Estimating Individual Radio Occultation Uncertainties Using the Observations and Environmental Parameters

Jeremiah Sjoberg aCOSMIC Program Office, University Corporation for Atmospheric Research, Boulder, Colorado

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Richard Anthes aCOSMIC Program Office, University Corporation for Atmospheric Research, Boulder, Colorado

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Hailing Zhang aCOSMIC Program Office, University Corporation for Atmospheric Research, Boulder, Colorado
bJoint Center for Satellite Data Assimilation, University Corporation for Atmospheric Research, Boulder, Colorado

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Abstract

Estimation of uncertainties (random error statistics) of radio occultation (RO) observations is important for their effective assimilation in numerical weather prediction (NWP) models. Average uncertainties can be estimated for large samples of RO observations and these statistics may be used for specifying the observation errors in NWP data assimilation. However, the uncertainties of individual RO observations vary, and so using average uncertainty estimates will overestimate the uncertainties of some observations and underestimate those of others, reducing their overall effectiveness in the assimilation. Several parameters associated with RO observations or their atmospheric environments have been proposed to estimate individual RO errors. These include the standard deviation of bending angle (BA) departures from either climatology in the upper stratosphere and lower mesosphere (STDV) or the sample mean between 40 and 60 km (STD4060), the local spectral width (LSW), and the magnitude of the horizontal gradient of refractivity (|∇ HN|). In this paper we show how the uncertainties of two RO datasets, COSMIC-2 and Spire BA, as well as their combination, vary with these parameters. We find that the uncertainties are highly correlated with STDV and STD4060 in the stratosphere, and with LSW and |∇ HN| in the lower troposphere. These results suggest a hybrid error model for individual BA observations that uses an average statistical model of RO errors modified by STDV or STD4060 above 30 km, and LSW or |∇ HN| below 8 km.

Significance Statement

These results contribute to the understanding of the sources of uncertainties in radio occultation observations. They could be used to improve the effectiveness of these observations in their assimilation into numerical weather prediction and reanalysis models by improving the estimation of their observational errors.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jeremiah Sjoberg, sjoberg@ucar.edu

Abstract

Estimation of uncertainties (random error statistics) of radio occultation (RO) observations is important for their effective assimilation in numerical weather prediction (NWP) models. Average uncertainties can be estimated for large samples of RO observations and these statistics may be used for specifying the observation errors in NWP data assimilation. However, the uncertainties of individual RO observations vary, and so using average uncertainty estimates will overestimate the uncertainties of some observations and underestimate those of others, reducing their overall effectiveness in the assimilation. Several parameters associated with RO observations or their atmospheric environments have been proposed to estimate individual RO errors. These include the standard deviation of bending angle (BA) departures from either climatology in the upper stratosphere and lower mesosphere (STDV) or the sample mean between 40 and 60 km (STD4060), the local spectral width (LSW), and the magnitude of the horizontal gradient of refractivity (|∇ HN|). In this paper we show how the uncertainties of two RO datasets, COSMIC-2 and Spire BA, as well as their combination, vary with these parameters. We find that the uncertainties are highly correlated with STDV and STD4060 in the stratosphere, and with LSW and |∇ HN| in the lower troposphere. These results suggest a hybrid error model for individual BA observations that uses an average statistical model of RO errors modified by STDV or STD4060 above 30 km, and LSW or |∇ HN| below 8 km.

Significance Statement

These results contribute to the understanding of the sources of uncertainties in radio occultation observations. They could be used to improve the effectiveness of these observations in their assimilation into numerical weather prediction and reanalysis models by improving the estimation of their observational errors.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jeremiah Sjoberg, sjoberg@ucar.edu

1. Introduction

Observations are the foundation for scientific understanding of atmospheric processes and applications, notably weather forecasting, which is based on numerical weather prediction (NWP) for forecasts beyond a few hours, as well as climate reanalyses, which provide consistent global atmospheric datasets widely used in research. Modern data assimilation systems assimilate many different types of observations, all with different bias and random errors. These data assimilation systems weight observations inversely to their estimated random error statistics (Desroziers et al. 2005; Fletcher 2017), so it is important to specify the uncertainties for the different observations as accurately as possible.

In the past 20 years Global Navigation Satellite System (GNSS) radio occultation (RO) observations of atmospheric bending angles associated with atmospheric refractivity have become one of the most impactful observational systems in global NWP and reanalysis (Cardinali and Healy 2014; Bonavita 2014). Anthes (2011), Bonafoni et al. (2019), Ho et al. (2020a), and Scherllin-Pirscher et al. (2021) provide reviews of RO and its applications. RO has small bias errors throughout most of the troposphere (above the atmospheric boundary layer), stratosphere, and lower mesosphere, but it has relatively large random errors (uncertainties) in the lower half of troposphere, upper stratosphere, and lower mesosphere. RO has the smallest bias and random errors between about 7 and 35 km (Poli et al. 2010; Harnisch et al. 2013).

These statistics of the errors describe the RO observations as a whole, and use of simple vertical profiles of RO uncertainty, such as those used in the European Centre for Medium-Range Weather Forecasts (ECMWF) assimilation system has been shown to be effective in reducing short-term forecast errors (e.g., Ruston and Healy 2021). But RO uncertainties vary with time and location (Semane et al. 2022), and accounting for this variability has been shown to lead to improved forecasts (Bowler 2020b).

In this paper we further investigate how the uncertainties of a large set of global RO bending angle (BA) observations vary with respect to four observational and environmental parameters. The selected dataset consists of observations from the Constellation Observing System for Meteorology, Ionosphere and Climate 2 (COSMIC-2, or C2 hereafter) constellation and Spire commercial RO data. We find that these parameters have a strong linear relationship to the variance of random errors at certain altitudes, and thus may be useful for adjusting prescribed observation error models used in data assimilation.

2. Sources of radio occultation errors and their indicators

Anthes et al. (2022) and references therein describe the sources of RO uncertainties. In the stratosphere and lower mesosphere (30–60 km), the uncertainties are caused primarily by orbit errors, ionospheric residual noise, receiver and transmitter clock errors, and receiver thermal noise. The latter is related to signal-to-noise ratio (SNR) at the receiver. In the troposphere, random errors are caused primarily by horizontal atmospheric inhomogeneities, or deviations of the atmospheric refractivity from spherical symmetry, which is a key assumption in the RO retrieval of bending angles and refractivity (Melbourne et al. 1994). In addition to horizontal inhomogeneities, large vertical gradients of refractivity alone can cause multipath and uncertainties in RO retrievals.

Spherical symmetry means that there are no horizontal variations of refractivity (N) along the ray path. This condition will be met if there are no horizontal gradients of refractivity in the vicinity (∼300 km) of the RO tangent point, i.e., |∇HN| = 0, where the horizontal gradient is defined on constant height surfaces. Refractivity is a function of temperature, pressure, and water vapor, so horizontal inhomogeneities of any of these three variables can violate the assumption of spherical symmetry. Departures from spherical symmetry can be caused by horizontal variations of N on any scale, ranging from random small-scale (1 km or less) turbulence or clouds to larger-scale (on the order of 50–500 km) horizontal gradients of N. The larger scales are resolvable by NWP and climate models (Healy 2001), while the smaller scales are currently not.

As discussed below, several parameters associated with the RO observation itself or its atmospheric environment are related to RO uncertainties. Above 30 km (stratosphere and lower mesosphere), the uncertainties are related to the standard deviation of BA departures from a chosen mean. In the lower half of the troposphere, RO uncertainties are related to the local spectral width (LSW hereafter; Gorbunov et al. 2006; Zhang et al. 2023). The horizontal gradient of N, ∇HN, as resolvable by models should also be related to uncertainties, although it does not capture small-scale horizontal inhomogeneities associated with turbulence and clouds nor possible large vertical gradients of N that cause multipath.

a. Standard deviation of bending angle departures from climatology above 60 km (STDV) and from the sample mean between 40 and 60 km (STD4060)

Bending angle uncertainties above 30 km can be characterized by the standard deviation of BA anomalies from climatology between 60 and 80 km, where the main error sources are thermal noise, ionospheric residuals, and GNSS clock errors (Schreiner et al. 2020). This parameter is reported by the UCAR COSMIC Data Analysis and Archive Center (CDAAC) as STDV. We show in this paper that a related parameter, the standard deviation of the normalized BA anomaly from the sample mean being studied between 40 and 60 km, which we designate as STD4060, is also closely related to the BA uncertainties above 30 km. This parameter is easy to calculate for any sample, whereas STDV requires a climatology and may not always be provided for different RO datasets. The appendix gives an alternative definition of STD4060 that produces highly consistent results for this study and may be preferable for operational settings.

b. LSW

Hocke et al. (1999), Gorbunov et al. (2006), Sokolovskiy et al. (2010), Liu et al. (2018), and Zou et al. (2019) describe the use of LSW as a parameter related to the uncertainty of RO observations, as well as their biases, in the troposphere. Departures from spherical symmetry can cause multiple rays from the GNSS transmitters to occur at the same impact height, a condition called multipath. When multipath occurs, the spectrum of the wave optics–transformed RO signals contain multiple spectral components, resulting in an increase in the overall width of the spectrum, or LSW, as indicated in Figs. 1 and 2 of Liu et al. (2018).

LSW depends mainly on fluctuations of the RO signal induced by propagation through horizontally inhomogeneous structures and inversion layers in the troposphere. Above the troposphere and in higher latitudes where fluctuations are weak, the LSW values are small. Where large horizontal inhomogeneities in refractivity exist, caused either by temperature or water vapor fluctuations, the LSW depends on the SNR. Large bending angles correspond to low signal amplitudes, which can be measured better by high SNR. Thus, higher SNR observations can resolve larger fluctuations and therefore higher LSW values, even in the same atmospheres (i.e., sampled by collocated observations) as those with lower SNR. Lower SNR also results in smoother vertical profiles of BA, even with the same processing and without explicit filtering.

Fractional LSW is the LSW divided by the BA, expressed as a percent. Liu et al. (2018) found that a quality control (QC) procedure that eliminated RO observations with large fractional LSW (greater than 35%) improved the water vapor analysis in the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2008) and the National Center for Atmospheric Research (NCAR) Data Assimilation Research Testbed (DART; Anderson et al. 2009). An alternative way of using LSW in NWP data assimilation is to base the assumed random error of individual RO observations according to their LSW magnitude, thus weighting observations with high LSW less than those with small LSW. Zhang et al. (2023) found that this approach improved the impact of RO observations in the National Centers for Environmental Prediction (NCEP) Gridpoint Statistical Interpolation (GSI) and Global Forecast System (GFS), reducing both the root-mean-square and bias errors. We consider this approach in this study.

c. Horizontal gradient of refractivity (|∇HN|)

Horizontal gradients of refractivity ∇HN on constant height surfaces cause a violation of spherical symmetry, which results in the variation of impact parameter along an RO ray path and a corresponding error (uncertainty) in the retrieved bending angle (Healy 2001; Zou et al. 2019; Yang and Zou 2021). Healy (2001) showed that both horizontal gradients tangent and perpendicular to the occultation plane can cause bending angle errors of 3%–10%. Chen et al. (2006) used a ray tracing model and a high-resolution numerical model to simulate the refractivity errors in a warm moist atmosphere and showed that the model-resolved ∇HN caused N errors of 2.5%–3.0% around 2–3 km. These errors decreased rapidly with height above 3 km. As noted above, ∇HN on scales resolvable by NWP models are one cause of RO uncertainties, which suggests that its magnitude |∇HN| evaluated from a model at the point of an RO observation may be an indicator of the observation error at that point.

3. Methodology and datasets

a. Estimation of statistical uncertainties

The statistical uncertainties of the RO observations are estimated by the three-cornered hat (3CH) method, which is described in detail by Sjoberg et al. (2021). The 3CH method provides estimates of the random error variances and standard deviations (STD) of three independent datasets simultaneously. Semane et al. (2022) and Todling et al. (2022) showed that the 3CH method is equivalent to the Desroziers et al. (2005) method when the same three datasets (observation, background, and analysis) are used in their calculated error estimates. The 3CH method does not require knowledge of “truth” to estimate the uncertainties of the three datasets, which may be any combination of observations or models. A key assumption of the 3CH method is that the random errors of the three datasets are uncorrelated.

b. Datasets

We perform this study for January 2021 using RO data from two constellations: C2 and the Spire commercial RO constellation. COSMIC-2 data (UCAR COSMIC Program 2019) have been shown to be state-of-the-art RO data in several studies (Schreiner et al. 2020; Ho et al. 2020b; Anthes et al. 2022; Weiss et al. 2022), and produced 5000–6000 profiles per day in the tropics and subtropics (45°S–45°N) for the study period.

Spire Global, Inc., produces occultations from multiple CubeSats and provided approximately 8000–10 000 profiles per day in January 2021. Spire data are summarized by Masters et al. (2019), and have been evaluated by Bowler (2020a), Marquardt et al. (2020), and Lonitz et al. (2021). These assessment studies have shown that Spire RO profiles are of generally high quality and are useful in NWP models. They have shown that Spire data have larger uncertainties than other RO missions above 35 km, due to their smaller satellites and antennas, lower SNR, and larger clock errors. Between 10 and 35 km, the Spire observations have been found to have slightly smaller uncertainties due to somewhat larger vertical smoothing in the Spire data (Bowler 2020a).

Several versions of Spire data are available depending on how the data are processed. Here we used the raw Spire open-loop RO measurements provided by NASA through its Commercial Smallsat Data Acquisition (CSDA) program, and computed BA and N using the standard UCAR CDAAC processing, which is also used for processing C2. This greatly reduces differences in uncertainties, as well as penetration depths, that may arise from processing (Weiss 2022). We refer to this dataset as SpireC hereafter.

The latitudinal distributions of the C2 and Spire profiles are quite different (Fig. 1a). Spire satellites are in polar orbits (52° inclination) and provide global coverage, while C2 satellites are in 24° inclined orbits and provide observations between 45°S and 45°N. To obtain global coverage with relatively uniform horizontal resolution, C2 and SpireC are combined into one dataset in this study. We use the full sample of C2 and SpireC data between 90°S and 90°N; we do not collocate the data nor correct for the different latitudinal distributions of the two datasets, as the primary purpose is not to compare the statistical properties of C2 and SpireC with each other. The combined dataset, denoted C2SpireC, provides global coverage with an average horizontal resolution (distance between observations) of approximately 228 km (Fig. 1b). The average horizontal resolution of 228 km was computed by assuming the RO profiles were evenly distributed in 5° overlapping bins with a spacing of 1°.

Fig. 1.
Fig. 1.

(a) Relative frequency of COSMIC-2 (red), SpireC (blue), and the combined dataset C2SpireC (black) for January 2021. The total number of profiles for COSMIC-2, SpireC, and C2SpireC is 118 227, 200 152, and 318 379, respectively. (b) The average daily horizontal resolution of the datasets. Sampling distributions here are calculated in 5° overlapping bins with a spacing of 1°.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

The 3CH method requires two independent datasets in addition to the RO data. Here we use ERA5 (Hersbach et al. 2020) and MERRA-2 (Gelaro et al. 2017) data. ERA5 began assimilating C2 BA on 25 March 2020 (Healy 2020). To avoid correlation of ERA5 errors with those of the RO observations they assimilate, we use short-range forecasts of ERA5 verifying at the time of the observations. We use 6-h-old MERRA-2 data in an effort to minimize possible error correlations between ERA5 and MERRA-2. Previous work with the 3CH method indicates that the estimated errors of a dataset such as C2SpireC are not sensitive to small differences in the two ancillary datasets (Anthes et al. 2022), so using slightly less accurate model datasets in the 3CH estimates does not noticeably change the 3CH estimates of the RO dataset.

The model BA profiles are forward modeled from the reanalyses following the one-dimensional forward operator described in Gilpin et al. (2019). Profiles of BA are collocated (interpolated in time and space) and quality controlled following Rieckh et al. (2021). We compute the RO relative error variances and standard deviations by normalizing the data in the 3CH triplets by the January 2021 sample mean of ERA5 collocated to C2SpireC.

CDAAC processing provides STDV and LSW for each RO observation as two of its data products. We calculate the STD4060 parameter against the January 2021 sample mean. We calculate |∇HN| as the mean magnitude of the local horizontal gradient at all model grid points within 300 km of the tangent point of the RO observation at each level. For a given grid point (i, j), the magnitude of the horizontal gradient is given by
|HN(i,j)|={[ΔNΔX(i,j)]2+[ΔNΔY(i,j)]2}1/2,
where ΔNX and ΔNY are the changes in refractivity (N) per unit distance in the east–west and north–south directions, respectively.

4. Uncertainty estimates of C2, SpireC, and C2SpireC

We first present some overall uncertainty estimates of BA for the three RO datasets for January 2021. The top row of Fig. 2 shows vertical profiles of 3CH estimates of the normalized BA uncertainties for C2, SpireC, and C2SpireC, together with the corresponding observational BA error model assumed in the ECMWF and ERA5 data assimilation system (Ruston and Healy 2021). This model is based on global statistics, and has been successfully used at the ECMWF for many years. The overall structure of the C2SpireC BA error profile is typical of other estimates of RO errors (e.g., Rieckh et al. 2021; Anthes et al. 2022; Semane et al. 2022). The uncertainties are smallest (less than 2%) between about 10 and 30 km above mean sea level (MSL) height. The small local maximum around 18 km is related to the transition from geometric optics to wave optics in the CDAAC processing (Anthes et al. 2022) and representativeness differences between RO and models (Rieckh et al. 2021). Above 30 km the relative uncertainties steadily increase as the magnitude of the bending angles decreases exponentially with height while the magnitude of the errors remains relatively constant. Below 8 km, the uncertainties increase toward the surface, reaching a maximum of between 10% and 14% at 2 km MSL height, which is in the vicinity of the top of the planetary boundary layer (PBL).

Fig. 2.
Fig. 2.

(top) Three-cornered hat estimates of normalized BA uncertainty for C2, SpireC, and C2SpireC, for (a) the entire sample (all latitudes) in each dataset and (b) the latitude-corrected sample. The ECMWF assumed uncertainties are shown in gray. (bottom) Differences of 3CH estimates for C2-SpireC, C2-C2SpireC, and C2-ECMWF for (c) the entire sample, (d) the latitude-corrected sample, and (e) the set of C2 and SpireC profiles collocated within 300 km and 3 h.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

The estimated uncertainties for the full datasets indicate that SpireC has smaller uncertainties than C2 below 8 km (Fig. 2c), which is mainly due to the higher percentage of SpireC observations in cool, dry high latitudes where the RO uncertainties are smaller than in the warm, moist tropics and subtropics (Ho et al. 2020b; Rieckh et al. 2021). However, when the C2 and SpireC datasets are sampled with the same latitudinal distribution (latitude corrected), their 3CH error estimates are more similar (Fig. 2d). We also compared the 3CH error estimates of C2 and SpireC BA that are collocated within 300 km and 3 h, and found that they are similar from 4 to 30 km (Fig. 2e). Below 2 km the C2 uncertainties are slightly smaller than the SpireC uncertainties (Figs. 2d,e). In agreement with the previous studies referenced above, from 30 to 60 km, the SpireC uncertainties are larger than those of C2 due to their lower SNR and larger clock and orbit errors. The differences between the two datasets are not dissimilar from those between variations of RO data within one mission due to different GNSS transmitters, different satellites, or other factors (e.g., Schreiner et al. 2020; Ho et al. 2020b), supporting our use of the combined dataset C2SpireC in this study.

Figure 3 shows the latitudinal variations of the 3CH estimates of BA uncertainties between 0 and 20 km MSL for C2 and SpireC in 10° latitude bands. A strong dependence of the uncertainties with latitude is evident, with estimated error STD varying from about 5%–15% in the lower troposphere for C2 and 3%–17% for SpireC. Highest uncertainties are present in the tropics and subtropics and lowest uncertainties are poleward of 45°. A seasonal asymmetry with latitude associated with the boreal winter is evident: the 35°–45°N uncertainties (orange profiles) are less than the 35°–45°S uncertainties (dark blue profiles) in Fig. 3.

Fig. 3.
Fig. 3.

Three-cornered hat estimates of normalized BA error STD in bins of latitude for (a) C2 and (b) SpireC.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

The strong dependence of the 3CH uncertainty estimates on latitude supports the use of a statistical error model in data assimilation that varies with latitude. However, atmospheric conditions and therefore RO uncertainties vary with longitude, season, and with daily weather variations, as well as latitude, and so it is desirable to further refine the statistical uncertainties of individual observations with parameters associated with the observations themselves, or with the atmosphere that the observations sample.

5. Variation of uncertainties with STDV, STD4060, LSW, and |HN|

In this section we present the C2SpireC error statistics and how they vary with STDV, STD4060, LSW, and |∇HN|. For this analysis, we divided the dataset of RO observations into 20 subsamples that approximately span the 5th–95th percentiles of the given parameter. The range and step size of the bins used for each parameter are given in Table 1. For clarity, we only show estimates from every other bin (10 total) in figures with vertical profiles of uncertainties.

Table 1.

Minimum, maximum, and step size of the 20 bins used for each parameter.

Table 1.

a. Standard deviation of bending angle differences from climatology and the sample mean

Figure 4 shows the vertical profiles of the uncertainties for bins of STDV and STD4060. A strong statistical relationship between the BA uncertainties with the magnitude of STDV and STD4060 above 30 km is evident, with a similar relationship in both. At 45 km, for example, the uncertainty varies from approximately 3% to 10% for both STDV and STD4060. For comparison, the error STD of the ECMWF error model at 45 km is approximately 10% (Fig. 2), indicating that it is conservative (overestimates the observation error) for many of the RO observations at this level, and a slight underestimate for the highest values of STDV and STD4060. The relationship is not present in the two models, as expected (not shown). Furthermore, there is little to no relationship between uncertainty estimates and either STDV or STD4060 below 30 km, indicating that these parameters are poor indicators of uncertainty below this level (cf. with the variation of uncertainties below 30 km with latitude in Fig. 3, for example).

Fig. 4.
Fig. 4.

Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of (a) STDV and (b) STD4060.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

Above 30 km the relationship is nearly linear between the uncertainty estimates and both STDV and STD4060, as shown by the examples at 45 km in Fig. 5. In Fig. 5, the normalized 3CH BA error STD for the 20 binned values of STDV and STD4060 are plotted against the mean value of STDV and STD4060 in each bin for the three RO datasets. The relationships for C2, SpireC, and C2SpireC are similar and show a nearly monotonic increase with increasing STDV and STD4060.

Fig. 5.
Fig. 5.

Three-cornered hat estimates of normalized BA error STD vs (a) STDV and (b) STD4060 at 45 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

b. LSW

The global distributions of LSW at 2 and 4 km for January 2021 from C2, SpireC, and C2SpireC are shown in Fig. 6, and the vertical cross section of the zonal mean LSW is shown in Fig. 7. We show global distributions of fractional LSW at these two levels because they are representative of the free troposphere and usually above the complicating effects of the PBL.

Fig. 6.
Fig. 6.

Global distribution of fractional LSW in % for January 2021 at (left) 2 and (right) 4 km from (top) C2, (middle) SpireC, and (bottom) C2SpireC.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

Fig. 7.
Fig. 7.

Zonal mean cross section of fractional LSW for January 2021 for (a) C2, (b) SpireC, and (c) C2SpireC. Fractional LSW values are averages in 5° latitude bins.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

The global patterns of LSW for C2 and SpireC between 45°S and 45°N (Fig. 6) are quite similar, and there is considerable longitudinal variation of LSW. At 2 km, lower values of LSW occur mostly over the continents. The highest values occur near the equator over the Pacific around the date line (180° longitude). The magnitudes of LSW decrease rapidly poleward of 30° latitude.

At 4 km there is greater geographic variation in LSW. A band of high LSW around the world indicates the intertropical convergence zone. Higher values occur in regions of moist convection. The South Pacific convergence zone is clearly visible between 180° and 120°W. Perhaps surprisingly, a region of high LSW occurs over Antarctica between 0° and 120°E. Because of the low water vapor values at these high southern latitudes even in January, these high LSW values are probably due to horizontal temperature inhomogeneities rather than water vapor inhomogeneities, which dominate in the moist tropics.

The zonal mean cross sections of fractional LSW (Fig. 7) show similar patterns in C2 and SpireC, although the LSW values below 4 km are noticeably higher for C2 because of its higher SNR. For C2SpireC, the zonal mean cross section shows a maximum just south of the equator at about 2 km. Strong meridional LSW gradients exist between 20° and 40°N and between 35° and 60°S, the asymmetry reflecting the hemispheric seasonal differences. The LSW values drop to small values above 8 km and there is relatively little variation with latitude at this level.

Figure 8 shows vertical profiles of 3CH BA uncertainties for the C2SpireC dataset between 0 and 20 km MSL in bins of fractional LSW at 2 and 4 km. The differences in uncertainties are small above 10 km. The 2-km uncertainty estimates for bins of LSW at 2 km (Fig. 8a) range from approximately 4% for the lowest LSW values, to a high of over 18% for the highest LSW values. For comparison, the ECMWF error model at 2 km is about 13%. A similar relationship is found for the magnitude of the BA versus fractional LSW, with the largest BA associated with the largest LSW values (not shown).

Fig. 8.
Fig. 8.

Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of fractional LSW taken at (a) 2 and (b) 4 km.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

For bins of LSW at 4 km (Fig. 8b), the 4-km 3CH error STD estimates vary from about 3.5% to 15% from the lowest to highest LSW. The ECMWF error model at 4 km is about 9%, near the middle of our uncertainty estimates. This again indicates that although it is a reasonable RO observation error model on the average, it overestimates the errors for some of the RO observations and underestimates the errors of others, provided the 3CH error estimates are accurate.

Figure 9 shows the number of profiles entering the 3CH calculations for the C2SpireC dataset sampled in bins of LSW at 2 and 4 km. The profile count at both levels is less than the number of C2SpireC profiles at each level primarily due to superrefraction detected in the model datasets. Data are masked below any superrefraction levels, decreasing the data entering the 3CH method.

Fig. 9.
Fig. 9.

Number of profiles entering the 3CH calculations as a function of height for the C2SpireC dataset sampled in bins of fractional LSW taken at (a) 2 and (b) 4 km.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

Several features of these count profiles are notable. First, the number of occultations with low values of LSW is much higher than the number with high values. Second, there are many more profiles that reach at least 4 km than 2 km: note the differences in the range of number of points given in the x axis.

Figure 10 shows plots of estimated 3CH BA uncertainties versus fractional LSW at 2 and 4 km for C2, SpireC, and C2SpireC. A strong quadratic relationship exists between the BA error STD and LSW at both levels; the relationship between 3CH BA error variance and LSW is nearly linear. Similar relationships occur at 1, 6, and 8 km (not shown). The y-axis intercept is around 4% and 3% for LSW bins at 2 and 4 km, respectively, which indicates that factors other than those revealed by LSW create BA uncertainties. These include representativeness differences, collocation errors, and other factors that cause RO retrieval uncertainties.

Fig. 10.
Fig. 10.

Three-cornered hat estimates of normalized BA error STD vs fractional LSW taken at (a) 2 and (b) 4 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

There is a noticeable difference in Fig. 10 between the values of uncertainty for SpireC and C2 as a function of LSW at the two heights. Higher uncertainties occur for SpireC for the same LSW at 2 km, but not at 4 km. Two factors explain this difference. At 2 km, the SpireC uncertainties are slightly higher than the C2 uncertainties (Fig. 2c). At the same time, the LSW for SpireC are smaller than those of C2 below 4 km (Fig. 7). As discussed in section 2b, in the moist lower troposphere where RO signal fluctuations are large, the LSW depends on the SNR. Because large bending angles correspond to low signal amplitudes, the higher SNR of C2 resolves larger fluctuations of bending angles than the lower SNR of SpireC, resulting in higher LSW for C2. The high bias of SpireC uncertainties together with the low bias of SpireC LSW compared to C2 explain the different relationships between uncertainty and LSW at 2 km seen in Fig. 10.

In summary, we find a robust relationship between LSW and 3CH estimates of uncertainty of C2SpireC RO bending angles below 8 km MSL. The relationship is approximately linear with 3CH error variance estimates. The high correlation of BA uncertainty and LSW indicates that relating the BA error model to LSW in the lower troposphere in data assimilation may have a positive influence on the analysis and forecast. Individual observations may still degrade the analysis and forecast, because in general only about 52% of all observations benefit forecasts, while the remaining 48% degrade the forecasts (Gelaro et al. 2010; Lorenc and Marriott 2014). With the specification of the observation error related to LSW, observations associated with high LSW, which are more likely to degrade the forecasts, will be weighted less in the data assimilation.

Although the focus of this subsection is on the relationship of RO BA random errors (uncertainties) to the magnitude of LSW, the atmospheric fluctuations that cause large values of LSW and random errors also cause biases (Sokolovskiy et al. 2010; Gorbunov et al. 2015). We computed the biases between the C2SpireC and the ERA5 BA as a function of LSW and indeed found a strong correlation, with negative biases of C2SpireC below 2 km and smaller positive biases between approximately 2 and 7 km MSL (not shown). The high correlation is likely related to other factors as well as LSW such as latitude, because both LSW and uncertainties are correlated with latitude. These biases—which are explained in Sokolovskiy et al. (2010) and Gorbunov et al. (2015), and are consistent in magnitude with those in Schreiner et al. (2020)—are not accounted for in the 3CH estimates of uncertainties. Data assimilation systems either ignore RO biases as being generally small, or treat them in ad hoc (empirical) ways such as by inflating the specified errors in the cost function or eliminating observations with large departures from the background through the quality control (QC). For example, Liu et al. (2018) and Zou et al. (2019) developed a QC based on LSW. The elimination of observations in regions of probable superrefraction also removes many RO observations with large biases. In any case, an error model that specifies larger values for high LSW will weight observations with high biases less than a statistical error model that does not vary with LSW, likely reducing the impact of high observation biases and uncertainties on the model, as found by Zhang et al. (2023).

c. Horizontal gradients of refractivity (|∇HN|)

Figure 11 shows 3CH estimates of C2SpireC BA uncertainty in bins of |HN| at 2 and 4 km. A strong relationship exists between the BA uncertainties and |HN| at the respective height of the bins, and this relationship diminishes rapidly with height. There is some indication of saturation at high values of |HN|.

Fig. 11.
Fig. 11.

Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of |∇HN| taken at (a) 2 and (b) 4 km.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

As shown in Fig. 12, the estimated BA uncertainties for both C2 and SpireC increase with increasing |HN| at 2 and 4 km in a similar way as they do with LSW (Fig. 10). However, the values of C2 and SpireC uncertainties for the same |HN| vary slightly. At 2 and 4 km the SpireC error STD are slightly lower than those of C2; this difference is also evident in Fig. 2c. These small differences are likely related to the different SNR values between C2 and SpireC, and how they affect the smoothness of the BA profiles in the moist lower troposphere (section 2b).

Fig. 12.
Fig. 12.

Three-cornered hat estimates of normalized BA error STD vs |∇HN| taken at (a) 2 and (b) 4 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

6. Summary and conclusions

This study investigates the statistical relationship between the three-cornered hat (3CH) estimated uncertainties of radio occultation (RO) observations of bending angles (BA) and four observational and environmental parameters. The four parameters include the standard deviation of BA departures from climatology in the 60–80-km altitude range (STDV) and from the sample mean between 40 and 60 km (STD4060), the fractional local spectral width (LSW), and the magnitude of the horizontal gradient of refractivity (|HN|).

The RO datasets include COSMIC-2 (C2) and Spire data processed by UCAR CDAAC (SpireC) RO observations, as well as a combined global dataset (C2SpireC) of over 300 000 profiles for January 2021. In the latitudes where the C2 and SpireC datasets overlap (45°S–45°N), C2 has smaller uncertainties above 30 km, slightly deeper penetration depths (not shown), and higher LSW values below 5 km. However, the uncertainties are similar from the surface to about 30 km. The geographic distributions of LSW for C2 and SpireC observations, and the relationship of the 3CH uncertainties to STDV, STD4060, LSW, and |HN| of the two datasets are also similar. Thus, we conclude that combining the two RO datasets into one large global dataset is justified for the purposes of this study, and that the combined datasets yields a large, relatively homogeneous global RO dataset that can be used for many operational and research purposes. The differences between the two datasets are not dissimilar from those between variations of RO data within one mission due to different GNSS transmitters, different satellites, or other factors.

The four observational and environmental parameters are shown to be robust indicators of the uncertainty of RO observations in certain altitude ranges. From 30 to 60 km the STDV and STD4060 are closely associated with the uncertainty of the BA observations. From 0 to 8 km, the fractional LSW and |HN| are closely related to the BA uncertainties. These are the two altitude ranges where the overall magnitude of the BA uncertainties are the largest, suggesting that these parameters would be useful in adjusting the assumed observational error in data assimilation.

Acknowledgments.

The authors have benefited from discussions with Neill Bowler, Ying-Hwa Kuo, Sergey Sokolovskiy, and Zhen Zeng. This work was supported by NSF Grant AGS-2054356, NASA Contract 80NSSC22K0658, and the National Oceanic and Atmospheric Administration (Cooperative Agreement R4310383). We thank Eric DeWeaver (NSF) and Will McCarty (NASA) for their support. This work utilized data made available through the NASA Commercial Smallsat Data Acquisition (CSDA) Program. We thank three anonymous reviewers for their detailed comments that led to an improved manuscript.

Data availability statement.

COSMIC-2 data are available from the UCAR COSMIC Program at https://doi.org/10.5065/T353-C093. ERA5 data are available from the ECMWF data catalogue at https://www.ecmwf.int/en/forecasts/datasets/browse-reanalysis-datasets. MERRA-2 data are available from the NASA GES DISC at https://disc.gsfc.nasa.gov/data-access. Due to its proprietary nature, Spire data cannot be made openly available. Further information about the data and conditions for Spire data access are available at the NASA Smallsat Data Explorer at https://csdap.earthdata.nasa.gov/.

APPENDIX

Sample-Independent STD4060

As defined in section 2a, STD4060 depends on the 40–60-km sample mean bending angle values. In operational data assimilation and other applications, it may be preferable to calculate this parameter independent of a defined sample. A profile-by-profile calculation of STD4060 may be done by fitting an exponential function to the 40–60-km values of each occultation profile, and then calculating the STD4060 of a profile as the standard deviation of the normalized BA anomaly from the exponential fit values. For the sample-independent STD4060, normalization is done by the exponential fit values.

Figures A1 and A2 are the same as Figs. 4b and 5b, respectively, but for this sample-independent STD4060. The values of the estimates differ only somewhat, primarily for STD4060 values greater than 25%. For example, the uncertainties at 45 km for these large values of STD4060 are up to 1.5% smaller for the sample-independent STD4060 than those for the sample-dependent STD4060. The overall high correlation between STD4060 and uncertainty, and the relationships between the datasets, remain the same between the two formulations.

Fig. A1.
Fig. A1.

As in Fig. 4b, but for the sample-independent STD4060.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

Fig. A2.
Fig. A2.

As in Fig. 5b, but for the sample-independent STD4060.

Citation: Journal of Atmospheric and Oceanic Technology 40, 11; 10.1175/JTECH-D-23-0029.1

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    • Export Citation
  • Anthes, R. A., 2011: Exploring Earth’s atmosphere with radio occultation: Contributions to weather, climate and space weather. Atmos. Meas. Tech., 4, 10771103, https://doi.org/10.5194/amt-4-1077-2011.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
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    • Search Google Scholar
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  • Bonavita, M., 2014: On some aspects of the impact of GPSRO observations in global numerical weather prediction. Quart. J. Roy. Meteor. Soc., 140, 25462562, https://doi.org/10.1002/qj.2320.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., 2020a: An assessment of GNSS radio occultation data produced by Spire. Quart. J. Roy. Meteor. Soc., 146, 37723778, https://doi.org/10.1002/qj.3872.

    • Search Google Scholar
    • Export Citation
  • Bowler, N. E., 2020b: Revised GNSS-RO observation uncertainties in the Met Office NWP system. Quart. J. Roy. Meteor. Soc., 146, 22742296, https://doi.org/10.1002/qj.3791.

    • Search Google Scholar
    • Export Citation
  • Cardinali, C., and S. Healy, 2014: Impact of GPS radio occultation measurements in the ECMWF system using adjoint-based diagnostics. Quart. J. Roy. Meteor. Soc., 140, 23152320, https://doi.org/10.1002/qj.2300.

    • Search Google Scholar
    • Export Citation
  • Chen, S.-H., F. Vandenberghe, and C.-Y. Huang, 2006: Error characteristics of GPS retrieved refractivity using a simulation study. J. Meteor. Soc. Japan, 84, 477496, https://doi.org/10.2151/jmsj.84.477.

    • Search Google Scholar
    • Export Citation
  • Desroziers, G., L. Berre, B. Chapnik, and P. Poli, 2005: Diagnosis of observation, background and analysis-error statistics in observation space. Quart. J. Roy. Meteor. Soc., 131, 33853396, https://doi.org/10.1256/qj.05.108.

    • Search Google Scholar
    • Export Citation
  • Fletcher, S. J., 2017: Data Assimilation for the Geosciences: From Theory to Application. Elsevier, 976 pp.

  • Gelaro, R., R. H. Langland, S. Pellerin, and R. Todling, 2010: The THORPEX observation impact intercomparison experiment. Mon. Wea. Rev., 138, 40094025, https://doi.org/10.1175/2010MWR3393.1.

    • Search Google Scholar
    • Export Citation
  • Gelaro, R., and Coauthors, 2017: The Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2). J. Climate, 30, 54195454, https://doi.org/10.1175/JCLI-D-16-0758.1.

    • Search Google Scholar
    • Export Citation
  • Gilpin, S., R. Anthes, and S. Sokolovskiy, 2019: Sensitivity of forward-modeled bending angles to vertical interpolation of refractivity for radio occultation data assimilation. Mon. Wea. Rev., 147, 269289, https://doi.org/10.1175/MWR-D-18-0223.1.

    • Search Google Scholar
    • Export Citation
  • Gorbunov, M. E., K. B. Lauritsen, A. Rhodin, M. Tomassini, and L. Kornblueh, 2006: Radio holographic filtering, error estimation, and quality control of radio occultation data. J. Geophys. Res., 111, D10105, https://doi.org/10.1029/2005JD006427.

    • Search Google Scholar
    • Export Citation
  • Gorbunov, M. E., V. V. Vorob’ev, and K. B. Lauritsen, 2015: Fluctuations of refractivity as a systematic error source in radio occultations. Radio Sci., 50, 656669, https://doi.org/10.1002/2014RS005639.

    • Search Google Scholar
    • Export Citation
  • Harnisch, F., S. B. Healy, P. Bauer, and S. J. English, 2013: Scaling of GNSS radio occultation impact with observation number using an ensemble of data assimilations. Mon. Wea. Rev., 141, 43954413, https://doi.org/10.1175/MWR-D-13-00098.1.

    • Search Google Scholar
    • Export Citation
  • Healy, S. B., 2001: Radio occultation bending angle and impact parameter errors caused by horizontal refractive index gradients in the troposphere: A simulation study. J. Geophys. Res., 106, 11 87511 889, https://doi.org/10.1029/2001JD900050.

    • Search Google Scholar
    • Export Citation
  • Healy, S. B., 2020: ECMWF starts assimilating COSMIC-2 data. ECMWF Newsletter, No. 163, ECMWF, Reading, United Kingdom, 5–6, https://www.ecmwf.int/sites/default/files/elibrary/042020/19508-newsletter-no-163-spring-2020_1.pdf.

  • Hersbach, H., and Coauthors, 2020: The ERA5 global reanalysis. Quart. J. Roy. Meteor. Soc., 146, 19992049, https://doi.org/10.1002/qj.3803.

    • Search Google Scholar
    • Export Citation
  • Ho, S.-P., and Coauthors, 2020a: The COSMIC/FORMOSAT-3 radio occultation mission after 12 years: Accomplishments, remaining challenges, and potential impacts of COSMIC-2. Bull. Amer. Meteor. Soc., 101, E1107E1136, https://doi.org/10.1175/BAMS-D-18-0290.1.

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  • Fig. 1.

    (a) Relative frequency of COSMIC-2 (red), SpireC (blue), and the combined dataset C2SpireC (black) for January 2021. The total number of profiles for COSMIC-2, SpireC, and C2SpireC is 118 227, 200 152, and 318 379, respectively. (b) The average daily horizontal resolution of the datasets. Sampling distributions here are calculated in 5° overlapping bins with a spacing of 1°.

  • Fig. 2.

    (top) Three-cornered hat estimates of normalized BA uncertainty for C2, SpireC, and C2SpireC, for (a) the entire sample (all latitudes) in each dataset and (b) the latitude-corrected sample. The ECMWF assumed uncertainties are shown in gray. (bottom) Differences of 3CH estimates for C2-SpireC, C2-C2SpireC, and C2-ECMWF for (c) the entire sample, (d) the latitude-corrected sample, and (e) the set of C2 and SpireC profiles collocated within 300 km and 3 h.

  • Fig. 3.

    Three-cornered hat estimates of normalized BA error STD in bins of latitude for (a) C2 and (b) SpireC.

  • Fig. 4.

    Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of (a) STDV and (b) STD4060.

  • Fig. 5.

    Three-cornered hat estimates of normalized BA error STD vs (a) STDV and (b) STD4060 at 45 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

  • Fig. 6.

    Global distribution of fractional LSW in % for January 2021 at (left) 2 and (right) 4 km from (top) C2, (middle) SpireC, and (bottom) C2SpireC.

  • Fig. 7.

    Zonal mean cross section of fractional LSW for January 2021 for (a) C2, (b) SpireC, and (c) C2SpireC. Fractional LSW values are averages in 5° latitude bins.

  • Fig. 8.

    Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of fractional LSW taken at (a) 2 and (b) 4 km.

  • Fig. 9.

    Number of profiles entering the 3CH calculations as a function of height for the C2SpireC dataset sampled in bins of fractional LSW taken at (a) 2 and (b) 4 km.

  • Fig. 10.

    Three-cornered hat estimates of normalized BA error STD vs fractional LSW taken at (a) 2 and (b) 4 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

  • Fig. 11.

    Three-cornered hat estimates of C2SpireC normalized BA error STD in bins of |∇HN| taken at (a) 2 and (b) 4 km.

  • Fig. 12.

    Three-cornered hat estimates of normalized BA error STD vs |∇HN| taken at (a) 2 and (b) 4 km. The C2, SpireC, and C2SpireC uncertainty estimates are indicated by red, blue, and black symbols, respectively.

  • Fig. A1.

    As in Fig. 4b, but for the sample-independent STD4060.

  • Fig. A2.

    As in Fig. 5b, but for the sample-independent STD4060.

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