Currently there are four types of hygrometers with state-of-the-art technology that can be flown on small meteorological rubber balloons for in-situ measurements in the UT/LS: (i) chilled mirror hygrometers such as CFH, FPH or SKYDEW (Vömel et al., 2016; Hall et al., 2016; Sugidachi et al., 2025); (ii) capacitive thin-film polymer sensors such as those in the Vaisala RS41 radiosonde (Survo et al., 2015; Vömel et al., 2022); (iii) Lyman-α fluorescence hygrometers such as FLASH-B (Lykov et al., 2011; Lykov and Khaykin, 2017); (iv) new developments of light-weight laser absorption spectrometers such as Pico-Light H₂O (Ghysels et al., 2024) and ALBATROSS (Graf et al., 2021; Brunamonti et al., 2023). The chilled mirror instruments are considered “best-in-class” and are used as a reference in several research and instrument intercomparison projects (e.g., Fahey et al., 2014). In this paper, we compare the performance of the CFH with that of the Vaisala RS41 capacitive polymer sensor and the FLASH-B Lyman-α hygrometer. In the remainder of this section, we provide relevant information on the chilled mirror instruments and refer to Appendices B and C for instrumental details on RS41 and FLASH-B.

Balloon-borne chilled mirror instruments are suitable for multi-decadal measurements with low drift at relatively low cost (Müller et al., 2016; Hall et al., 2016). The measurements taken regularly since 1980 in Boulder, Colorado (40° N) using the “NOAA Frost Point Hygrometer” (FPH) constitute the longest uninterrupted time series of stratospheric water vapor (e.g. Oltmans, 1985; Hurst et al., 2011a, 2016, 2023; Kiefer et al., 2023; NOAA, 2025). A close relative of FPH is the “Cryogenic Frostpoint Hygrometer” (CFH), manufactured by EN-SCI, with similar measurement and long-term monitoring capabilities (Vömel et al., 2007a, 2016; Hall et al., 2016; Davis et al., 2016; EN-SCI, 2025).

Balloon-borne chilled mirror hygrometry is the only known technique that can measure from the ground to the middle stratosphere with high long-term stability. The accuracy of the H₂O mixing ratios derived from the dew/frost points measured by CFH and FPH from the ground to an altitude of ∼ 28 km is specified as better than 10 %–12 % after downsampling (Vömel et al., 2007a; Hall et al., 2016). For the CFH and FPH data archived by NDACC, altitude-dependent uncertainty estimates are provided (NDACC, 2020). Under conditions of good mirror frost control and a cloud-free troposphere these instruments offer the potential to achieve the CIMO (but not GCOS) measurement goal of less than 4 % uncertainty in H₂O mixing ratio. For example, FPH can achieve a mean uncertainty < 4 % at 250 m resolution in the benign background stratosphere between 19 and 25 km altitude over Boulder, Colorado (Hall et al., 2016, see their Fig. 6b). The uncertainty increases to about 5 % at ∼ 28 km due to the growing influence of the radiosonde's pressure uncertainty (0.5 hPa for iMet-1-RSB), whereas FPH can satisfy < 4 % uncertainty up to ∼ 28 km if flown with RS41 (0.3 hPa pressure uncertainty). Above ∼ 28 km, measurements during balloon ascent often show an unrealistic increase in the H₂O mixing ratio caused by outgassing from the balloon skin and the payload train (Brunamonti et al., 2018). With clouds in the troposphere, the onset of contamination may occur at altitudes much lower than 28 km, in particular when the payload below the balloon performs a strong pendulum motion in mixed-phased clouds, which increases the likelihood of droplets colliding with the inner walls of the inlet tubes and freezing on them (Jorge et al., 2021).

In the absence of cloud contamination, FPH and CFH have their largest uncertainty in the UT/LS resulting from oscillations in the mirror temperature induced by the frost control scheme (Vömel et al., 2007a; Hall et al., 2016). The FPH and CFH controllers attempt to stabilize the voltage, which is a measure of the light reflected by the ice-covered mirror, i.e. the reflected light intensity (reflectance). The controllers attempt to maintain a constant reflectance and thus, a constant amount of condensed material, even when the water vapor concentration changes rapidly during a sounding. This results in oscillations of the mirror temperature around the ambient frost point temperature. For CFH with ice on the mirror this is nominally within less than 0.3 % of the reflectance (at 1σ and 1 s resolution), and similar for FPH (Vömel et al., 2007a; Hall et al., 2016). These damped oscillations of the frost layer are unavoidable with rapidly changing atmospheric conditions along the balloon trajectory. They are treated as an auto-correlated error component (i.e., error at time t is linearly related to error at t-1 s) in the case of CFH (Vömel et al., 2016) or as random (i.e., normally distributed, uncorrelated “noise”) in the case of FPH (Hall et al., 2016). Occasionally, undamped instabilities in the electronic control system or, in rare cases, the loss of condensate, may occur, leading to measurement artifacts that do not represent actual atmospheric features (Vömel and Jeannet, 2013).

Figure 1 is a schematic diagram of CFH with its various elements. The performance of a chilled mirror hygrometer is constrained by its sensitivity to the condensate (in mV (µg cm⁻²)⁻¹, see Eq. 2 in Sect. 4.1.1) and by the signal-to-noise ratio of its detection unit (in mV). The sensitivity to the condensate is given by the change in reflectance (in mV) per change in condensate coverage (in µg cm⁻²) and depends on the morphology of the condensate and on the implemented reflectance detection scheme (Vetelino et al., 1996; Vömel et al., 2016). In CFH, the mirror is illuminated by a near-infrared light-emitting diode (LED), whose light is specularly reflected by the mirror and weakened by ≈ 12 % by the thin condensate (with coating of a few µg cm⁻²). For simplicity's sake, we will often use the term “layer thickness” to refer to the ice coverage, even though this does not fully capture the nature of the condensate, which may consist of various types of droplets or ice crystals that can grow to different unevenly thick along preferred nucleation sites (surface irregularities) or along small temperature gradients on the surface. Trace gas interferences (such as co-condensation of HNO₃ or CO₂) can be excluded as a source for systematic errors at atmospherically relevant concentrations (Thornberry et al., 2011; Vömel et al., 2016). CFH performs cleaning cycles with subsequent forced freezing (i.e. cooling the mirror to about 30 K below the frost point) twice per sounding (see e.g. Jorge et al., 2021, their Fig. 1). These are performed at -15 °C (or -12.5 °C for early instruments) and at -53 °C, the former to ensure the transformation of dew to frost and the latter to prevent the formation of cubic or glassy (amorphous) ice (Vömel et al., 2007a). The phase of the condensate (liquid water or hexagonal ice) and associated saturation vapor pressure is therefore known at all times.

Under “good conditions”, i.e. for measurements with very stable PID control (the Proportional–Integral–Derivative controller used to regulate the mirror temperature), the total frost point uncertainty at the instrument's 1 s resolution is given by the deviation of the mirror temperature Tm and the frost point Tfp, namely |Tm-Tfp| < 0.2 K (Fahey et al., 2014; Vömel et al., 2016). However, under rapidly changing conditions, the resulting controller oscillations can be well in excess of 0.2 K. Traditionally, the oscillations are smoothed out through time averaging. For example, for CFH a 25 s boxcar filter was used (corresponding to a vertical resolution of ∼ 125 m), and for FPH a 50 s boxcar filter (∼ 250 m), resulting in a total uncertainty of the average frost point of typically |T‾m-T‾fp| < 0.51 K (Vömel et al., 2007a; Hall et al., 2016).

CFH uses a digital PID controller designed for fast response during ascent (Vömel et al., 2007a). The control (or “manipulated”) variable is the heating power (as a fraction of the maximum 12 W) applied to the heater coil wound around the mirror stem. This results in a change in mirror temperature through Joule heating of the cryogenically cooled cold finger. The process (or “feedback”) variable is the specular reflectance Um of the mirror (with setpoint 2.5 V corresponding to 88 % specular reflection). The mirror temperature Tm is the measurement output (in K), from which the amount of H₂O in the ambient gas phase is then derived. The feedback controller aims to maintain constant thickness of the condensate, which implies thermodynamic equilibrium with the gas phase (Barrett and Herndon, 1951; Mastenbrook and Oltmans, 1983). Assuming no instrumental problems, on a single measurement profile, the largest source of uncertainty is not the mirror temperature, but frost control stability (Vömel et al., 2016; Hall et al., 2016), which we describe and quantify in this work as nonequilibrium error. It can range from < 0.1 K during stable frost control to a few K in case of unstable frost control (Vömel et al., 2016; see also Fig. 7).

For long time series consisting of measurements from multiple balloon flights that have been downsampled to a coarse vertical resolution (e.g. for satellite comparison at a given pressure level as in Hurst et al., 2016), the error introduced by the controller oscillations is considered to be random (Vömel et al., 2016). During a single balloon flight, however, the frost point measurements (reported at 1 s resolution in the telemetry) are not randomly distributed around the true value, but exhibit some autocorrelation due to the oscillatory behavior of the PID controller (Vömel et al., 2016). Put simply, the reported mirror temperature periodically alternates between “swings” that are too warm or too cold. The period and amplitude of the oscillations depend on the controller's response to the transient dynamics of frost formation and evaporation, whose time scale is primarily governed by the difference between the ambient H₂O partial pressure and the ice vapor pressure (kinetic time scale τkin∼p/ΔpH2O). Further, it depends on the cryogen and ambient air temperatures, on the ventilation, and on the morphology of the frost coverage. This difficulty is only partially remedied by taking mean values of atmospheric layers, e.g. several 100 m thick, as done by Vömel et al. (2007a, 2016) and Hall et al. (2016).

CFH uses a small bead thermistor embedded into a gold-plated copper mirror disk of 7 mm diameter and 1.27 mm thickness, whose calibration is traceable to NIST (Vömel et al., 2007a). Details on the calibration procedure, accuracy and long-term stability are provided by Vömel et al. (2007a, 2016), see also Hall et al. (2016). We note that all CFH instruments analyzed in this work (Table A1) have serial numbers > 2L2825, such that a mirror temperature bias (see Vömel et al., 2016) is not expected. Adding all components in quadrature, the mirror temperature uncertainty is < 0.11 K for ice films and < 0.20 K for liquid films, as the random uncertainty of the thermistor measurement is largest in the lower troposphere and negligible higher up (Vömel et al., 2016; see also Hall et al., 2016). The systematic error, i.e. the error component that cannot be reduced by averaging over multiple measurements, is estimated to be less than 0.11 K (Vömel et al., 2016), and its main component is the inhomogeneity of the temperature across the mirror, reported to be less than 0.1 K (Oltmans, 1985; Vömel et al., 2007a; Hall et al., 2016). For routine measurements, it is good practice to perform a ground check before each launch (Immler et al., 2010; Vömel et al., 2016).

With a stable PID control, i.e. close to thermodynamic equilibrium between condensate and vapor phase, the accuracy of the CFH dew point or frost point is therefore excellent, with an overall uncertainty of < 0.2 K for ice and < 0.3 K for liquid condensate. This corresponds to an error in the H₂O mixing ratio of < 4 % up to the middle stratosphere (∼ 28 km), including the 0.3 hPa uncertainty of the RS41 radiosonde GPS-based pressure measurement (Vaisala, 2018b). This is also the basis for the temperature accuracy at the Golden Points presented in the next section. At the Golden Points all the oscillatory errors due to the PID controller, i.e. the main uncertainty of CFH, vanish. However, it is important to note that the accuracy of the frost point measurement depends not only on the instrument accuracy, but also on how well the position of a Golden Point along a profile is determined, which is particularly important for instruments with large amplitudes in the controller oscillations (see example in Sect. 5.7).

Manual operation and visual detection have since been replaced by automatic sensitive measurements with photo-electric detection of frost on a mirror and a mirror temperature control loop, but the fundamental principle of chilled mirror hygrometry remained the same: when the condensate grows, its temperature is too low, when it shrinks, its temperature is too high. A modern frost point hygrometer attempts to automatically control the cooling power of its mirror such that the condensate neither grows nor shrinks (Barrett and Herndon, 1951; Mastenbrook and Oltmans, 1983). However, in order to stabilize the condensate under rapidly changing conditions during a balloon sounding, the instrument must regularly overshoot and undershoot the frost point by oscillations in the mirror temperature. In a stable laboratory environment and with a well-tuned feedback controller, small perturbations may lead to controller oscillations that decrease over time and may reach final amplitudes of only a few milli-Kelvin.

While errors smaller than 0.1 K can also be achieved during balloon soundings, they may reach several K and not be symmetric about the frost point due to the difference in frost formation and evaporation (Vömel et al., 2016). A chilled mirror hygrometer is therefore unable to maintain the condensate in a continuous (or static) equilibrium with the partial pressure of H₂O in the ambient air (pH2O). Instead, the condensate undergoes intermittently moments in which the net exchange of water molecules with the gas phase vanishes. This occurs precisely when the condensate changes from growth to evaporation or vice versa, which corresponds to an extreme value in condensate thickness and, thus, in the mirror reflectance voltage Um. In such an equilibrium-like moment, the net flux of H₂O is zero because the vapor pressure of the condensate and the H₂O partial pressure are equal (pvap=pH2O). Therefore, only at these points does the chilled mirror hygrometer measure the true frost point.

For the sake of simplicity, in this article we refer to these equilibrium-like moments simply as “equilibrium,” even though the state with pvap=pH2O may only last for a fraction of a second. Specifically, all measurements with a net H₂O flux of zero between condensate and gas phase are referred to as the “Golden Points” of chilled mirror hygrometers. Formally, the relationship for the Golden Points reads 1Tm=Tfp⇔dUmdt=0 where Tm is the mirror temperature (an accurate representation of the condensate temperature), Tfp is the frost point of the ambient air, t is the time, Um is the feedback variable for the amount of condensate, and dUm/dt is the measure of condensate growth. In the case of CFH, Um is the mirror reflectance given by the voltage of the photodetector (in V), which in turn controls the mirror temperature Tm. We call Eq. (1) the “Golden Points equation”.

Golden Points occur in all chilled mirror hygrometer measurements, regardless of whether they are cooled cryogenically (e.g. Mastenbrook and Oltmans, 1983) or thermo-electrically (e.g. Fujiwara et al., 2003; Sugidachi et al., 2025), or whether their feedback measurement for condensate growth is specular reflectance (e.g. Vömel et al., 2007a), light scattering (e.g. Brewer and Dobson, 1951), phase/frequency shifts of a surface acoustic wave (e.g. Hansford et al., 2006) or energy attenuation of alpha radiation (e.g. Rohrbough et al., 1967). Automatic chilled mirror hygrometers usually try to maintain their feedback variable Um as close as possible to a predefined “setpoint” Uset, which is determined by a constant reflectance chosen to avoid a sudden loss of condensate while preserving good sensitivity to changes in condensate thickness. The effect of the feedback controller could lead to the misconception that Um=Uset is a sufficient condition for equilibrium, i.e. Tm=Tfp. This is generally not true, as Tm=Tfp applies if and only if dUm/dt = 0, which can occur far from the setpoint. Consequently, good chilled mirror instruments should have an optically sensitive ice film and measure mirror temperature and mirror reflectance simultaneously at high frequency, high accuracy, low noise and low reflex sensor drift. This is because excellent accuracy of the frost point measurement can only be achieved if the position of the Golden Point along the balloon trajectory is also precisely determined with well-synchronized temperature and reflectance time series. The accuracy is therefore not determined solely by the precision of the temperature sensor, but also by the accuracy of the position of the Golden Point. To meet the low noise condition, smoothing is often required. This concerns white noise from the electronics (on CFH, pre-launch clean mirror data indicates it contributes 10 % to the overall noise) and from the condensate itself (such as specular reflections of individual crystal faces), but above all spectral noise from problematic and unwanted signals that correspond to some eigenfrequency of the condensate-substrate-electronics system. In Fig. D1 we show that CFH with ice film sometimes appears to develop an eigenperiod of ∼ 5 s, which is not an atmospheric signal but needs to be removed by an appropriate smoothing. In this work we apply a smoothing of the CFH reflectance signal of 3 s for the liquid films and 15 s for the ice films (which takes account of changes in condensation rate and film morphology). Figures D1 and D2 suggest that a smoothing interval of 15 s is the optimal compromise between removing instrument artifacts and preserving natural features in the upper-air H₂O profile, and we examine the effects of using shorter or longer smoothing intervals. Smoothing complicates the determination of the exact timing of the Golden Point and must therefore be applied with caution in order not to reduce the accuracy of the frost point Tfp.

Reporting the Golden Points to achieve maximum accuracy is not new: the first accurate measurements of stratospheric water vapor performed by Dobson et al. (1946) were essentially the Golden Points of their manually operated hygrometer. In this respect, to our knowledge, they are the inventors of the Golden Points method. In the age of fully automated chilled mirror hygrometers, however, the method and its benefits fell into oblivion.

A first example of using the Golden Points method is shown in Fig. 2, which compares the humidity measurements in the midlatitude tropopause region of three instruments: the CFH chilled mirror, the FLASH-B Ly-α fluorescence instrument and the thin-film humidity sensor of the RS41 radiosonde. Measuring the fine structure of UT/LS water vapor mixing ratio (on a scale < 100 m) with high accuracy (< 10 %) is still a challenge even for state-of-the-art instruments, especially around the tropopause, where an abrupt transition from moist tropospheric air to very dry stratospheric air takes place. Near the tropopause, the differences between the instruments shown in Fig. 2 are more than 30 % in H₂O mixing ratio, which is far from the target of < 4 % set by CIMO, let alone the target of 2 % set by GCOS. We show in this section that we can reduce these discrepancies to < 15 % when applying physically meaningful corrections to the thin-film and Ly-α instruments, based on the Golden Points of the chilled mirror instrument. As we will show in Sect. 5.2, we can further reduce the discrepancies to ∼ 5 %, when CFH controller instabilities leading to strong nonequilibrium states of the mirror are corrected by means of a nonequilibrium correction of CFH.

Figure 2B and D show the original data of the three instruments as thick black lines for CFH and as green dashed lines for the other two instruments. Both FLASH-B and RS41 suggest a much drier upper troposphere and lower stratosphere than CFH. However, the extreme values of CFH mirror reflectance (dUm/dt = 0) indicate seven Golden Points in the selected altitude range, whose altitudes can be determined with sufficient accuracy (within better than ±5 m). CFH has been shown to measure the frost point with less than 0.2 K error under equilibrium conditions in the laboratory (Fahey et al., 2014) and under good flight conditions (Vömel et al., 2016), which we assume to also apply to the equilibrium situation of the Golden Points (see further Sect. 5.5 for a discussion of the uncertainty at the Golden Points). This corresponds to an uncertainty in the water vapor mixing ratio at frost point temperatures around 205 K and a pressure around 200 hPa of less than 0.5 ppmv (or 3 % in mixing ratio). Deviations in the original data of the other two instruments of several ppmv as shown in Fig. 2 therefore exceed the uncertainty at the Golden Points by almost an order of magnitude.

FLASH-B usually requires recalibration between flights. As Fig. 2 shows, large residual errors may be related to a constant offset in the fluorescence counts. Here, we determine the size of this offset from the CFH Golden Points. The FLASH-B instrument (“FI05/12”) in Fig. 2B can be corrected to match the Golden Points by assuming a constant offset (78.5 counts s⁻¹) in the fluorescence count rate (compared to typical count rates in the tropopause region of several hundred counts s⁻¹) prior to multiplication with the calibration constant (0.0256 ppmv (counts s⁻¹)⁻¹). This results in near-perfect agreement with the CFH water vapor mixing ratio at all seven Golden Points. See Appendix C for details on the FLASH-B offset.

RS41 typically requires corrections of the time-lag and bias of the thin polymer film measurement. In general, Vaisala's RS41 product, which is corrected for time lag but not for bias, shows good agreement with CFH in the lower and middle troposphere. Under the low relative humidities and low temperatures prevailing in the UT/LS, the RS41 generally still remains sensitive to moisture, but residual time-lag and bias errors, along with increasing sonde-to-sonde variability, reduce the accuracy of the RS41 measurement (Brunamonti et al., 2019; Poltera, 2022; see also Fig. B1 in Appendix B). Once the relative humidity above the tropopause drops below about 2 %RH, i.e., typically 1–3 km above the tropopause in midlatitudes, the RS41 lacks responsiveness to humidity changes (Survo et al., 2015; Vömel et al, 2022) and cannot be used as tandem hygrometer anymore. We apply our own processing to Vaisala raw data at the heated humidity sensor temperature (the Vaisala software provides the raw RH and T, at both sensor and air temperatures) by performing the following steps (see Sect. 4.2 and Appendix B for details):

we apply a time-lag correction to the raw data of relative humidity from RS41 according to Miloshevich et al. (2004), using a 2-parameter fit for a time constant of Arrhenius type (see Eq. B1);

While the agreement between CFH on the one hand and the offset-corrected FLASH-B or the time-lag and bias-corrected RS41 on the other hand is excellent at the Golden Points, there are still discrepancies up to ±0.9 K away from the Golden Points (blue and red lines in Fig. 2C, E). These remaining discrepancies are due to the CFH controller oscillations. As we will show in the next section, they can be further decreased to ±0.4 K by applying a nonequilibrium correction to the CFH data.

Under nonequilibrium conditions, if the mirror is too cold, the net H₂O flux towards the mirror is positive, the condensate grows, and the mirror reflectance decreases (Tm < Tfp ⇒ dUm/dt < 0), and vice versa for too warm conditions. Determining equilibrium according to Eq. (1) is straightforward, but is only a consequence of a more general nonequilibrium equation. Quantifying nonequilibrium relies on three assumptions, which we describe in the following subsections.

Throughout the troposphere and stratosphere, the CFH measurements are excellent at the Golden Points, i.e. Tfp accurate to ±0.2 K, corresponding to errors in H₂O vapor pressure or mixing ratio of less than 3.5 % (except where measurements are contaminated; Jorge et al., 2021). By contrast, nonequilibrium states of the controller may cause much larger errors just a few meters above or below the Golden Points. Realizing the CIMO and GCOS goals requires measurements of the atmospheric frost point with low uncertainty and high vertical resolution. For this, we need to correct the nonequilibrium errors between the Golden Points, which can be achieved by a second sensor on the same payload after recalibrating it using the Golden Points. This could be a sensor with higher vertical resolution than the Golden Points, but with lower accuracy than CFH. Sensors with sufficient capabilities include the humidity sensor of the RS41 or the FLASH-B hygrometer, while certain other sondes may not be suitable, because of a slow response in the upper troposphere and insufficient stratospheric RH sensitivity, such as the often-used iMet-1 radiosonde (e.g. Hurst et al., 2011b). As we will discuss in Sect. 4.3, under certain circumstances, correction can be achieved without the help of other sensors, i.e. by means of a self-correction of the frost point hygrometer. However, this is usually not possible when Golden Points are far apart. Instead, another well-calibrated sensor is needed to correct the measurement errors.

In Sect. 5.1 and 5.2, we present case studies of nonequilibrium correction with the auxiliary sensors RS41 and FLASH-B. The corresponding correction methodologies of RS41 and FLASH-B by means of the CFH Golden Points are described in Appendices B and C respectively.

Using an auxiliary sensor for this purpose, we apply the nonequilibrium correction according to the following scheme:

identify the Golden Points along the humidity profile measured by a chilled mirror instrument;

The effective sensitivity A can then be determined from a least squares fit using the scaled time derivative of the mirror reflectance B⋅dUm/dt as function of the difference of corrected auxiliary sensor data, Tfp,auxcorr and the CFH mirror temperature Tm: 8bB⋅dUmdt≈A⋅Tm-Tfp,auxcorr. Note that Eq. (8b) is equivalent to Eq. (8a), but with the still unknown true frost point Tfp approximated by the known corrected auxiliary sensor (Tfp,auxcorr). Figure 3D below shows a textbook example of the determination of A immediately before the launch of a balloon sonde using RS41 as auxiliary sensor (for the liquid film, i.e. with Tdp,RS41corr instead Tfp,auxcorr), resulting in A = 773 V s⁻¹. In Sect. 6, we estimated A in flight for 70 soundings from the tropics to the polar regions, and found for the liquid film and the first and second ice films a median value of A≈6000 V s⁻¹, albeit with a high degree of variability (700–34 000 V s⁻¹, see Fig. 11). An effective sensitivity of 6000 V s⁻¹ is an average value for a good ice film and translates to a morphological sensitivity A′ ≈ 71.4 V (mol m⁻²)⁻¹ ≈ 40 µV (ng cm⁻²)⁻¹ under typical balloon ascent ventilation conditions. With detectable changes (at S/N = 2) in CFH reflectance of about 20 µV s⁻¹ during nighttime flights, this means (using Eq. 8a with B = 3 × 10⁷ K and dUm/dt = 20 µV s⁻¹) that CFH can achieve measurements with a precision of 0.1 K in frost point or 100 ppbv H₂O in the lower stratosphere at 1 s temporal resolution (or 40 ppbv at 5 s resolution).

In addition to using a second instrument for non-equilibrium correction, there are other options. These are based on the fact that, for a given ice film, a reference between (at least two) Golden Points is required in order to evaluate the sensitivity parameter A of the ice film and then perform the nonequilibrium correction. These options are:

Use of the self-correction of the chilled mirror instrument when descent data of good quality are available in addition to ascent data, to identify features in humidity in the uppermost kilometers of a sounding. See example in Sect. 5.3.

Figure 3 shows CFH prelaunch data in outside air shortly before the balloon was released. During prelaunch, a small fan is temporarily installed on the upper inlet tube (Vömel et al., 2016), but the ventilation it creates is less than the airflow during the balloon ascent, resulting in pronounced oscillations in the CFH measurement. At the Golden Points, CFH compares well with the humidity measured by RS41 (after applying a 3.5 s synchronization shift Δtele discussed in Appendix B). In this example with a nearly constant ambient dew point, the strong oscillations observed in the signal are caused by mirror temperature oscillations resulting from a controller instability, i.e., “nonequilibrium excursions”, and not by a time-lagged response to changes in the dew point. In this special case without external dew point changes, the nonequilibrium errors in mirror temperature are nearly symmetric, but with a slight negative bias due to the nonlinearity of the Clausius-Clapeyron equation. If, instead of nonequilibrium correction, the H₂O mixing ratio or relative humidity is averaged over two periods (e.g., Vömel et al., 2007b), this would provide an accurate representation of the dew point, but the temporal resolution would still be about three times coarser than the Golden Points sampled with Eq. (1). A proper nonequilibrium correction using Eq. (8a) further increases the temporal resolution (here by a factor of 3 compared to using solely the Golden Points), enabling dew/frost point retrievals with similar measurement uncertainty but higher temporal resolution. The relative humidity of the nonequilibrium-corrected CFH agrees well with RS41; some residual differences might be due to both instruments actually measuring slightly different air due to small disturbances caused by the handling of the payload during the launching procedure.

Figure 3 illustrates that when the reflectance passes through the setpoint, this does not indicate that thermodynamic equilibrium is attained as discussed in Sect. 3. For example, the reflectance is at the setpoint at t = -20 s, yet the mirror temperature is at maximum distance from the dew point. Thermodynamic equilibrium is only reached when the reflectance remains constant or when it reaches a minimum or a maximum (e.g., at t = -40 s).

Figure 3D shows how the nonequilibrium correction term can be determined by means of a scatter plot of the change in reflectance (scaled by B) over the mirror temperature deviation from the RS41 dew point. The slope of the magenta line is the resulting effective sensitivity A = 773 V s⁻¹ with an estimated uncertainty of 10 %.

Figure 4 shows the results of the intercomparison sounding of CFH, FLASH and RS41 above Lindenberg, which we already discussed in Fig. 2, but now including the nonequilibrium-corrected CFH data. We applied the procedure outlined in Sect. 4.2 to the CFH original data, i.e., the nonequilibrium correction of CFH using the time-lag and bias corrected RS41 measurements of the same flight (Appendix B). The differences between the CFH original data (data points on black curve) and the nonequilibrium-corrected CFH data (blue curve) amount to 6 %. Although this correction is small compared to the recalibrations of RS41 and FLASH-B (dashed and solid green curves in Fig. 4), a 6 % change still exceeds the CIMO and GCOS goals, making it worthwhile to apply the nonequilibrium correction.

While the flight segment between 11.1 and 11.4 km could at first be confused with a time-lag error of CFH, the undershoots between 11.55 and 11.8 km and between 10.8 and 11.1 km show that the original CFH data suffer from controller/mirror oscillations, and not from a slow response of its mirror temperature signal. Conversely, the RS41 and FLASH-B hygrometers provide information about the actual humidity structure, but suffer from a dry bias in this flight section and have to be corrected first (Appendices B, C). Figure 4 also shows the profiles of the time-lag and bias-corrected RS41 (green curve) and the offset-corrected FLASH-B (orange curve). The discrepancies between these corrected profiles and the original CFH amount to ∼ 15 %, which is reduced to ∼ 5 % by the nonequilibrium correction of CFH (blue shaded area). This demonstrates the necessity and validity of the nonequilibrium correction of CFH for quantifying the frost point.

The two examples in this and the previous sections show that the use of mirror reflectance is a powerful diagnostic tool. Not only can the atmospheric frost point be determined at the Golden Points, but the time derivative of the reflectance also shows whether the mirror is too warm or too cold, allowing for appropriate correction. We therefore strongly recommend that chilled mirror instruments always provide the measured reflectance signal in addition to the mirror temperature.

Chilled mirror hygrometers are actively controlled instruments and typically follow a second order response with multiple overshoots (e.g. Jacobs, 1993), i.e., the mirror temperature satisfies a second-order differential equation of the form τ2d2Tm/dt2+2ζτdTm/dt+Tm=fext(t). Here, τ is the system's natural period of oscillation and ζ is its damping factor, which are defined, among other things, by the Joule heating of the mirror along with its thermal inertia, the PID settings, the H₂O gas phase kinetics, as part of the control system. Further, fext is the external forcing that results from the changing ambient humidity. Typically, the PID parameters are tuned such that the resulting system follows a damped oscillator response. This 2nd order differential equation with oscillatory behavior and the nonequilibrium equation (Eq. 5) are coupled and must be satisfied simultaneously. As dUm/dt satisfies the physical constraint given by Eq. (8b), this could be interpreted as 1^st order behavior (erroneously assuming the general relationship dUm/dt∝dTm/dt), unless one recognizes that the mirror temperature is actually a more complex function of the reflectance signal, Tm=Tm(Um), via the heating power applied by the controller (in response to ambient frost point changes).

Depending on the size of the damping factor ζ, the controller behavior can be overdamped (ζ > 1, sluggish approach to the instantaneous equilibrium without oscillations), critically damped (ζ = 1, faster approach to equilibrium without oscillations), underdamped (1 > ζ > 0, oscillatory approach to equilibrium with over- and undershoots), or unstable (0 > ζ, oscillations increase), assuming the forcing term to be constant. Figures 3 and 5 show that CFH often operates in the underdamped regime. Due to changes in the forcing term, it can happen that the mirror temperature oscillations are strongly underdamped for a few seconds and approach instability. Conversely, there are situations where the mirror temperature oscillations are overdamped and very close to a first order response. Under these latter conditions, a simple correction can be used to describe CFH, mimicking a time-lag similar to the behavior of RS41, and an exponential approximation to equilibrium is assumed (as in Miloshevich et al., 2004). This approach has been applied in the past (e.g. Vömel et al., 2007b, 2016; Hasebe et al., 2013) to match the measurements of stratospheric humidity features observed during both balloon ascent and parachute descent.

We revisit a CFH flight from Lindenberg in May 2014, where humidity features measured in the stratosphere during ascent and descent have been approximately matched in altitude using a time constant of 10 s (Vömel et al., 2016). This assumes a first-order response for the mirror temperature, Tfp(t)=Tm(t)+τCFHdTm/dt(t), where τCFH is the time constant for the apparent time-lag (here 10 s), or equivalently χH2O(t)=χH2O,m(t)+τCFHdχH2O,m/dt(t) in terms of water vapor mixing ratio.

Figure 5 compares the raw data (Fig. 5A) with the time-lag correction τCFH = 10 s (Fig. 5B), and the nonequilibrium correction, Eq. (8a) (Fig. 5C), which is based on the Golden Points and nonequilibrium correction (Fig. 5D). We determine the CFH nonequilibrium self-correction by using the CFH descent data to establish the a-priori for the ascent data and vice versa. Figure 5 reveals that at altitudes where both the ascent and descent exhibit simultaneous Golden Points, also both frost points agree well, and without any lag. The nonequilibrium correction provides a better match of the ascent and descent features in stratospheric humidity than the time-lag correction. This is because the nonequilibrium correction makes use of the mirror reflectance measurement through Eq. (8a) and corrects the main reason for the ascent/descent features mismatch, namely the nonequilibrium oscillations of the mirror temperature caused by the CFH controller, better than a time-lag treatment would. We have excluded the first 400 m of descent, like in Vömel et al. (2016), as they may be affected by an unstable airflow and short-term contamination from the payload. We have derived the effective sensitivity for the ascent, Aascent = 800 V s⁻¹ and for the descent, Adescent = 2300 V s⁻¹, by fitting for the best match between ascent and descent humidity features. Differences in sensitivity of this magnitude are to be expected due to the higher ventilation during descent (υdescent ∼ 20 m s⁻¹ and υascent ∼ 5 m s⁻¹ and A∼υ1/2).

The nonequilibrium correction is valid everywhere, whereas applying a time-lag correction is valid only within flight sections where the nonequilibrium excursions can be approximated by a first order response, such as in subsections of the flight shown in Fig. 5. To illustrate that the nonequilibrium error is qualitatively different from a time-lag error, but may appear locally as a time-lag error, we rewrite the nonequilibrium correction (Eq. 8a) in the mathematical form of a first order response: Tfp(t)=Tm(t)+τnoneqdTm/dt with the nonequilibrium “pseudo time-constant” τnoneq=B(t)/A⋅(-dUm/dTm). The term -dUm/dTm (in V K⁻¹) contains the reflectance measurement Um.

Therefore, τnoneq (in s) can take both positive and negative values (see Fig. 5E) and is highly variable over time, which is not a physically meaningful quantity. Only the median value of the pseudo-time constant τnoneq of 11 s agrees well with the 10 s used by Vömel et al. (2016), but with a standard deviation of 37 s around the median.

For these reasons, it is generally not appropriate to treat the response of chilled mirror instruments in the same manner as the response of polymer thin film sensors (as in e.g. WMO, 2024, their Table 12.7), because high-quality chilled mirror hygrometers do not suffer from time-lag of their mirror temperature (the thermistor measuring the mirror condensate temperature has a characteristic time in the order of 0.12 s for CFH, see Sect. 4.1.3), but from nonequilibrium errors.

The AquaVIT-1 intercomparison of atmospheric water vapor measurement techniques considered CFH as one of the “core” instruments to construct a metrological reference (Fahey et al., 2014), and the SPARC water vapor assessment II (WAVAS), which investigated biases and drifts of water vapor satellite retrievals with respect to frost point hygrometers, accepted CFH as a reference instrument (Kiefer et al., 2023), as did the WMO 2022 Upper-Air Instrument Intercomparison Campaign (Dirksen et al., 2024). Due to the generally high quality, it is often not necessary to apply corrective measures for errors of the CFH. Yet, averaging over extended sections (> 125 m in the stratosphere) of the balloon ascent profile by applying a Gaussian filter is part of the standard procedure for CFH data processing. This is intended to quantify and reduce uncertainty, at the expense of reducing vertical resolution, as described by Eqs. (1)–(5) in Vömel et al. (2016).

Figure 6 compares the averaging by Gaussian filtering with the nonequilibrium correction using the example of a CFH-RS41 tandem flight on 21 February 2020 in Ny-Ålesund on Spitsbergen (Svalbard, Norway). The figure shows a nearly isothermal profile section (208–210 K air temperature) in the lowest stratosphere in polar winter. Since the relative humidity is sufficiently high (Sice > 0.1), the RS41 radiosonde remains sensitive to moisture (e.g. Survo et al., 2015) and can be used as a reference for estimating the effective sensitivity parameter A in the stratosphere.

The effect of averaging and smoothing with the Gaussian filter can be seen by comparing the original CFH data (Tm,raw, gray crosses in Fig. 6B) and the pink “CFH NDACC” curve (with a vertical resolution provided in the NDACC data file ranging from 137 to 261 m for the flight section shown in Fig. 6). Gaussian filtering reduces here the noise in the mirror temperature, but does not correct the controller's nonequilibrium errors. The changes due to Gaussian filtering amount to ±0.2 K in the frost point temperature and about ±1 % in Sice. In contrast, the nonequilibrium errors actually caused by controller oscillations are about a factor of 5 larger, as the comparison of the original CFH data (gray crosses) and the black curve “Tfp” shows. The nonequilibrium corrected CFH profile, Tfp, is closer to the moisture characteristics measured by the RS41 than to Tm,raw or to the Gaussian filtered curve. This example also shows that the actual atmospheric humidity profile between the Golden Points can be highly variable (e.g. the two moist features at altitudes highlighted by gray shadings in Fig. 6B). In this case, Gaussian filtering of the mirror temperature profile fails to capture the structure of the atmospheric humidity field seen by both RS41 and the nonequilibrium corrected CFH, and its uncertainties appear to be underestimated at certain altitudes (see Fig. 6C). Similarly, a first-order time lag treatment cannot capture the detailed structures of atmospheric moisture, the same is true for a mere interpolation of the Golden Points. Finally, 250 m boxcar averages (as used by NOAA) of the mirror temperature with a center of 9.75 and 10.75 km would also lead to a dry bias of about 1 K at the frost point, and the associated uncertainty, i.e., the standard error of the 250 m linear fit (Hall et al., 2016) would be underestimated.

Good control of the frost on the mirror is a tradeoff between producing many Golden Points (dU/dt = 0) and minimizing the nonequilibrium error (small |dU/dt|). A strongly oscillating PID controller producing Golden Points at < 100 m vertical resolution, e.g. SKYDEW with its current PID implementation (Sugidachi et al., 2025), would be able to capture detailed humidity structures as in Fig. 6. However, PID controllers with strong oscillations tend to have a larger oscillation amplitude and thus a larger uncertainty (∼ 0.5 K) at the Golden Points (defined as the maximum mirror temperature difference ±1 s around Golden Points), as well as larger nonequilibrium errors. On the other hand, CFH has a well-tuned controller for background conditions of tropospheric and stratospheric humidity, with mostly small nonequilibrium errors. Nevertheless, rapid frost point changes can cause large nonequilibrium errors on CFH, with Golden Points more than 100 m apart. Thus, it is unclear whether PID control is the most appropriate approach to frost control of chilled mirror hygrometers. Control algorithms designed to “seek” Golden Points might be a valid alternative to PID control, but we restrain from discussing this further, as we are not aware of any implementation of this type yet. In any case, no matter the control approach, control-induced oscillations can be corrected with Eq. (5).

In this section, we describe two metrics to quantify the uncertainty of the nonequilibrium correction procedure described in Sect. 4.2. The first metric quantifies uncertainty by comparing the final (i.e., nonequilibrium corrected) CFH frost point estimate to the a-priori reference estimate. The second metric quantifies the uncertainty of the nonequilibrium correction method itself.

For the first metric, we quantify the uncertainty as the root mean square error (RMSE) between the a-priori reference profile (e.g., RS41 corrected for residual time-lag and bias errors using the CFH Golden Points), and the a-posteriori atmospheric frost point profile, i.e., CFH corrected for nonequilibrium errors. We term this uncertainty (at 2σ) uREF, with uREF = 2 × RMSE (e.g., in Fig. 6B, uREF = 2 × 0.36 K = 0.72 K). From the 70 CFH-RS41 tandem flights analysis in Sect. 6, we find a root mean square error < 0.16 K between the CFH corrected and RS41 corrected profiles at 125 m vertical resolution. Together with the mirror temperature uncertainty uTm = 0.11 K (at 2σ) for ice films (Sect. 2.5), this gives an overall uncertainty < 0.4 K (at 2σ) for the nonequilibrium corrected frost point profiles (when using, prior to correction, a 15 s half-width second-order Savitzky-Golay filter for sampling the Golden Points and calculating the derivative of the mirror reflectance signal, and a 25 s full-width boxcar filter for smoothing the raw relative humidity signals). An uncertainty in frost point < 0.4 K at 125 m vertical resolution reduces to < 0.2 K at 500 m vertical resolution (as 500 m = 4 × 125 m, i.e. the uncertainty is reduced by a factor 4^1∕2). Together with an uncertainty of 0.3 hPa in RS41 pressure from GPS (Vaisala, 2018b), this translates to less than 4 % uncertainty in water vapor mixing ratio (see also Hall et al., 2016).

For the second metric, we quantify the uncertainty of the nonequilibrium correction (B/A×dUm/dt) using a sensitivity analysis, where we vary the half-width of the Savitzky-Golay filter (used to smooth the reflectance signal) by a factor 0.5, 1 and 2 (i.e., 7, 15, 30 s), while keeping all other parameters constant. For each filter width, we sample the CFH Golden Points, find best fits for the RS41 time-lag and bias correction parameters, calculate the effective sensitivity parameter A, and finally calculate the nonequilibrium corrected CFH profile. We estimate the (2σ) uncertainty of the nonequilibrium correction as the maximal deviation in the final correct CFH profiles between all Savitzky-Golay filter widths: uSG=max⁡(Tfp,corr{7s,15s,30s})-min⁡(Tfp,corr{7s,15s,30s}). Adding in quadrature the mirror temperature uncertainty (2σ) uTm and the statistical uncertainty (2σ) um of the 25 s average (calculated as in Dirksen et al., 2014, their Eq. A2, then multiplied by 2), we calculate the total uncertainty u (2σ) of the nonequilibrium corrected frost point as: 9u=uSG2+uTm2+um2. The total uncertainty u is depicted as gray shaded area (±u) in Fig. 6C. The mean value for the total uncertainty for the flight section (8–10.2 km) is 0.32 K, while the standard deviation of non-equilibrium corrected CFH data (black lines in Fig. 6) with respect to time-lag and bias corrected RS41 data is 0.36 K.

The second metric u can be used to validate the first metric. The nonequilibrium corrected CFH data may be considered valid only in-flight segments where |Tfp,RS41corr-Tfp,CFHcorr|<u. This is the best ad-hoc method to validate the uncertainty of our retrieval in the absence of an SI-traceable sensor of similar or better accuracy as CFH. It also allows detection of measurement artifacts, such as contamination from the wake of the balloon or strong anomalies in airflow through the tube. We note that this (rather conservative) validation technique has not been applied to this work.

Filtering widths from ±7 to ±30 s give a reasonable range of structure-preserving smoothing filters for the mirror reflectance. Excessive smoothing (e.g., by ±300 s) would strongly smooth out features from the reflectance data, resulting in erroneous dUm/dt estimates and “fantasy” Golden Points (see Fig. D2). As a consequence, the fitted time-lag and bias corrected RS41 as well as the fitted effective sensitivity parameter A would not be meaningful. No smoothing would retain all noise from the reflectance data, which may lead to erroneous dUm/dt estimates as well as a large amount of incorrectly sampled Golden Points due to noise. Moreover, no smoothing or undersmoothing might pick-up high frequency oscillations with full anti-correlation of Tm and Um. Some CFH flights show an eigenfrequency of 5 s (see Fig. D1), which are not of atmospheric origin and would thus produce an incorrect correction.

For the Golden Points, we calculate uncertainty (2σ) as: 10uGP=utiming2+uTm2, where utiming is the ±1 s timing uncertainty in sampling the Golden Points, using the same uncertainty metric as Sugidachi et al. (2025, their Eq. 14, then multiplying by 2). From our analysis of 70 flights presented in Sect. 6, we find, at 2σ, utiming = 0.12 K for the liquid condensate, 0.16 K for the first ice condensate and 0.15 K for the second ice condensate. This translates, using Eq. (10), to a total (2σ) uncertainty uGP of the Golden Points of 0.23 K for the liquid film and 0.19 K for the ice films, i.e. < 0.3 K at the dew point and < 0.2 K at the frost point. Figures 2 and 7 showcase that the vast majority of CFH Golden Points remain within these uncertainty limits if shifted by ±1 s. This is because |dTm/dt| is generally well-constrained by CFH's PID controller, and heat equilibrates rapidly (∼ 120 ms) across the mirror disk (Sect. 4.1.3).

A well-adjusted CFH is one for which the combined effect of all sources of uncertainty in frost point (or dew point) measurements is better than 0.2 K, and systematic errors, which most likely affect long-term climate series, are demonstrably smaller than 0.1 K (Vömel et al., 2016). In fact, the nonequilibrium analysis of the present work confirms this performance for a majority of the analyzed CFH data points (for statistics see Sect. 6.2).

As mentioned earlier, not all instruments or flight legs are equally affected by nonequilibrium errors. As shown by Eq. (5) (1/A′⋅dU/dt∼χH2O(Tm)-χH2O(Tfp)), the reasons for this are the quality of the ice film, which is determined by the properties and cleanliness of the mirror, by the condensate thickness at the PID setpoint and by the cleaning cycles (all described by A′), as well as the properties of the vertical H₂O profile (χH2O), which determines the history in the integral part of the PID with a fixed setpoint (at low H₂O mixing ratios, a minute change in reflectance corresponds to larger deviations from frost point than at high H₂O mixing ratios). Subsequently, we discuss a few selected cases with widely varying performance and in Sect. 6 we discuss the statistics of the 70 analyzed soundings.

Figure 7 shows a collection of nonequilibrium errors obtained from six flights. The magnitude of the error in frost point temperature increases from left to right and varies between less than 0.1 K to several K, and from being symmetric to highly asymmetric in terms of frost point. Three of these six examples are subsequently discussed in more detail (Sect. 5.6.1 and 5.6.2).

As mentioned in Sect. 4.3, a dense array of Golden Points along the vertical profile of a balloon ascent can enable a self-correction of a chilled mirror instrument. Self-correction accomplishes independence of other instruments that have higher vertical resolution (at lower accuracy), such as FLASH-B or RS41. Recently, there has been a development in this direction in the form of the Peltier-based chilled mirror hygrometer SKYDEW (Sugidachi et al., 2025). While the Golden Points of CFH are typically separated by a few hundred meters (see multiple examples in Figs. 2–9), SKYDEW's Golden Points are typically 15 m apart in the troposphere and 20–25 m in the stratosphere. Figure 10 illustrates this feature of SKYDEW that has the potential to lead to a better constraint of the frost point. However, Fig. 10B also shows that the very large amplitude of the nonequilibrium excursions leads to strong temporal changes, dTm/dt, at the Golden Points. This renders the retrieval of precise timings of the Golden Points along the balloon trajectory difficult. Even sampling time differences between Tm and Um of less than 1 s can cause SKYDEW to suffer large uncertainties in the Golden Point timing, leading to incorrect Golden Points temperatures. This demonstrates that the perceived advantage of a denser Golden Point array, in terms of vertical resolution, can easily become a disadvantage in terms of frost point accuracy at the Golden Point on instruments with large nonequilibrium excursions.

In Fig. 10C, we have shifted the mirror temperature data by 0.4 s (about 2 m vertical distance) relative to the reflectance signal, which can remove most of the atmospherically meaningless oscillation of the Golden Points for this particular case (i.e., Tm is shifted such that the RMS difference between the maxima-only and minima-only polygonal lines is minimized). After this synchronization of Um and Tm, the Golden Points of SKYDEW can be used without further nonequilibrium correction, albeit with a vertical resolution of 15–25 m and a sampling uncertainty of ±1 s in the Golden Points (Sugidachi et al., 2025) associated with a steep slope dTm/dt. For finer structures to be resolved, SKYDEW requires a nonequilibrium correction, e.g. by means of a time-lag and bias-corrected RS41 in the same way as done for CFH (violet solid line in Fig. 10D).

Alternatively, SKYDEW can be self-corrected: by using the polygonal line defined by the neighboring Golden Points as a-priori reference in the nonequilibrium correction procedure, we can determine the sensitivity A and obtain the dashed red line in Fig. 10D. The good agreement (< 0.15 K) with the nonequilibrium corrected profile based on RS41 demonstrates the high potential of this procedure, which allows SKYDEW to be nonequilibrium corrected without a second independent sensor. However, in the absence of an additional sensor suited for the stratosphere, it remains unclear if the self-correction procedure presented here would work at 10 km higher altitudes. Also, due to the steep slopes dTm/dt at the Golden Points, even with synchronized Um and Tm, the noise in the measurements leaves SKYDEW's Golden Points with finite errors that can be quantified from Fig. 10C with ±1 s Golden Point timing uncertainties to be about ±1 K in this case (see Sect. 5.5). This calls for a better feedback control to reduce the amplitude of the nonequilibrium excursions, or for sub-second telemetry.

Figure 11 shows the effective sensitivity parameter A derived for the three condensate layer types (liquid film, first and second ice film) for each of the 70 flights. The values span about two orders of magnitude and vary from film-to-film and from flight-to-flight. This large variability does not allow the use of a fixed value of A for all flights, not even for instruments that have been recovered and flown again, using the identical mirror and optical hardware. This complicates the design of feedback controllers of consistently good performance, i.e. with the same sufficiently large A. Most likely, the variability is due to different film morphologies, themselves associated with the mirror surface properties (impurities, coatings, etc.), prevailing temperatures, ventilation, ambient H₂O molecule number density, cooling rate, supersaturation and possibly coalescence/sintering experienced during film formation. Variability in balloon ascent rate might play a minor role as well, as the boundary layer thickness of the airflow over the mirror is part of the effective sensitivity and depends on the flow conditions inside the inlet tubes. Over 95 % of the derived effective sensitivity values are larger than 1000 V s⁻¹ in the CFH data set we analyzed, which generally results in small mirror temperature deviations from equilibrium, as (pvap(Tm)-pvap(Tfp)∝1/AdUm/dt).

The first ice-film (triggered by the cleaning cycle at -15 °C mirror temperature) has the smallest effective sensitivity, likely due to an ice film with fewer and larger ice crystals, which is potentially “patchier”. Films with large ice crystals are known to deteriorate the detection limit and response time of optical frost point hygrometers (Pragnell, 1993; Vömel et al., 2016). The second ice film (formed at -53 °C mirror temperature) has the largest variability in effective sensitivity. This might be due to large flight-to-flight differences in prevailing conditions during condensate formation, mirror impurities, or increased variability in boundary layer thickness on the mirror, which increases as the Reynold's number decreases in the UT/LS (Jorge et al., 2021; see also Sect. 4.1.2). The liquid film (at mirror temperatures above -15 °C) has the highest effective sensitivity and the least scatter in effective sensitivity, possibly because of a higher consistency of dew morphology.

For each of the analyzed flight sections (liquid layer, first and second ice layer), the optimization solver described in Appendix B provides individual time-lag and bias parameters for the RS41, from which we calculate a corrected RS41 profile, the effective sensitivity A and the nonequilibrium-corrected CFH profile (and for liquid films also the sub-second time delay Δdelay, see Appendix B). In general, the corrected RS41 and CFH profiles agree well, although measurement noise, residual correction errors, contamination, and subscale atmospheric variability lead to a certain degree of uncertainty, with frost point differences > 1 K in some cases. Figure 12 shows the difference between individually corrected RS41 and CFH with and without nonequilibrium correction for all 106 814 data points. The data were retrieved with 1 s resolution between the ground and the tropopause, where the corrected RS41 provides more reliable measurements than in the stratosphere, as showcased in Fig. B1. In order to ensure flight-to-flight consistency in vertical resolution and improve the accuracy of the estimated nonequilibrium error when it is large, the boxcar filter is kept constant (5 s for liquid films, 25 s for ice films) and the nonequilibrium error is derived from the vapor pressure instead of the frost point (i.e. Eq. 5 instead of Eq. 6). The error due to the nonequilibrium of CFH (light gray points) is generally small, but there are cases with errors of 1–5 K or > 10 % in mixing ratio. These measurement errors benefit most from the nonequilibrium correction, which can reduce the error by more than 80 % (change from gray bars to black bars in Fig. 12B). Measurements with the largest nonequilibrium errors occur mostly in the presence of the second ice film (or UT/LS). They are clearly negatively biased (dry bias) due to the kinetics of the ice film with slow condensate growth and faster condensate evaporation (Clausius-Clapeyron equation). However, as Fig. 12A shows, the mean bias of the original CFH measurements for the second ice film in these 70 soundings is very low, namely only -21 mK, and reduces further to -6 mK after the nonequilibrium correction. The benefits of the nonequilibrium and Golden Points method are seen in the standard deviation, which reduces from ±305 mK for the original data to ±157 mK for the nonequilibrium corrected data and ±96 mK for the Golden Points.

The text insert in Fig. 12 lists individually for all three condensate types the mean bias and standard deviation, corresponding to the root mean square error (RMSE) between corrected RS41 and CFH (original, corrected and Golden Points). For the second ice layer (UT/LS) the nonequilibrium correction achieves about 45 % reduction in RMSE, for the first ice layer 30 % reduction, and for the liquid layer about 20 % reduction. Interestingly, there are a few Golden Points with |Tfp,CFH-Tfp,RS41corr| > 1 K, while one might expect this to be smaller than the uncertainty of 0.2 K. However, we first note the fitting procedure for each condensate film with two parameters for the time-lag correction and six parameters for the bias correction leaves some room for outliers, whereas the vast majority of Golden Points deviates less than 0.2 K (see logarithmic bar diagram in Fig. 12B). For the liquid film the nonequilibrium correction is the least efficient and the RMSE at the Golden Points is the largest. Despite the time-delay correction in the optimization routine described in Appendix B, the ±1 s uncertainty in sampling the Golden Points likely plays a bigger role for the liquid film, as CFH typically oscillates much more strongly (i.e. large |dTm/dt| at the Golden Points). In addition, the noise in the resistance measurement induces a small random uncertainty in the lower troposphere (Sect. 2.5).

Figure 12B depicts the logarithmically plotted distribution of CFH-RS41 errors across all condensate types, demonstrating the effectiveness of the nonequilibrium correction (change from gray to black histogram) and the superior quality of the Golden Points (yellow histogram). In particular, note the effectiveness in eliminating most of the extremely dry outliers, i.e. some layers of extreme-low bias can be identified and corrected, which is important for trend detection. The figure also reveals that nonequilibrium errors do not affect measurement accuracy when averaged over a large number of soundings (-6 mK bias over all soundings over all layers), which is indicative of appropriate PID control on average, as noted also in Vömel et al. (2016).

For CFH, the combined effect of all sources of uncertainty in frost point (or dew point) measurements is specified to be better than 0.2 K under good conditions, and systematic errors, which most likely affect long-term climate series, are smaller than 0.1 K (Vömel et al., 2016). Figure 12 confirms this performance for a majority of 89 % of the measurement points. For 11 % of the about 10⁵ measurement points, the nonequilibrium analysis revealed errors in the frost point estimated from the raw mirror temperature (i.e., |Tm-Tfp,RS41corr|) of more than 0.2 K, for 5.5 % of more than 0.3 K, for 2.2 % of more than 0.5 K, and for 0.6 % larger of more than 1.0 K. For the subset of points with the second ice film (near the tropopause, about 23 000 points), 37 % have errors in frost point larger than 0.1 K, 18 % larger than 0.2 K, 10 % larger than 0.3 K, 4 % larger than 0.5 K, and 1 % larger than 1.0 K. As mentioned earlier, because of different ice film qualities not all instruments or flight legs are equally affected by nonequilibrium errors. Since 40 of the 70 analyzed soundings have been launched from Lindenberg under the optimal launch conditions at the GRUAN Lead Center and the moderate midlatitude flight conditions, these statistics might be slightly worse for other stations and flight campaigns with lower H₂O mixing ratio, such as in the tropical UT/LS. In fact, for the subset of points with the second ice film (near the tropopause), only 32 % of all flights have errors in frost point < 0.2 K (for at least 95.4 % of their measurements points). All of these flights achieving this level of performance were launched from Lindenberg, where about 2/3 of the flights satisfy the < 0.2 K criterion.

A fundamental characteristic of chilled-mirror hygrometers that the actual dew/frost point is only measured when the condensate (liquid water or ice) is in equilibrium with the ambient gas phase, i.e. at the Golden Points (dUm/dt = 0). For CFH, the 2σ uncertainty is less than 4 % in the mixing ratio at the Golden Points. Away from the Golden Points, when the condensate is not in equilibrium (dUm/dt≠0), the measurement error is proportional to the change in mirror reflectivity per time, independently of the controller action. When the reflectance increases, the condensate is evaporating and the mirror is too warm. Conversely, when the reflectance decreases, the condensate is growing and the mirror is too cold. The Golden Points are a powerful feature of chilled mirror hygrometers as they provide physical confirmation of thermodynamic equilibrium (or equilibrium-like points in time). As such, they strengthen the use of chilled mirror hygrometers as reference instruments with high accuracy and negligible drift, suited for long-term records of H₂O in the upper troposphere and lower stratosphere.

Correcting the nonequilibrium states of CFH requires an estimate of the morphological sensitivity A′, which contains information about the frost (or dew) morphology on the mirror. When CFH is flown together with a second humidity sensor with potentially lower accuracy but a temporal resolution better than the spacing of the Golden Points, this sensor can first be recalibrated using the accurate CFH Golden Points, and then the recalibrated sensor can in turn be used to correct the nonequilibrium states of CFH. Such a second sensor can be a high-performance thin film sensor like the RS41 or an optical hygrometer like FLASH-B. Except for contaminated flights (hydrometeors in the inlet or outgassing of the balloon envelope), we found from 70 night time soundings that the non-equilibrium correction achieves a 60 %–100 % reduction in the frost point error for errors greater than 1 K, provides a significant improvement in measurement accuracy for 68 % of the flights, and results overall in a 2σ uncertainty of less than 0.4 K at a vertical resolution of 125 m or less than 0.2 K (i.e., 4 % in the mixing ratio) at 500 m vertical resolution from the ground to the middle stratosphere. This shows that the target uncertainty of < 4 % in the mixing ratio set by the WMO in 2024 is achievable when applying a nonequilibrium correction. The method achieves higher accuracy and vertical resolution, and is less prone to outliers, than traditional averaging techniques (that do not account for the mirror reflectance) for cooling mirror devices such as CFH, for which a 2σ uncertainty of 10 % in the mixing ratio is specified. Attempts to consider chilled mirror instruments as time-lagged (i.e. following a first-order response with time constants of several seconds), as found sometimes in literature, are generally inappropriate, because chilled mirrors are actively controlled devices and usually follow a higher order response with multiple overshoots. Furthermore, attempts to remove nonequilibrium errors with downsampling techniques or low-pass filtering over several hundreds of meters are prone to bias in case of large and/or asymmetric nonequilibrium errors, and reduce vertical resolution.

Self-correction: Under suitable circumstances, a nonequilibrium correction can be applied without the use of an auxiliary humidity sensor. One approach is to derive an a-priori profile from an array of closely spaced Golden Points with mirror temperature oscillations of sufficiently large amplitude in-between, given the Golden Points positions can be determined precisely. Another approach is to derive the morphological sensitivity by matching ascent and descent data in the stratosphere, given the presence of atmospheric/non-atmospheric features in regions of moderate enough flow rate.

Looking ahead, it would be interesting to investigate new hygrometer designs that make optimal use of the Golden Points concept. In particular, feedback control could be improved in two ways: minimize nonequilibrium errors such that a nonequilibrium correction is not required, or enable robust nonequilibrium self-correction.

Improved PID controller to minimize nonequilibrium errors: it is now clear that the error remains small, |Tm-Tfp|∝|dUm/dt|/A′ < 0.2 K (Eq. 8a), when the morphological sensitivity A′ is large and the change in reflectivity |dUm/dt| is small, ideally in conjunction with sufficiently frequent Golden Points and an average nonequilibrium error remaining close to zero along the flight path (see Fig. 7A–B for examples with Golden Points every ∼ 150 m). Maximizing A′ through an improved mirror design, such as a hydrophilic micropattern structure printed or deposited on the mirror, could increase the consistency and morphological sensitivity of the mirror condensate. Furthermore, the introduction of an adjustable setpoint could help reduce non-equilibrium deviations and thus improve the performance of chilled mirror hygrometers.

Improved PID controller for self-correction ability: a-priori profiles can be derived from dense arrays of Golden Points and quasi-symmetric mirror temperature oscillations of sufficiently large amplitude. The timing of the Golden Points is particularly important in such an approach. New designs with live detection of the Golden Points by a microcontroller (i.e., sampling Tm and Um at an internal rate ≫ 1 Hz and transmission of the data in telemetric batches) could reduce the uncertainty of the Golden Points positions and be useful for sensors with large |dUm/dt|, provided any time lag between Um and Tm can be well characterized. Enforcing mirror temperature oscillations at a fixed frequency (e.g., 1/60 Hz) and suppressing this artificial, non-atmospheric frequency in the non-equilibrium-corrected data could be another promising approach to the Golden Points.

Data archive: correction for nonequilibrium requires the composite Tm and Um chilled mirror signal, we therefore recommend reporting both Tm and Um in chilled mirror data products (e.g., from NDACC). The Um signal is also important for data quality check.