3.1 Atmospheric opacity

The majority of the incoming radiation measured by VESPA-22 comes from the troposphere. The tropospheric emission needs to be filtered in order to study the stratospheric signal. In order to simplify the radiative transfer equation, the following approximations can be adopted (e.g., Nedoluha et al., 1995).

• The troposphere is represented as an isothermal layer absorbing the signal to be measured. The first kilometers of the atmosphere produce the greatest part of this layer emission which has a line width larger than VESPA-22 bandwidth. This contribution is treated as an emission constant in frequency.

• The contribution of the stratospheric water vapor absorption to the opacity τ is small. This approximation was tested by calculating the atmospheric opacity by means of the radiative transfer simulation software ARTS (Eriksson et al., 2011) and using water vapor vertical profiles with and without their stratospheric component. At the frequency of maximum absorption, the contribution of the stratospheric profile to the overall atmospheric opacity is between 2 and 5 % depending on the season. When averaged over the VESPA-22 frequency range, the difference between the opacity calculated using a full water vapor vertical profile and a profile with no water vapor above the tropopause is between 0.4 and 1.5 %.

• The opacity τ_ν can be substituted by its mean value τ. The maximum difference between τ_ν and its mean value τ is between 1.6 and 3.6 % depending on season.

• The only signal coming from outside the atmosphere, T₀, is the cosmic background radiation with a constant brightness temperature of 2.73 K.

With these approximations the radiative transfer equation describing the radiation received at the ground from an elevation angle θ can be written as

The left term is the radiation received by the spectrometer when pointing at the elevation angle θ. The first term on the right side is the extra-atmospheric emission, the second term is the solution of the radiative transfer equation for an isothermal domain where the tropospheric emission is indicated with T_trop and represents the mean tropospheric temperature weighted with the water vapor concentration. The third term on the right is the emission coming from the stratosphere and mesosphere, T(ν), proportional to the air mass factor μ and attenuated in the troposphere by a factor e^−μτ. The air mass factor at the elevation angle θ is computed according to the formula presented in the work of de Zafra (1995).

3.2 Beam-switching technique

The technique that VESPA-22 employs to measure the stratospheric signal is called the beam-switching technique or Dicke switching technique (e.g., Parrish et al., 1988). The instrument compares the emission coming from an observation angle θ (signal beam) with a reference signal with the same mean power over the passband. The observation angle depends on the atmospheric opacity and for VESPA-22 it varies from 12 to 25^∘ above the horizon. The reference signal used by VESPA-22 is the sky emission at the zenith (reference beam). In clear-sky conditions, the emission at the zenith is smaller than the emission at a much larger zenith angle. Therefore, in order to ensure that the reference beam has the same mean power of the signal beam, a thin sheet of delrin is inserted in the reference beam. The delrin sheet acts as a grey body so that

where T_R is the radiation observed by VESPA-22 coming from the zenith, partially absorbed by the delrin sheet, T_d is the physical temperature of the sheet and τ_d its mean opacity value over the spectral passband. In this equation, the air mass factor μ is equal to 1.

During data-taking operations, VESPA-22 alternates reference and signal observations. The instrument constantly checks if the two beams have the same mean power and continuously changes the signal angle to minimize the difference between them. When the two beams have the same intensity, the frequency independent terms of Eqs. (3) and (2) can be equated, obtaining

where the stratospheric contribution to the mean beam intensity (about 1 %) is neglected (de Zafra, 1995).

The stratospheric signal T(ν) is obtained by subtracting reference from signal. Using Eqs. (2), (3) and (4) it is possible to write

Three delrin sheets with different thicknesses (3, 5, and 9 mm) and opacities can be employed, depending on the season, in order to maintain the signal angle between 12 and 25^∘ above the horizon. Spectra collected are smoothed using a 50-channel moving average. This smoothing process is, however, not performed in a 6 MHz interval centered around the emission line to maintain the maximum frequency resolution near the line peak.

In order to estimate τ_d, the mean opacity value of delrin over the spectral passband, during normal data-taking operations the signal angle is locked to its balanced position and the delrin sheet is removed from the reference beam. τ_d can then be calculated using

where ${\stackrel{\mathrm{‾}}{T}}_{{\mathrm{R}}_{\text{nod}}}$ is the mean intensity of the reference beam without the sheet and ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{S}}$ is the mean value of the signal beam. The brightness temperatures ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{S}}$ and ${\stackrel{\mathrm{‾}}{T}}_{{\mathrm{R}}_{\text{nod}}}$ are calculated using parameters obtained by means of a calibration performed using liquid nitrogen (LN₂; see Sect. 3.3) which is always carried out before estimating τ_d. In order to run these operations, qualified personnel must be at the observatory and therefore τ_d was measured only in July and November 2016 and February 2017 (Table 1). A total of seven measurements of the compensating sheet opacity were performed, two in July, three in November, and two in February 2017. Half of the difference between the minimum and maximum τ_d  values obtained during the same period is used as measurement uncertainty. The mean value of the delrin opacity changes with time, possibly due to a certain level of degassing, i.e., the property of absorbing/releasing water vapor molecules from/to the environment, which depends on atmospheric humidity and is often noticed in plastic materials. During winter, as the air is drier, the compensating sheets appear to release some water vapor and lower their opacities. VESPA-22 spectral data are calibrated using a linear interpolation between the τ_d values measured over time.

3.3 Calibration scheme

The broad-band response of VESPA-22 to the signal coming from the sky can be written as

In this equation, V is the incoming signal in count numbers of the FFT back-end spectrometer, T is the signal brightness temperature, α the gain of the instrument, T_rec the receiver noise temperature and V₀ the “zero” signal of the FFTS. All the quantities represented in Eq. (8) are frequency dependent. The “zero” signal, which amounts to approximately 0.5 % of the incoming signal to the FFTS, is measured approximately every 15 min and it is subtracted to each signal and reference 15 min integration spectra which are eventually saved on the control and acquisition PC (see Fig. 1). Using Eq. (8), Eq. (6) can be written as

VESPA-22 measures the gain parameter α by means of the noise diode. As mentioned before, VESPA-22 has two different noise diodes, one used to perform regular calibrations and the second inserted to ensure the stability over time of the first one, as described by Gomez et al. (2012). During regular measurements, the calibration noise diode is switched on and its emission is added to the reference signal. The noise diode is then switched off and VESPA-22 measures only the radiation coming from the zenith. The gain parameter can be obtained by subtracting these two measurements according to

where V_R+nd and V_R are the signals expressed in count numbers measured with the noise diode turned on and off, respectively, and T_nd is the noise diode emission temperature at the various frequencies, estimated during a calibration procedure. The calibration consists in measuring the emission from two sources at two different known emission temperatures. The first source is a black body at ambient temperature made with an Eccosorb CV-3 panel by Emerson and Cuming; the second source is mainly the sky at an angle of 60^∘ above the horizon. Every 3 or 4 months, approximately, the sky emission can be replaced by a second CV-3 panel immersed in LN₂. The use of LN₂ likely grants more accurate results, as the physical temperature of the emitting body has a smaller uncertainty with respect to the estimated sky temperature at 60^∘, but the LN₂ calibration cannot be performed automatically by the instrument and has so far been carried in July and November 2016 and in February 2017.

Table 1Mean values and SDs (standard deviations) of the measured opacity for the two used delrin sheets during different seasons.

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The general calibration equations are described in what follows, whereas the tipping curve procedure that allows the use of the sky as calibration source is explained in Sect. 3.4. Knowing the emission temperature of two sources α, T_rec, and T_nd can be obtained by using the following equations:

T_hot and T_cold and V_hot and V_cold are the emission temperatures and the recorded count numbers of the two sources, respectively. For the black body, the emission temperature is assumed to be equal to its physical temperature.

The noise diode produces a signal that is measured to be quite stable in frequency. In fact, single-channel T_nd values are always within 1.5 % of the spectral mean of the diode temperature brightness ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$. The spectra originated from black-body measurements (especially those from the CV-3 immersed in LN₂) can be affected by standing waves and in order to avoid to transfer them in the calibrated sky spectral measurements, T_nd is averaged over the central 11 000 channels of the FFTS, as suggested by Gomez et al. (2012). Therefore, it can be written

where V_{cold + nd}(ν_i) is the value of channel i with the noise diode turned on, while V_cold(ν_i) is the value of the same channel with the noise diode turned off.

3.4 Tipping curve procedure

In this section, the tipping curve calibration technique employed to calibrate the noise diodes using the sky signal is described. The tipping curve procedure is performed twice every hour as it is also used to measure the atmospheric opacity needed in Eq. (9). During a tipping curve, VESPA-22 collects the radiation coming from different elevation angles, approximately every 5^∘ from 35 to 60^∘ above the horizon. The measured spectra are averaged using the 11 000 central channels of the spectrometer. Radiation from the stratosphere contributes less than 1 % and can be neglected, so the signal intensity can be described by means of the following equation:

The atmospheric temperature T_trop can be estimated from the surface temperature T_surf:

where the value of d can be affected by seasonal variations. In order to characterize this parameter several radiosoundings were launched during July, November, and December 2016 and February 2017. The value of the parameter d as a function of time is obtained from a linear interpolation between the mean values of T_surf−T_trop measured during these four periods, as described in Table 2. T_trop is the average temperature obtained from radiosonde data by weighting the tropospheric temperature vertical profile with the water vapor concentration profile. Using Eq. (15), it is possible to explicitly give the relation between the opacity and the mean brightness temperature of the received signal:

A linear regression of the opacities $\ln \left(\frac{T_{\mathrm{0}}-T_{\text{trop}}}{\stackrel{\mathrm{‾}}{T}\left({\mathit{\theta }}_i\right)-T_{\text{trop}}}\right)$ observed at θ_i and the air mass factors μ(θ_i) allows us to retrieve the opacity at the zenith, τ (Nedoluha et al., 1995). Substituting for $\stackrel{\mathrm{‾}}{T}\left({\mathit{\theta }}_i\right)$ using Eq. (8) and approximating all the spectral quantities with their mean values (indicated by a bar) over the central 11 000 channels, Eq. (17) can be written as

where it appears that the mean values ${\stackrel{\mathrm{‾}}{T}}_{\text{rec}}$ and $\stackrel{\mathrm{‾}}{\mathit{\alpha }}$ are needed to perform the calculation.

Table 2Mean values and SDs of T_surf−T_trop obtained from the radiosoundings.

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The emission from two calibration sources at different temperatures is used to calculate $\stackrel{\mathrm{‾}}{\mathit{\alpha }}$ and ${\stackrel{\mathrm{‾}}{T}}_{\text{rec}}$ according to Eqs. (11) and (12). However, since the sky emission temperature $T_{\text{cold}}^{\text{sky}}$ is not known, an iterative procedure is used to obtain both τ and $T_{\text{cold}}^{\text{sky}}$. An initial opacity value, τ₀, is adopted as a first guess to obtain $T_{\text{cold},\mathrm{0}}^{\text{sky}}$ using Eq. (21) with $\mathit{\theta }=\mathrm{60}{}^{\circ }$. $T_{\text{cold},\mathrm{0}}^{\text{sky}}$ is then used to obtain α₀ and $T_{{\text{rec}}_{\mathrm{0}}}$ by means of Eqs. (11) and (12); μ(θ_i)τ is calculated for different elevation angles using Eq. (18), and ultimately a linear fit allows us to calculate a new estimate for τ, τ₁. The iterative procedure goes on until the intercept value is minimized. Figure 3 shows the results of a tipping curve measurement carried out on 10 December 2016.

Figure 3(a) The values of $\ln \left(\frac{T_{\mathrm{0}}-T_{\text{trop}}}{\stackrel{\mathrm{‾}}{T}\left({\mathit{\theta }}_i\right)-T_{\text{trop}}}\right)$ (indicated with τ_sig on the y axis of panel a) as function of the air mass factor μ(θ_i) (blue points) measured during a tipping curve on 10 December 2016, and the fit result (green line). (b) The residuals (measurements minus fit) as a function of the elevation angle.

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Figure 4Time series of the noise diodes mean emission temperature calculated by means of the tipping curve procedure (blue and cyan solid circles) compared with values obtained by means of LN₂ calibrations (orange and red stars). The green dots and the right y axis display the ratio of the emissions from two noise diodes.

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The value of $T_{\text{cold}}^{\text{sky}}$ measured with this procedure is used in Eqs. (11) and (13) to estimate ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$. In order to avoid the use of data acquired during inhomogeneous sky conditions, all measurements producing linear fits (see Fig. 3a) with a root mean square larger than 0.4 are discarded. Figure 4 shows the time series of both noise diodes' mean emission temperatures (blue and cyan) and their ratio (green dots and right y axis) from July 2016 to July 2017. The noise diode in blue is the one used as calibration diode. In the same plot, ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$ values obtained using a LN₂ cooled CV-3 as the cold load are also depicted (orange and red stars). The mean relative difference between ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$ values calculated with LN₂ and with tipping curves carried out immediately before or after LN₂ calibrations is (0.4±0.4) and (0.2±0.3) % for the calibration and the backup diodes, respectively. The ratio between the two noise diodes can provide insights into potential drifts of one diode with respect to the other. Since the estimated uncertainty on such a ratio is 0.05 (3.4 %), there appears to be no drift in the time frame discussed in this work.

4.1 Forward model and a priori profiles

Figure 6 displays the a priori profiles used by the VESPA-22 retrieval algorithm during 10 out of the 12 months of the year and based on local climatology (3 years' worth of data of Aura/MLS v4.2). These profiles are identical below 48 km and diverge above, due to the large difference in polar regions between summer and winter water vapor mesospheric profiles. The summer a priori (red line) is used during the period from 1 June to 15 September, while the winter a priori (blue line) is used during the period from 16 October to 30 April. During the month of May and from 16 September to 15 October, there are transition periods in which a linear daily interpolation from one a priori to the other was used for continuity.

The matrix K and the y_a spectrum are calculated using the radiative transfer simulation software ARTS (Eriksson et al., 2011), adopting a Voigt–Kuntz lineshape and the line intensity provided by the JPL 2012 catalogue (Pickett et al., 1998, and https://spec.jpl.nasa.gov/). Following the work of Seele (1999) and Tschanz et al. (2013), the line described by the JPL 2012 catalogue is divided into three emission lines indicating the hyperfine splitting of the 22.235 GHz water vapor line. The employed pressure broadening and self-broadening parameters are those reported by Liebe (1989). Table 3 summarizes the spectroscopic parameters used for the analysis of VESPA-22 spectral measurements which, in what follows, are indicated as the “reference” model.

The profile x_ARTS is the profile used in the forward model calculations to compute the a priori spectrum and the weighting functions. x_ARTS matches the a priori profile from 12 km of altitude upward and, below 9 km, it is consistent with the measurements of precipitable water vapor (PWV) collected by the HATPRO radiometer (Rose and Czekala, 2009; Pace et al., 2015) installed at the THAAO. This lower part of x_ARTS is calculated according to

where x_Eu is a water vapor mixing ratio profile obtained by daily smoothing monthly averages calculated from the radiosoundings launched from the Eureka station, PWV_Eu is the associated water vapor column content, and PWV_Hatpro is the column content measured by the HATPRO located at the THAAO. Data from Eureka were chosen (instead of those from Alert, Canada, for example) because they show the closest resemblance to the tropospheric profiles measured at Thule by local radiosoundings, when the latter are available. In order to avoid discontinuities in x_ARTS, values at altitudes between 9 and 12 km are obtained with a linear interpolation between x_ARTS(9 km) and x_ARTS(12 km).

The pressure and temperature profiles needed to run the forward calculation are built merging NASA Goddard Space Flight Center (GSFC), Aura/MLS and climatological temperature and pressure profiles. The NASA GSFC profiles obtained through the Goddard Automailer Service (Lait et al., 2005) are used to build the tropospheric meteorological state, from the ground up to 9 km of altitude. Between 10 and 87 km altitude, the MLS temperature and pressure profiles collected during VESPA-22 observations, in a radius of 300 km from the observation point of VESPA-22, are averaged together to produce a single set of daily meteorological vertical profiles. The VESPA-22 observation point coordinates are chosen to be 74.8^∘ N and 73.5^∘ W, and represent an estimate of the geographical coordinates of the air mass that is observed by VESPA-22 (which points southwest, at about 220^∘) at 60 km altitude when the instrument aims at an elevation of 15^∘ above the horizon. Daily temperature and pressure profiles from 97 to 110 km of altitude are obtained by daily smoothing zonal monthly averages from the COSPAR International Reference Atmosphere (Fleming et al., 1990).

The absence of vertical discontinuities in the temperature and pressure daily profiles is assured by a smoothing process performed at the altitudes where the three different datasets (GSFC, MLS, and COSPAR) are stitched together, between 9 and 12 km and between 87 and 97 km.

The software ARTS is employed to simulate the emission from the zenith, ${\stackrel{\mathrm{̃}}{\mathit{y}}}_{\mathrm{r}}$, and from an angle close to the horizon, y_s. The emission ${\stackrel{\mathrm{̃}}{\mathit{y}}}_{\mathrm{r}}$ is then rescaled using Eq. (22), therefore simulating the effect of the delrin compensating sheet:

whereas the mean difference y_s−y_r is

The opacity τ_d used in Eq. (22) is the opacity of the compensating sheet; the temperature T_d is the temperature of the sheet, measured by a sensor installed next to it; and the signal beam observing angle, $\stackrel{\mathrm{̃}}{\mathit{\theta }}$, is chosen in order to minimize the mean difference y_s−y_r, as it is in fact attained by VESPA-22 in its data-taking process. The a priori spectrum is calculated according to the same equation used for the measured signal (Eqs. 5 and 6),

where $\stackrel{\mathrm{̃}}{\mathit{\mu }}$ and $\stackrel{\mathrm{̃}}{\mathit{\tau }}$ are the air mass factor associated to the simulated signal beam and the zenith opacity calculated from the x_ARTS profile.

Figure 7(a) An example of VESPA-22 measured spectrum (blue) collected on 23 December 2016, with the a priori spectrum in green and the fit spectrum in red. The second-order polynomial is indicated with a cyan solid line. (b) The residual y_fit−y. The central part of the spectrum is unsmoothed in order to maintain the maximum spectral resolution near the peak and its residual is larger.

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Deriving Eqs. (24) and (22), and using the K matrix definition (Rodgers, 2000), the retrieval weighting function matrix can be written as

where K_s and K_r are the weighting function matrices that ARTS calculates for the simulated signal and reference beams. As a first approximation, the dependence of $\stackrel{\mathrm{̃}}{\mathit{\tau }}$ on the stratospheric water vapor profile in Eq. (25) is neglected.

4.2 Retrieval example

Figure 7a shows a VESPA-22 spectrum integrated for 24 h (blue line) on 23 December 2016, its corresponding synthetic spectrum y_fit (red line) and the a priori spectrum (green line), while the residual (defined as the difference between fit and measured spectrum, y_fit−y) is plotted in Fig. 7b. The cyan line is the second-degree polynomial retrieved by the inversion algorithm. Figure 8a shows the result of the inversion of the measured spectrum depicted in Fig. 7 with the a priori profile and retrieval 1σ uncertainty. The details on the uncertainty calculation are discussed in Sect. 4.3.

4.3 Retrieval uncertainty

The uncertainty characterizing VESPA-22 retrieved profiles can be divided into four major contributions: (1) the uncertainty due to the linear approximation used in the optimal estimation, (2) the uncertainty due to the various parameters used in the spectra calibration and pre-processing, (3) the uncertainty due to spectral noise and potential artifacts, and (4) the uncertainty introduced by the use of the second-order polynomial in the retrieval process. One additional error source is the limited vertical resolution inherent to concentration vertical profiles obtained by means of this ground-based observing technique. This leads to solution profiles that can be considered a smoothed version of the real atmospheric concentration profiles. In discussing the optimal estimation method, Rodgers (2000) suggests that this error, called “smoothing error”, should be estimated only if accurate knowledge of the variability of the atmospheric fine structure is available. This approach is used here and the smoothing error is not included in the error estimate.

The first contribution can be evaluated observing the difference Δy_lin between the fit spectrum without the addition of the second-order polynomial, ${\mathit{y}}_{\text{fit}}^{\ast }$, and the spectrum obtained using ARTS to calculate the emission expected from the retrieved profile x (in the calculation, ARTS does not perform any linear approximation):

The uncertainty Δx_lin that Δy_lin causes on the retrieved profile can be calculated with

where G is the gain matrix (Rodgers, 2000). The uncertainty Δx_lin has a negligible contribution to the total uncertainty (with a maximum of 0.1 % at 70 km altitude).

Figure 8(a) The retrieved VESPA-22 profile (blue solid line) correspondent to the spectrum showed in Fig. 7. The a priori profile is indicated with a green solid line. The two red dashed lines describe the uncertainty of this VESPA-22 retrieval (for details on the estimated uncertainty of VESPA-22 mixing ratio vertical profiles see Sect. 4.3). (b) Rows of the A matrix multiplied by a factor of 10 as a function of altitude (some A functions are highlighted in colors). The vertical profile of the sensitivity is shown in red.

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Table 4Uncertainties on the various parameters used in the calibration process. When the uncertainty is a function of altitude the minimum and maximum values of the uncertainty are reported.

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Figure 8b shows the corresponding averaging kernels (AKs, black and colored solid lines). AKs are multiplied by a factor of 10 and the sensitivity in indicated with a red solid line.

In order to evaluate the second contribution listed above, the effects on the retrieved profile due to the variation of each single parameter used in the measurements calibration and pre-processing was investigated. The difference between the profile retrieved using the “correct” value of a specific parameter and the retrieval obtained by changing such a value by the estimated relative uncertainty of the parameter is considered to be the contribution σ_i of this parameter to the total calibration and pre-processing uncertainty. The total uncertainty from these sources is defined as calibration uncertainty. Table 4 summarizes the uncertainties on the various parameters involved in the calibration and pre-processing of VESPA-22 spectra. When the uncertainty is a function of altitude the minimum and maximum values of the uncertainty are reported. The total calibration uncertainty σ_cal is given by

Figure 9Relative contributions to the calibration uncertainty defined by Eq. (28) and indicated with a red solid curve. The contributions are the signal beam angle (yellow), the noise diode temperature (green), the sky opacity (blue), the MLS meteorological profiles (orange and brown), the compensating sheet opacity (magenta), and the spectroscopic parameters (cyan).

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Figure 10(a) Mean retrieved water vapor profiles obtained using different spectroscopic models (the blue line is the model eventually adopted for the analysis of VESPA-22 spectra, called reference model, or RM). The spectroscopic parameters for the additional models tested are obtained from different versions of the JPL catalogue (years 1985, 2001 and 2012; Pickett et al., 1998) with the pressure broadening parameters taken from Liebe (1989) and Cazzoli et al. (2007). Two models include values from HITRAN 2004 and 2012 (Rothman et al., 2013). Models are indicated in the legend with their abbreviations: JPL2012Liebe1989 (JPL12L89), JPL1985Liebe, 1989 (JPL85L89), JPL1985Cazzoli, 2007 (JPL85C07), JPL2001Liebe, 1989 (JPL01L89), HITRAN2012 (H12), and HITRAN2004 (H04). (b) Vertical profiles of mean relative differences between VESPA-22 profiles obtained using the reference model and the different spectroscopic models. Data represented here range from 4 October 2016 to 22 May 2017. The average retrieval sensitivity range is marked by dashed horizontal green lines.

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In Fig. 9, the relative contributions of the various parameters needed in the calibration and pre-processing procedures are shown. The yellow line shows the σ_i contribution due to the uncertainty on the signal beam angle. In order to minimize this contribution, the choked antenna and the parabolic mirror are aligned using a He-Ne laser. During the period from April to October, the VESPA-22 pointing offset can be verified by scanning rapidly at around the sun elevation angle and comparing the measured position of the center of the sun against the known ephemerides. For the measurements discussed here an offset of $\mathrm{\Delta }\mathit{\theta }=\mathrm{0.2}{}^{\circ }\pm \mathrm{0.1}{}^{\circ }$ was estimated. The green solid line shows the potential relative error on the water vapor mixing ratio vertical profile due to the uncertainty on the estimated noise diode temperature ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$. In order to evaluate the uncertainty on ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$, the 0.4 % difference between the values obtained by means of the LN₂ calibrations and by means of the skydips immediately following or preceding the LN₂ calibrations (see Sect. 3.4 and Fig. 4) is considered. On top of this, the fluctuations of the signal produced by the calibration diodes must also be taken into account. These can be evaluated by using the standard deviation (SD) of the difference over time between the ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$ values of the two noise diodes, measured to be 1.1 %. The tipping curve procedure allows for calculation of the noise diodes brightness temperature averaged over the 11 000 central channels of the spectrometer. Since the noise diodes brightness temperatures do have a small frequency dependence, using their mean values introduces a source of uncertainty which is estimated to be 0.9 %. An additional source of uncertainty is due to sky inhomogeneities, and amounts approximately to 1 %. All these sources of uncertainty for ${\stackrel{\mathrm{‾}}{T}}_{\mathrm{nd}}$ are added in quadrature to obtain a total uncertainty of 1.8 %. The blue line shows the contribution due to the uncertainty Δτ on the sky zenith opacity τ. In order to estimate this uncertainty, the uncertainties introduced by sky inhomogeneity and by the estimation of the effective tropospheric temperature T_trop are considered. The first contributes to about 2 % and is estimated using the uncertainty on the slope of the linear fit of the tipping curve (see Sect. 3.4 and Fig. 3). The second contribution can be evaluated observing the daily fluctuations of T_trop measured at Eureka (Canada), Aasiaat (southern Greenland), and Alert (Canada). The natural day-to-day temperature fluctuations and the lack of tropospheric meteorological data at Thule during long intervals of time led us to estimate an uncertainty on T_trop of about 6 K, which then makes the total uncertainty Δτ∼5 %.

The brown and orange lines show the σ_i's due to temperature and geopotential height profiles used in the forward calculation. The uncertainties on these parameters are obtained from the MLS data quality and description document (Livesey et al., 2015). In Fig. 9, the magenta line shows the contribution of the uncertainty on the compensating sheet opacity, Δτ_d. Although the uncertainty on the measurements of τ_d is about 2 % (see Table 1), the use of a linear interpolation suggests the use of a more conservative estimate, eventually set at 10 %. The cyan line shows the contribution due to uncertainties in the employed spectroscopic parameters, and it has the largest impact on the calibration uncertainty. Following Straub et al. (2010), the emission line intensity and the pressure broadening coefficient were assigned an uncertainty of $\mathrm{8.7}\times {\mathrm{10}}^{-\mathrm{22}}$ m² Hz and 1014 Hz Pa⁻¹, respectively.

In addition to these uncertainties, Fig. 10 shows the results of retrieving VESPA-22 data from October 2016 to May 2017 using different sets of values for the spectroscopic parameters (see also Haefele et al., 2009). In the figure, the mean profiles retrieved using the different models listed in the legend are depicted. Such models include values taken from HITRAN (Rothman et al., 2013) and JPL catalogues (Pickett et al., 1998), for different years, and from Liebe (1989) and Cazzoli et al. (2007). The blue line represents the mean profile obtained using VESPA-22 reference model (RM). In Fig. 10b, the mean relative difference profiles between retrievals obtained using the reference model and the other spectroscopic models are drawn (see legend). The water vapor mixing ratio retrieved profiles show a strong dependence on the spectroscopic model of choice, with a relative difference between profiles obtained using different models reaching a maximum of 8 % at the top of the sensitivity range.

Figure 11 shows the uncertainty calculated for the retrieval of 23 December 2016, shown in Fig. 8. The overall calibration uncertainty is shown with a red curve. The uncertainty due to spectral noise and artifacts can be evaluated using the S uncertainty matrix (Rodger, 2000) obtained as

The square root of the diagonal elements of S represents the spectral uncertainty of the retrieved profile at different altitudes and is indicated in Fig. 11 with a blue line. This term depends on the S_e value of the retrieval and its value increases during summer and with poor weather conditions. As mentioned in Sect. 4, the diagonal values of the S_e matrix are held constant and do not take into account the spectral noise reduction obtained with the 50-channel smoothing process which is applied to most of the spectrum (Sect. 3.2). This may lead to a small overestimation of the spectral uncertainty depicted in Fig. 11 in the altitude range 20–50 km (with a maximum potential overestimation of 0.3 % at 25 km altitude). The green curve displays the estimated uncertainty due to the use of the second-order polynomial in the retrieval process. The total uncertainty is calculated as

and is represented with a purple line in Fig. 11.