Challenges in retrieving stratospheric aerosol extinction and particle size from ground-based RMR-LIDAR observations

We report on the retrieval of stratospheric aerosol particle size and extinction coefficient 1 profiles from multi-color backscatter measurements with the Rayleigh-Mie-Raman lidar operated at 2 the Arctic Lidar Observatory for Middle Atmosphere Research (ALOMAR) in northern Norway. 3 The retrievals are based on a two-step approach. In a first step the median radius of an assumed log4 normal particle size distribution with fixed width is retrieved based on the color ratio formed from 5 the measured backscatter ratios at wavelenghts of 1064 nm and 532 nm. An intrinsic ambiguity of 6 the retrieved aerosol size information is discussed. In a second step, this particle size information is 7 used to convert the measured lidar backscatter ratio to aerosol extinction coefficients. The retrieval 8 is currently based on monthly-averaged lidar measurements covering the period from the year 2000 9 to present. A sensitivity study is presented that allows establishing an error budged for the aerosol 10 retrievals. Assuming a log-normal aerosol particle size distribution with a geometric width of S=1.3, 11 median radii on the order of 100 nm are retrieved. The median radii are found to generally decrease 12 with increasing altitude. The retrieved aerosol extinction profiles are compared to observations with 13 various current and past satellite instruments. 14

mation and growth of sulfate aerosol particles. Stratospheric sulfate aerosols scatter incoming solar 23 radiation and also absorb and re-emit terrestrial thermal radiation. The net effect of an enhanced 24 stratospheric sulfate aerosol loading is generally a surface cooling. 25 Stratospheric sulfate aerosols also provide surfaces for heterogeneous chemical reactions. For an 26 anthropogenically enhanced stratospheric halogen loading, an increase of the aerosol surface area 27 leads to a net catalytic destruction of stratospheric O 3 . This effect will be reversed, once the strato-28 spheric halogen load has returned to background levels (e.g. Tie and Brasseur, 1995). In polar re-29 gions stratospheric aerosols provide condensation nuclei for polar stratospheric clouds (PSC) which 30 facilitate heterogeneous chemical reactions that lead to chlorine activation, which in turn leads to 31 catalytic ozone loss.  The number of experimental studies on the size of stratospheric sulfate aerosols is quite limited 40 and the published aerosol sizes cover quite a large range of values, even under background aerosol 41 conditions. A major advantage of this new method is that the lidar ratio does not have to be assumed, 42 but is calculated from the measurements themselves. For most other lidar studies on stratospheric 43 aerosols the value of the lidar ratio is determined based on a priori assumptions of the aerosol particle 44 size distribution. In addition, the lidar ratio is usually assumed to be independent of altitude, which 45 is generally not true. 46 To our best knowledge, the approach employed here -i.e. the retrieval of aerosol particle size 47 information in a first step, followed by calculating extinction coefficients -has not yet been applied 48 to lidar measurements of stratospheric sulfate aerosols. A similar approach, however, has been 49 employed by Blum et al. (2006) and Jumelet et al. (2008) for investigating polar stratospheric clouds 50 (PSCs). The paper is structured as follows. Section 2 provides a brief description of the lidar system whose 53 measurements are used in the present study. In section 3 we describe the steps of the retrieval ap- 54 The backscatter ratio for a given wavelength is the ratio of the detected signal originating from 66 scattering processes on aerosol particles (Mie-scattering) and air molecules (Rayleigh-scattering) 67 normalized by the contribution of molecular scattering only (see eq. (3)). Therefore it contains 68 information about the aerosol load in the scattering air volume. These backscatter ratios will be used 69 here for the retrieval and further explained in the next section. The main goal of this study is to describe an approach for the retrieval of vertical extinction and 72 particle size profiles of stratospheric sulfate aerosol from ground-based multi-color lidar observa-73 tions. Since backscatter ratios at three wavelengths are available, a method to simultaneously infer 74 the log-normal distribution width and median radius from two color ratios could in principle be em- up to about 100 nm, but is not generally applicable to stratospheric aerosol particles, whose median 80 radii may well exceed 100 nm. Therefore, this approach is not utilized here. Instead, a simplified 81 approach, as outlined by Yue and Deepak (1983), is necessary which sets one of the two distribution 82 parameters to a constant value (here, the distribution width) and retrieves the remaining one (the 83 median radius).

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The retrieval is performed in two steps. First, aerosol particle size is found by comparing the 85 measured and modelled color ratio of the backscatter ratio profiles at two different wavelengths.

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This can be done because the color ratio for the wavelengths used depends on the median radius of 87 the assumed log-normal size distribution. Second, the inferred particle size is employed to calcu- late aerosol extinction coefficient profiles from the measured backscatter ratio profiles. Finally, the 89 aerosol particle density is determined, once particle size and extinction coefficient are known. Some general assumptions were made prior to the retrieval. Backscatter ratios at 1064 nm and 92 532 nm were chosen for the retrieval instead of 355 nm, because measurements at this wavelength 93 essentially serve as a measure for Rayleigh scattering (Brand et al., 2019). The aerosol is assumed 94 to consist of 75% sulfuric acid (H 2 SO 4 ) and 25% water which defines its refractive index. For the 95 assumed aerosol composition the real part of the refractive index is roughly 1.43 at the wavelength 96 of 532 nm and 1.42 at 1064 nm (Palmer and Williams, 1975). Aerosol absorption turned out to have 97 only negligible influence on the retrieval, therefore the imaginary part of the refractive index was set 98 to zero. The particle size distribution (PSD) is assumed to be log-normal, with N A as the number density of the aerosol particles, S the geometric standard deviation (distri-  3.2 Retrieval of aerosol particle size information 107 In the first step the particle size is retrieved, which is a necessary requirement for further computa-108 tions. The lidar backscatter ratio R(z,r m ,λ) at altitude z and wavelength λ is given by which can be simplified to with the aerosol and Rayleigh volume backscatter coefficients β M ie (z,r m ,λ) and β Ray (z,λ) which 111 are defined as and 113 β Ray (z,λ) = k sca M ie (z,λ) · P Ray (Θ) with N A (z) as the aerosol particle density. For Rayleigh scattering an analogous relationship is All values used are summarized and explained in Table 1.

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Using lidar backscatter ratio measurements at two different wavelengths, a color ratio C can be with λ 1 = 1064 nm and λ 2 = 532 nm. Considering the last factor of eq. Rayleigh scattering cross sections (Bucholtz, 1995): Here, N 0 denotes the number density of air for standard conditions, n 0 (λ) is the refractive index for 129 the given wavelength, also for standard conditions, with n 0 (532nm) = 1.0002782, and γ(λ) is the 130 depolarisation factor of air. Finally, the color ratio from eq. (8) becomes Beside the dependence on altitude and wavelength the color ratio in eq. (10) depends only on 132 the assumed PSD and the aerosol refractive index which implicitly affects the aerosol backscatter 133 coefficient.

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For the forward model a log-normal PSD was assumed (eq. (1)). As already mentioned, our 135 retrieval approach uses a single color ratio from measurements at 1064 nm and 532 nm, so only a 136 single particle size parameter can be retrieved, e.g., the median radius of an assumed PSD with fixed 137 width. In this study, the standard deviation of the PSD is set to a fixed value of S = 1.3 and the 138 median radius r m is retrieved using eq. (10).  can be divided into intervals which can be labeled as branch 1, branch 2, branch 3 and so on. For 154 example, the first branch would extend from the first local minimal value at r=0 nm to r=146 nm, 155 whereas branch 3 would range from roughly r=290 nm to r=400 nm.  The division into separate branches is important to illustrate the ambiguity of the radius retrieval 157 employing this method. Since the aerosol radius is not known in advance, the correct branch for the 158 retrieval has to be chosen using physical considerations.

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First, it should be noted that by increasing the assumed distribution width the computed color 160 ratio curve changes its shape. This change has two consequences -branch 1 shifts toward smaller 161 radii and the eye-catching minimum at roughly 290 nm for S = 1.1 rises fast. This rise leaves an 162 ever growing portion of measured data points below its minimal value, hence without a possibility to 163 assign a radius using other branches than branch 1. Above a distribution width of around S = 1.75 164 any ambiguity vanishes since no minima are found on the computed color ratio curve. A subdivision 165 into several branches is then not possible anymore. Therefore only branch 1 allows for a radius  Retrievals based on other branches lead to a radius distribution with a very high minimal value of 173 several hundred nanometer without any transition to smaller values along the vertical profile. Figure   174 2 shows the issue. For color ratios of C < 2.4 (see Figure 1), branch 3 reproduces the measured color 175 ratio only for distribution widths smaller than S = 1.1. In this cases the particle size would be larger

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The other radius profiles shown in Figure 2 were obtained by evaluating branch 1 of the color 182 ratio function for different distribution widths. In the shown example the radius retrieval evaluating 183 branch 2 only was discarded because the resulting radius profile is inverted with a steady growing 184 median radii with altitude. So branch 2 leads to obviously implausible profiles.

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The other branches of the color ratio curve become important, if the particle size distribution 186 extends to median radii beyond the local extrema of the curve (Figure 1). This may happen, for 187 example, after volcanic eruptions (Deshler, 2008). In such cases the retrieval using single branches The possibility to compute the Mie phase function, and therefore the lidar ratio, from the retrieved 208 median radius is an advantageous feature of this method.  account for its dependence on the radius, and therefore on its implicit dependence on altitude. The 215 lidar ratio profile in Figure 4 shows that the assumption of a constant lidar ratio can be an good ap-216 proximation for a certain altitude range, here between 15 and 23 km. But for altitudes above 23 km 217 the lidar ratio changes significantly, therefore leading to inevitable errors if its value is assumed con-  Together with the extinction profiles retrieved in the previous section all information is available 225 to estimate the aerosol number density N A utilising the relationship between extinction coefficient, 226 scattering cross section and particle number density. With the mean cross section of a given PSD with n A 0 as the refractive index of the aerosol, the particle density is given by For both extinction profiles retrieved one step earlier (eq. (11)) density profiles are computed 229 which, of course, are identical for the two wavelengths as can be seen in Figure 5.

Error estimation 231
A key point of the retrieval approach employed here is a correct radius determination, since all 232 inferred quantities depend on it. Therefore, the input parameters, i.e. the assumed distribution width  For the first step of the retrieval procedure, i.e. the radius determination, the error contribution of 243 each parameter is shown in Figure 6. Since the radius is derived from the computed color ratio it 244 does not depend on temperature and pressure because those values cancel out (see eq. (10)). The 245 most significant impact on the radius determination comes from an incorrect assumption on the dis-246 tribution width. A difference of ∆S = ±0.2 leads to relative error of slightly below ∆r/r = ∓40%.

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If the assumed distribution width is increasing, then the retrieved median radius is decreasing, and comes from an erroneous assumption of the distribution width. This contribution is, however, not as 253 big as in the case of the radius retrieval shown in Figure 6 because the impact on the Mie phase func-254 tion -which is needed to compute the extinction coefficient (eq. (11)) -associated with a high/low 255 bias in the distribution width S is partly compensated by a low/high bias in the retrieved value of 256 r m . Therefore, the inferred extinction coefficients seem to be rather robust against variations of the 257 distribution width as seen in Figure 9, which shows extinction coefficient profiles together with the

Comparison of particle size retrievals 268
The number of available studies on the size of stratospheric sulfate particles is rather limited. Here, 269 we limit the considerations to qualitative comparisons of non-collocated measurements.    The wavelengths, at which the aerosol extinction coefficients are provided, the measurement ge- with theÅngström exponent α(z) and λ i denoting the considered or desired wavelength. Since 335 our retrieval provides extinction coefficients profiles at two wavelengths theÅngström exponent is 336 obtained by 337 α(z) = ln(k ext (z,λ 2 )) − ln(k ext (z,λ 1 )) ln(λ 1 ) − ln(λ 2 ) .
A comparison between the satellite observations and the lidar extinction profiles converted to the 338 corresponding satellite wavelength is shown in Figure 11.    shows a pronounced dip-like structure. This behaviour is probably caused by a sporadically appear-343 ing low bias at high latitudes and altitudes of the device, which is a known issue (Rieger et al, 2015).

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It is worth pointing out that the measurements are not performed at exactly the same location. In this work we present an approach for retrieving particle size and extinction coefficient profiles 347 of the stratospheric aerosol layer from multi-color measurements with the ALOMAR-RMR lidar 348 in northern Norway. The retrieval approach is based on comparing measured and modelled color 349 ratios of the wavelengths 1064 nm and 532 nm. In a first retrieval step profiles of the aerosol median 350 radius -assuming a log-normal particle size distribution with fixed width -are obtained. These 351 are used in a second step -together with temperature and density profiles -to calculate the desired 352 aerosol extinction profiles. Although assumptions on the aerosol properties have to be made, the 353 inferred extinction coefficients are relatively robust against variations of the assumed distribution 354 width, which constitutes the dominant error source. Additionally, aerosol density profiles were 355 computed, though with larger uncertainties.

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The median radii of the aerosol size distribution obtained with this approach are in good overall 357 agreement with other independent particle size measurements (except those by SAGE II) which 358 confirm that our assumptions are valid. The consequence of a potential low bias in aerosol size is a 359 high bias in aerosol density. In the case of density profiles, however, the broad error ranges should 360 be kept in mind.

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Finally, the direct comparison with extinction coefficient profiles obtained by satellite-borne mea-362 surements show a significant relative difference of roughly 40% at altitudes around 20 km which 363 in case of OSIRIS can reach over 100% at higher altitudes. As pointed out, this high deviation is 364