Supplement of: Remote sensing of aerosol water fraction, dry size distribution and soluble fraction using multi-angle, multi-spectral polarimetry

Remote sensing of aerosol water fraction, dry size distribution and soluble fraction using multi-angle, multi-spectral polarimetry Bastiaan van Diedenhoven1, Otto P. Hasekamp1, Brian Cairns2, Gregory L. Schuster3, Snorre Stamnes3, Michael Shook3, and Luke Ziemba3 1SRON Netherlands Institute for Space Studies, Leiden, the Netherlands 2NASA Goddard Institute for Space Studies, New York, New York, USA 3NASA Langley Research Center, Hampton, Virginia, USA. Correspondence: B. van Diedenhoven (b.van.diedenhoven@sron.nl)

Here we analyze refractive indices of binary aqueous mixtures of organics and ternary mixtures with organics and inorganic salts as a function of volume water fraction. Refractive index and density data is obtained from Lienhard et al. (2012) (their Table 5). Figure S1 shows refractive indices for aqueous mixtures of levoglucosan and three aqueous mixtures of both levoglucosan and 1) ammonium sulfate, 2) ammonium nitrate and 3) ammonium bisulfate as a function of volume water fraction.

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In the ternary mixtures, the molar ratio of levoglucosan and salts were 1:1. The original data is given as a function of mass fraction of solute (f m,s ). Here we convert those fractions to volume water fractions (f w ) through where ρ mix is the density of the binary or ternary aqueous mixture given by Lienhard et al. (2012) et al. (2012). For reference, the refractive index of pure ammonium sulfate as provided by Tang and Munkelwitz (1991) is also given in Fig. S1 (red line). Here a dry density of 1.76 g/cm 3 is used (Erlick et al., 2011) to convert f m,s to f w using Eq. 1. Note that this line is only plotted up to the critical mass fraction at which efflorescence occurs, assumed to be f m,s = 0.8.
3 Effective refractive indices of external mixtures.

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Remote sensing observations as presented in the main text, as well as in situ light scattering probes yield the effective refractive index of the observed population of aerosol particles, which may be externally mixed with several modes. A reasonable assumption may be that the effective refractive index is the optical depth-weighted average of those of the separate modes.
In turn, as fine mode aerosol extinction roughly scales with its volume, the volume mixing mixing rule may yield a good approximation of the effective refractive index of external mixtures. To test this assumption, we calculate the single scattering 35 properties of an external mixture of two aerosol modes with different refractive indices and then determine the refractive index of a single mode aerosol that is radiatively most equivalent to the external mixture, which can be considered as the effective refractive index of the mixture.
For this approach, Mie calculations are performed to calculate the phase matrices of non-absorbing aerosol modes with varying refractive indices. Subsequently, the phase matrices of two modes are added, weighted by the volume fractions and 40 volume scattering efficiencies. The P 12 element of the phase matrix is divided by the P 11 element to obtain the degree of linear polarization (DoLP). A radiatively equivalent single mode aerosol is then retrieved by finding the lowest root-mean-squared difference (RMSD) between the scattering properties (i.e. P 11 and DoLP) of the external mixture and that of a single mode of which the refractive index is varied. Here, the RMSD is calculated for scattering angles between 90 • and 165 • .
To focus only on the real part of the refractive index, the size distribution is the same for both modes in the mixture and the 45 single mode assumed for the effective refractive index retrieval, with a effective variance of 0.2 and effective radii of 0.12, 15, or 0.18 µm. One of the modes is assumed to have an refractive index of 1.54, while the refractive index of the second mode is varied between 1.33 and 1.54. These modes may be interpreted as respectively a insoluble mode and a soluble mode with a water fraction ranging from zero to unity assuming a volume mixing rule (cf. main text). The effective refractive index of the external mixture estimated using the volume fraction mixing rule is compared with the retrieved effective refractive index to 50 assess the accuracy of using volume fraction mixing for the interpretation of the retrieved effective refractive index. Here, we focus on an equal fraction of the two modes (i.e., f mode1 = 0.5), since the maximum errors are expected for that case. Figure S2 shows the differences between the estimated refractive index according to volume fraction mixing and that of the radiatively equivalent single mode aerosol. Volume mixing is shown to mostly underestimate the effective refractive index of the mixture. The absolute errors are shown to decrease as the difference between the refractive indices of the two modes 55 decreases. Furthermore, the error depends on effective radius with largest errors for the small sizes. This behavior can be explained by the variation of the scattering efficiencies with size and refractive index, i.e. the error in the effective volume mixing refractive index increases as the difference in scattering efficiency of the two modes increases.
The errors in effective refractive index from assuming the volume mixing are relatively large at 0.035, 0.026 and 0.018 for the extreme case with an equal mixture of a purely dry aerosol mode and purely water aerosol mode, for effective radii 60 of 0.12, 15, and 0.18 µm, respectively. In this case, the total volume water fraction estimated from the total refractive index under the assumption of the volume mixing rule would be 0.17, 0.13 and 0.09, respectively. However, we stress that these uncertainties must be considered as maximum values. For example, for an equal mixture of a purely dry aerosol mode and an aerosol mode containing 85% water (with refractive index of 1.365), errors in effective refractive index drop to 0.021, 0.013 and 0.006 for the three sizes, respectively, with representative errors in retrieved water volume fraction of less than 0.10, 0.06 65 and 0.03 for the three sizes, respectively. Furthermore, errors further decrease for mixing fractions deviating from 0.5. Hence, we can reasonably assume that the effective refractive index retrieved using multi-angle polarimetry is generally within about 0.02 of the volume-weighted refractive index of the externally mixed aerosol. Moreover, for our purpose of inferring volume water fraction from the retrieved effective refractive indices, the mixing state of the aerosol is generally irrelevant. Figure S1. Refractive mixture of binary solutions of levoglucosan and water and three ternary solutions of levoglucosan, water and inorganic salts, as indicated by the different colors. Dots are derived from observations of Lienhard et al. (2012), while dashed lines are approximations using the volume mixing rule. The red line represents the parameterized refractive index of pure ammonium sulfate provided by Tang and Munkelwitz (1991). Blue and grey areas are ranges obtained by applying the volume mixing rule to n dry = 1.54 ± 0.02 and n dry = 1.54 ± 0.04, respectively. Figure S2. Error in effective refractive index of a two-mode aerosol mixture from applying the volume fraction mixing rule as a function of the refractive index of one of the modes. The other mode is assumed to have a refractive index of 1.54. An equal fraction of the two modes is assumed. Results are shown for aerosols with size distributions with a effective radius of 0.12 (orange), 0.15 (green) and 0.18 (purple) µm.