All RO profiles are provided by the COSMIC Data Analysis and Archive Center (CDAAC), which can be found at http://cdaac-www.cosmic.ucar.edu (last access: 25 April 2018). RO refractivity, temperature, and water vapor pressure profiles are taken from the wetPrf files provided on a uniform 100 m mean sea level height grid. All available processed, high-quality profiles from the COSMIC-1, GRACE 1 and 2, and Metop-A and B missions during the time periods of comparisons are used. Summaries of these missions can be found at the CDAAC website above. The temperature and water vapor pressure profiles are computed using a 1D-VAR method for moisture retrievals. Details on the 1D-VAR wet temperature and water vapor retrievals can be found at http://cdaac-www.cosmic.ucar.edu/cdaac/doc/documents/1dvar.pdf (last access: 15 March 2018).

All RS profiles are provided by the Global Climate Observing System (GCOS) Reference Upper-Air Network GRUAN; seefor more information on the GRUAN project and downloaded from the National Oceanic and Atmospheric Administration (NOAA) National Climatic Data Center. GRUAN RS were chosen for their reference-quality observations and well-documented uncertainties , allowing for a better analysis of the reduction of sampling errors associated with each co-location method.

We chose four stations in different climates for this study for the time periods of 2014, 2013, 2012, and 2011–2013, respectively: Lindenberg, Germany (LIN); Ny-Ålesund, Norway (NYA); Tateno, Japan (TAT); and Nauru, Nauru (NAU). (Nauru is the only station in which the full period of activation was used; this is due to the low number of RS launches during 2011 through late August 2013.) Figure  is a map of the GRUAN RS stations with the locations of the four stations used in this study labeled and marked in red. All four stations use the Vaisala RS92 radiosonde instruments for details on GRUAN RS processing see.

For refractivity comparisons, RS refractivity is computed under the assumption of a neutral atmosphere : N=77.6pT+3.73×105eT2, where N is refractivity (N-units), p is dry air pressure (hPa), T is temperature (Kelvin), and e is water vapor pressure (hPa).

We generate two data sets: an unfiltered data set which contains all original RO and RS profiles and a filtered data set containing the vertically filtered versions of the original RO and RS profiles.

Representativeness errors result from two different aspects of the RO and RS observations. Firstly, GRUAN RS have a temporal resolution of 1 s, with vertical resolution of 5–10 m on average , which is much finer than the 100 m vertical resolution of the RO wetPrf profiles. Secondly, RS observations are a series of point measurements, whereas RO observations are weighted averages of a cylindrical volume of atmosphere with horizontal scales of 150–300 km . Vertical filtering of both RO and RS profiles should decrease the representativeness errors caused by differences in observation type and vertical resolutions , resulting in a more meaningful comparison between profiles. Removal of structures with very small vertical scales also has the effect of reducing representativeness errors associated with different horizontal scales (footprints) of the observations . Structures with very small vertical scales in the RS profile are likely associated with horizontal scales too small to be resolved by the RO observations. Though the majority of previous RO–RS comparison studies do not filter the RO or RS profiles before comparison, filtered the profiles to remove structures with vertical scales less than 1 km before comparison with model analyses in an effort to minimize vertical representativeness errors.

To remove small-scale, unrepresentative structures in both the RO and RS profiles, we applied the Savitzky–Golay low-pass filter . We first linearly interpolated profile variables (refractivity, temperature, and water vapor pressure) to a 10 hPa (∼ 100 m) uniform vertical grid, then filtered the full profile using a fixed 40 hPa (∼ 400 m) filter window and quadratic fitting polynomial. We tested various combinations of filter windows and number of passes to determine the effects of filtering on the RS and RO profiles (not shown). The number of passes of the same filter result in minor smoothing effects relative to different filter windows, and filter windows larger than 40 hPa cause too much smoothing of the RS profile. RO profiles, due to their lower resolution relative to the RS profiles, showed little change under different filter windows. Therefore, 40 hPa was chosen as a sufficient window to remove small-scale features in the RS profile while preserving the overall structures of both the RS and RO profiles. To accommodate for the differences in vertical resolution, RO profiles underwent a single pass of the 40 hPa filter and RS profiles underwent three passes of the 40 hPa filter.

We co-located RO and RS observations using the ellipse method every 10 hPa between 1000 and 10 hPa at each RS location. We included RO occultations taken within 3 h of the RS launch at Ny-Ålesund, Tateno, and Nauru and within 1 h at Lindenberg due to the high number of RS launches (at least four times daily). We considered three different geometric constraints to co-locate RO profiles: (1) large circle with 6∘ latitude radius (∼ 666 km); (2) small circle with 2.6∘ latitude radius (∼ 300 km); and (3) ellipse parallel to the wind direction, 6∘ latitude semi-major axis and 1.2∘ latitude (∼ 133 km) semi-minor axis, each centered at the RS location. (The ellipse and small circle are circumscribed by the large circle, and the small circle was chosen such that the area within the small circle and ellipse are approximately the same; see Fig. .) The X and Y coordinates of the points on the circles and ellipses are constructed using the following parameterization (a is semi-major axis, b is semi-minor axis; for circles, a=b= radius, where both a and b are in kilometers): X(s)=acos⁡(s)cos⁡(θ)-bsin⁡(s)sin⁡(θ),Y(s)=acos⁡(s)sin⁡(θ)+bsin⁡(s)cos⁡(θ), where s is a parameter that varies between 0 and 2π and is stepped in increments of 0.01, yielding a series of 629 points that approximate the circle or ellipse, and θ is the wind direction in radians (converted from meteorological to polar coordinates). The X and Y coordinates of the circle or ellipse are first computed at the Equator, where 1∘ of latitude and longitude equals 111 km, and then the X and Y coordinates are adjusted to the latitude and longitude of the RS station according to the following: Lat(Y,s)=Y(s)111+lat(RS),Long(X,s)=X(s)111cos⁡(lat(Y,s))+long(RS). The ellipses change orientation such that the semi-major axis is parallel to the wind direction at each pressure level and time per RS. The circles remain fixed and unaffected by the change in wind direction, pressure level, or time.

Figure  illustrates the three geometries centered at Tateno at 500 hPa for 2 days in February 2012, noting that the ellipse adjusts its orientation as wind direction changes with time and at a pressure level. Preliminary testing of the ellipse co-location method with ERA-Interim model refractivity fields confirmed that refractivity isopleths tend to follow wind flow (as illustrated in Fig. ); thus, orientation of the ellipse along the direction of the wind flow should increase atmospheric homogeneity relative to the large circle.

For each RS at a given time and pressure level, RO profiles are co-located with the RS profiles under the time and geometric constraints discussed above. There can be (and often are) multiple co-location pairs with the same RS at a given time and pressure level. We computed differences for each co-location pair: XRO-RS=XRO-XRS, where X refers to refractivity, temperature, or water vapor pressure. We computed the RMS of the differences for each pressure level over the full time period and used the RMS to quantify the reduction in sampling errors. In certain cases we also computed the percent difference between two quantities Xi and Xj: %diff=100×Xi-XjXj.

Applying a spatial–temporal sampling correction to the RO–RS differences is an alternate method of reducing sampling errors in the presence of an auxiliary data set. This method has been applied in previous studies and is not restricted to RO–RS comparisons. used this approach to homogenize RS records. used ECMWF forecasts for double-differencing to reduce the sampling differences between Atmospheric Infrared Sounder (AIRS) and RS co-located pairs. calculated an observed–background (O–B) double difference to estimate the mean RS temperature bias using co-located RO profiles and Met Office model background fields. These studies use a double-difference correction, but do not verify or discuss how the correction reduces sampling errors.

Here, we apply a spatial–temporal sampling correction double difference computed with model data and assess its effects on reducing sampling errors. We use ERA-Interim data to subtract the model background from both the RO and RS observations, removing spatial and temporal sampling differences and isolating the observational errors associated with the RO and RS pair. As shown in Fig. , we use the ERA-Interim profiles interpolated in time and space to the RO time and location (eraPrf files provided by CDAAC) and the ERA-Interim grid point nearest the RS location at the RS launch time to compute the sampling correction. Due to the coarser vertical resolution of the ERA-Interim grid relative to the RS and RO vertical resolutions, comparisons with the sampling correction are conducted on a common 50 hPa uniform pressure grid from 1000 to 100 hPa to avoid further vertical interpolation.

The spatial–temporal sampling corrected differences (Xsc) for the co-location pairs are computed as follows for the comparison variables (refractivity, temperature, and water vapor pressure): Xsc=(XRO-XERA(RO))-(XRS-XERA(RS))=(XRO-XRS)-(XERA(RO)-XERA(RS)), where XERA(RO) is the ERA-Interim model value interpolated in time and space to the RO location, XERA(RS) is the ERA-Interim model grid point value closet to the RS at the RS launch time, and their difference, XERA(RO)-XERA(RS), is referred to as the model correction term. We computed the RMS differences per pressure level on the 50 hPa grid.

The spatial–temporal corrected difference (Xsc) between the two data sets (Eq. 6a and b) can be interpreted in two ways. When the two data sets are close to each other in time and space, one interpretation is that the RO–RS differences are corrected for local spatial and temporal variability by subtracting these sampling differences, which are estimated by another data set (in this case ERA-Interim) from the measured differences (Eq. 6b). This is the interpretation used here and by . However, as the equivalent Eq. (6a) shows, the corrected difference is also the difference of the departures of the two data sets from a common, reference data set, or the “double-differencing” method . In this interpretation, the two data sets are not necessarily required to be close in space or time; the method is valid as long as the biases of the reference data set do not vary over the spatial and temporal scales of the comparison. We tested the sensitivity of the sampling correction method to the spatial and temporal scale of the comparisons in Appendix  for RO–RS pairs over a much larger spatial scale (within circles of radius 15∘ latitude) and longer time window (24 h) at Lindenberg. We find that the RMS, mean, and standard deviations of the comparisons are insensitive over this range of spatial and temporal scales, allowing for a large increase in the number of co-located pairs of data.

Filtering both the RO and RS profiles has a small, positive impact on reducing RMS differences in refractivity, temperature, and water vapor pressure. Compared to RMS differences computed using the unfiltered profiles at Lindenberg, filtering both the RO and RS profiles before co-location reduces RMS differences by about 1 % on average, up to almost 8 % in some instances (see Table ). Within the large circle, filtering has less of an impact on reducing vertical representativeness percent errors since sampling errors tend to dominate (mostly due to large spatial differences). The RO–RS percent differences in the small circle and ellipse are more affected by vertical representativeness errors since the sampling errors are relatively small. Based on these results, we filtered all RO and RS profiles before computing their differences to reduce representativeness errors.

The relationship between the wind direction and horizontal variability (and sampling error) of refractivity is expected to break down for light wind speeds – indeed we found that when wind speeds are low, the effectiveness of orienting the ellipse along the direction of wind flow is significantly reduced. We separated co-located RO–RS pairs at Lindenberg into two groups based on the reported wind speed of the RS at a given time and pressure level: (1) wind speeds less than 5 ms-1 and (2) wind speeds greater than or equal to 5 ms-1. Figure  shows the RMS differences in refractivity for each group.

As shown in Fig. , the clear distinction between RMS profiles for the ellipse and two circles at wind speeds greater than 5 ms-1 essentially vanishes when wind speeds are less than 5 ms-1. Similarities in RMS refractivity differences when wind speeds are less than 5 ms-1 are likely associated with greater atmospheric homogeneity and weaker gradients in the region, resulting in similar observations within the area containing both circles and the ellipse. When wind speeds increase to larger than 5 ms-1, there is a greater separation between the circles and ellipse with respect to the RMS difference. Under moderate to high wind speed conditions, the ellipse reduces RMS refractivity differences relative to both the large and small circle, particularly below about 700 hPa. In the upper troposphere, both the ellipse and the smaller circle give similar results and smaller RMS differences than the large circle (Fig. ). In the case of Lindenberg, since the majority of wind speeds are greater than 5 ms-1 (as noted by the differences in co-location counts under the two wind constraints in Fig. ), low wind speeds do not have a significant effect on the overall RMS differences when co-location pairs are not separated by wind speed (see Fig. ).

We carried out two RO–RS comparisons at four different RS locations (Lindenberg, Ny-Ålesund, Tateno, and Nauru): first, we compared pairs with RO observations co-located within the large circle, small circle, and ellipse centered at the RS station, and second, we applied the sampling correction to the RO–RS pairs within the ellipse and two circles. We then computed RMS differences in refractivity, temperature, and water vapor pressure.

Figure  illustrates the RMS differences for refractivity, temperature, and water vapor pressure with and without the sampling correction at Lindenberg. Considering the ellipse method only (solid lines), the ellipse reduces RMS differences relative to the large circle for all three variables at all pressure levels, having the most significant reduction in the temperature RMS differences. Generally, the small circle and ellipse have similar RMS differences, but there are pressure layers in which the ellipse reduces the RMS relative to the small circle. When the sampling correction is applied to the circles and ellipse (dashed lines), the RMS differences are significantly reduced and converge with minimal differences between the ellipse and circles. The RMS differences at Lindenberg are mostly affected by spatial sampling errors; temporal sampling errors are minimized by the reduced time window of 1 h due to frequent (four times daily) RS launches at this station.

The results at Ny-Ålesund and Tateno are very similar to those at Lindenberg, with some minor differences in the lower troposphere (Fig. ). Again, using pairs in the ellipse only (solid, magenta) results in significantly reduced RMS differences relative to the large circle (solid, orange) and equal to or smaller RMS differences compared to the small circle (solid, grey). Including the sampling correction shows the largest reduction in RMS differences such that the differences from all three geometric types converge to similar values. At both Ny-Ålesund and Tateno, the reduction of RMS differences in the ellipse compared to the large circle becomes less in the lower troposphere (below 800 hPa), where the frequency of wind speeds less than 5 ms-1 increases and the relationship between wind direction and RO–RS differences breaks down. For example, at Ny-Ålesund and Tateno the percentages of RO–RS pairs for which the wind speed is less than 5 ms-1 below 800 hPa are 36.1 and 32.7 %, respectively. The small sample size in the ellipse compared to the large circle (Fig. a, e) may also play a role, allowing a few outliers to dominate the statistics.

The results at the tropical location of Nauru (Fig. ) generally confirm the findings at the other three RS locations. The ellipse alone (solid, magenta) decreases RMS differences relative to the large circle (solid, orange) and produces RMS differences that are comparable to the differences of the small circle (solid, grey). There is less distinction, however, between the ellipse method only (solid) and sampling correction (dashed) RMS differences, particularly for temperature. All six geometric and sampling correction combinations have overlapping RMS temperature differences and show little separation between co-location methods, unlike the results at Lindenberg, Ny-Ålesund, and Tateno. This is most likely caused by the relative horizontal homogeneity in temperature at Nauru, which is located in the deep tropics. Thus, there is little to no distinction between RMS temperature differences using various geometrical constraints or sampling correction.

Refractivity and water vapor pressure RMS differences in the lower troposphere (1000–700 hPa) at Nauru are the largest of the four stations, caused primarily by atmospheric conditions at its location in the deep tropics. RO is known to have negative refractivity biases in the lower troposphere, particularly in moist tropical regions where large water vapor and associated refractivity gradients often result in super-refraction . Overall, this results in larger RMS refractivity differences at Nauru. Since super-refraction is an error characteristic of RO retrievals and not related to horizontal sampling errors, the sampling correction has no impact on the RO–RS refractivity differences.