Defining space-time collocation criteria for the validation of measurements requires the information about natural variability of geophysical parameters. In this paper, we analyzed the variability of the small-scale structure of temperature fields in the stratosphere using temperature profiles from radio-soundings at Sodankylä (vertical resolution ∼10 m) with a small time difference between sonde launches. We found that the small-scale structures in temperature profiles become different when the horizontal separation of measurements exceeds 20–30 km. The set of the collocated temperature profiles has allowed obtaining experimental estimates of the horizontal structure function of temperature fluctuations. The spectral analysis of the profiles has shown that vertical wavenumber spectra of temperature fluctuations are similar, even for profiles separated significantly in space and time (a few hundreds of kilometers, a few hours). Implications of these results for validation of high-resolution profiles are discussed.
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 Temperature profiles in the stratosphere contain mesoscale fluctuations that are caused mainly by gravity waves and, to a lesser extent, by advection of air masses and turbulence. Modern instruments are able to resolve the fine structure in temperature profiles. However, validation of high-vertical-resolution temperature profiles is a complicated task, because the small-scale structure is not constant. In order to define the time-space window for selecting collocated measurements, the expected difference between temperature profiles in close collocation and with a small time difference should be known. To find out, we analyzed the temperature profiles from the radio-sondes launched with small time difference at Sodankylä station (67°N, 27°E).
 We selected data from soundings at Sodankylä with a time difference of Δt < 420 min (7 hours) between balloon launches. During the years 2006–2007, altogether 172 profile pairs were found. Sodankylä station launches operational PTU radiosondes on regular basis at noon and at midnight (twice per 24 hours). Additional special sonde launches mostly related to various atmospheric sounding campaigns are also frequently performed, especially at winter/springtime, the majority of them being ozonesondes. The ozonesondes include radiosonde in the payload for data transmission and for the measurements of pressure, temperature, humidity and wind parameters. The majority of the collocated launches were made during the installation and testing of the automated sonde launching system in Sodankylä during 2006–2007 and during ozonesonde campaigns in March–April 2006 and in February–March 2007. The information about the collocated soundings is collected in Table 1.
Table 1. Number of Radio Sounding Pairs at Sodankylä With a Time Difference of Δt < 420 Minutes
Per Time Separation
 All analyzed measurements were performed with the Vaisala RS92-SGP radiosonde with DigiCora III ground equipment. Data were recorded with a 2 second time resolution, which corresponds to the vertical resolution of ∼10 m, provided the average sonde ascent velocity is 5 m/s. Information about the geolocation was obtained by a global positioning system (GPS), which has been integrated in each individual radiosonde of that type. The estimated random error in RS92 measurements is less than 0.3 K in the stratosphere according to the World Meteorological Organization radiosonde intercomparison [Nash et al., 2006].
3.1. Simple Comparison of Temperature Profiles: Structure Function
Figure 1 shows several profiles ordered according to an increasing delay in launch time of the sondes. The profiles measured on 11.04.2007 (1 min time difference between sonde launches) are practically identical. Visual agreement between profiles rapidly drops with increasing time separation. For example, the time difference for the 01.11.2006 profiles is only 22 min, but the small-scale structures in these two profiles are significantly different. The spatial separation at each altitude level with respect to the Earth dEarth = computed using information about sonde geolocation is shown in the right panels (green lines). It is small, not exceeding 10–15 km for all examples shown in Figure 1. However, air parcels probed at a given place with a time difference Δt become separated in the atmosphere due to displacement by wind, . The distance with respect to air, dair (z) = | (z) + (z)Δt(z)|, which takes into account advection of air masses, characterizes the horizontal separation of measurements in the atmosphere at each altitude level z. dair is indicated by blue lines in Figure 1 (right). It was calculated using the information about wind speed and direction provided in the radiosonde data. Variations in wind speed and balloon ascent velocity with time cause changes in dair. Comparing the information about wind speed in collocated profiles, we found that the uncertainty in dair is ∼2–5%. In calculation of dair, we took into account possible variations of Δt(z) at different altitudes caused by slightly different ascent velocities of the balloons. As seen in Figure 1, the small-scale structure in temperature profiles can differ significantly when the separation exceeds ∼20 km.
 Furthermore, for each collocated profile pair, the variance of temperature difference 〈(T1 − T2)2〉 was computed, for three altitude intervals: 10–15 km, 15–20 km and above 20 km. Being presented as a function of the corresponding separation distance, this gives an experimental estimate of the structure function:
where r = is the horizontal separation with respect to air. The separation distance is assigned as the mean one corresponding to each altitude range. In the calculation of the structure function DT, we assumed that the fluctuations are homogeneous and isotropic in the horizontal plane, thus DT depends only on the separation of the measurements. Stars in Figure 2 (top) show the values of DT(r) computed using individual profile pairs. Although each profile pair yields three points in DT estimates corresponding to the selected altitude layers, we do not distinguish different altitudes and assume that the three DT estimates represent the structure function for the lower stratosphere. The division into three layers reduces the uncertainty in r (for the considered dataset, it is ∼10%) and increases the statistics for DT(r) estimates.
 A relatively rapid growth of the variance of temperature difference (note the logarithmic scale) with increasing separation is observed. The smallest rms of temperature difference, which corresponds to the smallest spatio-temporal separation, is ∼0.3 K. This is in agreement with the estimates of the precision of temperature measurements with the Vaisala RS92 radiosonde [Nash et al., 2006]. The values of DT(r) exhibit a rather large scattering. This is not surprising: the data correspond to different seasons and meteorological conditions. The red line in Figure 2 (top) represents the median values 〈DT (r)〉 in the bins, whose mid-points are indicated by open circles.
 The black line in Figure 2 (top) shows the fit of DT(r) values corresponding to separation smaller than 300 km by a power law (performed as a robust linear fit in logarithmic scale). Analyses of aircraft measurements of temperature and wind have shown that horizontal spectra of temperature and horizontal wind fluctuations have a slope close to −5/3 in the wavenumber range from ∼3 · 10−6 cy/m to ∼8 · 10−4 cy/m [Nastrom and Gage, 1985]. However, the corresponding horizontal structure function of the form r2/3 covers a significantly smaller interval of scales [Lindborg, 1999]. Our estimate of the slope of the structure function 0.56 is not so far from the value 2/3. In fitting, we used only the points corresponding to a separation <300 km, in order to exclude temporal dependence in the data to some extent. However, changes in the structure function at r > 300 km are not clearly visible (note that there are not so many data with large spatial separation).
Figure 2 (middle) shows rms of temperature difference obtained using the collocated radiosonde profiles in original resolution (∼10 m), and for profiles smoothed down to 200 m and 1 km vertical resolution, on a linear scale. Solid lines denote the median value and dashed lines indicate the inter-quartile range. The rms of temperature difference is smaller for coarser vertical resolution, as expected, but the general behavior of the experimental structure function estimates is similar.
 An analogous study based on collocated Formosat-3/COSMIC radio occultation data (RO) has shown a similar dependence of rms of temperature and refractivity deviations on horizontal separation distance [Anthes et al., 2008]. Anthes et al. [2008, Figure 9] shows the mean standard deviation of refractivity difference (in %) for profile pairs with tangent point separation smaller than dmax, presented as a function of dmax. Hence, the corresponding variance is the integral of the structure function weighted by the separation distance. Figure 2 (bottom) shows the same characteristic computed using the collocated radiosonde data smoothed down to 1 km resolution (approximate vertical resolution of radio-occultation measurements). We take that the variance of relative temperature fluctuations is equal to that of relative refractivity fluctuations, if pressure fluctuations are ignored. The integrated structure functions are in very good agreement for separations <100 km, consistent up to 250 km within variability intervals, and they diverge slowly at large separations. Seasonal difference, longitudinal and sampling effects might contribute to this divergence. Both integrated structure functions, which were estimated using collocated radiosonde profiles and collocated radio-occultation data, grow non-uniformly with increasing separation distance. They have intervals of rapid nearly linear growth for separations smaller than 30 km (seen especially well in RO data because of the significantly smaller rms corresponding to r ≈ 0 km), and nearly flat intervals for separations of 30–70 km, followed by a steady increase for separations >100 km.
3.2. Comparison of Spectra
 Although the small-scale structure in temperature profiles becomes visually different when the spatial separation exceeds 20–30 km, the spectral properties of temperature fluctuations are very similar even for profiles separated by few hundred kilometers (as expected). This is illustrated in Figure 3, which shows examples of the vertical wavenumber spectra of relative temperature fluctuations δT = in time-collocated radiosonde temperature profiles. The altitude range above 10 km (the stratospheric part) was used for the analysis. The reference temperature profile was obtained by smoothing the original profiles by the Hanning window with a cut-off scale of 4 km. The information about the time differences and the mean distance with respect to air dair is provided in legends. The rms of temperature fluctuations σ1 and σ2 in the collocated profiles (σ1 corresponds to the first profile, while σ2 corresponds to the second one) are also indicated. As seen in Figure 3, the spectra of temperature fluctuations look similar even for profiles significantly separated in time and space (a few hundreds of kilometers, several hours).
Figure 4 compares rms of temperature fluctuations in the collocated radiosonde profiles (σ1 versus σ2) in the original resolution (Figure 4 (left)) and smoothed down to 200 m vertical resolution (Figure 4 (right)), for the whole dataset. The rms of temperature fluctuations in the collocated profiles are found to be very close to each other: the smaller the distance, the smaller the difference in rms of fluctuations, despite relatively large overall variability of σ.
 For temperature profiles separated in the stratosphere by up to 500 km, the rms of temperature fluctuations are within the ±40% interval in the majority (>95%) of cases.
4. Discussion: Implications for Validation
 The first and obvious conclusion from the performed analysis is that taking into account advection of air masses is important for computing the separation of measurements in the atmosphere and thus for defining collocation criteria.
 The analysis of collocated radiosonde temperature profiles has shown that their small-scale structures become different if the spatial separation of measurements exceeds 20–30 km. This implies a rather pessimistic conclusion for direct comparison (direct validation) of high-resolution profiles: for similarity of small-scale structure, the profiles should be almost exactly collocated in time and in space. For profiles located at a larger distance from each other, the natural variability can contribute significantly to the observed temperature difference. Our analysis has shown that the rms of temperature difference rapidly grows with increasing distance: for high-resolution profiles (vertical resolution <200 m), the rms of temperature difference is ∼0.5 K for 40 km horizontal separation, ∼0.7 K for 80 km separation and it is ∼1–1.5 K for separations 200–1000 km. These natural variation values are comparable with the accuracy of the best ground-based and remote sensing measurements having relatively high vertical resolution, such as lidars, stellar occultation satellite instruments using star scintillation for temperature profiling [Dalaudier et al., 2006], and GPS radio occultation temperature profiling [Schreiner et al., 2007; Schmidt et al., 2005].
 Let us consider how much information can be obtained by estimating statistical parameters of differences between temperature profiles Tval that are to be validated and reference ones Tref. By computing the mean of profile deviations, it is possible, of course, to detect a bias b = 〈Tval − Tref〉 (we assume hereafter that Tref and Tval have the same vertical resolution). The variance of the observed temperature differences between validated and reference profiles σ2 = 〈(Tval − Tref)2〉 − b2 is defined not only by the measurement precision, but also by the natural variability of the temperature field. In validation experiments, the natural variability is often assumed to be small within the time-space collocation window and is therefore ignored. Our analysis has shown that the natural variability is small (much smaller than the measurement precision) only within a very narrow time-space window. Therefore, σ2 should be compared with σref2 + σval2 + σnat2, where σref2 and σval2 characterize measurement accuracies and σnat2 characterizes the natural variability in the chosen collocation window. The analysis presented in the current paper provides estimates of σnat2 at Sodankylä, which can be also used for locations at Northern Hemisphere high latitudes. Provided the accuracy of reference profiles is known, it is possible, in principle, to get also experimental estimates of accuracy of validated profiles as σval,est2 = σ2 − σref2 − σnat2. In practice, such an approach should work if there are sufficiently many collocated pairs of validated and reference measurements and if the expected uncertainty, σval, is not negligible compared to σref and σnat. Variations in σnat should also be taken into account.
 The spectral analysis of collocated radiosonde temperature profiles have shown that the spectral properties of temperature fluctuations remain similar even for profiles significantly separated in space and in time (a few hundreds of kilometers, a few hours). This gives an opportunity for checking whether small-scale fluctuations have realistic amplitude and spectral shape. For the spectral validation, the time-space collocation window can be significantly broader than for direct comparisons. This approach is attractive for spectral validation of small-scale structure in temperature profiles retrieved from satellite stellar scintillation measurements [Dalaudier et al., 2006] (resolution ∼150–200 m) and retrieved from GPS radio-occultation measurements using wave-optics inversion [e.g., Jensen et al., 2003] (resolution better than 300 m). However, the comparison of vertical wavenumber spectra of temperature fluctuations from nearly instantaneous (e.g., satellite) measurements and balloon ones, advected by wind, should be performed with care. The vertical wavenumber spectra of temperature fluctuations in the ground-based and gravity wave intrinsic reference frames can be different, as a result of a wind-shifting effect [e.g., Eckermann, 1995]. Gardner and Gardner  estimated the influence of horizontal winds on vertical wavenumber spectra from balloon soundings and found that the distortion is only significant in case of very strong horizontal winds. The cases where very strong spatial inhomogeneities (e.g., above regions of intense deep convection) are expected should also be excluded from validation.
 The authors thank the Sodankylä team for the radiosonde data. The work of V.F. Sofieva was supported by the Academy of Finland. The authors thank two anonymous reviewers for their very valuable comments.