Lidar retrievals of atmospheric temperature and water vapor mixing ratio profiles using the optimal estimation method (OEM) typically use a retrieval grid with a number of points larger than the number of pieces of independent information obtainable from the measurements. Consequently, retrieved geophysical quantities contain some information from their respective a priori values or profiles, which can affect the results in the higher altitudes of the temperature and water vapor profiles due to decreasing signal-to-noise ratios. The extent of this influence can be estimated using the retrieval's averaging kernels. The removal of formal a priori information from the retrieved profiles in the regions of prevailing a priori effects is desirable, particularly when these greatest heights are of interest for scientific studies. We demonstrate here that removal of a priori information from OEM retrievals is possible by repeating the retrieval on a coarser grid where the retrieval is stable even without the use of formal prior information. The averaging kernels of the fine-grid OEM retrieval are used to optimize the coarse retrieval grid. We demonstrate the adequacy of this method for the case of a large power-aperture Rayleigh scatter lidar nighttime temperature retrieval and for a Raman scatter lidar water vapor mixing ratio retrieval during both day and night.

Lidars have high temporal and spatial resolution compared to passive remote sensing instruments, coupled with high signal-to-noise (SNR) ratio measurements over much of their dynamic range and thus have averaging kernels close to unity for the majority of their retrievals, with a much finer grid spacing than passive instruments. At most retrieval altitudes, the majority of the information comes from the lidar measurements. However, near the top of the lidar retrieval range, and in other regions where the SNR is low, the a priori contribution to the retrieval increases and consequently the amount of information from the measurement decreases. The a priori influence at the top of the retrieval should be considered when comparing OEM lidar measurements, particularly if different a priori profiles are used.

An estimate of the measurements' contribution to the retrieval, otherwise known as the “measurement response”, can be calculated by taking the sum of the averaging kernel functions. The measurement response is calculated by multiplying the averaging kernel matrix,

An example of the a priori's influence is shown in Fig. 1 of

Distribution of the differences in temperatures retrieved at the altitudes where the sum of the averaging kernels (

The mean value of the histogram at the altitude where

As

Several methods for reducing the a priori's influence on the retrieval have been suggested by

We have used two lidars in this study, whose specifications are discussed in more detail in Sect. 2. Section 3 summarizes some fundamental material of the OEM which will be referenced throughout the paper. Section 4 discusses the a priori removal methodology with a simple example. The method is then applied in Sect. 5 for three cases: Raman water vapor daytime, Raman water vapor nighttime, and Rayleigh nightly temperature retrievals. Section 6 discusses the differences between our practical application and the method in vCG and some of the proposed method's advantages. Sections 7 and 8 are the Summary and Conclusions respectively.

Two lidars were used in this study, the Raman Lidar for Meteorological Observation (RALMO) in Payerne, Switzerland, and the Purple Crow Lidar (PCL) in London, Ontario. RALMO was used for the water vapor daytime and nighttime retrievals, and the PCL was used for the Rayleigh temperature retrievals.

RALMO is located at the MeteoSwiss research station in Payerne, Switzerland (

The Purple Crow Lidar is located at the Environmental Sciences Western Field Station (

The optimal estimation method (OEM) is an inverse method based on Bayesian statistics which calculates the maximum a posteriori solution by minimizing a cost function involving both the fit residual and the difference between the result and the a priori information. The measured signal

The OEM assumes Gaussian probability density functions (PDFs) to maximize the a posteriori probability of the atmospheric state, given the value of the measurements (

One of the advantages of the OEM is that, in addition to obtaining a retrieval/solution vector, the method also provides diagnostic tools and a full uncertainty budget. The primary diagnostic tool is the averaging kernel matrix (

A maximum likelihood (ML) solution is an inverse technique which does not
make use of a priori information and finds a solution which is solely based
on the measurement information. If a Gaussian probability distribution of
measurement errors is assumed, the maximum likelihood solution is the
solution which minimizes the squared covariance-weighted differences between
the measurements and the forward model (Eq.

We see that it is possible to arrive at the maximum likelihood solution mathematically through the OEM solution by setting

Our objective in this study is to find a practical method to remove the a priori information from the retrieval vector. We have based our work upon the methodology of vCG and have developed a quick and straightforward method to remove the a priori information from the lidar retrieval. vCG proposed removing the effect of the a priori information by using an information-centered grid approach. Each level of the retrieval on the information-centered grid contains one degree of freedom, and therefore the number of degrees of freedom of the signal is the same as the number of retrieval levels. In this condition, the formal a priori information can be removed without destabilizing the retrieval.

To create an information-centered grid that contains close to one degree of freedom per level requires the averaging kernel of the fine-grid retrieval. For a lidar, this is either the raw measurement spacing or a grid found by integrating some number of raw measurements into larger bins. Therefore, the first step is to run the OEM retrieval following the same procedures as in

To illustrate the method, we will give a simple example with the fine-grid levels, diagonal components of the averaging kernel matrix, and the cumulative trace of the averaging kernel, as shown in Table

A simple example for demonstrating the averaging kernel matrix's role in finding the coarse grid which resembles the typical structure of a lidar temperature retrieval averaging kernel. The first column is the retrieval level which is typically in units of altitude for lidar OEM retrievals. The second column is the elements along the diagonal of the averaging kernel matrix

We then use the triangular representation from vCG to create the information-centered grid using the fine-grid averaging kernel. First, the cumulative trace of the averaging kernel matrix is used to determine the amount of information needed for each grid point on the coarse grid using Eq. (

The coarse-grid levels are shown for the example case as a function of the cumulative trace of the averaging kernel matrix. The total degrees of freedom for the retrieval is 8.2, which is spread over the entire retrieval grid such that each point has roughly one degree of freedom. As the SNR of the measurements decreases, more fine-grid points are used in the coarse grid, and the distance between points generally increases with altitude.

The resulting coarse grid is then used as the retrieval grid for a second retrieval run. In this paper we will refer to a “run” as one retrieval which typically requires 10 iterations to converge to a solution. However, before running the retrieval again we remove the regularization term in Eq. (

We now apply our information-centered approach, using the triangular representation from vCG, to lidar OEM retrievals in order to minimize the effect of the a priori information. We will examine the method's effectiveness with RALMO daytime and nighttime water vapor retrievals, as well as with a PCL Rayleigh temperature retrieval. This method is also applicable in general and can be applied to other lidar retrievals. First, we will discuss the results from the triangular representation and the creation of the coarse grid and how it is used as the new retrieval grid. Then we will discuss its effect on the retrieval, vertical resolution, uncertainty budgets, and averaging kernel for a case study for each type of retrieval. We will then discuss the results of the method using representative data sets for all water vapor and temperature retrievals.

The daytime water vapor case study retrieval is a 30 min integration obtained in conjunction with a Vaisala RS92 radiosonde launch from the Payerne station on 22 January 2013 at 12:00 UT. This date was chosen because it shows the large impact our method has on low signal-to-noise ratios, which occur during the daytime due to the high solar background or in dry layers (regions with relative humidities less than 25 %). The input data grid for this case was binned to 50 m to remove numerical features in the retrieval due to the high background noise levels.

The diagonal values of the daytime case fine-grid averaging kernels (Fig.

The clear-sky daytime water vapor averaging kernel matrix for 22 January 2013 at 12:00 UT

The vertical resolution profile on 22 January 2013 12:00 UT. The vertical resolution will decrease on the coarse grid as the points are used to reach one degree of freedom. The last two points have vertical resolutions of several hundred meters but are not considered meaningful points as they have total uncertainties larger than 60 %.

The daytime water vapor fine- and coarse-grid retrievals are shown in Figs.

The three main components of the uncertainty budget are shown in Fig.

Since the measurement response of the unconstrained coarse-grid retrieval is unity everywhere by definition, this quantity is not an adequate criterion for determining the last useful altitude of a retrieval. Therefore, we use the uncertainty of the retrieval as a criterion instead. A relative uncertainty of 60 % was chosen as the largest acceptable error, which resulted in a cutoff height of 4.5 km altitude. We found this height to correspond with the altitude at which the signal-to-noise ratio decreases below 1 and noise begins to dominate the retrieval. However, the choice of the critical uncertainty is a matter of preference, and depending on the goal of the research it may be more preferable to cut the retrieval at a lower uncertainty. It is also important to take the presence of dry layers into account to avoid cutting the profile too low if the uncertainty threshold is lowered. It may also be more useful to determine a threshold based on absolute errors instead of relative, particularly for the case of dry regions with low signal. To maintain consistency with

Finally, we compare the fine- and coarse-grid retrievals with the radiosonde profile in Fig.

The relative percent difference between the radiosonde and the fine- and coarse-grid retrievals on 22 January 2013 12:00 UT. The

The a priori removal technique was tested on 5 additional days to study the differences between the fine- and coarse-grid cutoff heights as well as their agreement to the radiosonde (Fig.

The daytime water vapor OEM fine- and coarse-grid profiles show similar differences to the radiosonde profile within their respective uncertainties. For each case, with the exception of 5 May 2009, there are very few differences between the fine- and coarse-grid retrievals from the radiosonde. On 5 May 2009, the coarse-grid retrieval was shifted with respect to the fine-grid OEM retrieval, possibly due to poor calibration on that day.

The daytime fine- and coarse-grid retrievals agree with radiosonde measurements within their respective uncertainties, and the coarse-grid retrievals significantly increase the final meaningful retrieval altitude by an average of 1.5 km. Daytime water vapor retrievals are often limited in altitude due to the high solar background in both the water vapor and nitrogen channels. Increasing the final meaningful altitude by up to 2 km is highly valuable for forecasting and validation purposes.

Daytime water vapor mixing ratio retrievals for 5 additional nights. Black lines are the original OEM retrieval on the fine grid, red solid lines are the ML coarse-grid retrievals, and the dashed green lines are the radiosonde mixing ratio measurements. The black dashed line is the original 0.9 measurement response cutoff height, and the red dashed lines are the coarse-grid cutoff heights which were chosen as the last altitude whose measurements had less than 60 % total uncertainty.

To confirm our choice of cutoff heights for the fine- and coarse-grid retrievals, we looked at the SNR profiles for the digital water vapor signal for each of the daytime comparisons (Fig.

It stands to reason that as the SNRs of the measurements drop, the OEM dependence on the measurements should also decrease (and the a priori's increase) due to the increase in noise. Typically, the SNR level drops below between 3 and 4 km altitude for daytime measurements due to the high solar background. The 0.9 measurement response cutoff height used for the fine-grid OEM results is shown by the blue dashed line in Fig.

Daytime water vapor SNRs (black). The various cutoff heights are shown in dashed lines. The 0.9 measurement response cutoff is blue, the 0.8 measurement response cutoff is green, and the coarse-grid cutoff is in red.

The nighttime case study retrieval uses a 30 min integration on 24 April 2013 00:00 UT which coincides with the time of radiosonde launch. The fine retrieval grid for the RALMO water vapor retrieval is 50 m.

The averaging kernel matrix for the fine- and coarse-grid retrievals is shown in Fig.

The averaging kernel matrix for the nighttime water vapor retrieval on 24 April 2013 00:00 UT.

Unlike the daytime case, the nighttime vertical resolution between the fine- and coarse-grid retrievals is very close up to 5 km where they begin to diverge (Fig.

Figure

The vertical resolution for 24 April 2013 00:00 UT. The vertical resolution on the coarse-grid retrieval decreases as more points are added to ensure that each bin has one degree of freedom. The coarse-grid resolution is shown in red and each point is marked. The fine grid has points every 50 m; therefore they are not shown individually.

The uncertainties for the nighttime retrievals are shown in Fig.

The fine- and coarse-grid retrievals do not change very much with respect to each other until 9.1 km where the averaging kernels begin to drop off significantly. They both produce similar differences with the radiosonde (Fig.

The percent difference from the radiosonde for both the fine- and coarse-grid retrievals. Both show similar differences with the radiosonde and the last valid height is 9.7 km.

The a priori removal method was applied to eight additional nighttime retrievals (Fig.

In all cases, the water vapor nighttime OEM fine-grid and ML coarse-grid retrievals produced profiles which agreed with the radiosondes within their respective uncertainties. Differences larger than 0.4 g kg

All nighttime water vapor retrievals. The radiosonde is shown by the green dashes, the fine-grid retrieval in black, and the coarse-grid retrieval in red. The 0.9 cutoff height for the fine grid is shown by the black dashed line, while the coarse-grid cutoff height is the horizontal red dashed line.

Using the a priori removal technique for nighttime retrievals may be helpful when trying to improve water vapor measurements of the upper troposphere and lower stratosphere (UTLS) region. However, in this case, because the nighttime measurements have large SNRs and a rapid change from high to low signal values, we do not see as large of a difference between the coarse- and fine-grid retrievals as we do in the daytime retrievals. For nighttime retrievals, the coarse grid may not provide an operational advantage but can still be used to homogenize a data set for trend analysis or climatological studies which would require no a priori influence. This will be discussed further in Sect.

Similarly to the daytime water vapor measurements, we have also compared the SNR values with the fine-grid and coarse-grid cutoff heights (Fig.

In all cases, the 0.9 measurement response cutoff corresponds to a SNR of 2. When we compare the 0.8 measurement response cutoff height with the 0.9 cutoff height, we see that the 0.8 cutoff is typically between a few hundred meters to 1 km higher. However, unlike the daytime measurements, the 0.8 cutoff and the coarse-grid cutoff are very close and are either close to 1 or at the boundary where the SNR starts to be noise-dominated. Therefore, we would suggest when using fine-grid nighttime OEM water vapor retrievals to use the 0.9 measurement response as a cutoff height since the 0.8 cutoff height may be in the region where noise dominates, which would lead to larger amounts of the a priori entering the retrieval.

Nighttime SNR calculations for each nighttime water vapor OEM retrieval. The dashed lines are the corresponding cutoff heights: 0.9 measurement response (blue), 0.8 measurement response (green), and coarse grid (red).

We picked a sample night, 12 May 2012, from the Rayleigh temperature climatology in

The averaging kernels for the fine-grid and coarse-grid retrievals are shown in Fig.

The PCL averaging kernels for the temperature retrieval on 12 May 2012 on the fine grid

The vertical resolution for both grids is similar up to 85 km altitude (Fig.

The PCL vertical resolution for 12 May 2012 on the fine and coarse grid. The vertical resolution is similar up to 85 km on both grids. Above this height the vertical resolution decreases until it is 10 km in resolution above 100 km altitude (dotted red line). We consider 100 km to be the highest meaningful point on the coarse grid due to large uncertainties above that height.

Figure

In this case, it cannot be concluded if the HC result is closer to the fine- or coarse-grid result. In order to investigate, we used nine additional nights randomly picked from PCL measurements, and the percent difference between the fine- and coarse-grid retrieval with the HC method was calculated (Fig.

A consequence of applying this method is that the uncertainties in the retrieval increase where the coarse grid is not equal to the fine grid. Figure

The percent difference between the fine-grid retrieval with the HC method (blue line) and coarse-grid retrieval (a priori removed) with the HC method (red line). Below 80 km the retrievals are identical, as the coarse and fine grid are identical.

To illustrate that the a priori information is in fact being removed, we compared the temperature retrievals using two very different a priori temperature profiles, one calculated by CIRA-86 and one calculated by the US Standard Atmosphere (Fig.

The HC method considers the fact that the atmosphere consists of isothermal layers and uses a seed pressure (or temperature) at the top of each measurement profile to calculate the temperature in the lower layers. The maximum height at which there is enough information in the signal is at SNR equals 2. Therefore, the seed value usually is chosen at the altitude that the SNR of 2 and 10 km from the top of the temperature profile is removed due to the seed value uncertainty.
We also examined the relationship with the Rayleigh temperature retrieval and the SNR of the Rayleigh channel signal to determine if there was a similarly consistent value associated with the measurement response cutoff height as there was for the water vapor retrievals. However, based on the examination of all

PCL temperature difference between the OEM retrieved temperature profiles using values from the US Standard Atmosphere and CIRA-86 as the a priori temperature profiles.

We have developed a method to remove the influence of the a priori temperature and water vapor profiles on the retrieval based on the method discussed in vCG. These authors presented a method to re-regularize the retrieval in a way that the original a priori information is removed and the regularization on the fine grid emulates a coarser grid. These re-regularized profiles can then be resampled on a coarse grid without additional loss of information. The optimal coarse grid is determined from the averaging kernel matrix of the original retrieval. This method effectively removes the prior information from the retrieval while keeping the retrieval stable by the use of the coarser final grid. This independence of a priori information can be diagnosed by the averaging kernel matrix, which is unity on the coarse grid.

vCG presented two approaches, a “staircase” representation and a “triangular” representation, to transform the retrieval from the fine to the coarse grid. The cumulative trace of

Our method differs from vCG in that we do not re-regularize the retrieval to remove the a priori information. Instead, after the initial retrieval, we remove the regularization term from the retrieval and rerun the retrieval using the coarse grid. This second run of the retrieval is then equivalent to a maximum likelihood retrieval whose results are solely based on the information provided by the measurements. Both the proposed method and that of vCG are equally effective; however, our method is more of a brute-force technique but easier to practically implement since it is trivial to rerun the retrieval a second time.

For lidars, the triangular coarse-grid calculation results in a grid that is very close to the original OEM retrieval at the lower retrieval altitudes where there is more signal and the averaging kernels of the OEM are close to unity. However, at higher altitudes, where the OEM averaging kernels decrease, the information is spread over more altitudes, and therefore the coarse-grid spacing becomes larger to compensate for the lack of information. An information-centered regridding approach is important for a ML retrieval because it is not guaranteed that any inhomogeneous grid will produce a stable a priori-free retrieval. Additionally, a statistical gridding approach is easily automated and creates a grid that represents the physical conditions of the atmosphere.

We have shown how the a priori removal method works for three sample retrievals: water vapor during both daytime and nighttime, and a nighttime Rayleigh temperature. The a priori removal technique is most useful when the SNR is low, such as for daytime water vapor measurements. The method can increase the daytime retrieval altitude by up to 2 km, which is highly beneficial for meteorological studies that rely on accurate tropospheric measurements. The nighttime water vapor retrieval was provided for contrast to illustrate how the a priori removal technique does not provide significantly more information when the signal level falls off rapidly.

For Rayleigh temperature retrievals, we used measurements from the PCL in London, Ontario.

An advantage of our method over OEM is that the entire coarse-grid profile is a priori-free, in the sense that the regularization term does not contribute to the retrieval. In regions where the SNR is low or the averaging kernel is significantly less than 1, the a priori removal method improves the validity of the retrieval. An a priori-free profile is especially useful for trend analyses and climatological studies which must not include prior information and must be wholly based on measurements. The advantage of an information-centered grid for a typical measurement may be used for multiple retrievals. A grid which is optimal for one atmospheric state will in most cases be close to optimal for a similar atmospheric state. With this consistent grid choice, the altitude resolution of a multiyear time series will be consistent, which is important when working with data over long time periods or conducting trend analyses. Varying information content of the individual measurements will lead to error bars of different size. The coarse grid allows time series analysis or trend analysis for single altitudes without problems caused by varying vertical resolution.

The important trade-off with this technique is that the uncertainties of the retrieval increase when moving from an OEM fine-grid retrieval to a ML coarse-grid retrieval. Both the systematic and statistical uncertainties in the second ML retrieval increase due to the removal of the inverse of the a priori covariance matrix from the gain equation (Eq.

We have developed a practical and robust method which removes the effect of a priori information in lidar OEM retrievals. The method utilizes an information-centered coarse grid which is derived using the averaging kernels from the initial fine-grid retrieval. The resulting coarse grid is then used, alongside setting the inverse of the a priori covariance matrix to zero, to create the final ML retrieval without any a priori information. The method has little computational cost; the OEM retrieval is extremely fast even on a laptop computer, so having to do the retrieval twice for each profile is not critical. We illustrated the method using a simple example in Sect.

Figure

The a priori removal technique increased the maximum altitude of the water vapor daytime retrieval by an average of 1 km and up to a maximum of 2 km; however, the maximum altitude is on the same order of the fine-grid retrieval height if a lower uncertainty threshold is adopted. Both OEM fine-grid and ML coarse-grid retrievals produced similar differences with respect to the radiosonde which agreed within their respective uncertainties (Fig.

Applying the method to the PCL temperature retrieval showed useful retrievals above the

In all cases, the vertical resolution of the OEM retrieval decreases after a priori removal.

The systematic uncertainties after a priori removal increase roughly by a factor of 2 but remain on the same order of magnitude as before the a priori removal. The values of the systematic uncertainties also remain significantly smaller than the statistical uncertainties.

The temperature difference between the PCL retrieved temperature profiles using two different a priori profiles was used to show the effectiveness of the a priori removal method. The temperature difference before removal around the 0.9 cutoff height was more than 2 K; however, this value was zero for the entire range after a priori removal.

The water vapor measurement response values of 0.9 consistently corresponded to a SNR of 2 for the nighttime retrievals and between 1 and 2 for the daytime retrievals. Therefore, it is our recommendation that traditional water vapor retrievals be cut at a SNR of 2 to compare with the OEM water vapor retrievals. Additionally, measurement response values of 0.8 or higher corresponded to SNR values of 1 or less than 1; therefore we would not suggest cutting the water vapor retrievals at heights above which the measurement response is less than 0.9.

The Rayleigh temperature measurement response 0.9 cutoff height was also compared to the SNR of the Rayleigh signal. However, no correlation could be found between the cutoff height and the SNR value. In fact, removing 10 km below a SNR of 2 tended to correspond to measurement response values of less than 0.8, which suggests that it may be more appropriate to remove 15 km from the altitude at which the SNR equals 2 to achieve results more consistent with the OEM.

When designing an OEM retrieval, it is often desirable to understand the effect of the chosen a priori parameters or profiles. This effect has been explored in detail for satellite-based and passive ground-based instruments but not for the new area of applying OEM to active-sensing measurements such as lidar. Lidars are high-resolution instruments with significant amounts of information available from their measurements, as evidenced by the retrieval averaging kernels. The OEM helps to illustrate the robustness of the lidar data products with the advantage of providing diagnostic tools, such as the averaging kernel and a full uncertainty budget.

The a priori removal technique may be helpful for checking the a priori's influence on the retrieval and in determining the appropriate a priori. It is also important to note that the differences between the fine-grid OEM retrieval and the coarse-grid ML retrieval may be smaller if one uses an a priori closer to the true atmospheric state. Often, reanalysis model profiles are used as a priori for OEM retrievals because they are closer to the atmospheric true state than a climatological profile. However, the nature of the a priori profile should depend on the design of the instrument and the goal of the work.

In this study, the US Standard Atmosphere water vapor profile was chosen as the a priori profile to accommodate the operational nature of RALMO lidar water vapor measurements, which requires a minimal number of dependencies in the code as possible and preferably no need for internet. The CIRA temperature profile was used for the temperature a priori because there are very few model temperature a priori profiles above 80 km for the PCL and coincident satellite measurements are not always available. Additionally, when conducting trend analyses or climatological studies it may be more useful to use a consistent a priori profile throughout the analysis to avoid inducing trends or biases into the results.

The removal method is most operationally useful for lidar measurements with low signal to noise and a slow transition from regions of high signal to low signal. The method is less effective at increasing the maximum retrieval altitude when signal strength changes rapidly, such as when the nighttime water vapor measurements quickly enter the dry upper troposphere or lower stratosphere. However, the method is most useful for homogenizing large data sets for trend analyses. One representative coarse grid would be applied to an entire data set and a ML retrieval would be run to remove a priori information from all measurements, thereby making them suitable for trends.

In the future, this method will be applied to the entire 10 years of RALMO measurements to retrieve the water vapor daytime and nighttime measurements and create a water vapor climatology. We anticipate that this technique will increase the altitude of the daytime water vapor retrievals by several kilometers. It is also our hope that this method may provide statistically significant measurements in the UTLS region. Finally, the RALMO water vapor climatology will be used to find trends.

RALMO data are available upon request from Alexander Haefele by email: alexander.haefele@meteoswiss.ch. PCL data are available upon request from Robert J. Sica by email: sica@uwo.ca. GRUAN radiosonde data from Payerne can be downloaded via the GRUAN website (

AJ was responsible for developing the a priori removal method and code as well as manuscript preparation, including the following sections: Introduction, Theoretical background, Methodology, Purple Crow Lidar Rayleigh temperature a priori removal, Summary, Discussion, and Conclusions. He also applied the method to the Rayleigh temperature analysis. This work is the second component of his doctoral thesis. SHJ applied the removal method to the water vapor daytime and nighttime analyses as well as helped with manuscript preparation, including the following sections: Introduction, Theoretical background, Methodology, Daytime RALMO water vapor a priori removal, Nighttime RALMO water vapor a priori removal, Summary, Discussion, and Conclusions. This will serve as a component of her doctoral thesis. RJS was responsible for supervising the doctoral theses and contributed to manuscript preparation. AH was also responsible for supervising the doctoral thesis of SHJ and contributed to manuscript preparation and scientific discussions. TvC significantly contributed to the scientific discussions which resulted in this paper, helped develop the method based on his original work, and contributed to manuscript preparation.

The authors declare that they have no conflict of interest.

We would like to thank the Federal Office of Meteorology and Climatology, MeteoSwiss, for its support of this project and providing the water vapor lidar measurements. We would like to thank the GRUAN support team for providing the GRUAN-processed radiosonde measurements.

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

This paper was edited by Andrew Sayer and reviewed by three anonymous referees.