The aircraft legs undergo an automatic segmentation algorithm that creates 4.5 min segments (≈ 27 km) by minimizing the heterogeneity of fluctuations of the three wind components simultaneously. This quantification allows the identification of sufficiently homogeneous segments for reliable statistical moment estimates. The fluctuations of the corrected wind in geographical coordinates are used for the computation of the heterogeneity score.

For a segment with wind perturbations from the three components in the geographical frame of reference {Ugeo,Vgeo,W}, the heterogeneity score H is defined as: 2H=13⋅n∑p∈{Ugeo,Vgeo,W}∑i=1nin-Cp(i)2 where Cp(i) is the normalized cumulative variance at sample index i: 3Cp(i)=∑j=1iPp(j)2nσp2 with Pp(j) the perturbation value at index j for component p, n the total number of data points in the segment, σp2 the variance of p over the segment, and i/n representing an ideal uniform accumulation of variance. H is dimensionless and independent of segment length, so that values remain comparable across segments of different duration or resolutions. The design of this score is based on work done by . While the original definition only used the vertical wind component, we added both horizontal components to account for heterogeneity in these directions.

H quantifies how unevenly the variance of the wind fluctuations is distributed along a segment. In the rest of the article and in the dataset, H is expressed in %. A homogeneous segment in which variance accumulates steadily throughout will have H≈ 0 %, while a segment containing an abrupt regime change will exhibit a markedly nonlinear cumulative variance profile and therefore a large H. A single dirac step of variance results in the maximum heterogeneity score H = 50 %. The score is invariant to time reversal but sensitive to the shuffling of time steps, which is the desired behavior for detecting regime transitions rather than simply measuring total variance. H has two purposes: (i) to be used in conjunction with the segmentation algorithm to optimize the partitioning of flight legs into segments representing single turbulent regimes, (ii) as a quantitative description of the wind field homogeneity for each sample.

Flight legs are divided into segments of 270 s (4.5 min), using an alternating bi-directional algorithm to minimize overlap and optimize coverage. For each flight leg, the number of fitting segments (nseg) is determined. The remaining duration R is distributed evenly across all segments to obtain an adjusted segment duration: Δt=270s+R/nseg.

Two boundaries are initialized at the start and end points of a given leg: the left boundary tL=tstartleg and the right boundary tR=tendleg<y. Segments are then extracted alternately from the left and right sides of the remaining leg portion. For even iterations (n=0,2,4,…), a search window [tL,tL+Δt] is defined; the starting position tmin that minimizes H over a 270 s segment is identified, the segment is set to [tmin,tmin+270s], and the left boundary is updated to tL=tmin+270s. For odd iterations (n=1,3,5,…), a search window [tR-Δt,tR] is defined; tmin is found analogously, the segment spans [tmin,tmin+270s], and the right boundary is updated to tR=tmin. After each iteration, R and Δt are computed again. This process continues until all nseg segments are extracted, working progressively from both ends toward the center. This heuristic balances positioning constraints across all segments while ensuring maximum coverage of each leg.

Figure  presents the histograms of H obtained for each segment category throughout the entire MAESTRO campaign. As expected, low-level segments are generally more homogeneous than those collected at higher altitudes. Visual inspection of the time series guided the choice of indicative thresholds to distinguish homogeneous, quasi-homogeneous, and heterogeneous segments; these thresholds are not strict quality gates but serve as guidance for users. Using an upper threshold of 10 % to define homogeneity (green shaded area in Fig. ), 8 (89 %) S-type and 180 (95 %) L-type segments qualify as homogeneous and are suitable for boundary-layer statistical moment computation. In total, 355 boundary-layer segments (all S-, L-, and B-type segments with H< 10 %) are included in the MAESTRO turbulence dataset as homogeneous.

Segments with H between 10 % and 20 % (orange) can be considered quasi-homogeneous. These likely contain intermittent cloud coverage or alternating air from the boundary layer and free troposphere (particularly for B-types). In the case of L- and S-types, quasi-homogeneous segments typically result from localized atmospheric events such as cold pools, rainfall, or regime transitions. These segments may be used for analysis, but moment estimates are less reliable.

Segments with H exceeding 20 % (red) should not be used for usual turbulence statistics, as such calculations rely on the assumption of quasi-homogeneity. These high scores indicate laminar, non-turbulent, heterogeneous, or intermittent flow, and predominantly occur in high altitude segments classified as H-types. The time-series data can still be used to study gravity waves below the filtered length scales.

The accurate measurement of high-frequency humidity fluctuations from an aircraft platform is a longstanding technical challenge. Fast sensors often degrade over time when exposed to cloudy, dusty, or precipitation environments, or may be affected by noise due to imperfect mounting, while slow sensors are relatively stable but lack high-frequency fidelity.

We emphasize that this dataset is optimized for turbulence analysis. The primary objective is to provide accurate turbulence statistics. Consequently, the proper scaling of fluctuation amplitudes and minimal instrumental noise are critical, whereas absolute offsets in mean values are of secondary importance. For example, a 1 gkg-1 bias in mean mixing ratio does not affect the variance q′2‾, but a 10 % error in the calibration slope increases the variance by 21 %. Our approach therefore prioritizes the accurate reproduction of fluctuation amplitudes across scales relevant to turbulence (sub-kilometer to a few meters), rather than absolute mean accuracy. Users requiring accurate absolute values of water vapor mixing ratio, for instance to compute precise relative humidity values, should be aware that the mean value of each calibrated FAST-WAVE signal is that of the WVSS-2 reference, whose absolute accuracy relative to the 1011C is characterized in Appendix .

It is common practice to calibrate a fast instrument against a slower, more stable reference by maximizing the low-frequency agreement between the two signals, quantified by the coefficient of determination R2 after low-pass filtering . This corrects the primary amplitude scaling and instrumental drift. The calibration is performed independently for each segment rather than once per flight or per campaign, for two reasons. First, fast sensors on airborne platforms can experience gradual or abrupt changes in their response characteristics during a flight, due to exposure to clouds, precipitation, salt, and wide ranges of temperature and humidity; a segment-scale calibration accounts for these slow drifts and environment-induced changes. And second, the MAESTRO flight legs sample distinct meteorological regimes at very different altitudes: temperature ranges from approximately +28 °C near the surface to -10 °C at 6000 m, and water vapor mixing ratio spans nearly an order of magnitude between the boundary layer and the mid-troposphere. Capacitive and spectroscopic sensors are known to exhibit environment-dependent sensitivities: the sensitivity of capacitive sensors to relative humidity is a nonlinear function of temperature, while laser absorption spectrometers can be affected by pressure and temperature variations. A segment-scale calibration ensures that the retrieved slope and offset are locally appropriate for the thermodynamic regime actually sampled, rather than imposing a single correction averaged over heterogeneous conditions.

Figure  illustrates the procedure for an example segment. For each segment and fast sensor, the fast and reference signals (Fig. a) are low-pass filtered with a cutoff at 1/6 Hz (Fig. b). The optimal lag between sensors is determined by maximizing their cross-correlation, and a linear regression is fitted to align amplitude and offset (Fig. c). Finally, the amplitude and offsets are applied to the original fast sensor signal to obtain calibrated time-series (Fig. d). Segment-sensor pairs with R2>0.9 indicate satisfactory agreement on large-scale structure.

We acknowledge that this approach has a limitation in segments where the thermodynamic field is nearly horizontally homogeneous,for instance in weakly turbulent or laminar layers. In corresponding segments, the low-frequency variability available for the regression is small, the fitted slope is uncertain, and the calibration is less constrained. For these cases, the R2 criterion (R2>0.9) indicates where the low-frequency variability is potentially insufficient to produce a reliable regression. This is consistent with the general observation that weakly turbulent H-type segments, where horizontal homogeneity is expected, also show the lowest R2 values in the calibration results.

Although technical limitations of the 1011C and chilled-mirror hygrometer and the HUMAERO capacitive sensor have been highlighted in Appendix  and reported in the instrumentation Sect. , we included them in the calibration procedure to further confirm our choice. The outcome of the calibrations of fast humidity sensors is presented in Fig. , categorized by segment type and each possible reference-fast sensor couplings. FAST-WAVE demonstrates the best agreement across all segment types and for each reference sensors, with only rare exceptions in a few T and H segments. Among the reference sensors, WVSS-2 consistently shows the best agreement across all segment types and fast sensors, except for the T-segments, where UCAP aligns better with HUMAERO than WVSS-2. Overall, the results demonstrate poor compatibility between UCAP and any reference sensor. FAST-WAVE and WVSS-2 are the most reliable combination for humidity calibration, achieving effective performance across 94 % of the segments. This highlights their robust compatibility and suitability for diverse conditions. On the basis of these results and to provide a consistent dataset, FAST-WAVE fluctuations, calibrated against the WVSS-2 are selected as the base signal for humidity fluctuations and computations of turbulent statistics.

The 25 Hz fluctuations are used to compute segment-scale statistics. To compute these estimates, the only horizontal wind component used are those in the wind-oriented frame of reference. Second-order moments (variance for every of the five variables: Ul, Vt, W, θ, q and covariance of each possible variable couple) provide information about the energy contained in the fluctuations. In the moment dataset, variances are identified by the VAR_ prefix, and covariances by the COVAR_ prefix. For any two variables x and y and a segment of duration ΔT, the eddy-correlation method gives: 4x′y′‾=1ΔT∫0ΔTx′(t)y′(t)dt where the overbar denotes a temporal average over the segment. The variance of a single variable x is the special case x′2‾.

Among covariance estimates, those of direct physical relevance include the vertical turbulent fluxes of momentum (U′W′‾, V′W′‾), heat (W′θ′‾), and moisture (W′q′‾). The momentum fluxes quantify the vertical transport of horizontal momentum and drive the wind shear structure of the boundary layer. W′θ′‾ controls buoyancy-driven convection and is a key driver of boundary-layer growth. W′q′‾ governs the supply of moisture to the cloud layer, important to boundary layer convection and associated cloud formation. U′V′‾ quantifies horizontal shear production of TKE. And finally, the temperature-humidity covariance (θ′q′‾) is related to buoyancy effects in moist air and can be related to the partitioning between heat and moisture contributions to the buoyancy flux.

Turbulent kinetic energy per unit mass (TKE) is defined as: 5TKE=12U′2‾+V′2‾+W′2‾ and is provided in the dataset under the variable name TKE. It gives a global measure of the intensity of turbulent motions and is a key input for turbulence parameterization schemes in numerical models.

Third-order moments and skewness are also computed for the five variables. The third-order moment x′3‾ and the normalized skewness Sk(x′)=x′3‾/(x′2‾)3/2 characterize the asymmetry of the probability distribution of fluctuations. Non-zero skewness of vertical velocity (Sk(W′)) can be indicative, for instance, of meteorological elements such as narrow updrafts and broad downdrafts, typical of convective boundary layers . In the dataset, third-order moments are identified by the M3_ prefix, and skewness by the SKEW_ prefix. All second- and third-order moments, as well as skewness values, are provided both from high-pass filtered fluctuations and from detrended fluctuations (identified by the _DET suffix), to allow users to assess the sensitivity of the statistics to the choice of mean-field removal method.

The integral length scale Lx is defined as the distance over which fluctuations of a variable x remain significantly correlated. It is obtained as the integral of the autocorrelation function up to its first zero crossing (Fig. a and b): 6Lx=Ua∫0τ0Rxx(τ)dτ where Rxx(τ) is the normalized autocorrelation function, τ the time lag, τ0 the time lag corresponding to the first zero crossing of Rxx, and Ua is the true airspeed used to convert from time lag to length. The integral length scale gives a characteristic size of the largest energy-containing eddies and indicates the average spatial extent over which turbulent structures persist (Fig. c). It is a key parameter for characterizing the dominant turbulent structures in the boundary layer, and for quantifying sampling adequacy through the random error estimates (Sect. ).

Integral length scales are computed for each of the five variables (LU, LV, LW, Lθ, Lq) as well as for all ten cross-correlation pairs (e.g. LWθ, LWq, etc.), and are provided in the segment statistics files. In particular, LW is used as input to the TKE dissipation rate computation described below. For each variable, the inertial-range spectral slope is also extracted. The power spectral density is computed using a Fast Fourier Transform on the fluctuation time series, and a linear regression in log-log space is performed over the inertial subrange, defined from the wavenumber corresponding to the integral length scale kIL=2π/Lx up to the Nyquist wavenumber (as illustrated in Fig. c).

The TKE dissipation rate ε measures the rate at which turbulent kinetic energy is converted into heat through viscous effects at small scales. In the case of aircraft measurements with a spatial resolution of 4 m, the dissipation range lies beyond the scope of direct observations.

Following standard practice for aircraft turbulence measurements , ε is computed from the vertical wind component W, which is less affected by mesoscale horizontal variability and more reliably captures isotropic fluctuations in the inertial subrange. Assuming locally homogeneous and isotropic flow , the one-dimensional power spectrum of W in the inertial subrange takes the form: 7SW(k)=43αε2/3k-5/3 where α=0.52 is the Kolmogorov constant for the vertical-velocity spectrum, taking the values obtained by and . This expression assumes that the spectral slope in the inertial subrange is exactly -5/3. Using the formula of , and defining the inertial subrange from the integral length scale wavenumber kIL up to an upper bound kmax (discussed below), ε is computed as: 8ε=σf32αkIL-2/3-kmax-2/3-3/2 where σf=∫kILkmaxSW(k)dk is computed by integrating the W power spectral density over the inertial subrange.

In the absence of instrumental noise, kmax is set to the Nyquist wavenumber (12.5 Hz ≈ 0.8 radm-1 at 100 ms-1). However, the accuracy of ε estimates is fundamentally limited by instrumental noise at high wavenumbers, which becomes relevant in low-turbulence conditions where W fluctuation amplitudes are small. Following , we assessed the spectral noise floor of the vertical wind measurements. Analysis of SW(k) across all segments revealed that instrumental noise begins to influence the spectrum when spectral density values fall below approximately 2 × 10⁻⁶ m1s-2. Such contamination can be observed exclusively in some H-type segments at mid-tropospheric altitudes; for all boundary-layer segment types (S, L, B), no spectral contamination by instrumental noise was detected.

To prevent noise from biasing ε estimates, kmax is therefore defined as the minimum of the Nyquist wavenumber and the first wavenumber at which SW(k) falls below 1 × 10⁻⁵ m1s-2, which is a conservative threshold approximately five times the observed noise floor. In practice, this truncation affects primarily weak-variance H-type segments; boundary-layer segments retain their full inertial subrange.

Let us consider the uncertainty of ε associated with the assumption SW(k)∝k-5/3 throughout the integration range in Eq. (). When the observed spectral slope deviates from -5/3, the integrated vertical wind variance σf2 changes, introducing a bias in the retrieved ε. To quantify this sensitivity, we numerically evaluated Eq. () for a spectrum of the form SW(k)=43αε2/3ks, where the slope s=-5/3+δ is varied. ε is kept constant, and the retrieval of ε uses Eq. (), still assuming the -5/3 slope and meaning the error associated with a slope deviation is not only sensitive on δ, but also slightly on the integration range as σf is still evaluated on the s-scaled spectrum.

For parameters representative of L-type segments (integral length scale LW = 50 m corresponding to kIL≈ 0.13 radm-1, kmax equal to the Nyquist wavenumber ≈ 0.8 radm-1), a slope deviation of δ=+0.1 (i.e., an observed slope of -1.57 instead of -5/3) leads to an underestimation of ε by approximately -18 %, while a deviation of δ=-0.1 (slope of -1.77) causes an overestimation of approximately +23 %. For parameters representative of H-type segments (integral length scale LW = 200 m corresponding to kIL≈ 0.03 radm-1, kmax equal to the Nyquist wavenumber ≈ 0.8 radm-1), a slope deviation of δ=+0.1 (i.e., an observed slope of -1.57 instead of -5/3) leads to an underestimation of ε by approximately -30 %, while a deviation of δ = -0.1 (slope of -1.77) causes an overestimation of approximately +43 %. Segments where the parameter corresponding to the extracted spectral slope of the vertical wind (Slope_W) deviates substantially from -5/3 should be treated with caution when interpreting absolute values of ε.

Unlike fixed-scale approaches e.g., which use a fixed wavelength range, our formulation dynamically adapts the spectral range to the integral length scale of each segment. In weakly turbulent H-type segments, LW typically reaches 200 m, so the integration extends over a substantial wavenumber range before encountering the noise floor, yielding meaningful ε estimates even in relatively quiescent conditions. In highly turbulent boundary-layer segments, the integral length scale is tens of meters and the full inertial subrange up to the Nyquist limit is retained.

Similar methodological choices have been made in comparable aircraft studies , enabling direct comparison with prior field campaigns. Users requiring precise absolute values of ε at low turbulence levels are advised to consult the corresponding spectral slopes (SlopeW) and integral length scale (LW) included in the segment statistics files as additional quality indicators that can be used to test the validity of the assumptions used.

Beyond instrumental limitations, the reliability of statistical moment estimates depends on sampling conditions. Following , the random error ϵr is inherent to the finite segment length relative to the integral length scale. For any two variables x and y: 9ϵr=2LxyL1+1rxy2 where Lxy is the integral length scale of x′y′, L the segment length, and rxy the correlation coefficient between x and y. For the variance of a single variable x: 10ϵr=2LxL

In both cases, reliable estimates require L≫Lxy (or L≫Lx).

Figure  shows the histograms of ϵr for the whole campaign and for S-, L-, B-, and H-type segments. For the three boundary-layer categories (S, L, B), most segments show variance random errors below 20 %, with positively skewed distributions. For the H category, the only segment type with a statistically large sample size outside the boundary layer, the distributions are negatively skewed and centered around 20 %–30 %, consistent with the larger integral length scales expected in the free atmosphere. The heterogeneity score coloring confirms overall agreement between H and ϵr: homogeneous segments (green) exhibit low random errors, while the high ϵr values in H-type segments with elevated H (red, orange) reflect the mismatch between the fixed segment length and the large integral scales of laminar or intermittent flow.

The definition of the mean field directly influences the magnitude of the fluctuations, and the choice of a 5 km high-pass cutoff removes variance at scales larger than 5 km. To quantify this filtering effect, which is called systematic error (ϵS) the covariance of the high-pass-filtered series (Ffil) is compared to that of the detrended series (Fdet): 11ϵf=Fdet-FfilFdet

This quantity is computed for every second-order moment (variances and covariances) and provided in the dataset. It is not a systematic error in the classical sense cf.who define systematic error as the inherent bias from approximating ensemble means with finite-sample temporal averages. Instead, it quantifies the variance suppressed by the choice of cutoff scale.

Figure  shows the histograms of ϵf for all five variables and each segment type. For the boundary-layer types (S, L, B), the distribution of ϵf for vertical wind variance is much narrower and confined entirely below 25 %, reflecting the small contribution of large scales to the vertical wind in the mixed layer. For the other four variables, substantially more variance is removed by the 5 km filter, revealing the relatively stronger mesoscale influence on thermodynamic fields. The heterogeneity score coloring shows a strong link between H and ϵf: non-linear trends in the large-scale component, associated with regime transitions or mesoscale structures, produce both high heterogeneity scores and large filtering effects.