EARLINET lidar quality assurance tools

This paper describes the EARLINET quality assurance (QA) check-up tools for the hardware of lidars, developed in the recent years to monitor and improve the quality of the lidar systems and of their products. These check-up tools are the trigger-delay test, the Rayleigh-fit, the lidar test-pulse generator, the dark measurement, the telecover test, and the polarisation calibration.


Introduction
The introductory paper of this special issue (Pappalardo et al. 2014) gives an overview of the development of the European Aerosol Research Lidar Network, EARLINET, since its foundation in 2000, and a detailed introduction to the present paper in section 3.1, wherefore it will not be repeated here. More than 20 lidar groups work together since 2000 with a very heterogeneous field of lidars and calibration procedures. For lack of standardised equipment and common quality assurance procedures -not only within EARLINET -there was a need for standardisation in order to make the lidar products of the different systems comparable and to be able to provide quality-assured data sets of network products for the characterisation of the European aerosol conditions as function of height. Because an atmospheric standard target doesn't exist, apart from the far range Rayleigh calibration described below, and because the main target of EARLINET was the tropospheric and boundary laser aerosol in the near range of lidars, which was (and is) a critical range for old-style lidars developed in the age and with the money of stratospheric ozone research, we started to develop standardised tests for the lidar's subsystems which should help to characterise and finally homogenise the performance of lidars also in the near range. Although a direct lidar intercomparison with a reference lidar is one possibility, the reference lidar itself has to be characterised first. Furthermore, such intercomparisons are expensive, need a mobile reference system, or the lidar itself must be mobile, which always bears the risk of damage and misalignments during the transport, and cannot be done frequently enough. Nevertheless, a direct lidar intercomparison with a reference system bears a high credibility and often reveals to date unknown problems, wherefore it still is considered the ultimate test after all others are passed. Several intercomparisons were conducted in the frame of EARLINET, over which Wandinger et al. (2016) give an overview. In this paper we focus on self-testing check-up tools with the emphasis that the tools should be cheap, simple, everywhere to use, and for a variety of lidar systems with comparable results. Of course, the tools should address the main problems of near range and tropospheric aerosol lidars. The uncertain trigger delay between the outgoing laser pulse and the start of the data recording, especially with low resolution transient recorders, causes large uncertainties in the near range (section 2).
PλR is the power received from distance r at the Raman wavelength λR, CλR is the lidar constant that contains the system parameters for this Raman channel, βλR is the Raman backscatter coefficient, and αm(λ0,R,r) and αp (λ0,R,r) are the extinction coefficients of air molecules and aerosol particles at the laser and the Raman wavelengths λ0 and λR, respectively. For the wavelength dependence of the particle (p) and molecular (m) extinction coefficients the Angström-approach (usually with k = 1) is used and the parameters fp and fm are introduced as: 4.085 For λLaser = 355 nm, λRaman = 387 nm, and k = 1 follows fp = 0.917. Eq. 1 can now be rewritten as The molecular (Rayleigh) extinction coefficient αm(λ0,R,r) can be calculated from actual radio soundings as shown below in App. 9.2. Assuming that the Raman backscatter coefficient βλR(r) = const(λ0, λR) αm(λ0,r) is proportional to the molecular extinction coefficient because both are proportional to the air density, the particle extinction coefficient αp(λ0,R,r) can be determined from the derivation of the logarithm of Eq.
wherefore we can differentiate Eq. (5) with respect to rd and see from Eq. (6) that the absolute error of the particle extinction coefficient due to an error in the zero-bin depends only on rd and fp if we neglect the relative small contributions of the terms containing m. Below 1 km range this error becomes quite large even for small zero-bin errors (see Fig. 2).  Below a range of 1.0 km the deviation of the range corrected signals can be seen clearly, increasing with decreasing range. Figure 4 shows the derived profiles of the particle extinction coefficient with errors due to the zero-bin uncertainty very close to the theoretically estimated values in Fig. 2.

How to measure the trigger delays
In case pre-trigger samples are recorded, the zero-bin can easily be detected due to the signal peak from straylight diffusely reflected from the laboratory walls. As the distance to the laboratory walls is not well defined, a diffuse scattering target blocking the laser path (see Fig. 5 top) can be used together with a small hole aperture above the telescope to decrease the signal height to within the detection range of the detectors.
In case no pre-trigger samples are recorded, the zero-bin can be detected by means of a near range target with a known distance to the lidar. Alternatively the sufficiently attenuated outgoing laser pulse can be fed into an optical fibre with sufficient length s and the fibre output positioned at the aperture of the telescope (see   Note, although we can only measure the range in steps (rangebins) with the resolution of the transient recorder, i.e. 3.75 m or 7.5 m for the LICEL systems TR40 and TR20, respectively, the uncertainty rd can take any value as it results from several electronic delays independent from the transient recorder. averages are distributed between two rangebins. The statistical properties of this distribution can be used to determine the mean trigger delay and its uncertainty with a resolution better than a rangebin. This is important for transient recorders with low spatial resolution. Figures 7 shows also that the peaks of the analogue signal are about eleven rangebins delayed with respect to the photon counting (pc) signal (analogue-pc-delay), which is typical for the LICEL transient recorders. Note that this delay can be different for different LICEL transient recorder modules, wherefore the analogue-pc-delay must be determined for every module individually. The analogue-pc-delay can also be determined by means of the cross correlation of the two range corrected signals. Fig. 8 A shows atmospheric signals with the same setup and at the same date as in Fig. 7, plot B shows the cross correlation of the whole signals and plot C the cross correlation of the featureless signals between 600 and 1000 rangebins. Both exhibit a distinct peak at −11 rangebins. The small correlation peak stems from the photon noise of the atmospheric backscatter of the laser pulse, which should be the same in the analogue and in the photon counting signal and detectable in ranges where the signal noise dominates the background noise. The near range peaks of atmospheric lidar signals (see Fig. 9) seem to be unsuitable to determine the zero bin, which can bee seen in Fig. 10 where the analogue and pc near range peaks of three LICEL TR-40 recorders are shown (dashed lines) together with the zero-bin peaks (solid lines) measured with a diffuse laser block (Fig. 5 top). The atmospheric near range peaks are one to two rangebins delayed with respect to the zero-bins. Possible reasons for this delay are multiple scattered light from the near range atmosphere and diffuse reflections from laboratory walls which are further away. It is remarkable that this delay is less in the cross-polarisation signal, which should be further investigated.

Rayleigh fit
The comparison of lidar signals in clean air ranges with the calculated signals from air density is the only absolute calibration of lidar signals. To be able to calibrate lidar signals with Rayleigh (molecular) backscatter, the optoelectronic detection systems must have a high dynamic range. Fig. 11 shows a POLIS-6 532 nm signal from a Hamamatsu R7400-P03 photomultiplier recorded with a LICEL TR40-160 transient recorder with 12 bit A/D resolution and an averaging time of five hours. average of 500 m (green) with the Rayleigh signal calculated from a local radiosonde is shown with Rayleighfits at 7.6 km (black dash-dotted) and between 12 and 16 km (blue dash-dotted). The uncertainty of the fit of the unsmoothed range corrected signal between 13.3 and 16.3 km (not shown) is less than 1%, i.e. the standard error of the mean of the residuals relative to the mean Rayleigh backscatter coefficient. This uncertainty is about 4e-6 times the signal maximum in the near range. The relative deviation of the analogue signal, smoothed with a gliding average of 500 m (light magenta line in Fig. 11), from the smoothed pc-signal is shown as grey line with the right y-scale. This deviation corresponds to the Rayleigh-fit uncertainty of the analogue signal.
The Rayleigh-fit is a normalization of the range corrected lidar signal to the calculated attenuated molecular backscatter coefficient ( βm attn , Rayleigh signal) in a range where we assume clean air without aerosols and where the calculated signal fits the lidar signal sufficiently good. Fig. 12 shows an example of an analogue signal where an aerosol free range is assumed between 5 km and 6 km. Although there are several aerosol signatures below 11 km, the deviation plot at right indicates that the lidar signal can be used up to about 11 km, above which the analogue signal distortions become too strong. Because of the signal noise the normalisation of the range corrected signal r 2 P(r) (Eq. (7)) to the calculated, attenuated molecular backscatter coefficient βm attn has to be performed over a range (rmin, rmax) with the reference range r0 as centre (Eq. (9)).
Considering the discrete range-bin resolution this results in the normalized, range corrected signal Note that for the numerical integration (Fernald-Sassano-Klett inversion etc.) the signal value at the reference range r0 , where the integration starts, must be replaced by the normalized value in order to avoid a noise error, which means: In first approximation, for small aerosol extinction αp, the normalized, range corrected signal of a total backscatter signal is close to the backscatter ratio around r0 (Eq. 12) and the relative deviation from the Rayleigh signal can serve as an estimation of the particle backscatter coefficient βp(r) (Eq. 13).  Because the analogue signals of the LICEL transient recorders of POLIS are optimised for the near range, they start to deviate much earlier from the Rayleigh signal than in Fig. 12 where the MULIS analogue signals are optimised for the far range. Besides the analogue signal distortions, the Rayleigh-fits and the comparison between the different wavelengths can reveal errors as, e.g., wrong background subtraction, too high discriminator level setting of the photon counters (so called hyper-counting), and differences in the receiver optics.

Lidar test-pulse generators
Analogue signals suffer from distortions from multiple sources, which cannot be unambiguously recognized and identified in normal atmospheric lidar signals.

Purpose and description
The accuracy problems of analogue detection channels are clearly visible in section 3 about the Rayleigh-fit above. Signal induced distortions, interspersions of the laser flash lamp and Q-switch triggers and also of the recorder trigger itself as well as the limited bandwidth of the analogue amplifier and its supplementary electronic circuits for range and offset settings etc. are some possible reasons for that. Ground loops in the laser-detector -recorder assembly may add to the signal distortions.
The dynamic range is determined by the ratio of the signal peak value to the amount of noise and signal distortion in the low signal range where the reference value for the Rayleigh calibration is taken. Therefore special emphasis has to be put on low frequency accuracy, i.e. about 30 to 100 µs after the laser pulse (4.5 to 15 km lidar range), where in general Rayleigh calibration is applied.
One of the several sources of errors are the preamplifiers of the A/D converters. Their response to pulses cannot be tested with commercial pulse generators because the accuracy of these pulse generators cannot be guaranteed  The LISIG test pulse generators produce a selectable negative or positive square pulse with 10 µs length, corresponding to a boundary layer with 1.5 km top, and fixed pulse heights of nominal 10 mV, 50 mV, 150 mV, and 750 mV, with the base at 0 V. The corresponding measured output voltages of LISIG2 with new batteries are 8.8 mV, 44 mV, 148 mV, and 616 mV. The emphasis of these pulse generators is not on the perfect square pulse shape and rise/fall times, but on the reliable, return to ground level after about 30 µs with subsequent low frequency ripples well below 10 µV amplitude. To avoid ground loops as well as cross talking of digital electronics into the analogue data line, the LISIG is split into two separate devices. The first device generates a rectangular pulse of approximately 10 µs length. For this a simple integrated circuit of type 555 is used. Its output is coupled to a separate analogue electronics box by using an optical link based on a plastic fiber system.
The analogue electronics only contains a fast RF switch (ZYSWA-2-50DR by Mini-Circuits) that switches on and off a battery voltage on the analogue signal output according to the optical input signal. This way, the signal is guaranteed to return to zero (the negative pole of the battery) in a few nanoseconds. The analogue electronics box has an electrical connection to the A/D converter only through the analogue signal cable. To ensure this, the analogue electronics are placed in an isolating platstic box. Therefore the LISIG can be placed in the real measuring environment simply by replacing the detector.

Measurements with the test pulse generators
Within  While the pulse response of the PCI412 is not bad for a commercial recorder, the MI4022 responses are much improved with a distortion to pulse height ratio of 1e-5. As a first step of optimisation all electronic input circuits of the preamplifiers of both recorders had been removed by the manufacturer except the electronic switch for three fixed range settings. The further improvement in the MI4022 has been achieved by individual tuning of the amplifier input stages of each channel using an earlier version of the test pulse generator as a reference.
Further results of our test measurements are that the distortions have a component which increases about linearly with the pulse length, and together with the shown increase with pulse height shown in Fig. 16 we can see an about linear increase with pulse area. We also saw considerable cross talk between the channels of the MI4022, which might be present also in other models. Such cross-talks for example between cross-polarised signal channels and the Raman-or parallel-polarised channels can cause considerable signal distortions in ranges with high depolarisation ratio. The results for the presented LICEL transient recorders are typical for all the several other LICEL transient recorders tested by several EARLINET groups.

4.3
Linearity of the analogue output of MB-01/02 photo-receiving modules with respect to the optical input LISIG tests only the electronic part of the photo receiver. It cannot be used to test the integrated photo-receiving modules MB-01 and MB-02 (Fig. 18), which are widely used in the lidar systems of CIS-LiNet, because the photo-detector and the whole A/D conversion electronics as well as the amplification and power supply and transient recorder are included in one module in order to avoid ground loops and external EM-interspersions.  The analogue pulse height linearity of these modules has been tested with a pulsed, external light source as shown in Fig. 19. Figure 20 shows the normalised output/input pulse height ratio of an MB-01 module with an FEU-84 photomultiplier at F-in levels spanning four decades with two different background light levels. The measurements show that the output/input ratio non-linearity is less than ±2% over four decades of input level.

Dark measurement
If signal distortions are independent of the lidar signal, they can be determined with so-called dark-measurement.
The measured dark-signals without atmospheric backscatter can be subtracted from the normal lidar signals just as the skylight background or the analogue DC-offset, but as range dependent offset. The dark measurement is like a normal measurement with laser and Q-switch trigger etc., but with fully covered telescope, so that no light from the atmosphere and from the backscattered laser pulse is collected by the detectors. In such signals we can see EM-interferences from the electro-magnetic laser pulses or other electronic interferences which are synchronous to the laser trigger, but also rests of analogue low frequency noise, which can never be completely removed by means of spatial or temporal averaging (see Fig. 21 C). As there are different sources of such disturbances with different effects on averaged lidar signals, we currently don't have a standardised procedure for the dark measurements and cannot use them for the evaluation of the lidar signal quality in a standardised way. However, if after sufficient temporal averaging of the dark measurement the signal distortions are stable, which means not changing by further temporal averaging, the dark signals can be subtracted from the atmospheric signals to improve their accuracy. Figure 21 A shows an example of a 1064 nm LICEL APD analogue signal where the subtraction of a dark signal from the raw signal could be applied, which is verified by a Rayleigh fit and the Klett inversion backward and forward from the fitting range, chosen between 5 to 10 km (Fig. 21 B).  (Fig. 21 C) could not be removed. We therefore recommend to not smooth the dark signal in the near range and to start smoothing only when it would increase the signal noise. Furthermore, we found that the near range interspersions can change quite fast. Hence it is necessary for each channel to test the temporal stability of the dark signal regularly before using it for signal correction. In contrast to the far range, where we can use the Rayleigh-fit in clear air ranges, we don't have a calibration method for a lidar system in the near range, where almost never clean air conditions can be assumed. But shortcomings of the optical and opto-mechanical design or misalignments have their largest effect in the near range. A test for this range is based on the fact that the backscattered photons collected by different parts of the telescope of a lidar system must give the same range dependency of the signal, and if not, the range dependency of the whole signal is uncertain. With ray tracing simulations we see that ray bundles collected by different telescope parts reach the signal detector in different paths through the optical receiver and hit the optical components under different incident angles (see Fig. 22), with possibly different transmission. Possible causes for the differences are laser tilt, telescope misalignments, displacement of field and aperture stops (vignetting, defocus), optical coating effects of, e.g., beam-splitters and interference filters with spatial inhomogeneity or angle dependency of the transmission (see Fig. 23), or spatial inhomogeneity of the detector sensitivity (Simeonov et al. 1999). The geometrical overlap function, which is mainly determined by the size and location of the telescope's field stop, is just the most obvious feature producing differences in different telecover signals.

Figure 23: Optical elements in a typical lidar receiver optics which can influence the transmission of the ray bundles due to vignetting (red arrows) or angular transmission dependency (blue arrows).
In a first attempt the telescope can be covered in a way that just quarters of the telescope are used, which we call the Quadrant-test (see Fig. 24), or using only an inner and outer ring of the telescope, i.e. the In-Out-test. Using In-Out sections of the quadrants is called the Octant-test.

parts (plot at right) with respect to the laser position at North (biaxial systems) or any prominent orientation of the receiver optics (mono-axial systems). Using the four quarters N,E,S, and W in the left picture is called the quadrant test. Using the outer and inner parts of the quadrants is called the octant test. The pictures above show (from left to right) the sectors North (N), North-Out (NO), North-In (NI), Full-Out (FO), and Full-In (FI) on a telescope, assuming the laser on top.
With an ideal lidar system the normalized signals from all different telecover tests must match -apart from the overlap range, which can be therewith assessed, and assuming constant atmospheric conditions during the test. Figure 25 shows an example of quadrant telecover signals from three POLIS-6 channels. The cyan N2 signals are taken with the same telecover sector as the N signals, but at the end of the temporal sequence of the measurements. Deviations between N and N2 signals indicate the influence of the changing atmosphere during the measurements, which are here visible in the cross polarised 355xcg signals between about 200 m and 500 m range, but much less pronounced in the other channels. The relative differences between the normalised signals are well below 5% in the near range and mainly due to signal noise, except for atmospheric disturbances. The distance of full overlap is not below about 100 m. In this case of a well aligned lidar system with well designed eyepieces, the raw signal differences between the telecover sectors indicate just the different sensitivities of the different areas of the photomultipliers, which are different in the three channels. The measured telecover differences can be compared to paraxial (see also Kokkalis (2017)) and exact ray tracing (ZEMAX) simulations of the system including apertures and optical coatings to narrow down possible causes. Figure 26 shows the paraxial near range simulation of the field of view for the full telescope aperture (left), and for the N (blue) and S (magenta) telecover sector with a circular field stop in the focal plane of the telescope (neglecting the obscuration of the secondary mirror). The distance of full overlap between the full telescope and the laser (grey) is only reached when the laser beam is fully in the light green inner core of the field of view of the full telescope. The distance of the full overlap is reached earlier for the N sector than for the S sector ( Fig.   26, right). This difference is very small for the POLIS-6 in Fig. 25, because there a tilted slit field stop is used (Freudenthaler 2003). Figure 27 shows the paraxial simulations as in Fig. 26 but for different defoci of the field stop and for the far range (top rows) and the near range (lower rows). A negative/positive defocus results in a later/earlier distance of full overlap and generally in a loss of full overlap in the far range. Comparing the telecover simulations in the near range (Fig. 27, lowest row), we see that at +20 mm defocus the distance of full overlap is reached earlier for the S than for the N sector, which is in contrast to the situation with defoci smaller than about +10 mm. In the far range the full overlap is earlier lost for the N sector than for the S for negative defoci and vice versa for positive defoci. An exact ray-tracing simulation (ZEMAX) of the relative telecover deviations in Figs. 26 and 27 is shown in Fig.   28 for defoci of 0 mm (top) and +10 mm (bottom) with a Gaussian laser beam with full width divergence of 1 mrad encircling 86% of the energy. As expected from the paraxial simulation in Fig. 27, the decrease of the distance of full overlap from about 300 m to about 150 m between 0 and +10 mm defocus can be identified while closing the gap between the N and S-signal. There is also the concurrent loss of full overlap of the S-signal in the far range with deviations in the 5% range, which are almost hidden by the signal noise and due to the noise in the normalisation range (2 to 4 km). But it shows that already relative deviations in the 5% range reveal a critical situation of the optical setup, wherefore the measured signals should have an as good as possible signal to noise ratio.  In Fig. 29 we see a disadvantage of the telecover test. It shows the effect of a S-N tilted, small bandwidth interference filter together with a laser S-tilt of 0.25 mrad. At +1.5° filter tilt (bottom) the relative deviations from the mean (right) are clearly visible even above 3 km, but at -1.5° filter tilt (top) the relative deviations from the mean seem to vanish above about 600 m -in contrast to the deviations of the normalised signals, which differ up to at least 1 km from the signals without filter consideration (lines with open circles). But this information we have only in the simulations, not in the real world. This shows that if the misalignments/distortions of the lidar setup affect all telecover signals in a similar way, the relative deviations can't show it, and in the normalised signals we can't distinguish such distortions from the aerosol signature in the near range. Therefore we must consider a perfect telecover test as a sine qua non and not as a sufficient condition for an ideal lidar setup. However, the difference between Fig. 29 top and bottom shows clearly the combined effect of the interference filter and laser tilts. Considering the collimator lens (see Fig. 23) as a Fourier-transform lens for the angular transmission function of the interference filter, the latter can be transformed in a spatial transmission function at the location of the lens' focal plane, i.e. the location of the telescope's field stop, and the combined overlap function of the receiver optics is the convolution of interference filters' and the field stop's spatial transmission functions.  As the incidence angles at the telescope aperture are magnified by the telescope-collimator combination by a factor of (-f_telescope / f_collimator) = -1200 / 70 = -17, the +0.25 mrad S-tilt of the laser is equivalent to a -17 x 0.25 mrad = -4.25 mrad = -0.25° tilt of the interference filter. Furthermore, a difference between the laser wavelength λ and the centre wavelength of the interference filter λ0 causes the same effect as a difference between the laser and interference filter tilts (α) according to the relation between the centre wavelength shift and incidence angle in the interference filter (Eq. (14)).

Figure 26: Schematic field-of-view (paraxial simulation) of a two-channel lidar system with a laser with beam expander and steering mirror, a telescope, and the receiving optics. The laser beam (grey) with a small divergence is tilted towards the axis of the telescope. The left plot shows schematically the limits of the field of view of the full telescope and their overlap with the laser beam. The full overlap of the laser beam and the filed of view of the telescope is only reached when the laser beam is fully in the inner light green core of the field of view. The right plot shows the same but for the N (blue) and S (magenta) sector of the telescope. The full overlap with the N sector of the telescope is reached earlier than with the S sector.
A simulation example for a coaxial lidar (same as above but without laser-telescope axes distance and interference filter) is shown in Fig. (30)    Further examples of what the telecover test can reveal are shown in Figs. 31 and 33. In Fig. 31 the Quadrant telecover test signals of two channels of a lidar are plotted, with very different near-range deviations in the two channels. The small Hamamatsu photomultiplier R5600 and its successors as the R7400 exhibit a strong inhomogeneity of the detection sensitivity across the detector surface as shown in Fig. 32 plot A and B (Simeonov et al. 1999).   In case the laser beam is focused on the PMT, the laser spot moves over the detector area with the lidar range, especially in the near range, and the measured signal reflects rather the detector inhomogeneity than the atmospheric structure. A simulation of the possible relative deviations from the mean of the Quadrant signals (Freudenthaler 2004)is shown in Figure 32 D together with the measured ones in plot C. Plot A and B show the simulation conditions for the location and movement of the beam spot on the PMT. A possibility to countercheck the influence of the PMT is to rotate the PMT by 90° and to compare the signal features of both Quadrant tests. Figure 33 (left) show a loss of signal intensity of the N-signal in the far range. The first assumption would be a misalignment of the laser with a S-tilt (compare Fig. 27 for -5 and 0 mm defocus), but the comparison with the signal simulations (right) with a good agreement in the onset of the overlap indicated no misalignments of the laser, defocus of the telescope, or tilt of the interference filter. The second signal from the E-sector (E2, cyan) coincides with the first signal (E, red) and shows that atmospheric changes didn't influence the test. A closer inspection of the receiver optics by means of a CCD-camera was done, and looking through the receiver optics at an image of the telescope aperture we saw strong distortions in the Nsector of the telescope (at NW in the right image plot of Fig. 33), probably due to stress in the thin secondary mirror of the Cassegrain telescope. This lead to the decrease of signal intensity from the near to the far range in the N-signal. The preliminary solution was to mask the N-sector permanently, and the final solution to replace the telescope.

Polarization calibration with error analysis
For the calibration of the relative sensitivity of the two polarisation channels and for the compensation of the effect of polarizing optical elements in lidar systems the EARLINET QA requires a Δ90 calibration (or similar) and a comparison of the corrected linear depolarisation ratio (LDR) in clean air ranges with the calculated molecular LDR -similar to the Rayleigh calibration in section 3. The theoretical background of the Δ90calibration is described in detail in Freudenthaler (2016a). Figure 34 shows the corresponding measurements of POLIS-6 during an episode with rather clean atmosphere and their analysis. The top left plot with a log-y-scale shows the range corrected and smoothed calibration signals (IT for transmitted and IR for reflected intensities behind the polarising beam splitter) at 532 nm with a rotated calibrator, which is in this case a mechanical rotation of the whole receiver optics behind the telescope ) (Freudenthaler 2016a;section 7.2) . Also shown are the normal so-called Rayleigh signals (the transmitted ITRayleigh and the reflected IRRayleigh), which should contain ranges without aerosol for the comparison with the calculated molecular depolarisation ratio. While the calibration should be done as often as possible, also during daylight if necessary, and therefore cannot be too long, the Rayleigh measurement should be done at night and as long as possible to decrease the signal to noise ratio in the aerosol-free region. They don't have to be at the same day or night, but with the same system settings as the calibration measurements. However, we recommend to do the Rayleigh measurement as close as possible to the calibration and if possible without switching off the lidar or the data acquisition, because already small changes in the PMT high voltage supply or of the PMTs themselves due to environmental or system temperature or temperature changes of the second and third harmonic crystals of the laser can influence the accuracy. Even the data acquisition electronics doesn't necessarily settle in the same state at each power-on.
The measurements in Fig. 34 were done at the same night (15.09.16, 20:38) as the calibration measurements (16:59). The following references F16 refer to Freudenthaler (2016a). The top, right plot in Fig. 34 shows the range dependence of the calculated calibration factors at the ±45° positions and the geometric mean using Eqs.
which can also be written using the King correction factor F k describing the anisotropy with Eq. 17.
We get the molecular scattering coefficient σm ,which in absens of absoption is equal to the extinction coefficient, by multiplying the total Rayleigh scattering cross section with the number density N (molecules / m^3) of air, which depends on pressure p(z) and temperature T(z) at height z in the atmosphere As the mean polarizability a is a property of an individual molecule, the term on the right hand side does not depend on the density of the gas, i.e. pressure and temperature, at least for atmospheric conditions. It is usual to indicate this independence by calculating the mean polarizability at STD air conditions (subscript s , => Ns , ns ).
Considering additionally that the mean polarizability and the anisotropy depend on the wavelength λ of the incident light, we get with Eq. 18 the well known equation for the molecular scattering coefficient σm The symbols ε and RA for the square of the relative anisotropy are frequently used, e.g. by Kattawar et al. (1981) and She (2001). The wavelength λ represents the energy of the photons and is thus the wavelength in vacuum.
The King factor has been determined very early by measuring the polarization of light scattered perpendicular to the incident light and using the theoretical relations in Eq. (15) assuming a "mean diatomic air molecule" (see Young (1982)).
With measurements of the individual air constituents, the King factor can also be determined from a weighted sum of Fk-values according to Bates (1984). Tomasi et al. (2005) compiled the latest fits to measured values of the wavelength dependent refractive indices and King factors and include the contributions of CO2 and water vapour in their formulas. These values together with Eqs. (15) and (21) are usually used to calculate the extinction and backscatter coefficients and the linear depolarization ratio of air for the full Rayleigh scattering, i.e. the Cabannes line plus the rotational Raman lines, for atmospheric research.
In order to explain the rotational Raman lines, we need a better model. Manneback (1930) derived the basic formulations already, but unfortunatelly the paper is in German. In the quantum mechanical concept the air molecules are visualized as rotors, consisting mainly of two (O2, N2, H2) or three (CO2, H2O) atoms. Such rotors can rotate around different symmetry axes, can vibrate, and the electrons also have momentums and spins.
Incident photons are re-emitted, with or without changing the original state of rotation, vibration, and spins of the molecule. The quantized energy transfers to the molecules result in discrete wavelength shifts of the emitted light, i.e. rotational and vibrational Raman scattering (see e.g. Long (2002)). In the quantum mechanical theory The following is equivalent to Hostetler and Coauthors (2006) in order to use this paper as reference. We write using Eq. (18) is wrong. However, in the actual CALIPSO data analysis the correct conversion factors are used according to Powell et al. (2009) Kaminskii (1990). In order to enable the comparison of the accuracy of the calculations by the readers, more decimal digits are shown than certified by the accuracy of the model and the assumtions.