistics of COSMIC profiles relative to ECMWF reference
profiles, with the same layout of panels as for Fig.

Third, the BLB-corrected L1 excess phase profile and this profile perturbed by the total estimated systematic uncertainty profile are processed again through the standard (forward) FIO CT2-wave-optics retrieval in order to obtain a BLB-corrected retrieved bending angle profile, for a consistency check with the original BLB-corrected bending angle profile, as well as the total estimated systematic bending angle uncertainty profile, from the difference of the two CT2-retrieved bending angle profiles. The systematic bending angle uncertainty profile at the second (F2) frequency is finally obtained from also processing the L2 excess phase profile perturbed by its associated systematic uncertainty through the wave-optics retrieval and estimating it from the difference of the resulting perturbed bending angle profile to the one originally retrieved from the unperturbed L2 excess phase.

Despite of the complexities from the nonlinearities involved, we obtain in
this way a consistent set of excess phase and bending angle profiles together
with their estimated systematic and random uncertainties, which are
BLB-corrected at the L1 frequency in the lower troposphere. The extra
computational expense for the uncertainty propagation due to the
nonlinearity is reasonably limited to one additional forward and inverse FIO
operation at L1 frequency, which is required for the perturbation approach to
systematic uncertainty propagation. This is similar to the uncertainty
propagation work of

Here we evaluate the consistency of the BLB-corrected bending angles and
their associated retrieved refractivities by (i) using the original
BLB-corrected bending angles and (ii) back-projecting the original
BLB-corrected retrieved bending angles to obtain BLB-corrected excess phases and then retrieving the
bending angles again.
This provides a basic validation of our procedure as
described in Sect.

We investigated the BLB correction of an independent ensemble of
COSMIC-retrieved bending angles employing our BLB model, as in
Sect.

Deviation statistics based on BLB-corrected retrieved bending angles
(after back projection of original BLB-corrected bending angles to excess
phases and in turn retrieving the bending angles again): COSMIC–ECMWF
difference statistics with the same layout and using the same COSMIC and
ECMWF data as for Fig.

Cross-checking these results with results from COSMIC and ECMWF ensembles
using the 16th day of every month (not separately shown), we find them
practically indistinguishable in terms of their difference statistics. This
indicates the statistical homogeneity of the data sets and the robustness of
the BLB model. Furthermore, from comparing Fig.

Figure

Figure

Deviation statistics based on original BLB-corrected bending angles:
latitude–longitude map (10

In this study we developed a regression-based approach for modeling and propagating the atmospheric boundary layer biases and associated (residual) systematic uncertainties within the wave-optical retrieval chain of the reference occultation processing system, which is a new RO processing system with integrated uncertainty propagation that focuses on calibration–validation and climate applications.

Currently, there is no quantitative physical model describing BLB in RO
data, although there was a series of studies discussing different mechanisms
resulting in BLB. The starting point encouraging and informing our BLB model
design was the fluctuation-based explanatory modeling of the well-known
negative refractivity bias problem in the boundary layer. We showed that
it is possible to achieve a reasonable agreement with observed bending angle
and refractivity biases by modeling fluctuation statistics consistent with
reasonable tropospheric profiles of the refractivity structure constant

Based on this understanding we can robustly assume that reliable modeling of the bending angle BLB, and subsequent use of the model for BLB correction, will also effectively mitigate biases in the retrieved refractivity profiles and further-derived atmospheric profiles. However, given the highly variable refractivity fluctuations affecting individual RO events in reality, which implies a complex dependence of the bending angle BLB on the location and the data characteristics of individual RO profiles, we found it necessary to implement a BLB model with a very flexible functional behavior in order to reliably serve its purpose.

We therefore have chosen a versatile empirical regression-modeling approach and found suitable predictors of the BLB in the lower-tropospheric bending angle, including the bending angle and its standard deviation, CT amplitude and its fluctuation index, impact altitude and its trigonometric functions, and trigonometric functions of latitude. Degrees and cross products of these predictors were used to form a set of flexible adaptive functions that served as the basis for the BLB model, which was then obtained by regression to a large ensemble of COSMIC and ECMWF profile differences. Also, a simple (residual) systematic uncertainty model was formulated, applying to the bending angles after BLB correction. For any given RO event, the BLB model profile can be computed based on the predictors that purely depend on the event location and the characteristics of the bending angle and CT amplitude profiles.

Together with the linearized wave-optics (random) uncertainty propagation
approach described by

Our bias model uses ECMWF fields as a reference; therefore, it involves the biases that are intrinsic to these ECMWF fields. However, the same approach can be applied together with an independent estimate of the ECMWF biases. In this study, we assumed that ECMWF biases form a small fraction of the observed systematic COSMIC–ECMWF differences.

These results are encouraging for follow-on work in the near future that can
provide a refined BLB model design and a detailed inspection and validation
of the complete wave-optical retrieval and uncertainty propagation as
introduced in this study. In this way, the rOPS geometric-optical bending
angle retrievals

The code used in this study does not belong to the public domain and cannot be distributed.

COSMIC radio occultation data are freely available. To get
access to them, it is necessary to sign up at the website of the COSMIC Data Analysis and Archiving Center (CDAAC):

Both authors formulated the initial approach to integrating wave-optical uncertainty propagation into the reference occultation processing system (rOPS) and the overall study design. MG conceived and developed the bias-modeling approach and the boundary layer bias model, performed the computational work and the analysis, prepared the figures, and wrote the first draft of the manuscript. GK provided input on the bias and uncertainty modeling design and feedback during the work and significantly contributed to the writing of the manuscript. Both authors contributed to consolidating the manuscript for submission and publication.

The authors declare that they have no conflicts 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.

Work on Sects. 1 and 2 was supported by the Russian Foundation for Basic Research (grant No. 16-05-00358-a). Work on Sects. 3 and 4 was supported by the Austrian Research Promotion Agency FFG within the Austrian Space Applications Programme ASAP (ASAP-9 project OPSCLIMPROP and ASAP-10 project OPSCLIMTRACE). We acknowledge Taiwan's National Space Organization (NSPO) and the University Corporation for Atmospheric Research (UCAR) for providing the COSMIC RO data via the COSMIC Data Analysis and Archiving Center (CDAAC). We acknowledge the European Centre for Medium-Range Weather Forecasts (ECMWF) for providing the atmospheric analysis fields. Edited by: Axel von Engeln Reviewed by: two anonymous referees