Radiosondes are indispensable for monitoring the upper air as they provide high vertical resolution in situ observations of temperature, pressure, and water vapour between the surface and the upper troposphere–lower stratosphere. Determining long-term temperature trends from radiosonde measurements is challenging because changes in instrumentation can, among other things, introduce discontinuities in the measurement time series (see Fig. ). Since radiosonde measurements are primarily made to provide the data needed to constrain weather forecasts and not to detect long-term changes in climate, little attention has been paid to ensuring the long-term homogeneity of the measurement record when changing from one instrument to another. As a result, radiosonde data records typically fall short of the standard required to reliably detect changes in climate. Another cause of inhomogeneities in the record is undocumented changes in data processing . While much effort has been spent attempting to remove discontinuities in radiosonde data records e.g., lack of confidence in the long-term homogeneity erodes confidence in derived trends. used upper-air temperatures from the NCEP-NCAR reanalysis to investigate the effects of sampling frequency, changes in observation schedule, and the introduction of inhomogeneities on the radiosonde climate data record. Their results indicate that introducing inhomogeneities into a temperature time series provides the most significant source of uncertainty in trend estimates. Maintaining the temperature measurement stability to within 0.1 K for periods of 20 to 50 years avoids uncertainties in trend estimates in at least 99 % of cases . With a weaker stability requirement of 0.25 K, the uncertainty in a 50-year trend estimate increases by about 5 % for twice-daily sampling. showed that inhomogeneities in temperature measurements can cause spurious memory, leading to larger uncertainty for statistics derived from these series. The results of these studies demonstrate the need to account for any inhomogeneities in the measurement time series prior to any trend analysis.

The GCOS (Global Climate Observing System) Reference Upper-Air Network (GRUAN) was established to provide reference-quality measurements of atmospheric ECVs suitable for reliably detecting changes in global and regional climate on decadal scales. To avoid compromising the integrity of the long-term climate record, it is essential that any change, e.g. in the instrumentation or data processing, is adequately assessed before the change is implemented. For example, when transitioning from one radiosonde type to another, inter-comparison between the two radiosonde types is required to assess a potential systematic difference between the radiosondes and to correct for it, ensuring a continuous homogeneous data set without any introduced discontinuities. Typically, inter-comparisons of measurements from dual or quadruple (two of each instrument type) radiosonde flights are used to robustly detect systematic differences between the instruments e.g.. Results presented in indicated that temperature biases often increase significantly with increasing altitude, particularly in the lower stratosphere. In the past, WMO conducted several radiosonde inter-comparison campaigns e.g. with the objective of investigating the performance of operational radiosonde systems. The results of these campaigns are used in part to improve the accuracy of daytime operational radiosonde measurements and the associated correction procedures to provide temperature and relative humidity accuracies currently possible with night-time measurements. The knowledge of the performance that can be expected from various radiosonde systems allows the users to make a well-informed decision on the choice of future equipment. For a measurement network like GRUAN, it is essential to have more than one good-quality radiosonde type for operations. Instrument biases are also influenced by clouds as shown in who found systematic differences in temperature measurements greater than 2 K between the Vaisala RS92 and RS41 radiosonde when exiting cloud layers. This large difference in temperature measurements between the two radiosondes was attributed to the wet-bulb effect, in which the temperature sensor gets wet while passing through a cloud layer and is subject to evaporative cooling after entering drier parts of the atmosphere. Below 28 km of altitude, found a mean systematic difference between the temperature measurements of the two radiosondes of 0.13 K. For radiosonde measurements performed at GRUAN sites, it is suggested that sites conduct dual sonde launches for at least 6 months when changing from one instrument type to another . However, analysis of data from dual sonde launches conducted at the GRUAN Lead Centre suggests that at least 200 dual flights over a period of 1 year are required to accurately assess the systematic difference between the two sonde types . The number of dual sonde flights required may be site dependent, and therefore site-specific analysis is likely required to determine the required number of dual flights at any site. Furthermore, it is possible that instrument biases at one site may not be the same in different atmospheric conditions at other sites, though this has not been extensively evaluated. Therefore, it would be ideal if all GRUAN sites could complete thorough radiosonde inter-comparisons by performing dual radiosonde launches for at least 6 months prior to any instrument change. However, the costs of such a measurement campaign can be significant, preventing some stations from performing extensive dual launches.

In this study, we investigate the feasibility of quantifying the difference in biases of two instrument types by alternating between the two different instruments and then applying a statistical model to infer any systematic biases between the two instruments. For this study, we conduct the investigation by applying the statistical model developed to synthetic data sets, in which the persistence of weather conditions is a controllable parameter, that represent such interlaced radiosonde flights. Specifically, we investigate (i) whether a combination of interlaced measurements together with an appropriate statistical model can be used to estimate the differences in biases of two instrument types and, (ii) if so, how effective the approach is. This method, if feasible, could reduce the financial burden for sites seeking to manage such a transition, since an interlacing approach would not require additional measurements above what is needed for normal daily operation.

The box plots in Fig.  summarize the distribution of M=1000 bias estimates Δ^ for a varying number of interlaced flights N. Figure a is based on the simulated temperature time series with an AR[1] coefficient a=0.5, being similar to the autocorrelation coefficient found for temperature measurements at 300 hPa above Lindenberg. Figure 3b and c are examples for stronger persistence, i.e. a=0.8 and a=0.9, respectively. All panels show that the spread in the estimated difference in bias between instruments A and B (Δ^) converges towards the true value (Δ=-0.3) for increasing N in all cases. The rate at which this converges with increasing N depends on the persistence (i.e. autocorrelation) in the underlying time series. Weak persistence (small a) leads to slower convergence (Fig. a), while strong persistence (a approaching 1) shows faster convergence.

The SD of Δ^ (see Fig. ), representing the uncertainty with which the difference in the bias between instruments A and B can be estimated, depends on the number of interlaced flights and on the AR[1] coefficient a (coloured lines in Fig. ). The SD can be used to construct asymptotic confidence intervals for the estimates using the standard normal assumption e.g.chap. 5; i.e. for a 95 % confidence interval, the estimated bias needs to be within 1.96 times the SD. For all a, the SD decreases with increasing N; however, the SD is generally larger for weak persistence (small a∈(0,1)) and smaller for strong persistent (large a∈(0,1)).

The synthetic time series of dual flights performed with instrument A and B simultaneously at N times (i.e. 2N measurements, solid black line in Fig. ) provides the most reliable estimate of the biases between the instruments; i.e. the SD is smallest for any N. To provide a robust comparison of the results from the dual flights to the results from N interlaced measurements, the results from the dual flights need to be compared to the results of doubled N interlaced flights. For a time series with an autocorrelation coefficient of a=0.5, at least 2000 days of consecutive interlaced daily measurements would be required to estimate the difference in instrument biases with a SD of 0.22 K. Consider the following example: a station operator seeks to detect the difference in bias between two radiosondes in a temperature time series showing an autocorrelation coefficient of 0.95. The station operator requires a SD of Δ^≤0.05 K, which leads to a 95 % confidence interval of about 0.1 K (≈0.05×1.96). Then, from Fig.  it can be inferred that 500 interlaced measurements are required to achieve this. Furthermore, we conclude that if an operator has a given amount of two types of radiosondes available from which the difference in instrument biases needs to be estimated, it is clear from Fig.  that dual flights result in better estimates (i.e. smaller SD in Fig. ) than interlacing the instrument types from one day to the next. The results presented here (from dual and interlaced flights) also depend on the variance of the signal; for a higher measurement noise, the number of required days will increase and vice versa (not shown).

The results indicate that for typical difference in biases between radiosonde types, the presented method on interlaced measurements is unlikely to provide a robust estimate of the difference in biases for a reasonable length of the measurement period (reasonable is considered as 2 years here). That said, there might be cases of larger instrument biases and/or larger persistence in which the interlaced method could provide an alternative method to dual measurements, requiring fewer resources. Vertical profiles of autocorrelation coefficients as calculated from temperature data obtained from ERA5 reanalyses (https://www.ecmwf.int/en/forecasts/datasets/archive-datasets/reanalysis-datasets/era5, last access: 4 April 2018) are shown in Fig. . Temperature data were interpolated to the locations of six GRUAN sites, including sites in the tropics and the middle and high latitudes. Here we calculated the autocorrelation coefficient from ERA5 data rather than from radiosonde measurements, as long-term continuous measurements are required to obtain a robust estimate of the seasonal cycle of the temperature time series before calculating the autocorrelation coefficients. Such continuous observations, covering at least 2 years of daily radiosonde flights, are currently only available at a small subset of GRUAN sites, which does not cover all latitude bands. ERA5 is the latest reanalysis provided by the ECMWF and the calculated autocorrelation coefficients are expected to provide a good estimate of the autocorrelation coefficient at each of the selected sites. Figure  shows that the persistence varies strongly with altitude, and if the interlacing method is used, it has to be applied at different altitudes separately. For lower altitudes (pressure levels above 250 hPa), the autocorrelation coefficients vary between 0.4 and 0.8, with the lowest coefficients at the southern middle latitudes (e.g. Lauder, New Zealand). The persistence increases at higher altitudes (below 250 hPa), ranging from 0.7 in the tropics to 0.95 at higher latitudes. The results indicate that the interlacing method may be able to provide an estimate of the difference in biases for high altitudes at e.g. Ny-Ålesund, a GRUAN site showing the highest autocorrelation coefficients. However, a detailed case study needs to be performed to investigate potential benefits; this is beyond the scope of this study, which focuses on describing and presenting the methodology.

We have used synthetic time series representing temperature measurements to investigate the possibility of using interlaced measurements performed with two different instruments types together with generalized additive models to obtain an estimate of the difference in the bias between the two instrument types. Performing dual radiosonde flights with both instrument types is costly, and therefore we investigated the feasibility of using interlaced flights to obtain an estimate of the difference in the bias. This would be more sustainable and less costly. Information about typically small differences in instrument biases can be obtained from non-simultaneous measurements using a persistence assumption; i.e. some information from the day's measurement is carried over to the next day. As atmospheric temperatures tend to be autocorrelated in time e.g., the persistence assumption is justifiable. However, the strength of the autocorrelation depends in part on the geographical location of the measurement site and on altitude. Here we investigated how a statistical approach to estimate the difference between two instrument biases is affected by the persistence of a time series.

The results presented here indicate that while it is in principle possible to estimate the difference between two instrument biases from interlaced measurements, the number of interlaced flights required to obtain a satisfying accuracy is very large for reasonable values of the autocorrelation coefficient. Strongly autocorrelated signals require fewer data for an accurate estimate of the difference in biases and therefore fewer interlaced flights than time series with low autocorrelation. The results show that for very strong persistence (e.g. an AR[1] coefficient of 0.99) about twice the number of measurements is needed compared to parallel measurements to obtain a comparable uncertainty in estimates for interlaced measurements. Hence, the described approach may be used for measurements with very strong persistence or for which the costs for sufficient parallel measurements exceeds the costs for sufficient interlaced measurements to confidently infer the difference in the instrument bias. However, if, for example, it were possible to derive a robust estimate of the difference in instrument biases from interlaced measurements in some reasonable time period (e.g. 2 years) and even if this period was more than 2 or 3 times longer than would be required from a dual measurement strategy to achieve the same level of confidence, the interlacing approach would provide a cost-saving alternative to an approach that would start with dual flights and then continue with flights using only the new instrument.