The terms random uncertainty and systematic uncertainty are used as defined in the Guide to the Expression of Uncertainty in Measurement (GUM; ). We adopt the concept of error covariance (matrix) and error correlation (matrix) from NWP terminology to describe vertically correlated random uncertainties, and we use the terms error variance and error standard deviation to refer to the diagonal of an error covariance matrix and its square root.

The term vertical footprint of a data set is used here in the same way as in : The vertical scale that an observation value represents. The word resolution may be used to describe this property, but we shall avoid this term because in the NWP community it is used in the meaning of sampling density – the number of data points per spatial interval (e.g. in ). The vertical footprint will typically be larger than the distance between the vertical height levels which the data values refer to.

For a given refractivity data profile, x, we consider the observation error ε as the deviation from the unknown truth t, ε=x-t. The G3CH does not make any assumptions about exactly what the true profile t is. t may be thought of as defined with respect to a given but unknown finite vertical footprint, which may differ from the vertical footprints of all three data sets.

The quantity ε is a sum of the measurement error εI and a representativeness error εR. Both terms may contain random and systematic error components, but for each subset of collocated triplets being analysed, we remove systematic error differences between the three involved data subsets prior to the analysis. The measurement error εI acts as a superimposed noise, possibly correlated in space and time. εI may for instance include instrument errors, radio noise from external sources, and also some errors arising during data processing steps. The εR component represents the distortion of the underlying truth in a data set, as it is being mapped to the vertical observation grid. εR contains errors associated with, for instance sampling, interpolation, and mismatch between the observation grid and measurement resolution in time and space. Especially εR contains a geometric error component, εG, representing the departure of the ERA5, RO, and RS92 vertical or skewed profiles in time and space from the unknown true profile at the RO reference coordinates. The RO reference coordinate is the point at which a straight line between the GNSS satellite and the receiving Low Earth Orbiter tangents the Earth ellipsoid. The ERA5 profile is strictly vertical, interpolated to the RO reference time and position, while the RO profile is a weighted average of the 3D atmosphere in the plane of occultation , and the radiosonde follows the balloon trajectory. The forward operator used estimates refractivity along a 1D assumed vertical line, and this has an impact on the uncertainty estimates. Thus, the RO observation errors estimated by the 3CH method in this paper are applicable for variational assimilation with a 1D operator, but not for 2D or 3D operators. The time scale of an RO profile is in the order of 1 min and the timescale of an radiosonde profile is in the order of 1 h. The skew trajectories of the RO tangent points and RS92 balloons are assumed not to be correlated with each other. Hence the εG term can be assumed to contain no cross correlations. However, there are potentially error cross-correlation components arising from spatial correlations between the data sets, which we cannot assess. This could, for example be the case for ERA5 and RO, because these are sampled on similar horizontal scales.

Given the definition of t to be the actual profile at the RO reference location and time, there are, in addition to εR and εI, errors induced by the methods applied in this paper. These are a collocation error, εC, due to the distance in time and space between the radiosonde and the reference coordinates, and a cross-correlation error, εX, representing error cross correlations induced by the finite vertical footprints of the three data sets. The raw G3CH uncertainty estimate for one of the data sets will not represent the observation error, but it will represent a combination of the observation error ε and error components added by the G3CH: 1εG3CH=εI+εR+εC+εX. We are able to remove εC and the εX components of the three data sets, by adding the following additional analysis steps to the G3CH. The εC covariance matrix, CC, is eliminated by first calculating G3CH estimated covariance matrices Ci for a series of collocation subsets, sampled from areas of decreasing size around the RO reference coordinates. Next, the sequence of decreasing covariance estimates is extrapolated to the virtual zero-area case C0, by assuming that CiC=Ci-C0. Subsequently the εX covariance matrix, CX, is eliminated by smoothing all three data sets such that they have the same vertical footprint, and then for each data set a covariance matrix Cs is calculated with G3CH from the smoothed data sets, such that CX=C0-Cs. Thus, the observation error covariance matrices that we estimate in the end include only measurement error εI and representativeness error εR. 2ε=εI+εR. The final estimate of ε will be stated with reference to a common vertical footprint of the three data sets, which is determined by the data set with the largest vertical footprint, ERA5. These general definitions of measurement error and representativeness error are thought to be applicable for all three data sets.

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 . A straightforward generalization of the method allows us to also infer internal error correlations for each data set. In the generalized 3CH (G3CH) it is assumed that we have three independent variables x,y, and z, which are composed of four stochastic vectors; the truth t, and three independent error terms εx, εy, and εz, such that 3x=t+εx,y=t+εy,z=t+εz. In the present paper x,y, and z may represent atmospheric refractivity profiles obtained from different sources. εx,εy, and εz represent the random observation error vectors. In the following the bracket notation, 〈〉, is used to denote expectation values.

The error vectors εx,εy, and εz may have internal correlations, expressed as error covariance matrices X=〈εxεxT〉, Y=〈εyεyT〉, and Z=〈εzεzT〉, but we assume no cross-correlation components; i.e. 〈εxεyT〉=〈εxεzT〉=〈εzεyT〉=0. We may allow that the error is correlated with the physical property t; e.g. 〈tεxT〉≠0. Besides this, no assumptions are made about the particular shape of error distribution functions. In the present paper we only estimate the random uncertainties. In practice we remove biases in each subset of collocations where G3CH is to be applied by subtracting the subset mean of each of the three data sets prior to the analysis. Thus, in the following derivation we can assume that all data are bias-free. In the absence of bias, the covariance matrices of each subtraction pair can be written as 4〈x-yx-yT〉=〈xxT+yyT-xyT-yxT〉,〈x-zx-zT〉=〈xxT+zzT-xzT-zxT〉,〈y-zy-zT〉=〈yyT+zzT-yzT-zyT〉. Expanding the right-hand side of, for instance the first line of Eq. () we obtain 5〈x-yx-yT〉=〈εxεxT-εxεyT-εyεxT+εyεyT〉. If we keep in mind that error cross correlations between data sets are set to zero, the three subtraction pair covariances reduce to 6〈x-yx-yT〉=〈εxεxT+εyεyT〉,〈x-zx-zT〉=〈εxεxT+εzεzT〉,〈y-zy-zT〉=〈εyεyT+εzεzT〉. Finally, by solving these three equations for the error covariance matrices X=〈εxεxT〉, Y=〈εyεyT〉, and Z=〈εzεzT〉 for the variables x,y, and z, we get X=〈εxεxT〉=12〈x-yx-yT+x-zx-zT-z-yz-yT〉, Y=〈εyεyT〉=12〈y-xy-xT+y-zy-zT-x-zx-zT〉, 7Z=〈εzεzT〉=12〈z-xz-xT+z-yz-yT-x-yx-yT〉. The above G3CH model is applied to the three data sets described in Sect. . In this analysis the mean is subtracted from each data set prior to applying the G3CH. The biases are not the focus here, but for reference the global means of RS92 and RO refractivity differences to ERA5 for all collocations used in the analysis are plotted in Fig. .

In order to compensate for the impact of collocation uncertainty εC on the obtained refractivity error covariance matrices, the G3CH analysis is applied to a series of data subsets with increasing collocation distances between 50 and 300 km. The collocation uncertainty is removed from the uncertainty estimates by extrapolating the covariance matrices to zero collocation distances. This procedure, which is also performed by in another context, also allows one to track how the G3CH method partitions the collocation uncertainty among the three data sets (see Sect. ).

The 3CH algorithm cannot distinguish between true physical variability and mutual positive error correlations . In cases where errors of two data sets (x and y) are positively correlated, the discrepancy between the third data set, z and (x,y) will be attributed as an uncertainty of z, because the term 〈(x-y)(x-y)T〉 would be reduced in such cases.

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.

The three data sets differ in their vertical footprints. The RS92 radiosonde has a vertical footprint of around 50 m . This vertical footprint is increased through the interpolation to the common grid , and through the procedure for correcting for collocation error. The RO refractivity has been shown to have a vertical footprint of about 200 m under optimal conditions in the lower troposphere . In the RO data used here, the processing has removed some small-scale information, and thus the RO vertical footprint is expected to be larger than 200 m. In Fig.  two examples of refractivities of triple collocations are shown. The plots illustrate the ability of resolving vertical structures in the middle troposphere and lower stratosphere of the three data sets. Even though the highly resolved ERA5 has 137 vertical levels, shown on the right vertical axis, it provides a somewhat smoother representation of the vertical structures compared to the RO. The RO profiles and RS92 profiles show more vertical structure than ERA5.

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. .

Figure  shows the raw estimates of the mid-latitude refractivity uncertainty expressed as error standard deviation of the three data sets, obtained by applying the G3CH directly to the raw data sets. Generally the G3CH attributes a big part of the collocation error (εxC) to the RS92 uncertainty. The reason is that the collocation is performed by interpolating ERA5 to the RO reference point, such that ERA5 and RO are closely collocated, while RS92 is chosen such that it is within 300 km of the RO reference point; thus, naturally RS92 will stand out from the two other data sets in many cases.

The most striking feature in Fig.  is the bulge of RS92 around the tropopause. The main part of this bulge is removed along with the collocation uncertainty by the procedure described in Sect. . We calculate the G3CH estimates of covariance matrices for a sequence of collocation criteria (between 50 and 300 km) and use these to extrapolate all covariance matrices to 0 km collocation distance, with a linear fit to the full covariance matrices as a function of the squared collocation distance. The effect of varying the temporal collocation window is small, and therefore we have excluded that from the analysis.

The impact on RS92 of changing collocation distance is shown as an example in Figs.  and  where a few examples of extrapolations are presented. The result of this procedure is summarized for all three data sets in Fig. . The RS92 uncertainty estimate is reduced considerably, while the uncertainty estimates for the two other data sets are slightly changed. In the subsequent analysis the 0 km estimates of covariances are used for evaluation of covariance matrices and difference terms in the G3CH equations.

In Fig.  the impact of smoothing on error standard deviations estimated with G3CH (Eq. ) is shown for middle latitudes. The smoothing is applied as a sequence of Gaussian filters of increasing widths. For each data set, filtering has been performed, not on the data set itself, but on the two other complementary data sets (see figure legends). The idea is basically to probe the vertical footprint of one data set with two other data sets of varying vertical footprint. The G3CH analysis was performed at the sequence of such prepared triplets of data sets with increasing filter width. The impact of applying sequences of Gaussian filters is best viewed near the tropopause. We note that all variances eventually start to grow at some filter width, but the ERA5 error standard deviation drops in most cases at small filter widths, and does not start to increase until the width of the filter, applied on the RO and RS92 data, exceeds a certain threshold. We interpret this threshold as the ERA5 vertical footprint. The ERA5 vertical footprint was estimated for each altitude, as the minimum of a second-order polynomial, fitted to σERA5 as a function of filter width. These vertical footprints are plotted in Fig. , for middle and high latitudes. At low latitudes the result is unstable, and therefore that plot has been omitted. We use these results to identify a common ERA5 vertical footprint to be applied globally as the mean of the middle- and high-latitude vertical footprints, shown as a dashed line in Fig. .

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. : The impact of smoothing RS92 on the ERA5 variance is shown in Fig. a. Generally σERA52 decreases as the RS92 data are brought closer to the ERA5 data by smoothing, consistent with RS92 having a smaller vertical footprint than ERA5. The RO error variance, σRO2, on the other hand increases as a result of smoothing the RS92 data (see Fig. b). This is consistent with the RS92 vertical footprint being close to the RO vertical footprint and the RS92 data moving closer to the ERA5 data as smoothing is applied to RS92. In Fig. c σERA52 is seen to decrease as the RO refractivity is brought closer to ERA5 refractivity, as smoothing is applied to RO.

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.  the final G3CH inferred uncertainties are shown for each data set for low, middle, and high latitudes. For all data sets the unfiltered uncertainty is also plotted, for later discussion.

In Figs. and  the G3CH-based error covariance matrices for ERA5, RO, and RS92 are shown for rising and setting occultations for middle and high latitudes. These matrices have been calculated without any vertical filtering applied. The tropics are not shown because of insufficient number of data in that region. The fine-scale off-diagonal structures must be attributed to statistical noise, but there are certainly larger-scale vertical correlation structures especially in the RO and RS92 data.

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.  and panel (b) with panel (e) in Fig. . This is believed to be due to the ionospheric correction in the RO processing for rising occultations, where the L2 GPS signal is often not available below 20 km, and extrapolation from above is necessary. In the CDR v1.0 data set, it is in particular the rising occultations for Metop after instrument firmware upgrades in 2013 that suffer from missing L2 data below 20 km , and consequently there are broader vertical error correlations in the retrieved refractivity profiles for Metop after 2013 (not shown).

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. : Here the vertical error correlation function of RO refractivity is exemplified for two heights, approximately 5 and 20 km, at low, middle, and high latitudes. The correlation functions are slices of the RO refractivity error correlation matrix at these altitudes. For instance at high latitude there are pronounced long-range correlations at these two heights. In the tropics the data are too sparse to get an estimate of the correlation function. In Fig.  the dashed curves show the three kilometre exponential correlation which is assumed in the current ROM SAF 1D-Var analysis . Given that the finer correlation structures, around the 1 km scale, are influenced by sparseness of data, the current correlation function appears to be reasonably adequate at high latitudes at the selected altitudes. At middle latitudes there is a potential for decreasing the error correlation length in future applications.