We introduce a new dynamic statistical optimization algorithm to initialize ionosphere-corrected bending angles of Global Navigation Satellite System (GNSS)-based radio occultation (RO) measurements. The new algorithm estimates background and observation error covariance matrices with geographically varying uncertainty profiles and realistic global-mean correlation matrices. The error covariance matrices estimated by the new approach are more accurate and realistic than in simplified existing approaches and can therefore be used in statistical optimization to provide optimal bending angle profiles for high-altitude initialization of the subsequent Abel transform retrieval of refractivity. The new algorithm is evaluated against the existing Wegener Center Occultation Processing System version 5.6 (OPSv5.6) algorithm, using simulated data on two test days from January and July 2008 and real observed CHAllenging Minisatellite Payload (CHAMP) and Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) measurements from the complete months of January and July 2008. The following is achieved for the new method's performance compared to OPSv5.6: (1) significant reduction of random errors (standard deviations) of optimized bending angles down to about half of their size or more; (2) reduction of the systematic differences in optimized bending angles for simulated MetOp data; (3) improved retrieval of refractivity and temperature profiles; and (4) realistically estimated global-mean correlation matrices and realistic uncertainty fields for the background and observations. Overall the results indicate high suitability for employing the new dynamic approach in the processing of long-term RO data into a reference climate record, leading to well-characterized and high-quality atmospheric profiles over the entire stratosphere.

Global Navigation Satellite System (GNSS)-based radio occultation (RO) is a robust atmospheric remote-sensing technique that provides accurate atmospheric profiles of the Earth's atmosphere (Kursinski et al., 1997; Hajj et al., 2002; Kirchengast, 2004). This technique has several distinctive advantages in terms of high accuracy, high vertical resolution, global coverage, and self-calibration (Anthes, 2011; Yu et al., 2014). GNSS RO data are now widely used in numerical weather prediction, climate monitoring, and space weather research (e.g., Healy and Eyre, 2000; Cucurull and Derber, 2008; Le Marshall et al., 2010; Anthes, 2011; Steiner et al., 2011; Carter et al., 2013).

Although the RO technique has been rather successful, it still suffers from some weaknesses. For example, the RO observations are affected by higher-order ionospheric effects and observation errors at high altitudes (> 30 km) (Bassiri and Hajj, 1993; Danzer et al., 2013; Liu et al., 2013, 2015). These errors propagate downward from bending angles to refractivity through the Abel integral and also degrade the accuracy of the retrieved temperature and other atmospheric profiles (Healy, 2001; Rieder and Kirchengast, 2001; Gobiet and Kirchengast, 2004; Steiner and Kirchengast, 2005). Therefore, it is very important to have a best-possible initialization of the ionosphere-corrected bending angles at high altitudes for more accurate climate monitoring.

Statistical optimization is a commonly used method to initialize RO bending
angles at high altitudes (e.g., Sokolovskiy and Hunt, 1996; Gorbunov et al.,
1996; Hocke, 1997; Healy, 2001; Gorbunov, 2002; Gobiet and Kirchengast, 2004;
Gobiet et al., 2007). It is a generalized least-squares approach that
combines an observed RO bending angle profile with a background bending angle
profile (Turchin and Nozik, 1969; Rodgers, 1976, 2000). The weights of the
two types of bending angles are determined by the inverse of their error
covariance matrices. The statistical optimization equation used is (Healy,
2001; Gobiet and Kirchengast, 2004)

In statistical optimization, the more accurately the error covariance matrices represent the error characteristics, the more accurate is the optimized bending angle profile. However, it is not straightforward to obtain such suitable error covariance matrices, especially for the background bending angle since they are not supplied together with common climatological models nor is the construction a straightforward task. Therefore, previous approaches have usually simplified the calculation of the error covariance matrices.

A typical approach is to estimate the background error covariance matrix by assuming a constant relative standard error of the background bending angle and a simple error correlation structure like exponential fall-off over an atmospheric scale height (Healy, 2001; Rieder and Kirchengast, 2001; Gobiet and Kirchengast, 2004) or disregarding correlations (Sokolovskiy and Hunt, 1996; Hocke, 1997; Gorbunov, 2002; Lohmann, 2005; Gorbunov et al., 1996, 2005, 2006). Similarly, the observation error covariance matrix is formulated from estimating the observation error at a defined mesospheric altitude range (where the RO signal is weak) and using simple exponential fall-off error correlations (Healy, 2001, Gobiet and Kirchengast, 2004) or again just ignoring the latter. These rough estimations generally result in inaccurate error covariance matrices and therefore result in inaccurate optimized bending angles that degrade the accuracy of subsequently retrieved atmospheric profiles. More details on the various schemes are provided by Li et al. (2013), Sect. 2.1 therein.

Improved accuracy in optimized bending angles was obtained when using an improved statistical optimization algorithm to initialize ionosphere-corrected bending angles (Li, 2013; Li et al., 2013). Li et al. (2013) used European Centre for Medium-Range Weather Forecasts (ECMWF) short-range (24 h) forecast fields as background bending angles. Their background error covariance matrix was accurately and realistically estimated using large ensembles of ECMWF short-range forecast, analysis, and RO observed bending angles. It was constructed using daily global fields of estimated background uncertainty profiles and a daily global-mean correlation matrix. The background uncertainty profile was dynamically estimated taking into account its variations with latitude, longitude, altitude, and day of year. They not only calculated the random errors of background bending angles using large ensembles of ECMWF and RO data but also empirically modeled the potential systematic background uncertainty and finally combined these two uncertainties to formulate the background uncertainty. The global-mean correlation matrix was also calculated using large ensembles of ECMWF analysis and forecast fields. Finally, the biases in the background bending angles were corrected to avoid the potential effects on optimized bending angles.

Since this first-version dynamic statistical optimization algorithm dynamically estimated the background error covariance matrix only, it is hereafter referred to as the b-dynamic algorithm (“b” represents background) in this study. The b-dynamic algorithm was evaluated by Li et al. (2013) against the Occultation Processing System version 5.4 (OPSv5.4) algorithm developed by the Wegener Center for Climate and Global Change (WEGC) (Pirscher, 2010; Ho et al., 2012; Steiner et al., 2013). It was found that the b-dynamic algorithm significantly reduced random errors of the optimized bending angles and left less or about equal levels of residual systematic errors. The quality of the subsequently retrieved refractivity and temperature profiles was also improved. In addition, even the dynamically estimated background error correlation matrix alone was able to improve the optimized bending angles.

The aim of this study is to obtain even more accurate and reliable atmospheric profiles for optimal climate monitoring by advancing the b-dynamic algorithm to a complete dynamical estimation of both background and observation uncertainties and correlations. This is accomplished by employing a realistically estimated observation error covariance matrix in addition to the b-dynamic formulation. The observation error covariance matrix is constructed with dynamically estimated observation uncertainties for each occultation event and daily global-mean correlation matrices from large ensembles of data. The observation uncertainty is calculated using bias-corrected difference profiles of observed RO bending angle profiles relative to co-located ECMWF forecast bending angle profiles. The global-mean correlation matrix is calculated using multiple days of RO bending angle profiles and co-located ECMWF analysis bending angle profiles. In addition, the basic b-dynamic algorithm is updated to obtain even more robust background error covariance matrices. Finally, the stability of the new dynamic algorithm is evaluated using full months of RO data from CHAMP and COSMIC.

The structure of this paper is as follows. Section 2 introduces the methodology of the dynamic statistical optimization algorithm. Section 3 evaluates its performance using both simulated and observed RO data. Finally a summary and conclusions are given in Sect. 4.

Figure 1 shows the algorithmic steps of the dynamic statistical optimization algorithm. The new algorithm mainly includes two parts: (1) the dynamic estimation of the background error covariance matrix and the bias correction of background bending angles, and (2) the dynamic estimation of the observation error covariance matrix. Information on background/observation uncertainty and on the background/observation correlation matrix is needed for constructing complete background/observation error covariance matrices. The uncertainty at any vertical level is the square root of the diagonal value of the error covariance matrix at that level. The correlation matrix includes correlation functions for all vertical levels. Each such correlation function comprises the correlation coefficients of the error at the vertical level where it peaks to the errors at all other vertical levels. In summary, the background/observation uncertainties and the corresponding correlation matrix together formulate the background/observation error covariance matrices (Gaspari and Cohn, 1999).

Assuming that

The dynamic estimation of the background error covariance matrix includes three algorithmic steps: (1) construction of basic daily background fields (blue boxes in the left part of Fig. 1), (2) preparation of the derived daily background fields (green boxes), and (3) dynamic estimation of the background error covariance matrix (orange boxes).

In the first step, daily fields of the basic background variables are
prepared using a 10

Schematic illustration of the algorithmic steps of the dynamic statistical optimization approach; for description see Sect. 2.1 and 2.2.

The data used to calculate these basic background variables include ECMWF
analysis fields and corresponding 24 h forecast fields with a T42L91
resolution at 00:00 and 12:00 UTC, and observed RO bending angles. In
calculating the mean variables in each grid cell, time averaging over 7
days (from 3 days before to 3 days after the day of interest) and
horizontal averaging over geographic domains of at least 1000 km

The second step involves the preparation of the derived daily background
fields. These specific statistical quantities include (i) the
forecast-minus-analysis standard deviations

In the third step, the background error covariance matrix is calculated from
the fields obtained in step 2. Co-located profiles of

In order to effectively reduce the residual bias in the background bending
angle

Figure 2 illustrates the estimated relative uncertainty of the
forecast-minus-analysis standard deviation

Variability of relative standard deviations of forecast-minus-analysis bending angle differences (upper two panels) and of the systematic uncertainty of the mean analysis bending angle (bottom two panels) as function of latitude (left) and of day of month (right), respectively.

As a function of latitude (left), forecast-minus-analysis standard deviation
reveals largest errors at high altitudes over the Antarctic region. The
relative standard deviations are larger than 10 % near 80 km, decreasing
to

Regarding variations of

Global-mean error correlation functions from the background error covariance matrix (left), for the 5, 15, and 25 July 2008 at three representative impact altitude levels (30, 50, and 70 km), and estimated correlation lengths of the correlation functions (right) at all impact altitude levels from 20 to 80 km for the same 3 days.

Figure 3 shows exemplary global-mean correlation functions (left) and associated correlation lengths (right) for the 5, 15, and 25 July 2008. The correlation functions, plotted for three representative height levels (30, 50, 70 km), are rather similar over the month. They have a main peak of nearly Gaussian shape with negative side peaks at each side. Further outward, small secondary positive peaks occur, after which the functions essentially approach zero. The correlation lengths increase rather smoothly with altitude from about 0.8 km at 20 km to 6 km at 80 km. They also show little variation over the example month of July 2008.

Overall this behavior indicates that in months without larger atmospheric anomalies (such as for example sudden stratospheric warming at high latitudes; e.g., Manney et al., 2008) a daily update of correlation matrices is not necessarily needed. In a long-term application, however, it is not clear when and where some (transient) anomalies may occur, so a daily update of the background fields was used as a cautious baseline.

The error covariance matrix of the observed bending angle

Different to the OPSv5.6 and the b-dynamic algorithm, which estimate the
observation uncertainty

More specifically, the first step is to subtract the co-located forecast
bending angle profile

The difference profile

The next step is to subtract the smoothed difference profile

Finally the observation uncertainty at any impact altitude level

Figure 4 illustrates the observation uncertainty profile

Observation uncertainty and key intermediate variables for six representative RO events, two simMetOp events (top), two CHAMP events (middle), and two COSMIC events (bottom) from 15 July 2008. “delta” is the difference profile of the RO ionosphere-corrected bending angle to the co-located ECMWF forecast profile used as a reference, “deltadelta” is the delta-difference profile after subtracting a smoothed profile “deltasmooth” from the difference profile “delta”, and “Uncert” is the resulting observation uncertainty estimate; for detailed description see Sect. 2.2.

Global-mean error correlation functions from the observation error covariance matrix (left), for the 5, 15, and 25 July 2008 at three representative impact altitude levels (30, 50, and 70 km), and estimated correlation lengths of the correlation functions (right) at all impact altitude levels from 20 to 80 km for the same 3 days. For ease of intercomparison the layout is the same as in Fig. 3.

Estimated uncertainties are smallest (near 0.5

The global-mean observation error correlation matrix

Figure 5 illustrates representative observation error correlation functions
(extracted from

The intra-monthly variation is essentially negligible within the given July 2008 test month (applies also to January 2008, not shown), pointing to room for further improvement of the utility of the estimation for long-term processing, e.g., considering larger ensemble sizes and sub-global regions. These slow dynamics of the observation error covariance matrices, and of the background error covariance matrices as discussed in Sect. 2.1, enable reliable use also in near-real-time or fast-track processing (i.e., processing within 3 h or within follow-on day of observations). Instead of using 7 days centered about the day being processed (including 3 days before and after the center day) 7-day-history data (from the previous day to 7 days prior) may be used in these cases, with insignificant degradation in performance.

In the b-dynamic algorithm of Li et al. (2013), the statistical optimization
was applied exactly down to 30 km. However, for some noisy RO events
especially from CHAMP, ionosphere-corrected bending angles can be still
noisy around 30 km impact altitude. For these noisy events there can be a
sharp change of bending angle characteristics from the rather smooth
statistically optimized profile above 30 km to the rather noisy
purely observed profile below 30 km. On the other hand, some (simulated)
events may have very small observation uncertainties at the altitudes below
40 km, which may lead to degraded robustness of the matrix inversion

First, we gracefully adjust the observation uncertainty

Second, we apply the statistical optimization down to 28 km and then apply a
half-sine-weighted transition across 32 to 28 km between the
statistically optimized bending angles and purely observed bending angles.
That is, the weighting function over this transition altitude range,

Another issue requiring caution is the robustness of matrix inversions,
especially related to the weighting matrix in Eq. (2),

This could be overcome by replacing the main peak by a 5th-order
polynomial function as described by Gaspari and Cohn (1999), which
approximates a Gaussian shape and was also successfully used in the context
of matrix inversion by Steiner and Kirchengast (2005), Eq. (5) therein (note
that a typo leaked into the

The dynamic algorithm was implemented in the EGOPSv5.6 software (Fritzer et al., 2013) to enable a complete RO retrieval. The EGOPSv5.6 system was also used to simulate RO observations (simMetOp events) and to retrieve atmospheric profiles. The standard RO data processing chain within this system is the OPSv5.6 retrieval.

We evaluated the new dynamic algorithm against this OPSv5.6 algorithm, which is, in terms of statistical optimization formulation, the same as the OPSv5.4 algorithm used for comparison by Li et al. (2013). Briefly, the OPSv5.6 algorithm uses ECMWF short-range forecast bending angles as a background and employs exponential fall-off functions to express the correlations of both background and observation uncertainties. The background uncertainty is modeled as amounting to 15 % of background bending angles. The observation uncertainty is estimated as the standard deviation of observed bending angles relative to co-located MSIS model bending angles in the impact altitude range from 65 km to about 80 km. For more detailed information on OPSv5.6/v5.4 see Pirscher (2010), Steiner et al. (2013), and Schwaerz et al. (2013).

In addition to the OPSv5.6 intercomparison, the atmospheric profiles retrieved by the dynamic algorithm are compared with those retrieved by the b-dynamic algorithm (Li et al., 2013) and with those by the UCAR/COSMIC Data Analysis and Archive Center (CDAAC) Boulder.

The data sets used for the evaluation include simulated MetOp data (simMetOp) as well as real observed CHAMP and COSMIC data. simMetOp data were simulated in the same way as by Li et al. (2013), using moderate ionosphere conditions in the forward simulations and using observational errors representing MetOp/GRAS-type receiving system errors.

As a basis for the CHAMP and COSMIC retrievals, excess phase and orbit data
were downloaded from UCAR/CDAAC Boulder (CDAAC data version 2009.2650 for
CHAMP and 2010.2640 for COSMIC). CDAAC atmospheric profiles (atmPrf) used for
the comparison of retrieved profiles are mainly from the same CDAAC data
version. Recently reprocessed atmospheric profiles provided by CDAAC (version
2014.0140) are also used for comparison, but due to their very recent release only
CHAMP has been used so far. We denote the CHAMP data version 2009.2650 as
“CDAAC” in figure legends, and version 2014.0140 as
“CDAAC

Figure 6 illustrates the effects of statistical optimization on individual
bending angle profiles by a few representative RO events. The left panels
show the background and observation uncertainties as well as the
observation-to-background (Obs-to-Bgr) weighting ratio

Since bending angles increase roughly exponentially with decreasing altitude, as seen in the middle column of Fig. 6, differences among the various retrieved profiles seem to be small. The right panels, however, actually show the differences of optimized bending angle profiles relative to their reference. For the simMetOp event, bending angle differences from the dynamic algorithm are smallest over all altitudes, confirming the high utility of the algorithm, since here the “true” profile from forward simulation serves as a reference. The bending angle differences of the b-dynamic algorithm are similar and the values are also small. The differences from the OPSv5.6 algorithm are largest and significantly noisier.

For the CHAMP event the relative differences from the dynamic algorithm, the b-dynamic algorithm, and CDAAC are smaller than those from the OPSv5.6 algorithm below 50 km. Above about 50 km, the relative differences from the dynamic algorithm increase and are largest. For the COSMIC event, bending angles from the dynamic, b-dynamic, and OPSv5.6 algorithms are rather similar below 50 km. Above 50 km, differences from the dynamic algorithm are tentatively largest. For this event, the differences of CDAAC are generally larger than of the other three algorithms.

Background and observation bending angle uncertainty profiles as well as observation-to-background (Obs-to-Bgr) weighting ratio (left); statistically optimized bending angle profiles from the OPSv5.6, b-dynamic, dynamic, and CDAAC algorithms together with their reference profile (middle); and difference of the optimized profiles to the reference profile (right). Three example events from 15 July 2008 are illustrated, from simMetOp (top), CHAMP (middle), and COSMIC (bottom), respectively.

Statistically optimized bending angle profiles together with their reference profile (left) and their difference to the reference profile (right), of three example events from simMetOp (top), CHAMP (middle), and COSMIC (bottom) from 15 July 2008, using either the realistic global-mean correlation matrix of the new dynamic method (“full correlation”) or simple exponential fall-off correlation as existing in OPSv5.6 (“exp.falloff only”).

Inspecting further individual RO events (not shown) confirmed that the relative differences of simMetOp data from the dynamic algorithm are consistently smaller and smoother than those from the other approaches. This underlines the robust capability of the dynamic algorithm for improving the quality of the ionosphere-corrected bending angles. For CHAMP and COSMIC measurements, the relative differences from the dynamic algorithm are also generally smaller and smoother than those from other algorithms below 50 km. However, above 50 km, the differences from both the dynamic algorithm and from CDAAC are generally larger than those from the OPSv5.6 and b-dynamic approaches. This does not mean that bending angle profiles from the dynamic and CDAAC algorithms are not accurate at high altitudes, however; the result mainly depends on the determination of the weights of the background and observed bending angles in the statistical optimization.

In the new dynamic algorithm, the estimated relative background errors at
high altitudes are generally larger than those of the OPSv5.6 algorithm
(e.g., in the polar winter regions, the relative background errors of the
dynamic algorithm are usually larger than 30 % around 60 km, while the
OPSv5.6 algorithm assumes a constant error of 15 % at all altitudes). At
the same time the dynamically estimated observation uncertainties are
usually smaller than those of OPSv5.6, which often sets a large standard
value (22

Figure 7 shows the comparison results for these two types of correlation matrices, again using three exemplary events from simMetOp, CHAMP, and COSMIC, showing absolute bending angles (left) and differences to reference (right). The simMetOp event highlights that bending angle differences from the full correlation case are much smoother and smaller than those from the exponential fall-off correlation. For the real CHAMP and COSMIC events, the magnitudes of the differences from the two cases are similar, but also here it is clearly evident that the use of the full correlation leads to smoother differences than the use of exponential fall-off correlation. We conclude that the use of adequately realistic correlation matrices is preferable.

Systematic differences (SysDiff, light lines) and standard
deviations (SD, heavy lines) of statistically optimized bending angles,
relative to “perfect” simulated bending angles or co-located ECMWF analysis
bending angles used as a reference, of the global ensemble of simMetOp events
on 15 January and 15 July 2008 (upper two panels), and of CHAMP and
COSMIC events from the complete months of January and July 2008 (middle and
bottom panel, respectively). Statistics of the OPSv5.6 (black), b-dynamic
(blue), dynamic (red), CDAAC (version 2009.2650 for CHAMP and version
2010.2640 for COSMIC, green), and CDAAC

Systematic differences (SysDiff, light lines) and standard
deviations (SD, heavy lines) of statistically optimized bending angles,
relative to “perfect” simulated bending angles or co-located ECMWF analysis
bending angles used as a reference, of the global ensemble of simMetOp events
on 15 January and 15 July 2008 (upper two panels), and of CHAMP and
COSMIC events from the complete months of January and July 2008 (middle and
bottom panels, respectively). Results from three different bias coverage
factor choices in the dynamic algorithm – i.e.,

In this section the performance of the dynamic, b-dynamic, and OPSv5.6 statistical optimization algorithms are evaluated using simulated MetOp data from 15 January and 15 July 2008 and monthly CHAMP and COSMIC observations from January and July 2008. In addition, atmospheric profiles retrieved and provided by UCAR/CDAAC for the same time periods are used for comparison. The mean systematic differences between retrieved and reference profiles and the associated standard deviations are calculated and analyzed in bending angle, refractivity, and temperature profiles, similar to the statistical performance evaluation of the b-dynamic algorithm by Li et al. (2013).

In order to detect and exclude outlier profiles from the statistical profile
ensembles, the quality of retrieved profiles is checked as follows. Bending
angle profiles are checked from 25 to 80 km, and a profile is flagged as bad
if a bending angle exceeds a threshold at any impact altitude level, which
was defined based on careful sensitivity tests as the maximum of either
40

Refractivity and temperature profiles are checked in the same way as used for OPSv5.6/v5.4 (Schwaerz et al., 2013; Steiner et al., 2013; Pirscher, 2010), i.e., the deviation from co-located ECMWF analysis profiles at any altitude level must not exceed 10 % for refractivity between 5 and 35 km and 20 K for temperature within 8 and 25 km. These quality checks are performed on all profiles retrieved with the EGOPS software. For profiles provided by CDAAC, we check the CDAAC quality flag and use only profiles flagged to be of good quality.

Figure 8 shows the systematic differences and standard deviations of optimized bending angle profiles of the global ensembles of simMetOp from 15 January and 15 July 2008, and of CHAMP and COSMIC events from January and July 2008. For simMetOp (top), it is clear to see that the performance of the dynamic algorithm outperforms the b-dynamic algorithm and the OPSv5.6 algorithm, exhibiting smallest systematic differences and associated standard deviations. Compared to the OPSv5.6 algorithm, the best improvement is found between 40 and 60 km. These results are very encouraging and confirm the fundamental capabilities of the dynamic algorithm.

Comparison of the CHAMP (middle) and COSMIC (bottom) results from the dynamic algorithm with the OPSv5.6 and CDAAC algorithms shows that the bending angle standard deviation from the dynamic (and b-dynamic) algorithm is again generally smaller than that of the OPSv5.6 and CDAAC results. Above 45 km for CHAMP, and above 55 km for COSMIC, the standard deviations from the dynamic algorithm exceed those from the OPSv5.6 algorithm. This is due to increased weight of noisy RO bending angles in the mesosphere compared to OPSv5.6 as discussed in Sect. 3.1 above. Standard deviations from both CDAAC data versions are larger than those from the other methods, and particularly the new data version (shown for CHAMP) exhibits largest standard deviation already from about 35 km upwards.

Systematic differences of CHAMP and COSMIC data (differences are calculated against co-located ECMWF analysis profiles) are rather similar for the dynamic, b-dynamic, and OPSv5.6 algorithms. The systematic differences from CDAAC algorithms are also similar, in particular below about 35 km, but they are larger and feature somewhat different characteristics above about 40 km for July 2008. In particular the new data version (shown for CHAMP) exhibits different (oscillatory) behavior both in January and July. These results indicated that the new dynamic algorithm is robust and competitive in providing optimized profiles with biases minimized in a best-possible manner, as should be expected from its realistic account for both observation and background uncertainties and error correlation structures.

Furthermore it can be seen, in particular from the CHAMP results (CHAMP data have highest observational noise), that the improved treatment of the transition to purely observed data around 30 km has mitigated the sharpness of the change in standard deviation.

Systematic differences (SysDiff, light lines) and standard
deviations (SD, heavy lines) of statistically optimized bending angles,
relative to “perfect” simulated bending angles used as a reference, of
simMetOp events on 15 July 2008. Statistics for the OPSv5.6 (black),
b-dynamic (blue), and dynamic (red) statistical optimization algorithms are
shown for six different regions: Global (90

In order to discuss the effects of different choices of

For the CHAMP and COSMIC data, standard deviations are largest for the

Overall Fig. 9 demonstrates that the sensitivity to the detailed
quantitative choice of

Systematic differences (SysDiff, light lines) and standard
deviations (SD, heavy lines) of statistically optimized bending angles,
relative to co-located ECMWF analysis bending angles used as a reference, of
CHAMP events from July 2008. Statistics for the OPSv5.6 (black), b-dynamic
(blue), dynamic (red), CDAAC (version 2009.2650, green), and
CDAAC

Systematic differences (SysDiff, light lines) and standard deviations (SD, heavy lines) of statistically optimized bending angles, relative to co-located ECMWF analysis bending angles used as a reference, of COSMIC events from July 2008. Statistics for the OPSv5.6 (black), b-dynamic (blue), dynamic (red), and CDAAC (version 2010.2640, green) statistical optimization algorithms are shown for the same six regions as in Fig. 10. The figure layout is the same as for Figs. 10 and 11.

Systematic differences (SysDiff, light lines) and standard deviations (SD, heavy lines) of retrieved refractivity profiles, relative to “perfect” simulated refractivity or co-located ECMWF analysis refractivity used as a reference, for the global ensemble of simMetOp events on 15 January and 15 July 2008 (top panels) and of CHAMP events (middle panels) and COSMIC events (bottom panels) from the complete months of January and July 2008. Statistics of the OPSv5.6 (black), b-dynamic (blue), dynamic (red), CDAAC (version 2009.2650 for CHAMP and version 2010.2640 for COSMIC, green), and CDAACnew (version 2014.0140 for CHAMP, magenta) statistical optimization methods are shown. The figure layout is the same as for Fig. 8.

Systematic differences (SysDiff, light lines) and standard deviations (SD, heavy lines) of retrieved temperature profiles, relative to “perfect” simulated temperature or co-located ECMWF analysis temperature used as a reference, for the global ensemble of simMetOp events on 15 January and 15 July 2008 (top panels) and of CHAMP events (middle panels) and COSMIC events (bottom panels) from the full months of January and July 2008. The figure layout is the same as for Figs. 8 and 13.

Bending angle (left), refractivity (middle), and temperature (right) systematic differences (SysDiff, light lines) and standard deviations (SD, heavy lines), relative to their “perfect simulated” or co-located ECMWF analysis data used as a reference, of the global ensemble of simMetOp (top) and COSMIC (bottom) events from 15 July 2008, using either the realistic global-mean correlation matrix of the new dynamic method (“full correlation”) or simple exponential fall-off correlation as in the existing OPSv5.6 (“exp.falloff only”). The number of events (NoE) in each statistical ensemble is also indicated in the panels.

In order to evaluate the performance of different statistical optimization
algorithms in different latitude regions, the systematic differences and
standard deviations of optimized bending angles were calculated for five
latitudinal bands in addition to the global case (90

Figure 10 shows that the performance of the dynamic, b-dynamic, and OPSv5.6 algorithms are rather similar globally and in the Northern Hemisphere (NHSM, NHP). In the Southern Hemisphere, and in particular in the SHP region (Antarctic winter in July), the conditions are evidently more challenging, such that the OPSv5.6 algorithm accrues increased biases in the upper stratosphere above 50 km. The new dynamic algorithm underscores its good and reliable basic performance in all regions, both in terms of biases and standard deviations.

Figures 11 and 12 for the real bending angle data show that the performance of all algorithms in all latitude bands except SHP (Antarctic winter) is consistent and generally similar to the performance visible from the global ensemble, which we discussed along with Fig. 8 above. In SHP, both systematic differences and standard deviations are markedly larger. In particular the CDAAC results, and most so the new CDAAC data version, exhibit relatively large systematic differences at altitudes above 35 km (up to around 5 %) and also standard deviations closely reaching or exceeding 10 % even at altitudes near 50 km. Error characteristics for January (not shown) in the NHP region (Arctic winter) generally mirror the SHP July (Antarctic winter) error characteristics.

We consider the new dynamic algorithm in this context to confirm its robust performance also for real data in all regions, although the lack of a “true” reference in these cases does not allow for strong conclusions. In evaluating future long-term processing application of the algorithm, we will also include stricter validation against independent co-located data of high quality over the stratosphere and mesosphere from other sources such as the Envisat/MIPAS and TIMED/SABER satellite instruments (Remsberg et al., 2008; García-Comas et al., 2012).

Figures 13 and 14 show the global statistics results for refractivity (Fig. 13) and temperature (Fig. 14) for simMetOp (top), CHAMP (middle), and COSMIC (bottom). These refractivity and temperature results reflect the results for the bending angles in a filtered manner, after having passed through the Abelian integration (refractivity) and in addition the hydrostatic integration (temperature), which lead to smoothing and downward propagation of biases and to reduction of standard deviations (e.g., Gobiet and Kirchengast, 2004; Steiner and Kirchengast, 2005). Due to this downward propagation, the differences from the various algorithms become smaller, and results are closely similar below about 40 km and in most cases even above. The most notable differences from the consistent behavior of the different algorithms are those of the CDAAC processings above about 50 km, which exhibit the largest systematic differences and standard deviations, and the additional deviations of the new CDAAC version (shown for CHAMP) already from about 35 km upwards.

Again we consider the performance of the new dynamic algorithm robust and
encouraging for larger-scale applications, which may also include further
adjustments of parameters like

Figure 15 extends the view of Fig. 7 on the sensitivity to the choice of correlation modeling to a statistical view. It depicts the results of global statistics for simMetOp (top) and COSMIC (bottom), for bending angle (left), refractivity (middle), and temperature (right), from either operating the full dynamic algorithm or from using simplified correlation modeling with the exponential fall-off approximation. The simMetOp results show, in line with the results of Fig. 7, that the use of the realistically modeled full correlations is a superior choice, though the reduction of systematic difference relative to the “true” reference is small (after the Abel or hydrostatic integrations) in refractivity and temperature. The COSMIC results indicate that the choice of correlation modeling strongly impacts the standard deviation and to a more limited degree also the systematic differences. While this behavior does not itself imply a preference, it is very clear that the choice of the realistic full correlation modeling will be the physically more sound and more adequate approach also for real data.

This study presented a new dynamic statistical optimization algorithm to initialize RO ionosphere-corrected bending angle profiles at high altitudes for optimal climate monitoring throughout the stratosphere. This dynamic algorithm uses multiple days of ECMWF analysis, ECMWF short-range (24 h) forecast, and RO observation data to realistically estimate background and observation error covariance matrices. Both the background and observation error covariance matrices are constructed with geographically varying uncertainty estimation and with a global-mean correlation matrix estimated on a daily basis. The b-dynamic algorithm recently introduced by Li et al. (2013) was used as a starting point and provided for the estimation of the background error covariance matrix and the bias correction of background bending angles.

The main advancements of the new dynamic algorithm compared to this previous algorithm are that it (1) adds a dynamically estimated observation error covariance matrix with altitude-dependent observation uncertainty and a realistically calculated global-mean correlation matrix; (2) updates the algorithm of the calculation of basic statistical mean variables by using ECMWF and RO data from a longer time window and larger geographical regions for more accurate and reliable estimation; and (3) eliminates weaknesses that existed near the lower boundary of statistical optimization (30 km) by improving the uncertainty formulation and transition to purely observed data across this boundary.

We illustrated and discussed key variables of the dynamic background and observation error covariance matrices, including systematic and random uncertainties and correlation functions, in order to provide insight and show the realistic character of their behavior. Both the random and systematic background uncertainties appear to be largest in the polar regions of the winter hemisphere at mesospheric altitudes. The observation uncertainties capture variations with altitude, especially in the mesosphere, and can well represent the error characteristics of RO events, from high-quality simulated data to comparatively noisy CHAMP data. The observation error correlation functions show a similar shape to background data, but with less functional smoothness and with much smaller correlation lengths of about 0.8 km (while background error correlation lengths range from about 1 km near 20 km to about 6 km near 80 km). All uncertainty and correlation estimates were found to exhibit little sub-monthly variations during the test months January and July 2008. In the case of anomalous sub-monthly conditions (e.g., sudden stratospheric warming) that will occasionally happen during long-term processing periods, we expect more variation, however.

The new dynamic algorithm was evaluated mainly against the one currently used in the OPSv5.6 system, using simulated MetOp data on single days (15 January and 15 July 2008) and real observed CHAMP and COSMIC data from two full months (January and July 2008). The following was found for the new dynamic algorithm, in particular compared to OPSv5.6: (1) it can reduce systematic errors (biases) and standard deviations of optimized bending angles, as proven by simMetOp data including “true” reference profiles from end-to-end simulations, and subsequently also benefits the error characteristics of retrieved refractivity and temperature profiles; (2) it can reduce the random errors of optimized bending angles in the stratosphere for real data, as evaluated for CHAMP and COSMIC, still at the same time leaving less or about equal residual systematic error (bias) in the bending angles; (3) it can better account for the observational noise in the mesosphere, leading to larger standard deviations than OPSv5.6 there from greater weight of the observations in the optimized profiles, albeit without applying any artificial observation uncertainty values in the case of high noise levels.

Beyond the evaluation of the new dynamic algorithm against OPSv5.6, atmospheric profiles from UCAR/CDAAC were also intercompared, including use of very recently released CHAMP data from the newest (2014) CDAAC data version. It was found that CDAAC bending angles generally exhibit markedly higher standard deviations above about 35 km and that in particular the new data version shows comparatively large systematic differences and standard deviations. The reasons for this new-version behavior deserve further study.

Overall, compared to previous simplified approaches of statistical optimization, the dynamic algorithm presented here, which realistically estimates both background and observation error covariance matrices, contains high capabilities for future large-scale implementation. The evaluation of the algorithm provided clear evidence that it can deliver reliable and accurate atmospheric profiles for atmosphere and climate applications. The results therefore indicate high suitability for employing the new dynamic approach in the processing of long-term RO data into a climate record, leading to well-characterized and high-quality atmospheric profiles over the entire stratosphere.

We thank M. Bonavita and S. B. Healy (ECMWF Reading, UK) for valuable advice related to ECMWF's analysis and forecast and associated error characteristics, and we are thankful for fruitful discussions within the IGMAS group at the IGG (Wuhan, China) and to Suqin Wu at RMIT. We also thank UCAR/CDAAC for access to their RO data as well as ECMWF for access to their analysis and forecast data. Furthermore, we acknowledge the funding support by China Natural Science Funds (no. 41231064, no. 41321063), National 973 (no. 2012CB825604), China Scholarship Council and CAS/SAFEA International Partnership Program for Creative Research Teams (KZZD-EW-TZ-05) at the IGG side; the funding support by the Australian Space Research Program (ASRP2), the Australian Antarctic Science (AAS) Grant project (AAS 4159) and the Australian Natural Disaster Resilience Grant Scheme (NDRG) of Victoria at the RMIT side; and the funding support by the European Space Agency (ESA) project OPSGRAS, the Austrian Research Promotion Agency (FFG) project OPSCLIMTRACE, and the Austrian National Science Fund (FWF) project DYNOCC (T620-N29) at the WEGC side. Edited by: A. von Engeln