Methodology for deriving the telescope focus function and its uncertainty for a heterodyne pulsed Doppler lidar

Doppler lidars provide two measured parameters, radial velocity and signal-to-noise ratio, from which winds and turbulent properties are routinely derived. Attenuated backscatter, which gives quantitative information on aerosols, clouds, and precipitation in the atmosphere, can be used in conjunction with the winds and turbulent properties to create a sophisticated classification of the state of the atmospheric boundary layer. Calculating attenuated backscatter from the signal-to-noise ratio requires accurate knowledge of the telescope focus function, which is usually unavailable. Inaccurate assumptions of the 5 telescope focus function can significantly deform attenuated backscatter profiles, even if the instrument is focused at infinity. Here, we present a methodology for deriving the telescope focus function using a co-located ceilometer for pulsed heterodyne Doppler lidars. The method was tested with Halo Photonics Streamline and Streamline XR Doppler lidars, but should also be applicable to other pulsed heterodyne Doppler lidar systems. The method derives two parameters of the telescope focus function, the effective beam diameter and the effective focal length of the telescope. Additionally, the method provides uncer10 tainty estimates for the retrieved attenuated backscatter profile arising from uncertainties in deriving the telescope function, together with standard measurement uncertainties from the signal-to-noise ratio. The method is best suited for locations where the absolute difference in aerosol extinction at the ceilometer and Doppler lidar wavelengths is small.


Editor
I appreciate your clear answers to the reviewers remarks but I would suggest to answer some minor remarks from the reviewers directly in the manuscript (and not only in their answers). Could you please take into account the following corrections to be brought to the manuscript before publication: 1. p5, line 11 : as you included a new section about the impact of the turbulent parameter (rho). Could you just refer to section 4.2 to mention that it will be discussed later in the manuscript ?
We have modified the manuscript as suggested: We first assume that we can neglect ρ 0 , and describe a method for estimating f and D, together with their uncertainties, which can then be propagated to obtain the uncertainty in the attenuated backscatter coefficient. The impact of the ρ 0 parameter is discussed in section 4.3.
2. p6, ligne3 : Could you please clarify why the interpolation is needed to follow reviewer 1, minor remark number 3.Instead of writing "when necessary" you could re-use the answer you made to the reviewer We have modified the manuscript as suggested: Both ceilometer and Doppler lidar data were averaged to a common 30minute, 30 m vertical resolution grid, using linear interpolation where necessary (only for one period from Darwin).
Both ceilometer and Doppler lidar data were averaged to a common 30minute, 30 m vertical resolution grid. If the Doppler lidar vertical resolution was larger than 30 m (as was the case for one period from Darwin), linear interpolation was used to match resolutions.
3. p6, line3 : additionally to the reference for the SNR threshold used in the paper, could you also include one part of your answer that you did to reviewer 1 and 2 to explain the choice of this threshold.
We have modified the manuscript as suggested and included an additional sentence: The threshold is based on the expected noise floor for the instruments considered here (Halo Streamline and Streamline XR) and should probably be modified for different instruments.
4. P10, legend of table 3 : can you clarify what is included in the total estimates to answer reviewer 1, major remark number 1 ? It could be something like : total estimates is the number of profiles without clouds, precipitation and multiple aerosol layer, good estimates is the number of good estimates after outlier removal using MAD.
We have modified the table caption as suggested: Total estimates refers to the number of ceilometer and Doppler lidar profiles suitable for comparison. Good estimates refers to the number of estimates remaining after outlier filtering with MAD.

p12, figure4 a) : what does arbitrary unit means ?
Uncalibrated lidar quantities are often plotted with units of [a.u.] or arbitrary units, see for example the units for normalised signal given in this glossary for E-PROFILE: http://www.eumetnet.eu/wp-content/uploads/2016/10/ALC_glossary. pdf 6. P19, line 22: please correct "should included" by "should be included" We have modified the manuscript as suggested: 7. In line with reviewer 1, major remark number 3, could you mention somewhere in the manuscript that thanks to your method, after deriving the telescope focus function from a short-period with a co-located ceilometer, the Doppler lidar can be moved to another place not equipped with a co-located ceilometer but still able to provide both winds profils and attenuated backscatter profiles.
We have included the following sentence in the conclusion: The method does not necessarily require a permanently co-located ceilometer as the estimates of f and D can be made from a short timeseries with a co-located ceilometer; the Doppler lidar can then be moved to another location without a co-located ceilometer using the T f (R) estimated previously, assuming that there has been no modification to the instrument parameters. has been added to the conclusions.
Methodology for deriving the telescope focus function and its uncertainty for a heterodyne pulsed Doppler lidar 1 Introduction Coherent Doppler lidar systems are capable of providing accurate radial Doppler velocities at high temporal and spatial res-15 olution, and have been employed across a wide range of scientific and operational fields. Meteorological applications include the retrieval of turbulent properties to determine the strength, location and source of mixing in the atmospheric boundary layer, and, with many systems having scanning capability, the retrieval of winds. Information on the targets responsible for the radial Doppler velocities measured by the Doppler lidar (e.g. aerosol, cloud, precipitation), would greatly aid the interpretation of both the velocities and the products derived from them, but this requires quantitative use of the signal power received by the 20 instrument.
The performance of a Doppler lidar depends on the signal-to-noise ratio, SNR, of the system, as SNR determines the radial velocity uncertainty (Rye and Hardesty, 1993;Pearson et al., 2009). The outgoing laser beam can be focused to improve the SNR at ranges close to the focal length (Pearson et al., 2002), and this is often used to improve the Doppler lidar velocity data quality and data availability, particularly in the atmospheric boundary layer. The optimal choice of focus will depend on the atmospheric conditions at the deployment location (Hirsikko et al., 2014). 5 Knowledge of how the choice of instrument parameters, such as the effective focal length of the telescope, impact the SNR profile is necessary in order to obtain profiles of attenuated backscatter coefficient (Zhao et al., 1990). A comprehensive overview of the theoretical considerations in determining the performance of coherent Doppler lidar systems was given by Frehlich and Kavaya (1991), who provided analytical expressions for deriving the expected signal measured by the coherent detector for a given target for a range of instrument configurations, including analytical expressions for the telescope focus 10 function (also termed coherent responsivity). Most analytical expressions assume ideal Gaussian beams, which may not always be appropriate (Hill, 2018), hence experimental approaches have also been used to determine the impact of beam aberrations (Hu et al., 2013).
The profile of attenuated backscatter coefficient has the potential to be used in real time by weather forecasters (Illingworth et al., 2019), as it can be used in the same manner as for ceilometers. This includes the detection of liquid, supercooled-liquid, 15 mixed-phase and ice clouds (Hogan et al., 2003;Van Tricht et al., 2014;Tonttila et al., 2015), aerosol layer and mixing-height determination (Flentje et al., 2010;Kotthaus and Grimmond, 2018), and retrieving precipitation parameters (Lolli et al., 2018).
In addition to providing velocity estimates for wind and turbulence, the inclusion of the profile of attenuated backscatter coefficient is advantageous for Doppler lidar boundary layer classification schemes (Tucker et al., 2009;Harvey et al., 2013;Manninen et al., 2018) by enhancing the discrimination between aerosol, cloud and precipitation, and can be used for tracking 20 elevated aerosol plumes (Hannon et al., 1999). The combination of attenuated backscatter profiles from coherent Doppler lidars with other profiling instruments permits additional retrievals; for example, together with a ceilometer (Westbrook et al., 2010b), or with a cloud radar (Träumner et al., 2010), can yield drizzle drop size and precipitation rate. There is also an advantage to obtaining attenuated backscatter and Doppler velocity measurements in the same measurement volume, since this will simplify the calculation of cloud or aerosol mass-fluxes (Engelmann et al., 2008).

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Therefore, an accurate profile of attenuated backscatter coefficient requires confidence in the parameters used to generate the telescope focus function. The parameters may not be known a priori, or may differ from what is assumed, and incorrect values can result in artefacts and very large biases in attenuated backscatter coefficient. We present a methodology for deriving the parameters of the telescope focus function experimentally from co-located Doppler lidar and ceilometer observations, together with the uncertainties in the function parameters. The ceilometer, for which the overlap function is known, provides 30 our reference attenuated backscatter profiles. This methodology is relevant for coherent Doppler lidars designed for meteorological applications with maximum ranges suitable for observing the full extent of the boundary layer and beyond. Note that a calibration constant may still need to be determined and applied after implementing the calculated telescope focus function to retrieve the profile of attenuated backscatter coefficient (Westbrook et al., 2010a;Chouza et al., 2015).
The theoretical description of the telescope focus function is outlined in Sec. 2. In Sec. 3, we introduce the instruments and the methodology for deriving the parameters of the telescope focus function experimentally. An iterative least-squares regression using weighted-Mean-Square-Error (MSE) is used to find the best solution for the telescope focus function, where the weights represent the measurement uncertainties in both instruments. The use of long time periods (one year or more) also provides an estimate of the uncertainties in the parameters for the telescope focus function, which can then be propagated 5 through to uncertainties in the retrieved attenuated backscatter coefficients. The methodology is applied to different instruments in multiple locations in Sec. 4 and the validation of the method is presented in Sec. 5.

Telescope focus function
Following Frehlich and Kavaya (1991), the coherent Doppler lidar equation can be expressed as where SNR is the signal-to-noise ratio, varying as a function of range, R, from the instrument, β is the attenuated backscatter coefficient, η is the detector quantum efficiency, c is the speed of light, E is the beam energy, h is Planck's constant, ν is the optical frequency, B is the receiver bandwidth, and A e is the effective receiver area. For a monostatic system emitting a circular Gaussian beam, using a circular aperture, and having matched filters, the effective 15 receiver area is given by (Frehlich and Kavaya, 1991;Henderson et al., 2005) A e (R) = πD 2 where D is the 1/e 2 effective diameter of a Gaussian beam, λ is the laser wavelength, f is the effective focal length of the telescope for the transmitter and receiver, and ρ 0 is a turbulent parameter, also termed transverse field coherence length.
Collecting the range-dependent terms, we obtain a unitless telescope focus function which is also termed the coherent responsivity (Frehlich and Kavaya, 1991).
The profile of attenuated backscatter coefficient is then obtained by rearranging (1) (4) Figure 1a shows how T f (R) depends on the telescope focal length, f , and show that the apparent focus -i.e range to the T f (R) maximum -is always closer than f , and that decreasing D shortens the apparent focus. This makes estimation of the parameters by eye in T f (R) prone to errors, since the apparent focus cannot be translated into f without knowledge of D.  Figure 1c shows that even if the telescope is focused at infinity, knowledge of the D is essential to derive attenuated backscatter coefficient profiles. While the gradient of T f (R) may be independent of D at the near and far ranges, the relative magnitude is not, and the potential variation is high in the range of the profile that is commonly of most interest.

Uncertainty in attenuated backscatter coefficient
Assuming that the parameters T f (R) and SNR are independent, and have uncertainties that can be described as Gaussian, the 5 relative random uncertainty in attenuated backscatter coefficient is where σ S is the relative uncertainty in the Doppler lidar SNR, and σ T f is the relative uncertainty in T f (R). An expression for deriving σ S is given by Manninen et al. (2018), and we describe our method for obtaining σ T f in Section 4.2.
3 Application to data 10 There are 3 range-dependent unknowns in (2): f , D, and ρ 0 . We first assume that we can neglect ρ 0 , and describe a method for estimating f and D, together with their uncertainties, which can then be propagated to obtain the uncertainty in the attenuated backscatter coefficient. The impact of the ρ 0 parameter is discussed in section 4.3.

Instruments
We  Table 1). The focus for the Streamline version can be set by the operator, whereas the Streamline XR has the focus set by the manufacturer; however ARM has had some instruments upgraded from their original specification.
The ceilometer at all sites was a Vaisala CL31 ceilometer, which has a coaxial design and full overlap before 100 m and a temporal resolution of 30 s (more specifications given in Table 2).

Telescope focus function parameter estimation
The methodology for deriving the parameters of the telescope focus function compares profiles from co-located Doppler lidar and ceilometer using an iterative least-squares regression to find the best solution. The method follows the process diagram given in Fig. 2. Before input, the Doppler lidar SNR data had a background correction applied to reduce bias (Manninen et al., 2016).
Both ceilometer and Doppler lidar data were averaged to a common 30-minute, 30 m vertical resolution grid. If the Doppler lidar vertical resolution was larger than 30 m (as was the case for one period from Darwin), linear interpolation was used to match resolutions. After averaging, data below a minimum threshold of -22.2 dB (Manninen et al., 2018) was discarded. The threshold is based on the expected noise floor for the instruments considered here (Halo Streamline and Streamline XR) and 5 should probably be modified for different instruments.
The data was then filtered to select only those portions of the profiles that are considered reliable for comparison. Ceilometer data below 195 m was discarded to ensure that only data with full overlap was used.
Due to the wavelength difference between the Doppler lidar and the ceilometer, it cannot be assumed that the atmospheric backscattering properties are the same at both wavelengths. However, we are only interested in the profile shape, not the 10 absolute values, so profiles from the Doppler lidar and the ceilometer can be compared as long as they contain only one type of scatterer, and one which can be assumed to be distributed homogeneously throughout the portion of the profile used for comparison. Hence, the portion of a profile selected for comparison should contain only one aerosol layer, no clouds, and no precipitating hydrometeors.
We removed clouds by identifying the range gate 150 m below the cloud base detected by the ceilometer and excluding all data beyond this. Elevated aerosol layers and precipitating hydrometeors were filtered out by identifying layers using a convolution of the ceilometer profile with a haar-wavelet to detect changes in the gradient. The base of the second layer 5 was identified where the gradient was increasing over 2 range gates and all data above this was discarded. This process also eliminates noisy profiles with low SNR.
The T f (R) parameter estimation is performed on a profile-by-profile basis for each profile where the filtering process leaves 8 successive range gates present. From equation 4, dividing the Doppler lidar SNR profile with the appropriate T f (R) will generate a Doppler lidar attenuated backscatter profile whose shape should match that of the ceilometer attenuated backscatter 10 profile.
We use a brute-force approach to iterate through a range of reasonable f and D values, generating a corresponding T f (R) and Doppler lidar attenuated backscatter profile for each combination of values. The ceilometer profile and resulting Doppler lidar attenuated backscatter profiles are normalised so that the integral value of the unfiltered portion is unity. We then use a least-squares regression using weighted-MSE to find the best solution (smallest MSE), where the weights represent the measurement uncertainties in both instruments. Collecting results over many profiles results in a bi-variate distribution; the 5 peak of this distribution is chosen as the best estimate of f and D, and hence the best estimate of T f (R) using (3).

Outlier removal
Occasionally, data of poor quality passes the filtering step in Fig. 2. The most common issues are noisy ceilometer data, and a bias in the Doppler lidar SNR profiles. If not screened, these occasional profiles result in significantly altered T f (R) estimation.
Any noise in the ceilometer data is magnified by the profile length often being relatively short, and hence large uncertainty 10 in even a single range gate can skew the regression. Doppler lidar SNR bias will impact the normalisation process, changing the T f (R) selected by the method due to the now incorrect profile shape. Due to the non-linearity of the T f (R) parameter estimation process, these issues result in regression solutions wildly inconsistent with the estimates based on good data. These outliers, which do not fall within the normal uncertainty observed in good data, are then removed from the bi-variate distribution of solutions before calculating the uncertainty estimates. 15 We used the median absolute deviation, MAD (Huber and Ronchetti, 2009;Leys et al., 2013), to distinguish outliers in the bivariate distribution of estimated f and D. MAD can be calculated using where b = 1.4826 when the distribution excluding the outliers is normal. However, the distribution of f and D may not meet this criterion due to the non-linearity of T f (R) and the computational T f (R) estimation process. We expect the distributions 20 of D and f −2 to be close to normal and will use f −2 rather than f to determine outliers. Additionally, the peak of the bivariate distribution may not always coincide with the medians of the uni-variate D and f −2 distributions, and, hence, we use a modified form of (6), We selected 3 MADs as the threshold for flagging outliers: Assuming the distribution excluding the outliers to be normal, 3 MADs correspond to 3 standard deviations of the underlying distribution. In cases where f is at infinity, all estimates with a finite f will be flagged as outliers.

Parameter estimation
We applied the T f (R) estimation method to Doppler lidars at 5 ARM atmospheric observatories. Figure 3a shows the distri-  is clearly seen in the change in D, whereas the best estimate for f did not change. However, inspecting the data by eye would suggest a significantly shorter apparent focus, and the T f (R) calculated using the best estimates for f and D also exhibits a significantly shorter apparent focus. Consequently, the Streamline XR in SGP has been noted to suffer from poor SNR at the boundary layer top.

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The bi-variate distributions of f and D show notable variations in how tight they are around the peak, and is likely a result of differences in data quality between the instruments. The best estimates of f and D and their uncertainties for all sites are presented in Table 3. The Doppler lidar measurements at Darwin were split into two periods, as there was a two month break in the measurements between these two periods. We performed the T f (R) parameter estimation separately for both periods. The best estimates from these periods differ from each other, which is expected as some adjustments were made to the instrument. The telescope focal length for this instrument is directly adjustable by any operator while the beam diameter is set by the manufacturer and is not modifiable by the operator. We note that the D estimates are the same for these two periods within the margin of error calculated.
For the sites and instruments selected here, only the Doppler lidar at NSA had f set to infinity. In fact, all Streamlines have D in the vicinity of 25 mm, whereas D for the Streamline XR versions is about half this. Nevertheless, the variation between 5 instruments of the same version is not negligible and should be taken into account when calculating T f (R) and then attenuated backscatter.
The final step to obtain attenuated backscatter profiles is to apply a calibration constant, which can be achieved using the liquid cloud calibration method (Westbrook et al., 2010a;O'Connor et al., 2004).
The parameters f and D calculated for period 1 in Darwin have been used to derive T f (R) and the results applied in Fig. 4. 10 This shows the utility of the method, able to provide reliable Doppler lidar attenuated backscatter profiles in Fig. 4b that show no over correction below 1 km and display similar in-cloud values to the ceilometer in Fig. 4c. It is expected that the aerosol attenuated backscatter coefficients will differ due to the different scattering properties of aerosol at the different wavelengths; the scattering properties of cloud droplets remain similar at the two wavelengths (O'Connor et al., 2004;Westbrook et al., 2010a).

Uncertainty
A computational method was used to calculate the uncertainty in the estimated T f (R) as it is a non-linear function of f and D.
We used Monte Carlo simulation (MCS) (Morgan and Henrion, 1990) where a distribution of input values is fed into a model, here the effective receiver area equation (2), and the uncertainty is obtained from the distribution of the output. The input values can be created either from observed statistics, or by bootstrapping, i.e. re-sampling the data. We created three different sets of 20 input values for our MCS: 1. Re-sampling the individual estimates of f and D provided directly by the T f (R) estimation method (i.e. those displayed in Fig. 3) after excluding outliers. Table 3, assuming that D and f −2 are normally distributed and independent, N (f −2 , D).

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3. Generating the values from statistics presented in Table 3, assuming D and f to be normally distributed and independent, For each set of input values, the relative uncertainty in T f (R) is calculated as  where σ T φ (R) is expressed in terms of the mean-squared deviation of the MCS-simulated telescope focus function, T φ (R), from the best estimate of T f (R), to avoid underestimating the uncertainty resulting from the asymmetry in T φ (R). This also allows us to estimate the impact of refractive turbulence on the uncertainty estimate.

5
Examples of the three input parameter distributions are presented in figure 5. Re-sampling (Fig. 5a) is the most accurate method as it does not require assumptions about the parameter distributions and their independence. We recommend re-sampling as the primary method for uncertainty calculation. Using the N (f −2 , D) distribution (Fig. 5b) produces a set of input values that appear to be a reasonable approximation, except that the distribution is not as tight around the peak. Using the N (f, D) distribution (Fig. 5c) produces a set of input values that tend to over-emphasise shorter values of f , and under-10 emphasise higher values. We also note that the central bin of the re-sampled distribution contains 50% more samples than the central bin of the statistically-generated distributions do. We presume that this is a consequence of the variation in SNR not being necessarily normally distributed.  Table 4. σ T f (R) uncertainty envelopes generated using MCS with three different sets of input values.  The largest value of σ T f (R) provides the uncertainty envelope value for each site, which is presented in Table 4. Re-sampling provides values ranging from 0.12 for the updated instrument at SGP, to 0.32 at NSA. MCS values created using N (f −2 , D)

Location
provided similar values, whereas MCS using N (f, D) often provided much larger uncertainties.

Impact of refractive turbulence
So far we have neglected the potential impact of turbulence on T f (R) arising from the refractive turbulent parameter, ρ 0 , in 10 (2). An expression for ρ 0 is given in Frehlich and Kavaya (1991), where H = 2.914383, k = 2π/λ, and C n 2 (z) is the refractive turbulence at range z. We chose 3 profiles with constant C n 2 (z), and a realistic vertical profile based on the most turbulent case presented by Roadcap and Tracy (2009). Figure 7 shows the impact that these different profiles have on T f (R), and the resulting re-sampling calculation of σ T f (R) for two Doppler lidar 15 instruments with different T f (R). Values of C n 2 up to 10 −14 m −2/3 have negligible impact on T f (R), and even the realistic profile only showed a slight increase in σ T f (R) for the instrument with a focus set closer than infinity. This suggests that the impact of turbulence can be safely neglected for low values of C n 2 , and for most applications, can also be neglected when operating in the vertical. Hence, turbulence has no significant impact on the methodology described here for deriving the parameters f and D and their uncertainties from vertical profiles, but can be included for completeness. 20 Figure 7. Impact of turbulent parameter, ρ 0 , on telescope focus function, T f (R) and relative uncertainties, σ T f (R), for different C n 2 profiles. a) selected profiles of C n However, it is clear that the turbulent impact should not be ignored when measuring at low elevation angles close to the horizon, where a profile similar to C n 2 = 10 −13 m −2/3 may be possible. Fig. 7 shows that such a profile has a major impact on T f (R), especially in the far range. In these cases, the parameters f and D obtained from vertical measurements are still applicable, but ρ 0 (R) must also be calculated or estimated in order to derive the profile of attenuated backscatter, β (R).

5
The liquid cloud calibration method (O'Connor et al., 2004;Westbrook et al., 2010a) is used to determine a calibration constant by integrating attenuated backscatter profiles containing fully attenuating liquid clouds, which have well-constrained apparent lidar ratio, ηS, where η is a multiple scattering factor and S is the lidar ratio. In the absence of multiple scattering, ηS can be assumed to be independent of the height of the cloud.
This calibration method can be used to evaluate the estimated T f (R) for Doppler lidar by checking whether the attenuated  confirming that the method of estimating T f (R) is valid. Table 3 shows that the proportion of data that can be used for the T f (R) parameter estimation varies significantly from site to site. Over a third of the available profiles from SGP are used, whereas only 0.3% pass the filtering for Ascension. The lack of 20 suitable profiles at Ascension is explained by the almost constant low cloud cover at this site, with very few profiles having a sufficient number of successive range gates.

Limitations
Data quality is also a limiting factor so at sites with very low aerosol optical depth, AOD, such as NSA, the Doppler lidar SNR decreases so rapidly that again there are few profiles having a sufficient number of successive range gates. Low AOD also impacts the performance of the ceilometer, with 48% of the estimates at NSA discarded as outliers even after the initial 25 filtering was performed. While the outlier removal can separate the good and the poor estimates, the largest uncertainty in D was at NSA. We attempted to perform the T f (R) parameter estimation on Doppler lidar from an ARM campaign in Cape Cod, but could not obtain reliable estimates due to the low SNR of the ceilometer data.
The T f (R) parameter estimation method is suitable only in situations where there is minimal difference in atmospheric extinction within the aerosol layer between the two instrument wavelengths of 910 nm and 1500 nm. Using AERONET AOD 30 measurements collocated at the ARM atmospheric observatories, the median difference in AOD at 870 nm and 1640 nm varied from 0.016 and 0.027, which should correspond closely to what might be expected for the difference between ceilometer and Doppler lidar. Very occasional periods of notable AOD differences were observed at some sites, but including these profiles in timeseries extending beyond a year will have negligible impact on the T f (R) parameter best estimate. However, there were breaks in the AOD measurements, and some periods experiencing a significant differential extinction may have gone unnoticed.
An additional filter using AERONET AOD measurements to remove profiles experiencing significant differential extinction could be included in Fig. 2 for those sites where this may be an issue.

6 Conclusions
We have developed a method for deriving the telescope focus function and its uncertainty for pulsed heterodyne Doppler lidars, and applied the method to Halo Photonics Streamline and XR Doppler lidars. The method compares profiles of the Doppler lidar SNR to profiles of attenuated backscatter coefficient from a collocated ceilometer, producing estimates for two parameters of the T f (R); the effective focal length for the telescope, f , and the 1/e 2 effective diameter of a Gaussian beam, 10 D. This method was developed because it also provides uncertainties in f , D and T f (R), necessary for quantitative use of the subsequently derived attenuated backscatter profiles. The method can be used to check the manufacturer specifications for these parameters, calculate them if not known, and also check their stability over time. The method does not necessarily require a permanently co-located ceilometer as the estimates of f and D can be made from a short timeseries with a co-located ceilometer; the Doppler lidar can then be moved to another location without a co-located ceilometer using the T f (R) estimated 15 previously, assuming that there has been no modification to the instrument parameters.
The method was applied to data from Doppler lidars with different configurations deployed at 5 ARM sites. Relative uncertainties in f for these instruments ranged from 6% to 17% with the median uncertainty being 10%; the relative uncertainty in D ranged from 2% to 12% with median of 3%. The uncertainty in T f (R) was calculated using Monte Carlo simulation, using 3 methods to prepare the input values. We recommend the direct re-sampling method, but reasonable results were obtained 20 used statistically-derived input values assuming a normal distribution. The envelope of relative uncertainties in T f (R) ranged from 13% to 32%, and depend on both the instrument configuration and the instrument location. We also show that, even for a Doppler lidar with the focus set at infinity, uncertainty remains in estimating T f (R) arising from the uncertainty in D. The method was validated by calculating the apparent lidar ratio of fully attenuating liquid clouds from T f (R) corrected profiles of Doppler lidar attenuated backscatter.

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The impact of turbulence on T f (R) was also investigated and was found to have no significant impact on the methodology described here for deriving the parameters f and D and their uncertainties from vertical profiles. However, the turbulent impact should not be ignored when measuring at low elevation angles close to the horizon, as it can modify T f (R) considerably, especially in the far range. In these cases, the parameters f and D obtained from vertical measurements are still applicable, but the turbulent contribution to T f (R) should be included when deriving the attenuated backscatter coefficient.