Random uncertainties and vertical error correlations are estimated for three independent data sets. The three collocated data sets are (1) refractivity profiles of radio occultation measurements retrieved from the Metop-A and B and COSMIC-1 missions, (2) refractivity derived from GRUAN-processed RS92 sondes, and (3) refractivity profiles derived from ERA5 forecast fields. The analysis is performed using a generalization of the so-called three-cornered hat method to include off-diagonal elements such that full error covariance matrices can be calculated. The impacts from various sources of representativeness error on the uncertainty estimates are analysed. The estimated refractivity uncertainties of radio occultations, radiosondes, and model data are stated with reference to the vertical representation of refractivity in these data sets. The existing theoretical estimates of radio occultation uncertainty are confirmed in the middle and upper troposphere and lower stratosphere, and only little dependence on latitude is found in that region. In the lower troposphere, refractivity uncertainty decreases with latitude. These findings have implications for both retrieval of tropospheric humidity from radio occultations and for assimilation of radio occultation data in numerical weather prediction models and reanalyses.

In variational estimation of geophysical parameters from satellite
observations, the obtained accuracy relies on the validity of the
underlying uncertainty and error correlation assumptions of the
observation and of the model background fields.
The three-cornered hat (3CH) method

In numerical weather prediction (NWP), the method developed by

To distinguish error correlations between data sets from vertical
error correlations within each data set, we will refer to the former as

Recently

In this paper the 3CH method is generalized to include off-diagonal
elements of the error covariance matrices. We apply the generalized
3CH (G3CH) to three data sets where the random error components can
be assumed not to be interdependent, meaning that their error cross
correlations are assumed to be negligible. The refractivity error covariance
matrices of RO measurements are estimated and compared with current
vertical correlation assumptions, used in 1D-Var retrieval of specific
humidity and temperature from RO refractivity. The main objective of
this study is to assess refractivity random uncertainty and vertical
error correlations, expressed as the refractivity error covariance
matrix, to be used in 1D-Var retrieval of temperature and specific
humidity

The remainder of the paper is organized as follows: Sect.

The terms

The term

For a given refractivity data profile,

The quantity

Given the definition of

Three data sets are combined in the analysis. The RO data set includes refractivity profiles from the Metop and
COSMIC-1 missions

The ERA5 forecast is prepared at model levels and
interpolated in time (3 h grid) and horizontal space (

The 3CH method has historically been applied to triplets of data
without considering vertical error correlations, meaning that the data
sets have effectively been treated as scalar properties

The error vectors

In order to compensate for the impact of collocation uncertainty

The 3CH algorithm cannot distinguish between true physical variability
and mutual positive error correlations

In this study the measurement error cross correlations between the chosen data sets are assumed to be negligible, since the three data sets at hand are obtained by completely independent techniques. In particular the ERA5 model forecast data are chosen such that no information from either a given RO or an RS92 profile can have been passed to the forecast being used in a given collocation triplet. However, if two data sets have similar vertical footprints, or if they are sampled at similar horizontal scales, these two data sets may have cross-correlated errors, and possibly biases. All biases are removed prior to application of G3CH, but the error cross correlations introduced by finite vertical footprints or similar horizontal scale may influence the result of G3CH.

Global biases of RS92 and RO refractivity,
with ERA5 used as reference, based on all collocation triplets
(collocated ERA5, RO, and RS profiles) used in this study,
evaluated at the RO reference location. See also Sect.

The three data sets differ in their vertical footprints. The RS92
radiosonde has a vertical footprint of around 50 m

Refractivity of two selected triple collocations
exemplifying differences in vertical footprint.

Uncertainty estimates for any vertically represented variable must refer to a specified vertical footprint to be meaningful. Thus, the G3CH analysis has to be accompanied by an assessment of the vertical footprints of the data sets. In our approach the data set with the largest vertical footprint determines the common vertical footprint to be used for all three data sets. In other words, if one of the data sets does not contain information below a certain length scale, there is not enough information in the data triplet to apply the G3CH method to estimate uncertainties related to variability below that length scale.

Because ERA5 is missing some fine-scale physical features, seen in the
better resolved RO and RS92 data set, we are forced to state the
uncertainty on the common scale determined by ERA5. This means that
the RO and RS92 data must be smoothed to match the ERA5 vertical footprint
prior to the G3CH analysis. If this smoothing is omitted, the G3CH may
give a biased estimate of uncertainties. By smoothing the data sets to
a common scale we remove both the physical features and the errors on scales
shorter than the common vertical footprint. Therefore the estimated
uncertainties of RO and RS92, which are correct on the common
scale found, may be viewed as lower uncertainty boundaries for these
variables on their native scales. The vertical footprints of the
three data sets are examined in Sect.

In this section the G3CH results are presented, first as raw
unfiltered uncertainty estimates, then with corrections for
collocation mismatch (

Figure

Raw estimate of refractivity random uncertainty (standard deviation) of ERA5, RO, and RS92 at middle latitudes.

The most striking feature in Fig.

The impact on RS92 of changing
collocation distance is shown as an example in Figs.

Estimate of refractivity uncertainty (standard deviation as percent of ERA5 mean refractivity) of RS92 for a series of collocation criteria. The dashed black line shows the STDV obtained by extrapolation of the variance to zero collocation distance.

Examples of extrapolation of refractivity error variance of RS92, i.e. diagonal elements of covariance matrix, to zero collocation criterion at five different altitudes. The variances are divided by the mean ERA5 refractivity.

3CH estimates of refractivity error
standard deviations are shown for 300 and 0 km collocation
criterion for each of the ERA5, RS92, and RO data sets at low

In Fig.

Effect of smoothing on error
standard deviations of ERA5

Estimates of the ERA5 vertical footprint
for middle and high latitudes. The effect of filtering of RO and
RS92 on the estimated ERA5 error standard deviation,

A similar analysis cannot be performed for the RO or RS92 data sets,
because these appear to have small vertical footprints which happen to lie
close to each other. There is not a finite filter length which
minimizes the refractivity error standard deviation for RO and RS92
(the filters being applied to the complementary data sets in each
case). Therefore the RO and RS92 vertical footprints cannot be inferred from
these three data sets alone, but it can be concluded that their
vertical footprints are smaller than the ERA5 vertical footprint since the ERA5 estimated
error standard deviation decreases if either the RO or RS92 is
smoothed. This is illustrated in Fig.

Estimates of middle-latitude refractivity
error variances based on smoothed data divided by refractivity
error variance based on un-smoothed data;

To estimate the final G3CH uncertainties with reference to the common
vertical footprint determined by ERA5, all three raw data sets were
smoothed with a Gaussian filter with the width of the ERA5 vertical footprint
prior to the G3CH analysis. In Fig.

Best estimate of refractivity
uncertainties, shown as standard deviations in percent of the
ERA5 refractivity for ERA5, RO, and RS92 at low

In Figs.

Generally the vertical correlations are divided into two separable
regimes: Close to the diagonal we see a short-range correlation with
standard deviation of approximately 0.5 km, and a long-range
correlation component of varying shape and amplitude. The short-scale
vertical correlations are very similar for all data sets. Rising
occultations are found to have larger vertical error correlations (and
slightly larger standard deviation where correlations are broader)
than setting occultations in this data set, which is seen when
comparing panel (b) with panel (e) in Fig.

G3CH estimate of ERA5

Same as in Fig.

It is worth noting that the estimated vertical correlations of RS92 are larger for setting than for rising RO at high latitudes, especially between 6 and 22 km. Thus the G3CH fails to give an independent estimate of the RS92 correlations. The RS92 is expected to have long-ranging vertical correlations due to corrections implemented in the GRUAN processing, but the G3CH fails to attribute these correctly when strong long-range correlations are also present in the RO data. The estimated RS92 diagonals (standard deviations superimposed vertically on correlation matrices) seem reasonably consistent for rising and setting occultations.

The relative magnitude of the off-diagonal covariance components can
also be viewed in Fig.

Estimate of RO vertical error correlations at approximately 5 and 20 km for low, middle, and high latitudes (full curves). The dashed lines show exponential correlation with three kilometre decay length. No smoothing was applied on any of the data sets before the G3CH was applied.

It is evident from the results in Sect.

The unfiltered G3CH RO uncertainty estimates, also seen in
Fig.

The estimated uncertainties of RO in the lower stratosphere
are only slightly above 0.2 %, the theoretical estimates found in

The RO uncertainty in the Upper Troposphere–Lower Stratosphere (UTLS),
where the uncertainty estimates are at a minimum, does not vary much
with latitude, except for a small increase at the tropopause. The
structure of the uncertainty profiles is quite similar for all
latitudes. The increase of uncertainty in the troposphere is smaller
at high latitude, but the crossover between high and low uncertainty
happens at approximately the same altitude, 5–7 km, for all
latitudes. For the tropics, the noisy uncertainty estimates, due to an
insufficient number of data, do not allow us to safely read a minimum
above the tropopause from Fig.

In

In the theoretical uncertainty analysis study by

The estimation of error correlation matrices with G3CH is a novelty introduced in the present study. The method is able to detect expected differences between rising and setting RO long-range vertical error correlations, and long-range correlations are also seen in RS92 data. The vertical correlation estimates have limitations since the estimate of long-range vertical error correlations of RS92 seems to be dependent on whether the RO data used represent rising or setting occultations. However, this inaccuracy seems relatively small compared to the difference between the RO vertical error correlation estimates found and the vertical error correlation estimates currently used in RO 1D-Var retrievals, and therefore the correlation estimates will be useful in this context. Especially at middle latitudes there is a potential for decreasing the vertical error correlation length in future applications.

The analysis of G3CH presented here may have consequences for uncertainty
parameterization in retrieval and assimilation of RO refractivity. In
particular the estimates of RO refractivity uncertainty reveal a
potential for deflating tropospheric refractivity uncertainty in the
ROM SAF 1D-Var configuration. A reduction of assumed refractivity
uncertainty is of particular interest in the troposphere, where it can
improve the information content of water vapour retrievals. There is
a need to establish tropospheric water vapour climate data records
for climate research, for instance as expressed in the
objectives of the GEWEX water vapour assessment (G-VAP;

The collocation-corrected G3CH random uncertainty estimate provides full refractivity error covariance matrices for three independent data sets. The method was applied to collocated refractivity profiles from ERA5 forecasts, RO, and radiosondes.

The RO refractivity uncertainty is between 0.2 % and 0.6 % in the UTLS at between 8 and 25 km at middle and high latitudes, and between 0.2 % and 1.4 % below 8 km. The G3CH method presented here also yields estimates of the vertical error covariance matrices for refractivity.

The achieved refractivity uncertainty estimates are lower than empirically determined uncertainties previously reported in the literature. The results can be used to model uncertainty assumptions used in NWP data assimilation and in 1D-Var calculations of atmospheric temperature and specific humidity based on RO refractivity data.

The analysis was performed in Jupyter Notebook. The code is kept in an internal git repository at the Danish Meteorological Institute. It is available from the corresponding author upon request.

The RO refractivity profiles and interpolated ERA5 temperature and humidity profiles on model levels (including surface pressure) are available at the ROM SAF web-page

The supplement related to this article is available online at:

JKN put together (and partly invented) the method, performed the analysis, and is the main author of the paper. SS, KBL, and HG contributed with substantial reformulations of every part of the manuscript, through multiple iterations, which led to changes in the basic rationale of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Analysis of atmospheric water vapour observations and their uncertainties for climate applications (ACP/AMT/ESSD/HESS inter-journal SI)”. It is not associated with a conference.

This work was carried out as part of EUMETSAT's Radio Occultation Meteorology Satellite Application Facility (ROM SAF) which is a decentralised operational RO processing center under EUMETSAT. Johannes K. Nielsen, Hans Gleisner, Stig Syndergaard, and Kent B. Lauritsen are members of the ROM SAF. We wish to thank the reviewers John Eyre, Paul Poli and the third anonymous reviewer for helping to improve the paper.

This research has been supported by the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT).

This paper was edited by Chunlüe Zhou and reviewed by John Eyre, Paul Poli, and one anonymous referee.