Journal topic
Atmos. Meas. Tech., 12, 3699–3716, 2019
https://doi.org/10.5194/amt-12-3699-2019
Atmos. Meas. Tech., 12, 3699–3716, 2019
https://doi.org/10.5194/amt-12-3699-2019

Research article 09 Jul 2019

Research article | 09 Jul 2019

# Calibration of a water vapour Raman lidar using GRUAN-certified radiosondes and a new trajectory method

Calibration of a water vapour Raman lidar using GRUAN-certified radiosondes and a new trajectory method
Shannon Hicks-Jalali1, Robert J. Sica1,2, Alexander Haefele2,1, and Giovanni Martucci2 Shannon Hicks-Jalali et al.
• 1Department of Physics and Astronomy, University of Western Ontario, London, Canada
• 2Federal Office of Meteorology and Climatology MeteoSwiss, Payerne, Switzerland

Correspondence: Shannon Hicks-Jalali (hicks.shannonk@gmail.com)

Abstract

1 Introduction

Water vapour is one of the main contributors to the greenhouse effect due to its ability to absorb infrared radiation efficiently. Water vapour has high temporal and spatial variability, making it difficult to characterize its influence on the atmosphere . When conducting climatological studies, ground-based lidars have an advantage over satellite-borne instruments in that they have the ability to provide frequent measurements from the same location. Lidar measurements are particularly useful for creating statistically significant water vapour trends throughout the troposphere, as they are able to make long-term and frequent measurements . Minimizing the uncertainty in the measurements is critical in order to establish a valid trend. A large component of a lidar measurement's uncertainty budget is its calibration constant. Water vapour lidars measure relative profiles and therefore require a calibration to convert the measurements into physical units (here mixing ratio). Refining the calibration process is critical to detect the small changes anticipated in the trend analysis. Several Raman lidar calibration techniques have been developed over the years, including internal, external, and a hybrid of internal and external methods.

Internal calibration techniques require no external reference instrument. They can account for the entire optical path in the lidar system to find the water vapour calibration constant. In essence, all optical transmittance, quantum efficiencies of the detectors, Raman cross sections, the geometric overlap, and their associated uncertainties must be quantified and accounted for. Some of these can be derived simultaneously using the white light calibration discussed in . The white light technique is advantageous in that it can accurately track changes in the calibration constant. However, the calibration is incapable of detecting shifts in spectral separation units and is not able to accurately detect the cause of calibration changes unless multiple lamps in different locations are used . Venable et al. (2011) improved the technique by using a scanning lamp instead of a stationary lamp. The limiting factor in the white lamp calibration technique is the degree to which we know the molecular cross sections, which have uncertainties on the order of 5 % . While internal calibration offers many advantages, it is impractical for many systems, such as lidars that use multiple mirrors or large-aperture mirrors such as the rotating liquid mercury mirror of the University of Western Ontario's Purple Crow Lidar .

The standard external method involves comparing the lidar and a reference instrument; typically the reference instrument is a radiosonde , but microwave radiometers or GNSS satellites may also be used . External calibrations are often preferable because there is no need to characterize every system component, and the uncertainties in the Raman cross sections do not contribute. However, the accuracy of the external calibration is dependent on the accuracy of the reference instrument. Radiosondes are widely used calibration instruments, as they have high spatial resolution, are routinely available, and are widely available. Uncertainties for the Vaisala RS92 relative humidity measurements vary between 5 % and 15 % depending on the time of day . To minimize the calibration uncertainties induced by biases in the radiosonde reference, the GCOS (Global Climate Observing System) Reference Upper-Air Network (GRUAN) has established a robust correction algorithm for the Vaisala RS92 radiosondes, as RS92 radiosondes are the most frequently used calibration radiosondes . GRUAN RS92 relative humidity profiles have been shown to be 5 % more moist than uncorrected RS92 relative humidity profiles while reducing the relative humidity uncertainties by up to 2 % .

Hybrid internal–external methods, which also attempt to minimize variations in the sampled air mass due to the balloon's horizontal motion, have also been implemented by and . In these hybrid techniques, the white light calibration lamp is used to monitor the efficiency of the lidar optical paths but is supplemented with radiosondes for the absolute calibration value. The hybrid technique will monitor relative changes in the calibration constant but must be supplemented periodically with an external calibration .

For any external calibration in which the lidar and the calibration instrument do not share a common field of view, variations in water vapour cause an additional uncertainty in the calibration that is often not quantified in the uncertainty budget. A portion of the calibration uncertainty when using radiosondes can occur from the radiosonde's lack of co-location with the lidar, hereafter the “representation” uncertainty. This paper attempts to resolve the co-location problem and minimize the representation uncertainty by using a tracking technique that expands upon those discussed in , , and . The co-location problem can be particularly acute for calibration via a radiosonde, as the radiosonde takes approximately 30 min to reach the tropopause at mid-latitudes, during which time the radiosondes in this study travelled a minimum of 4 km from the lidar's field of view (assumed here to be the zenith, which is typically how water vapour lidars are operated). The distance travelled by the radiosonde has little effect on a calibration measurement if the air mass being sampled is horizontally homogeneous. However, this is not necessarily the case, and when we calibrate while on the edge of an air mass, or the air mass simply is not horizontally uniform, then the water vapour field may change dramatically over the distances the radiosonde travels. Lidar stations which have the resources to use daily radiosondes may not see this as much of a hindrance; however, if the station relies on infrequent calibration campaigns, then the campaign calibration results are dependent on the air masses which are sampled.

As in any atmospheric calibration method, it is important that the instruments involved measure the same air mass. To improve the coincidence for periods in which calibration is required but the atmospheric water vapour content is changing, we have developed an improved lidar–radiosonde calibration technique that utilizes the position of the radiosonde and the wind speed and direction measured by the radiosonde. The wind speed and direction measurements allow us to track the air parcels as measured by the radiosonde with respect to the position relative to the lidar. If the air is within a 3 km radius around the lidar, we use the corresponding times and lidar scans for calibration. A lidar “scan” refers to a 1 min (1800 shots) raw measurement profile. We have implemented the technique using 76 night-time GRUAN RS92 radiosonde flights from 2011 to 2016. The GRUAN sondes represent the best characterized sonde measurements available in terms of calibration and uncertainty budget . Daytime calibrations were not tested due to the significantly reduced signal-to-noise ratio (SNR) in daylight measurements and the inability to reach above 5 km effectively with the lidar. We will illustrate the method using measurements from the MeteoSwiss RAman Lidar for Meteorological Observing (RALMO) in Payerne, Switzerland , on 22 July 2017 corresponding to the 00:00 UTC GRUAN RS92 radiosonde launch.

Section 2 will outline the measurements used in the study. Sections 3 and 4 discuss the methodologies of the traditional and trajectory methods. Sections 5 and 6 will compare the new trajectory method with the traditional calibration technique and their respective uncertainties. Sections 7 and 8 will summarize the results and discuss their implications and the next steps forward.

2 Calibration measurements

The radiosonde water vapour mixing ratios are calculated using the GRUAN-corrected relative humidity profiles and the Hyland and Wexler 1983 formulae for the saturation vapour pressure . By convention, the relative humidity measurements are assumed to be over water for all altitudes. A total of 76 GRUAN RS92 night-time flights were initially used to conduct this analysis; however, due to clouds and lack of coincident lidar measurements, only 24 flights were used for calibration.

## 2.2 Lidar measurements

Lidar measurements in this study were made using RALMO. RALMO was built at the École Polytechnique Fédérale de Lausanne (EPFL) for operational meteorology, model validation, and climatological studies and is operated at the MeteoSwiss Station in Payerne, Switzerland (46.81 N, 6.94 E; 491 m a.s.l.). RALMO was designed to be an operational lidar and therefore was designed to have high accuracy, temporal measurement stability, and minimal altitude-based corrections . RALMO operates at 355 nm with a nominal pulse energy of 300 mJ and a repetition frequency of 30 Hz. Measurements are recorded for 1 min (1800 laser shots) with a 3.75 m height resolution from both the nitrogen (407 nm) and water vapour (387 nm) Raman scattering channels. RALMO runs day and night with an average of 50 % uptime from 2008 to 2017. RALMO downtime is due to the presence of fog, clouds below 800 m, or precipitation (40 %) as well as repairs or routine maintenance (10 %).

The lidar measurements are processed for calibration in several steps. First, we select ±2 h of 1 min lidar profiles around the launch time of the radiosonde. While 2 h was chosen as an arbitrary time range to allow for scan selection, in practice the method rarely selects scans more than 30 min before or after the launch. The 1 min scans are filtered to remove scans with high backgrounds above 0.01 photon counts per bin per second. We assume clouds are present if the nitrogen SNR is less than 1 at 13 km. If a cloud is present, the scan is masked and removed from the calibration. The calibration is conducted at the lidar's native altitude resolution in order to provide as many data points as possible and to avoid smoothing out small features.

The traditional method for calibrating water vapour lidars is done by integrating a fixed number of lidar profiles as a function of height starting at a time which is coincident with the radiosonde launch and then calculating a linear weighted least-squares fit between the radiosonde and lidar measurements to determine the calibration constant . The altitudes over which the fit is conducted are either fixed (e.g. always 1–5 km), or the optimal altitude region may be determined by calculating the correlation between the radiosonde and the lidar measurements . For the purposes of this paper, we refer to the traditional method as using 30 min of integration with a weighted least-squares fit over altitudes determined by the correlation coefficient which minimizes the variance of the fit's residuals.

## 3.1 Calculation of the water vapour mixing ratio for RALMO measurements

The water vapour mixing ratio (w) for RALMO is calculated from the saturation- and background-corrected lidar signals using the water vapour Raman lidar equation :

$\begin{array}{}\text{(1)}& w\left(z\right)={C}_{\mathrm{w}}\frac{{N}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(z\right)}{{N}_{{\mathrm{N}}_{\mathrm{2}}}\left(z\right)}\frac{{\mathrm{\Gamma }}_{{\mathrm{N}}_{\mathrm{2}}}\left(z\right)}{{\mathrm{\Gamma }}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(z\right)},\end{array}$

where ${N}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O},{\mathrm{N}}_{\mathrm{2}}}\left(z\right)$ is the background- and saturation-corrected water vapour and nitrogen photon signals as a function of altitude (z), and ${\mathrm{\Gamma }}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O},{\mathrm{N}}_{\mathrm{2}}}\left(z\right)$ is the total Raman-backscatter transmissions for the water vapour and nitrogen channels, including molecular and particulate scattering. The molecular transmission values are calculated using the GRUAN-corrected temperature and pressure profiles from the corresponding radiosonde, and the Rayleigh cross sections are determined using the formulae from .

discussed the necessity of accounting for aerosol transmissions, as the presence of aerosols can create uncertainties in the lidar profiles of up to 4 %, depending on the aerosol load. Therefore, to minimize this effect on our calibration constants, we have calculated the aerosol extinctions using the RALMO backscatter ratio product, which is calculated by taking the ratio of the elastic backscatter signal to the sum of the pure rotational Raman signals (Whiteman2003). Similarly to the method followed in , we calculate the extinction profile (αaer(z)) using the following equation:

$\begin{array}{}\text{(2)}& {\mathit{\alpha }}_{\mathrm{aer}}\left(z\right)=\mathrm{LR}\left(z\right)\left({\mathit{\beta }}_{\mathrm{mol}}\left(z\right)\left(\mathrm{BSR}\left(z\right)-\mathrm{1}\right)\right),\end{array}$

where LR(z) is the assumed lidar ratio profile, βmol(z) is the molecular backscatter profile taken from the NCEP model, and BSR(z) is the backscatter ratio profile. The lidar ratio profile is a step function with a constant value of 50 in the boundary layer and 20 in the free troposphere. The height of the boundary layer is estimated using the backscatter ratio profile. The assumed lidar ratios are climatological values which have been based on the typical aerosols detected using the co-located Precision Filter Radiometer (PFR). The aerosol transmissions for water vapour and nitrogen were calculated using the following:

$\begin{array}{}\text{(3)}& {\mathrm{\Gamma }}_{\mathrm{aer},X}={e}^{-\mathit{\tau }}=\mathrm{exp}\left[-{\left(\frac{{\mathit{\lambda }}_{X}}{{\mathit{\lambda }}_{\mathrm{0}}}\right)}^{A}\int {\mathit{\alpha }}_{\mathrm{aer}}\left(z\right)\mathrm{d}z\right],\end{array}$

where Γaer,X(z) is the aerosol transmission profile for a given molecule X (e.g. N2 or H2O), and the optical depth is τX. The wavelength for a particular channel is λX, while λ0 is the reference extinction profile, which is 354.7 nm for the elastic channel. The Ångström exponent, A, is assumed constant with altitude. The Ångström exponent is also measured during the daytime using the co-located PFR. However, it is not calculated daily as it requires stable, cloud-free conditions to get an accurate calculation. Since it is not always available, we fit the sum of a 6- and 12-month sinusoid to the Ångström exponent time series over measurements from 1 January 2012 until 31 December 2015, with 2014 removed due to a faulty sensor. The fitted sinusoid was then used as the values for the Ångström exponents. The standard deviation of the residuals was ±0.34 and was used as the uncertainty for the Ångström exponents. The uncertainty in the calibration due to our assumptions of the aerosol extinction and the Ångström exponent is 0.1 % and 0.4 %, respectively, and is included in the uncertainty budget for the calibration constant discussed in Sect. 6.

RALMO uses a polychromator with a bandpass of 0.3 nm . The central wavelengths of RALMO's water vapour and nitrogen channels were chosen to minimize temperature dependence of the Raman cross section. showed that the nitrogen channel had a relative change in transmitted intensity of 0.4 % per 100 K, and the water vapour channel intensity changed by roughly 1 % when varied between −60 and +40C.

The calibration constant Cw is defined as

$\begin{array}{}\text{(4)}& {C}_{\mathrm{w}}=\mathrm{0.781}\frac{{M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{M}_{\mathrm{Air}}}\frac{{\mathit{\eta }}_{{\mathrm{N}}_{\mathrm{2}}}}{{\mathit{\eta }}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}\frac{{O}_{{\mathrm{N}}_{\mathrm{2}}\left(z\right)}}{{O}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(z\right)}\frac{{\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}}}\left(T\left(z\right)\right)}{{\mathit{\sigma }}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(T\left(z\right)\right)}\frac{{F}_{{\mathrm{N}}_{\mathrm{2}}}\left(T\left(z\right)\right)}{{F}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(T\left(z\right)\right)}\end{array}$

(Whiteman2003).

The calibration constant contains all scaling constants and unknown factors, such as the fraction of nitrogen molecules in air, 0.781, the molecular weights of water and dry air (${M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O},\mathrm{Air}}$), the system efficiency of the nitrogen and water vapour channels (${\mathit{\eta }}_{{\mathrm{N}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$), the overlap function for both channels (${O}_{{\mathrm{N}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(z\right)$), the Raman cross section for each molecular species (${\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(T\left(z\right)\right)$), and the temperature dependency of the Raman cross section (${F}_{{\mathrm{N}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\left(T\left(z\right)\right)$). In RALMO's case, the differential overlap is designed to be unity .

## 3.2 Correlated and weighted least-squares fitting of the lidar water vapour mixing ratio to the radiosonde water vapour mixing ratio

After calculating the uncalibrated mixing ratio profiles from the ratio of the two lidar signals (Sect. 2.2), we use a correlated and weighted least-squares fit to normalize the lidar profile to the radiosonde and find the calibration constant . The radiosonde relative humidity profile is transformed into water vapour volume mixing ratio using the standard WMO conversion of Hyland and Wexler 1983 saturation vapour pressure formulae . The calibration range extends from the surface (491 m a.s.l.) to roughly 7 km a.s.l. depending on the profile. The bottom limit is the first lidar altitude bin at the surface, and the final calibration altitude is determined by the SNR and integration limits we impose. We remove scans at all altitudes where the trajectory spent less than 5 min in the lidar region due to their SNR values being less than 2. This cut-off typically results in the calibration region extending to 7 to 8 km altitude. To ensure that the calibration constant is not biased by a vertical displacement of the air parcel between the lidar and the radiosonde volume, we require the resulting uncalibrated lidar and the radiosonde mixing ratio profile to have a correlation coefficient which minimizes the variance of the fit's residuals and must be higher than 0.75. Several fits are made using correlation coefficient thresholds between 0.75 and 0.9, and the fit with the minimum variance in the residuals is chosen for the final calibration constant.

A moving window of 300 m is run over both the radiosonde and lidar profile, and the cross correlation between the two profiles inside each window is determined. To reduce the effect of noise on the cross correlation, both profiles are smoothed beforehand with a boxcar filter of 101.5 m width. In less than one-third of the cases, when the radiosonde leaves the lidar region early, or the wind is such that the air is spending less than 5 min in the lidar region, a large portion of the profile may be cut off. If the correlation between the radiosonde and lidar mixing ratios within each window is higher than the correlation threshold, then that window's altitude range is accepted for calibration . If there is less than 900 m of data available for calibration at the end of the correlation process, then we do not use that night for calibration as it does not have enough data with which to accurately calibrate. This criterion caused 2 out of the 76 nights to be rejected. While the correlation is calculated on the smoothed profiles, the fit is done by using the native resolution of the lidar inside the accepted calibration windows with a requirement of at least 243 points. The least-squares fit is conducted over all of the points selected in the cross-correlation procedure. Each fitting point is weighted by the inverse of the sum of the variances of the water vapour mixing ratio percent uncertainty and the average radiosonde mixing ratio uncertainty. The lidar water vapour mixing ratio statistical uncertainty (σw,stat) is propagated from the water vapour and nitrogen channel statistical uncertainties using Eq. (1) (Melfi1972). The radiosonde mixing ratio percent uncertainties (σMR,Radiosonde) are calculated using the total relative humidity, temperature, and pressure uncertainties reported for each GRUAN flight and propagating through the equations for saturation vapour pressure to calculate mixing ratio while assuming the relative humidity to be over water . However, the GRUAN processing occasionally does not report pressure uncertainties below 15 km. Therefore, it was necessary to create a nightly average pressure uncertainty profile which was used on nights for which pressure uncertainties were not reported. The variation in the pressure uncertainties was on the order of 0.01 %; therefore this assumption is justified. The calibration constant is then determined by using a one-parameter weighted least-squares fit of the form shown in Eq. (1).

The radiosonde trajectory method begins with the same procedure as the traditional method, in which each of the scans is filtered for clouds or abnormally large background levels, as is discussed in Sect. 2.2. However, instead of choosing the first 30 scans after the radiosonde launch, the scans are chosen based on the radiosonde's movement with respect to the air mass and the wind direction measured by the radiosonde.

First, we use the latitude and longitude of the radiosonde, as calculated by the onboard GPS system, as the initial position for air parcel tracking. The air parcel is then tracked backwards from the radiosonde position and is assumed to have travelled in a straight line. We then transform the coordinates onto a Euclidean grid, with the lidar located at the origin using the local flat Earth approximation, which is appropriate when distances are shorter than 20 km (Daidzic2017; Smart1977). We do not consider the vertical movement of the air parcel in this method. Users may need to consider other distance conversion methods, such as the haversine conversion (Smart1977), if utilizing radiosondes which are not launched from the same site as the lidar station. Additionally, it may be more appropriate in such cases to use a wind field measurement or model to conduct the trajectory calculation.

RALMO's field of view projects to a circle of approximately 1 m diameter at 5 km altitude, an area too small for most trajectories to pass directly through. Therefore, it was necessary to construct a region of assumed horizontal homogeneity in which the water vapour mixing ratio is constant. In order to maintain significant lidar SNR, we defined the homogeneous region, hereafter called the “homogeneous lidar region”, to be a circle around the lidar. In order to maintain a SNR in the water vapour channel greater than 2 above 7 km altitude for the majority of the cases, we defined the homogeneous lidar region to be a circle of 3 km radius centred around the lidar. The size of the homogeneous region was chosen by varying the radius in a range of 1–25 km and finally increasing it to infinity. Radii below 3 km resulted in SNRs smaller than 2 below 7 km and in some cases halved the SNR of the water vapour channel at altitudes below 5 km, which decreased the altitude coverage for the calibration and increased the noise in the primary calibration region. While radii above 3 km resulted in SNRs larger than 2 above 7 km, the water vapour profiles started to exhibit biases due to using integration times that are too long at certain altitudes and losing small features which had previously been visible. The 3 km radius provided the most altitude coverage with profiles closest to the radiosonde measurements and is the best compromise.

Figure 1Trajectory calculation and scan selection hypothetical example. The purple circle around the lidar has a 3 km radius and represents the region in which we assume the humidity field is horizontally homogeneous. The green dot is the radiosonde position, the purple dot is the lidar position, and the red arrow is the air parcel trajectory. The variable z refers to altitude, t1 is the entry time, and t2 is the exit time from the 3 km radius. The integration time, t(z), is the total time that the air parcel spends inside the homogeneous region. When the air parcel trajectory does not intersect with the circle, then no data are available for calibration.

Figure 1 shows how air parcels will always be “seen” by the lidar if the radiosonde remains inside the 3 km radius, whereas any air measured outside the radius may not intersect with the lidar region. If the trajectories do not enter the region, we do not use these altitudes for calibration. The entry and exit times from the homogeneous region mark the first and final scans used to calculate the lidar water vapour mixing ratios, with a maximum of 30 min of integration in order to accurately compare with the traditional technique, which uses a standard 30 min summation across all altitudes . The standard 30 min integration is the average time it takes a radiosonde to reach the tropopause at mid-latitudes and therefore generally covers the primary calibration altitudes. If the total time spent inside the homogeneous region exceeds 30 min, we take ±15 min around the time of closest approach to the lidar. The variation of the integration length with altitude is shown in Fig. 2. The integration time will decrease with altitude for two reasons: the wind speeds increase with altitude, and the air parcel trajectories may intersect with the outer edges of the homogeneous region and are therefore inside for shorter time spans (Fig. 1). The decrease in integration time with altitude will also change depending on the rate at which the radiosonde moves away from the lidar. The majority of nights had integration times less than 5 min above 7 km. However, if the wind is strong at a particular altitude, sharp decreases in integration times may be seen as the radiosonde moves quickly away. It is also possible to see the integration times decrease and then increase again as the radiosonde drifts in and out of the homogeneous lidar region.

Figure 2Example integration times from 22 July 2015. The lidar water vapour integration period is determined by the length of time the air parcels spend inside the homogeneous region. The integration time will decrease with altitude due to higher wind speeds. The maximum integration time is 30 min, in order to properly compare with the traditional analysis.

Once the appropriate scans have been chosen by the trajectory analysis, they are integrated to form the raw water vapour and nitrogen profile. The same procedure as in the traditional method (Sect. 3.1 and 3.2) is then followed to calculate the ratio of the two channels' profiles, find the appropriate calibration regions, and derive the calibration constant. The final calibrated water vapour profile for 22 July 2015 is shown in Fig. 3. The correlation algorithm selected 84 % of the profile above 1.5 km to use for the calibration, while regions with high variability were excluded from the calibration. The calibrated profile closely follows the radiosonde profile, with differences fluctuating between 5 % and 20 % over all altitudes. The uncertainty of the slope from the weighted fit is the uncertainty in the calibration constant due to measurement noise. The accuracy to which we know the calibration constant will be discussed further in Sect. 6.

Figure 3(a) The final trajectory-calibrated profile for 22 July 2015. The lidar profile is in black, and the radiosonde is in red. The correlation calibration regions are shown by the overlaid green points. Panel (b) shows the least-squares fit of the green points in (a). The uncertainty of the calibration constant is the standard error of the slope calculated from the weighted least-squares fit.

For clarity on the similarities and differences between the traditional and trajectory methods, a flow chart of the calibration process for both methods is shown in Fig. 4. The main difference is the selection of the appropriate scans for calibration at the beginning. Both methods use the same process for choosing the appropriate calibration regions and calculating the calibration constant.

Figure 4Flowchart of the steps to calibrate the RALMO lidar by the trajectory method (left) and the traditional method (right).

5 Comparing the traditional and trajectory methods

We applied the trajectory technique to 76 nights between January 2011 and December 2016, in which 31 were removed due to a lack of lidar measurements during the radiosonde launch window, primarily due to precipitation or routine maintenance. From the 45 remaining nights, the trajectory calibration and traditional method automatically removed 8 nights due to abnormally high background values above 0.01 counts per bin per second. An additional 13 nights were removed from both the trajectory and traditional calibrations due to low signal-to-noise levels (below 1 SNR) and the presence of clouds. The filtering process removed all of the night-time flights from 2008 to 2011 due to significant cloud cover coincident with the radiosonde launch. A final list of the nights with their calibration constants is shown in Table 1.

While comparing the calibration constants from the two methods, it became apparent that we could separate them into two groups when observing the water vapour mass mixing ratio contours over the course of the calibration period. One set of nights exhibited water vapour fields which were horizontally homogeneous around the lidar over the course of the 30 min calibration period and were thereby dubbed “homogeneous” nights. The second set of nights showed movement of water vapour layers over 100 m in altitude over the course of 30 min and were called “heterogeneous” nights. Table 1 has been divided into the two categories and shows the calibration constants for each night and calibration technique.

Table 1A comparison of the calibration constants of all nights used in this study. The table is split into two sections – homogeneous and heterogeneous nights. Column 1 is the date on which the radiosonde was launched (yyyy-mm-dd). Column 2 or Ctrad is the traditional calibration constant with its total percent uncertainty. Column 3 or Ctraj is the trajectory method calibration constant with its percent uncertainty. Column 4 is the absolute value of the difference between the two constants. Column 5 is the absolute value of the percent difference of the two constants with respect to the traditional calibration constant. Column 6 is for comments regarding the differences.

For the homogeneous nights, we hypothesized that if the water vapour field is stable for long periods of time and experiences very little change over the distance travelled by the radiosonde, then the radiosonde and the lidar should measure roughly similar water vapour content. Therefore, we should see small differences between the traditional and trajectory methods' calibration constants. While both methods should produce similar profiles and calibration constants on homogeneous nights, the two may not share the same calibration constants due to using different lidar scans (Fig. 5). The traditional method uses all profiles from the radiosonde launch to 30 min after launch, which are all scans inside the two dashed red lines in Fig. 5. The trajectory technique will choose the appropriate calibration scans based on each air parcel's trajectory and its position of closest approach shown by the scans between the magenta dots. Consequently, the trajectory method will not include measurements from altitudes where the air parcel trajectories do not intersect with the homogeneous lidar region.

Figure 5Lidar water vapour mixing ratio measurements on 5 June 2013 at 00:00 UTC. The time axis is measured relative to the radiosonde launch. The traditional method uses all scans between the two dashed red lines. The trajectory method uses all measurements between the magenta dots. The white “x” markers show the height of the radiosonde with time.

When the water vapour field is horizontally heterogeneous, meaning water vapour layers moved over 100 m in altitude over the course of the 30 min traditional calibration period, the trajectory method should better represent the air sampled by the radiosonde than the traditional technique. Layers on the order of several hundred metres' thickness can change in altitude over this period, resulting in water vapour mixing ratios changing over 30 % at a given height.

Similarly to the homogeneous nights, a subset of the heterogeneous nights is shown in Fig. 7. The contour of the percent difference from the mean water vapour profile for each night shows water vapour layers which change rapidly over the course of the 30 min calibration period (column 1 of Fig. 7). These rapid changes produce large differences in the radiosonde and lidar mixing ratio profiles if the movement of the radiosonde with respect to the air mass is not taken into account. These differences can be on the order of 15 %–20 %, as is shown in the second column of Fig. 7, particularly on the night of 22 July 2015. Large differences on this night on the order of ±10 % are reduced to less than 5 % by the trajectory method (third column of Fig. 7). Both methods on that night produce sharp differences at 3.8 km due to the sharp change in water vapour content. Above 4 km there is a constant bias between the radiosonde and the lidar in the traditional method of 10 %, which is reduced to 5 % in the trajectory technique. On 27 July 2012 there are large difference features present throughout the entire percent difference profile for the traditional method on the order of 10 %. These are similarly reduced to less than 5 %, with the exception of the larger spike at 1.5 km caused by the sharp change in water vapour concentration. The night of 25 May 2012 shows less variation than the other two nights but does have large differences at 4.2 and 2 km. The higher feature is reduced from −25 % difference to 10 % by the trajectory method, while the percent difference in the lower feature changes from +20 % to +10 %.

Similarly to the homogeneous profiles, we do see an increase in noise at the higher altitudes of the trajectory method profiles. This is again caused by the large drift in the radiosonde's position, as well as wind speed. However, below 4 km we do see that the trajectory method reduces the differences between the radiosonde and the lidar by up to 15 %.

The differences between the calibration constants on the heterogeneous nights is larger than the homogeneous nights due to the difference in calibration regions (Table 1). The average difference in the calibration constants on heterogeneous nights is 1.82±1.02 from the traditional method calibration constant. One night out of the 11 in the heterogeneous nights showed very small differences in the calibration constant despite structural changes throughout the calibration period. This night used similar calibration regions that were also stable over the course of the calibration in both methods. If this night is not included, then the average difference becomes 1.92±0.93 %.

Figure 7A subset of the dates with largely heterogeneous conditions showing the differences between the traditional and trajectory calibration techniques. This figure follows the same format as Fig. 6 and shows that when the water vapour field changes over the 30 min traditional calibration period, the traditional water vapour profile can look significantly different from the radiosonde. The trajectory method produces a profile with a smaller percent difference with respect to the radiosonde.

The average and the standard deviation of all percent difference profiles with the radiosonde from the trajectory and traditional method profiles are shown in Fig. 8. The average trajectory bias oscillates around 1 %, but the variability increases above 4.5 km. This is due to the shorter integration times and smaller SNRs at higher altitudes (Fig. 8). The average traditional bias also oscillates around −0.7 %; however, the average profile deviates farther from the centre than the trajectory method (Fig. 8). The standard deviation of the ensemble of percent difference profiles between both calibration methods and the radiosonde shows that the trajectory method has 10 %–15 % less variability with respect to the radiosonde profile above 2 km. Below 2 km the traditional and trajectory methods produce similar profiles on average, with similar consistency. In summary, the trajectory method shows a similar absolute bias to the radiosonde but with the opposite sign compared to the traditional method. The variability of the differences between the lidar and the radiosonde is 10 %–15 % smaller in the trajectory method than it is in the traditional method between 2 and 4 km altitude but is the same below 2 km and above 4 km.

Figure 8(a) The average bias between the radiosonde and the trajectory method calibrated profiles at 25 m vertical resolution for both the trajectory (black) and traditional methods (red). Dotted lines are further smoothed results using a running average of 75 m for both methods. (b) The standard deviation of all trajectory percent difference profiles at 25 m resolution for both trajectory (black) and traditional (red) methods. Dashed lines are the further smoothed results by a running average of 75 m for both methods.

6 Lidar calibration uncertainties for trajectory and traditional methods

The standard practice for determining the uncertainty of the calibration constant has been to conduct extensive calibration campaigns and assume that the calibration value does not change over the campaign period and then measure the variability of the constant . The variability of the constant is then assumed to be the uncertainty, and the calibration constant is not changed until the next campaign when multiple radiosondes (or other reference calibration instruments) are available for calibration. The assumption that the calibration constant does not change over long periods of time introduces another source of uncertainty into water vapour measurements, which is often unknown until the next calibration period. Uncertainties calculated during a campaign period vary between 4 % and 5 % of the calibration constant during the calibration period but do not account for the individual sources of contribution, nor do they typically account for the variability in the calibration constant beyond the campaign period.

Accounting for drift or changes in the calibration constant and its uncertainty is extremely important for long-term trend analyses, since such a drift/change could easily be larger than the uncertainty of the calculated trend . Many systems have now taken this into account by conducting daily or semi-daily calibration measurements either using an internal, hybrid, or external calibration. Taking more frequent calibration measurements with uncertainties calculated for each calibration then turns a systematic uncertainty component of a trend analysis into a random uncertainty component, particularly if the uncertainty of the calibration constant is recalculated with each calibration.

Previous studies have shown that the largest uncertainty is typically the uncertainty of the reference instrument . It was not until recently that such detailed uncertainty budgets became available routinely for radiosonde measurements. The GRUAN radiosonde products are the first radiosonde profiles to have a published uncertainty budget for each measurement as a function of altitude . By using the GRUAN radiosonde product, we are now able to calculate the uncertainty in the calibration constant due to the radiosonde's uncertainties.

We investigated five major sources of uncertainty in the determination of the calibration constant for both methods: the lidar statistical, GRUAN radiosonde mixing ratio, dead time, aerosol extinction, and Ångström coefficient uncertainties. The uncertainty in the calibration constant, the lidar statistical uncertainties, and dead time were identified as the major sources of uncertainty in RALMO water vapour measurements by , who also retrieved aerosol extinction, Ångström exponents, and their associated uncertainties. In the traditional method, the mixing ratio requires the uncertainty due to aerosol extinction and the Ångström exponent to be evaluated, as discussed in and . The GRUAN radiosonde water vapour mixing ratio uncertainties were calculated using the reported GRUAN total uncertainties (combined statistical and systematic) for pressure, temperature, and relative humidity and by propagating through the Hyland and Wexler 1983 formula for saturation vapour pressure . We use an average pressure uncertainty profile calculated from all the nights when the pressure uncertainty is not reported for less than one-third of the nights. The radiosonde relative humidity uncertainties vary between 5 % and 10 % RH in the troposphere. The pressure uncertainties are on the order of 10−3 hPa in the troposphere, and the total temperature uncertainty varies between 0.1 and 0.3 K in the troposphere. The radiosonde mixing ratio uncertainties are linearly interpolated onto the lidar's 3.75 m resolution grid for the uncertainty determination.

The lidar mixing ratio statistical uncertainties are propagated through Eq. (1) using the random uncertainties from both the water vapour and nitrogen signals. The lidar statistical uncertainties from the trajectory method are smaller than the radiosonde uncertainties below 3 km but are larger than the radiosonde uncertainties, varying from 10 % to 20 % at and above 4 km from profile to profile.

Both the lidar statistical and radiosonde uncertainties were used as the weights for the least-squares fit performed in Sect. 3.2, defined by Eq. (5) .

$\begin{array}{}\text{(5)}& {C}_{\mathrm{w}}=\frac{{\sum }_{i=\mathrm{1}}^{K}\frac{{R}_{i}{L}_{i}}{{\mathit{\sigma }}_{i}^{\mathrm{2}}}}{{\sum }_{i=\mathrm{1}}^{K}\frac{{L}_{i}^{\mathrm{2}}}{{\mathit{\sigma }}_{i}^{\mathrm{2}}}},\end{array}$

where Cw is the calibration constant, K is the number of points used in the fit, Ri represents the radiosonde mixing ratio points used in the calibration, Li represents the saturation- and transmission-corrected ratio of water vapour and nitrogen signals, and σi represents the weights. Using the variance of the residuals of the least-squares fit, one can calculate the uncertainty in the fit, or “fitting uncertainty”. This fitting uncertainty is the result of the amount of photon counting noise in the lidar measurements and can be treated as the uncertainty in the calibration due to the lidar photon counting statistics. The fitting uncertainty is calculated using the standard equations for the slope of a line . The average trajectory method fitting uncertainty is 0.4 % of the average calibration constant. The average fitting uncertainty for the traditional method is 0.3 % of the average calibration constant. The traditional method has smaller fitting or statistical uncertainties than the trajectory method due to the larger number of scans used per altitude, on average, compared to the trajectory method. The fitting uncertainty does not encompass the entire uncertainty of the calibration constant, since it is due only to the photon counting noise.

The calibration of a lidar using a radiosonde is limited primarily by the accuracy of the radiosonde measurement. The uncertainty of the water vapour calibration constant due to the lidar's random uncertainty and the radiosonde's total uncertainty (both systematic and random) was determined using the uncertainty propagation in Eq. (6) (JCGM2008).

$\begin{array}{}\text{(6)}& {U}_{{C}_{\mathrm{w}}}=\sqrt{{\mathrm{\Sigma }}_{n=\mathrm{1}}^{N}{\mathrm{\Sigma }}_{m=\mathrm{1}}^{N}\frac{\partial {C}_{\mathrm{w}}}{\partial {\mathbit{X}}_{n}}\frac{\partial {C}_{\mathrm{w}}}{\partial {\mathbit{X}}_{m}}\mathrm{cov}\left({\mathbit{X}}_{n},{\mathbit{X}}_{m}\right)},\end{array}$

where X is the measurement vector including both the radiosonde and lidar measurements (e.g. $\mathbit{X}=\left[{L}_{i},\mathrm{\dots }Ri,\mathrm{\dots }\right]$) used to calculate the calibration constant from Eq. (5) with length N=2K. We make several assumptions in Eq. (6).

First, by definition, the covariance of a radiosonde or lidar measurement uncertainty with itself is simply the variance. Second, we assume that the lidar photon counting uncertainties are uncorrelated with each other. Third, we assume that the radiosonde measurement uncertainties are uncorrelated with lidar measurement uncertainties. Lastly, we assume that the radiosonde measurement uncertainties are correlated with each other with a correlation coefficient of r=1. Choosing r equal to unity implies that we are assuming complete correlation and therefore the maximum possible uncertainty. With these assumptions, Eq. (6) becomes

$\begin{array}{}\text{(7)}& \begin{array}{rl}& {U}_{{C}_{\mathrm{w}}}=\\ & \sqrt{{\mathrm{\Sigma }}_{i=\mathrm{1}}^{K}{\left(\frac{\partial {C}_{\mathrm{w}}}{\partial {R}_{i}}\right)}^{\mathrm{2}}{U}_{R}^{\mathrm{2}}+{\mathrm{\Sigma }}_{i=\mathrm{1}}^{K}{\left(\frac{\partial {C}_{\mathrm{w}}}{\partial {L}_{i}}\right)}^{\mathrm{2}}{U}_{L}^{\mathrm{2}}+\mathrm{2}{\mathrm{\Sigma }}_{i=\mathrm{1}}^{K-\mathrm{1}}{\mathrm{\Sigma }}_{j=i+\mathrm{1}}^{K}\frac{\partial {C}_{\mathrm{w}}}{\partial {R}_{i}}\frac{\partial {C}_{\mathrm{w}}}{\partial {R}_{i+\mathrm{1}}}{r}_{ij}{U}_{i}{U}_{j}}\end{array},\end{array}$

where UL,R represents the corresponding lidar and radiosonde mixing ratio uncertainties, and Ui,j represents the uncertainties corresponding to the measurement vector X. The derivatives are calculated from Eq. (5). Note that the second term in Eq. (7) is the uncertainty due to the lidar's photon counting uncertainty. This term is the same as the fitting uncertainty discussed in the previous paragraph, and the values agree with each other within a tenth of a percent. The combined uncertainty in the calibration constant due to the radiosonde and lidar uncertainties is an average of 4 % for both the trajectory and traditional techniques with signal levels below 15 MHz, which is to be expected since the traditional technique tends to sample the same volume of air as the trajectory as shown in Fig. 5.

The dead time uncertainty can be large for RALMO, particularly during the daytime. Thus, Eq. (6) must be modified to account for this contribution when it is present. The dead time uncertainty is propagated through Eq. (1) assuming a non-paralyzable system and using Eq. (6). For RALMO, we assume a dead time uncertainty (Uγ) of 5 % or 0.2 ns, which was the standard deviation of the retrieved dead times for all of these nights when using the optimal estimation method of . The average calibration constant uncertainty due to dead time uncertainty is then 0.3 % of the calibration value for both the trajectory and traditional techniques, about equal to the fitting uncertainty.

The uncertainty in the calibration constant due to the uncertainty in the extinction profile is calculated using Eq. (6). The uncertainties for the extinction were assumed to be 100 % to determine an upper-limit uncertainty contribution. However, the derivatives of the calibration constant with respect to the individual extinction values were so small that the uncertainty contribution from the extinction was consistently less than 0.01 % for all cases. The larger uncertainty component in the extinction is the calibration uncertainty due to the assumption of the Ångström exponent. The uncertainties in the Ångström exponent values were estimated by detrending the time series of these measurements from 2011 to 2015 using a summation of a 6- and 12-month sinusoid to the Ångström exponent measurements. The standard deviation of the fit's residuals was 0.34. The uncertainty in the calibration constant due to the uncertainty in the Ångström exponent was then calculated to be 0.4±0.5 %. While on average it is only an order of magnitude larger than the extinction uncertainty component, the Ångström exponent contributes more to the uncertainty when more aerosols are present. The maximum contribution from the Ångström exponent was 1.8 % on 23 March 2016 due to the presence of a stronger aerosol layer. The rest of the nights had either no aerosols present or weakly interacting aerosol layers, resulting in lower uncertainty contributions from the Ångström exponent.

Another possible contributor to the total calibration constant uncertainty is the overlap function. RALMO is designed to have no differential overlap in the water vapour and nitrogen channels and an overlap ratio between the nitrogen and water vapour signals of unity. However, a small differential overlap could result from chromatic aberration from the protective windows and edge filters . The average total uncertainty of both the trajectory and traditional calibration constant is 4.5 %, with the majority from the radiosonde's contribution.

We also compared the average of the nightly calibration results to the standard deviation of the entire 24-night RALMO calibration time series used in this study. The RALMO system is known to have differential ageing of its photomultipliers, which causes the calibration to drift . A linear fit was made to the calibration time series and then removed to calculate the standard deviation of the calibration over 6 years. The standard deviations for both the traditional and trajectory time series were 4.5 %, thereby agreeing with the average nightly uncertainty. The results from calculating the standard deviation of the time series show that the typical methods used in calibration campaigns will generally give the same result as taking the average uncertainty of the individual uncertainties. However, we would suggest that taking the individual uncertainties is a better approach for long-term analysis and maintaining consistency throughout a time series of measurements.

7 Summary

We have presented a new method, using GRUAN-corrected radiosondes, to calibrate Raman-scattering water vapour lidar systems that incorporates geophysical variability into the determination of the calibration constant. The trajectory method tracks the air parcels measured by the radiosonde and matches them with the appropriate lidar measurement time; thus, the integration time varies with height. We compared this method to the traditional lidar calibration technique, whereby we sum 30 min of lidar measurements and fit them to a radiosonde profile.

The difference between the traditional and trajectory method calibration coefficients is due to the difference in 1 min lidar scans selected by the methods, as well as the difference in correlation regions used to determine the calibration coefficient from these profiles. We found that when the water vapour field is homogeneous, the traditional method and trajectory method profiles will produce similar profiles, with slight differences due to the correlation regions included. The homogeneous nights had an average difference of 0.4 % from the traditional calibration constant value. In contrast, the heterogeneous nights, or nights with significant structural changes over the 30 min traditional calibration period, had an average difference of 2 % with respect to the traditional constant. We have also shown that using trajectories to track the air sampled by the radiosonde more accurately reproduces the radiosonde profile when the water vapour field is variable and decreases the percent difference between the lidar and radiosonde measurements by 5 %–10 %. In summary, we found the following:

1. The traditional and trajectory methods agree when the water vapour field is homogeneous during the radiosonde flight. The average difference between their calibration constants (when not considering the single outlier) was 0.43±0.21 %.

2. The trajectory method provides a better fit with the radiosonde when the water vapour field changes appreciably over the time of the radiosonde flight. For these cases the calibration constants calculated by the trajectory method resulted in an average of 1.92±0.93 % difference with the traditional method calibration constants.

3. The trajectory method produces a smaller average bias between the radiosonde and the lidar than the traditional method between 2 and 4 km (Fig. 8). Adding points above 4 km does not change the calibration constant significantly as the photon counting uncertainty becomes large at these altitudes.

4. The combined lidar statistical and radiosonde mixing ratio uncertainties contribute an average of 4.5 % uncertainty in the calibration constant determination for both calibration methods, for which the radiosonde mixing ratio uncertainty is the dominating factor.

5. The uncertainty in the calibration coefficient due to the uncertainty in dead time contributes an average of 0.3 % in the calibration coefficient for a 5 % dead time uncertainty.

6. The uncertainty in the calibration constant due to the uncertainty in the extinction is less than 0.01 %. The uncertainty in the calibration constant due to the Ångström exponent's uncertainty is larger and is on average 0.4 % but can reach higher than 1 % when strongly attenuating aerosol layers are present.

7. The average statistical uncertainty in the calibration constants produced by the trajectory technique is 0.4 %, as opposed to the traditional method uncertainty of 0.3 %. However, the fitting uncertainty is negligible relative to the uncertainty of the calibration constant due to the uncertainty of the radiosonde measurements.

8. The uncertainties calculated by the standard deviation of the trajectory and traditional method time series were both 4.5 %, which is consistent with the total uncertainties calculated using Eq. (6).

A summary of the uncertainty components for both methods in the calibration constant is shown below in Table 2.

Table 2Components of the calibration uncertainty, their inherent uncertainty, and their contribution to the uncertainty of the calibration constant for both the trajectory and traditional methods. The uncertainty contributions are the same for both methods since the only difference between the two methods is the selection of lidar measurements.

8 Discussion and conclusions

The trajectory calibration technique attempts to more realistically represent the physical processes taking place during a radiosonde–lidar calibration, by ensuring the radiosonde and lidar sample the same air mass. This tracking method was built upon the methods suggested in , , , and . Similarly to the techniques discussed in these studies, we match the measurements at each altitude with the radiosonde. However, assumed a horizontally homogeneous and uniformly translating atmosphere and did not consider varying wind speed and direction. In the integration time was varied with altitude in order to keep the random uncertainty below 10 %; however, the position of the air parcels was not considered. This technique was ultimately found to be not as accurate as other methods and was later improved upon using the correlation comparisons in . Our method does not assume a uniformly translating atmosphere; however, we do consider a homogeneous region around the lidar, and the integration time is varied as a function of the time the air parcels spend inside the homogeneous region. used four methods to match the radiosonde and the lidar measurements: (1) no matching, summing 2 h of lidar profiles; (2) using all lidar scans before the radiosonde reaches 10 km, about 30 min of scans, similar to our traditional method; (3) only using altitudes with minimum water vapour variability over 2 h to calibrate, and (4) only using scans which were coincident with the radiosonde altitude – similar to the “track” technique and our trajectory method. However, method 4 did not track the air parcels as we did. found that the second method provided the smallest variation in their calibration constant but did mention that the other methods produced very close results and could be used as well.

Another way to attempt to correct for the movement of the air mass or radiosonde is to follow the methods in and which look for regions of high correlation as in these regions it is more likely the radiosonde and lidar are sampling the same air mass. The traditional method, using the correlation algorithm, does provide similar calibration constants to the trajectory method on homogeneous nights. However, using the combined correlation algorithm with the trajectory tracking can provide more regions of high correlation for the calibration, particularly on heterogeneous nights when the air mass changes rapidly as it passes over the lidar. In some cases, of course, it may not provide more regions for calibration particularly if the wind speeds are high and the radiosonde quickly leaves the 3 km homogeneous lidar region.

Future studies using this technique could possibly show an improvement using a cone instead of a cylinder for the homogeneous lidar region. The trajectory method works better between 2 and 4 km but worse than the traditional method above. Using a cone could increase the integration time for the higher altitudes where fewer trajectories tend to intersect the homogeneous lidar region cylinder, improving the comparison. We initially tested a cone and set its radius using the horizontal correlation lengths of water vapour using the wind speeds measured by the radiosonde. However, this scheme produced a cone which was too variable in size to be useful for calibration. We explored using a cone the same size as the lidar's field of view, but it allowed so few trajectories to intersect the cone that no calibration could be performed. Varying the cylinder size did not significantly change the shape of the profile above 4 km but reduced the noise. Reducing the noise of the profile is important; however, as the lidar measurements only contribute on the order of 0.1 % of the uncertainty to the calibration constant, it would not provide much benefit for RALMO. Using a cone could prove advantageous for other sites which exhibit higher wind speeds than Payerne. However, for sites which use radiosondes which are not co-located with the lidar, it would not be as beneficial as a cylinder. We would encourage others who might implement this method to try a cone to see if it significantly improves their results.

A significant new aspect of our study is using calibrated GRUAN radiosondes whose analysis includes a complete uncertainty budget. The full uncertainty budget shows the radiosonde measurement is the dominant uncertainty source as compared to the uncertainty in the regression line on an individual night derived from the uncalibrated lidar measurements and sonde. The uncertainty in the lidar measurements, the dead time, and the extinction components contribute uncertainty that is an order of magnitude smaller than the radiosonde. However, the Ångström exponent can contribute uncertainty on the same order of magnitude as the radiosonde if there are strongly interacting aerosols present during the calibration. Using the GRUAN sondes allows a calibration to be determined with a full uncertainty budget on an individual night, as opposed to requiring a times series of nights to calculate a statistical calibration variation. The uncertainties in our calibration determinations could be reduced using the hybrid method of , as a refinement of our method would be to combine the trajectory calibration with an internal lamp source or some sort of internal calibration technique which could further reduce the variation in the calibration over time.

Eleven of the calibration nights in this study showed significant structural variations in water vapour over the 30 min traditional calibration period. These nights had an average of 2 % difference in the calibration constant, which is less than the average calibration uncertainty of 4.5 %. Therefore, the trajectory and traditional methods do not produce statistically different calibration values. However, the trajectory method does more accurately reproduce the radiosonde profile than the traditional method between 2 and 4 km, and above 4 km the methods do equally well (Fig. 8). The water vapour content below 4 km for the nights in this study was an average of 87 % of the total content measured by the radiosonde. Therefore, we believe that calibration should be limited to below 4 km where the signal is highest and the trajectory method performs best. Additionally, the points above 4 km do not make a significant difference in the calibration factor obtained.

RALMO has had an average of 50 % uptime over the last 10 years, making it an ideal database for the detection of water vapour trends in the free troposphere. In addition to frequent measurements, trend analyses also require minimal uncertainty and well-characterized retrievals. The aim of this work was to develop a calibration method that characterized the uncertainty of the calibration constant as well as making sure it was physically consistent with the reference instrument. The trajectory calibration technique will be used in conjunction with an internal calibration method to produce a 10-year water vapour climatology and upper troposphere–lower stratosphere (UTLS) trend analysis using RALMO measurements.

Data availability
Data availability.

All GRUAN data are accessible at http://www.gruan.org/ (last access: 4 April 2019), and access may be requested through them. MeteoSwiss lidar data may be requested by contacting Alexander Haefele (alexander.haefele@meteoswiss.ch).

Author contributions
Author contributions.

SHJ was responsible for developing the trajectory method technique code, applying the technique to the GRUAN radiosondes and RALMO measurements, comparing the method results to the traditional method, deriving and calculating the uncertainty budget for each calibration, and preparing the manuscript. This work will be used as part of her doctoral thesis. RJS was responsible for supervision of the doctoral thesis and made contributions to manuscript preparation. AH had the original idea for a trajectory method, was also responsible for supervision of the thesis, and helped with manuscript preparation. GM helped with manuscript preparation and provided RALMO and radiosonde data from MeteoSwiss as well as helpful guidance on the workings of the MeteoSwiss lidar code.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We would like to thank the GRUAN support team for providing the GRUAN-processed radiosonde measurements. Shannon Hicks-Jalali would also like to thank Ali Jalali who spent time reading the paper and providing helpful scientific discussions and suggestions throughout the entire process.

Financial support
Financial support.

This project has been funded in part by the National Science and Engineering Research Council of Canada through a Discovery Grant (Robert J. Sica) and a CREATE award for a Training Program in Arctic Atmospheric Science (Kimberly Strong, PI) and by MeteoSwiss (Switzerland).

Review statement
Review statement.

This paper was edited by Andreas Hofzumahaus and reviewed by David Whiteman and two anonymous referees.

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