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| 7 | <title>Algorithm and calculations of aerosol scattering indicatrices</title> |
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| 8 | </head> |
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| 9 | <body> |
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| 10 | |
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| 11 | <center><b><font size=+1>Algorithm and computation of aerosols phase functions</font></b> |
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| 12 | <p><i>by A.N. Rublev</i> |
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| 13 | <p><font size=-1>(Internal Note IAE-5715/16 of Russian Research Center |
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| 14 | " Kurchatov Institute ", Moscow, 51 pp., 1994).</font> |
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| 15 | <p><b>Extended abstract</b></center> |
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| 16 | |
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| 17 | <p>Aerosols are known to influence the propagation of the solar radiation |
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| 18 | in the atmosphere. Aerosols emission sources are numerous: e.g. dust storms, |
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| 19 | fuel combustion (soot), ocean sprays, etc... Stratospheric aerosols and |
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| 20 | tropospheric anthropogenic aerosols which play an essential role in climate |
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| 21 | forcing (Charlson et al.<sup>1</sup>) can be generated by atmospheric chemical |
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| 22 | reactions with sulfates, sulfuric acid and nitric acid. The volcanic eruptions |
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| 23 | are one of the important atmospheric aerosol generators, for example the |
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| 24 | eruption of the volcano Pinatubo, Philippines, June 1991 resulted in the |
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| 25 | emission 20 Mts of SO<sub>2</sub> (Gregs et al.<sup>2</sup>) which is a |
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| 26 | main source of sulfuric acid aerosol fraction. |
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| 27 | <p>Despite the large number of different aerosol sources, only some selected |
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| 28 | basic aerosol components have been considered in the development of various |
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| 29 | aerosol models (WMO publication<sup>3</sup>). Principal aerosol models |
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| 30 | (<i>e.g. continental, urban, maritime, stratospheric, volcanic, upper atmosphere, |
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| 31 | and cloudy</i>) and their basic components (<i>e.g. dust, water-soluble |
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| 32 | particles, soot, salt particles (oceanic), sulfuric acid solution droplets, |
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| 33 | volcanic ash, and water</i>) are listed in Table 1 (from Ref. 3) with the |
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| 34 | following entries: |
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| 35 | <ul> |
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| 36 | <li> |
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| 37 | the aerosol model name (first column) and the related basic aerosol components |
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| 38 | (second column);</li> |
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| 39 | </ul> |
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| 40 | |
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| 41 | <ul> |
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| 42 | <li> |
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| 43 | the aerosol fraction by volume in % (third column), and the related aerosol |
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| 44 | relative concentration <img SRC="/icons-geisa/alexei_n1n2.gif" height=41 width=25 align=CENTER>(fourth |
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| 45 | column), where <i><font size=+1>N<sub>i</sub></font></i> is the number |
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| 46 | of particles of the i-th component, and N is total number of particles |
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| 47 | in a given aerosol sample.</li> |
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| 48 | </ul> |
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| 49 | Main expressions for the aerosol integrated optical properties as given |
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| 50 | by Deirmendjian<sup>4</sup> are: |
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| 51 | <div align=right> the scattering coefficient: <img SRC="/icons-geisa/alexei_eq1.gif" height=50 width=224 align=ABSCENTER> |
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| 52 | (1) |
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| 53 | <p> the extinction coefficient: <img SRC="/icons-geisa/alexei_eq2.gif" height=50 width=225 align=ABSCENTER> |
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| 54 | (2)</div> |
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| 55 | the scattering phase function corresponding |
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| 56 | to the scattering angle |
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| 57 | <i><font face="Symbol"><font size=+1>q</font></font></i>: |
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| 58 | <div align=right><img SRC="/icons-geisa/alexei_eq3.gif" height=50 width=190 align=ABSCENTER> |
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| 59 | (3)</div> |
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| 60 | |
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| 61 | <div align=right> |
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| 62 | <br> the single scattering albedo: <img SRC="/icons-geisa/alexei_eq4.gif" height=45 width=81 align=ABSCENTER> |
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| 63 | (4) |
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| 64 | <p> the asymmetry factor: <img SRC="/icons-geisa/alexei_eq5.gif" height=51 width=154 align=ABSCENTER> |
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| 65 | (5) |
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| 66 | <p> the normalization factor: <img SRC="/icons-geisa/alexei_eq6.gif" height=51 width=124 align=ABSCENTER> |
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| 67 | (6)</div> |
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| 68 | where: |
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| 69 | <br><img SRC="/icons-geisa/alexei_k_2pi.gif" height=43 width=54 align=ABSCENTER> |
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| 70 | <br><i><font size=+1>x=kr </font></i>is the dimensionless size of the particles |
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| 71 | with radius <i><font size=+1>r</font></i> |
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| 72 | <br><i><font size=+1>m=p-iq</font></i><font face="Arial"> </font>is the |
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| 73 | complex index of refraction with the real (<i><font size=+1>p</font></i>) |
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| 74 | and imaginary (<i><font size=+1>q</font></i>) parts |
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| 75 | <br><i><font size=+1>n(x)</font></i>-is the aerosol particle size distribution |
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| 76 | function (i.e., <i><font size=+1>n(x)d(x)</font></i> is the number of particles |
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| 77 | per cm<sup>3</sup> with dimensionless radii <i><font size=+1>x</font></i> |
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| 78 | in the interval <i><font size=+1>dx</font></i> so that <img SRC="/icons-geisa/alexei_integ.gif" height=50 width=90 align=CENTER>is |
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| 79 | the total number of particles per cm<sup>3</sup>); |
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| 80 | <br><i>K<sub><font size=+1>sc</font></sub></i>(<i><font size=+1>x,m</font></i>)and |
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| 81 | <i>K<sub><font size=+1>ex</font></sub></i>(<i><font size=+1>x,m)</font></i> |
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| 82 | are dimensionless efficiency factors for scattering and extinction, respectively |
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| 83 | (Ref. 4) |
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| 84 | <br><i><font size=+1>i(x,m,<font face="Symbol">q</font>)</font></i> is |
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| 85 | the scattering intensity for non-polarized radiation (Ref. 4):<img SRC="/icons-geisa/alexei_i.gif" height=44 width=156 align=ABSCENTER> |
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| 86 | , where |
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| 87 | <i><font size=+1>S<sub>1</sub>, S<sub>2</sub></font></i> are dimensionless |
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| 88 | complex functions (see Ref. 4, 5 for explicit formulas) which give the |
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| 89 | complex amplitudes <img SRC="/icons-geisa/alexei_as.gif" height=24 width=44 align=ABSCENTER>of |
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| 90 | the scattered wave in terms of the complex amplitudes <img SRC="/icons-geisa/alexei_ai.gif" height=24 width=43 align=ABSCENTER>of |
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| 91 | the incident radiation resolved along the transverse and parallel directions |
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| 92 | with respect to the scattering plane, respectively (Ref. 5): |
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| 93 | <center> |
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| 94 | <p><img SRC="/icons-geisa/alexei_matr.gif" height=51 width=144 align=CENTER></center> |
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| 95 | |
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| 96 | <p>is a linear interpolation of the phase function <i><font size=+1>I<sub><font face="Symbol">q</font></sub>.</font></i>. |
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| 97 | <br>Eqs. (1-5) determine optical properties of the aerosols to be considered |
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| 98 | in non-polarized radiative transfer problems. In particular, the optical |
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| 99 | thickness <i><font face="Symbol"><font size=+1>t(l)</font></font></i> at |
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| 100 | the wavelength <i><font face="Symbol"><font size=+1>l</font></font></i> |
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| 101 | of an atmosphere including aerosols is expressed as the sum: <i><font face="Symbol"><font size=+1>t(l)=t</font></font><sub>gas</sub><font face="Symbol"><font size=+1>(l)+t</font></font><sub>aer</sub><font face="Symbol"><font size=+1>(l)</font></font></i>, |
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| 102 | <p>where <i><font face="Symbol"><font size=+1>t</font></font><sub>gas</sub></i> |
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| 103 | is the atmospheric gases optical thickness calculated using, for example, |
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| 104 | well-known spectroscopic " line-by-line " methods; |
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| 105 | <p><i><font face="Symbol"><font size=+1>t</font></font><sub>aer</sub></i> |
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| 106 | is the aerosol optical thickness calculated for an arbitrary non-homogeneous |
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| 107 | path <i><font size=+1>L</font></i>: |
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| 108 | <center> |
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| 109 | <p><img SRC="/icons-geisa/alexei_thick.gif" height=41 width=132 align=ABSCENTER></center> |
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| 110 | <i><font size=+1><font face="Symbol">s</font><sub>ex</sub></font><sub>(x;</sub><font size=+1><font face="Symbol">l</font>)</font></i> |
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| 111 | is the aerosol extinction coefficient at a point <i><font size=+1>x</font></i> |
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| 112 | of the path <i><font size=+1>L</font></i>. |
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| 113 | <br>It should be outlined, that the normalization factor of Eq.(6) (<img SRC="/icons-geisa/alexei_kn.gif" height=22 width=48 align=ABSCENTER>) |
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| 114 | has been calculated to check the reliability of the linear interpolation |
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| 115 | of the phase function of Eq.(3) used in the calculations of Eq. (5). It |
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| 116 | is aimed at the determination of a required number of angular mesh points |
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| 117 | providing an accurate interpolation of the phase function according to |
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| 118 | the following criterion: the closer K<sub>n</sub> is to 1, the better is |
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| 119 | the interpolation (see last column of Table 2 as an example). In Rublevs |
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| 120 | paper 204 angular mesh points (from 0 to 180 degrees) are used in the calculations. |
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| 121 | <br>The Mie theory (see, for example, Deirmendjian<sup>4</sup>, Van de |
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| 122 | Hulst<sup>6</sup>) based algorithm has been developed and a related computer |
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| 123 | code as well, providing a reliable accuracy for computations of the above |
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| 124 | mentioned aerosol optical properties (estimated relative error <font face="Symbol">£</font> |
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| 125 | 0.3%). |
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| 126 | <p><i>Main results presented in the publication are (see Table 2 as an |
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| 127 | example):</i> |
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| 128 | <p> Tables in the Appendix to the paper provide the computed values of |
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| 129 | the phase function for the principal aerosol models and their basic components |
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| 130 | as listed in Table 1. The calculations were made for 8 wavelengths in the |
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| 131 | UV, visible and IR regions, with an estimated relative error <font face="Symbol">£</font> |
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| 132 | 0.3%. |
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| 133 | <p> The principal optical properties of the basic aerosol components (column |
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| 134 | 2 of Table 1: soot, dust, water-soluble particles, etc...), namely <img SRC="/icons-geisa/alexei_sigmaext.gif" height=22 width=48 align=ABSCENTER>- |
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| 135 | the extinction coefficient (km<sup>-1</sup>) for a particle number concentration |
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| 136 | <i><font size=+1>N</font></i>=1 |
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| 137 | particle per 1 cm<sup>3</sup>; <i><font face="Symbol"><font size=+1>w</font></font></i>- |
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| 138 | the single scattering albedo; <i><font size=+1>g</font></i>- the asymmetry |
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| 139 | factor; <i><font size=+1>K<sub>n</sub></font></i>- the normalization factor |
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| 140 | and its values at 204 angles. |
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| 141 | <p> The same as above defined optical properties for non-cloudy basic |
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| 142 | aerosol models (column 1 of Table 1: continental, maritime, urban, etc...). |
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| 143 | As an example, results of the calculations for the urban aerosol model |
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| 144 | with basic components from Table 1 (water-soluble, soot, dust) are shown |
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| 145 | in Table 2. |
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| 146 | <p> The optical properties for a cloudy aerosol model with a particle |
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| 147 | number concentration <i><font size=+1>N<sub>0</sub></font></i>=353.678 |
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| 148 | cm<sup>-3</sup> corresponding to a typical cloud water content <i><font size=+1>W</font></i>=0.3 |
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| 149 | g m<sup>-3 </sup>(Ref. 7), with the modified Gamma function n(r) (Ref. |
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| 150 | 6, 7) as a particle size distribution function: |
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| 151 | <div align=right><img SRC="/icons-geisa/alexei_eq7.gif" height=45 width=220 align=ABSCENTER> |
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| 152 | (7)</div> |
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| 153 | with the following values of parameters (Ref. 4): <i><font face="Symbol"><font size=+1>a</font></font></i>=2; |
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| 154 | <i><font size=+1>r<sub>0</sub></font></i>=1.5 |
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| 155 | <font face="Symbol">m</font>m. |
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| 156 | <p>The software package AERCOMP (FORTRAN code) allowing the determination |
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| 157 | of the optical properties of more complex aerosol models has been developed. |
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| 158 | In particular, using optical properties of basic aerosol components, one |
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| 159 | can calculate (applying linear interpolation on wavelengths and cosines |
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| 160 | of scattering angels) the optical properties for more complex, composite |
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| 161 | aerosol models. Table 2 is an example of outputs of this program. |
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| 162 | <center> |
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| 163 | <p><b>References</b></center> |
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| 164 | |
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| 165 | <dir> |
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| 166 | <dir> |
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| 167 | <ol> |
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| 168 | <li> |
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| 169 | Charslon R.J., S.E. Schwartz, J.M. Hales, R.D. Cess, J.A. Coakley, Jr., |
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| 170 | J.E. Hansen, and D.J. Hofman, " Climate forcing by anthropogenic aerosols |
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| 171 | ", <i>Science</i>, <b>255</b>, 423-430 (1992)</li> |
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| 172 | |
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| 173 | <li> |
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| 174 | Gregs J.S., et al., " Global tracking of the SO<sub>2</sub> clouds from |
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| 175 | the June 1991 month Pinatubo eruptions " <i>Geophys. Res. Letters</i>, |
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| 176 | <b>19</b>, |
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| 177 | 151-154 (1992)</li> |
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| 178 | |
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| 179 | <li> |
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| 180 | World Meteorology Organization (WMO) publication: "<i>A preliminary cloudless |
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| 181 | standard atmosphere for radiation computation</i>", WCP-112, WMO/TD-NO. |
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| 182 | 24 (1986)</li> |
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| 183 | |
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| 184 | <li> |
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| 185 | Deirmendjian D., <i>Electromagnetic Scattering on Spherical Polydispersions</i>. |
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| 186 | Elsevier, 290 pp. (1969)</li> |
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| 187 | |
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| 188 | <li> |
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| 189 | Twomey S. <i>Atmospheric aerosols</i>. Elsevier, 302 pp. (1977)</li> |
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| 190 | |
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| 191 | <li> |
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| 192 | Van de Hulst, H.C., <i>Light scattering by small particles</i>, 470 pp., |
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| 193 | New York : Dover Publications, 1981.</li> |
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| 194 | |
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| 195 | <li> |
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| 196 | <i>Handbook: Clouds and cloudy atmosphere.</i> Leningrad, " Gidrometeoizdat |
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| 197 | ", 649 p., 1989 (in Russian).</li> |
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| 198 | </ol> |
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| 199 | </dir> |
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| 200 | </dir> |
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| 201 | |
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| 202 | <center><b>Table 1. Principal aerosol models.</b> |
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| 203 | <p>(from Ref. 3)</center> |
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| 204 | |
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| 205 | <p><br> |
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| 206 | <center><table BORDER CELLSPACING=2 CELLPADDING=4 WIDTH="642" > |
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| 207 | <tr> |
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| 208 | <td VALIGN=TOP WIDTH="34%"> |
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| 209 | <center><b>Aerosol model</b></center> |
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| 210 | </td> |
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| 211 | |
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| 212 | <td VALIGN=TOP WIDTH="34%"> |
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| 213 | <center><b>Basic aerosol components and their designation</b></center> |
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| 214 | </td> |
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| 215 | |
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| 216 | <td VALIGN=TOP COLSPAN="2" WIDTH="33%"> |
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| 217 | <center><b>Relative content</b></center> |
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| 218 | |
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| 219 | <p><b>volume (%) N<sub>i</sub>/N <sup>*)</sup></b></td> |
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| 220 | </tr> |
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| 221 | |
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| 222 | <tr> |
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| 223 | <td VALIGN=TOP WIDTH="34%"> |
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| 224 | <center>Continental</center> |
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| 225 | </td> |
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| 226 | |
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| 227 | <td VALIGN=TOP WIDTH="34%"> |
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| 228 | <center>dust (Dust-Like) |
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| 229 | <p>water-soluble (W-S) |
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| 230 | <p>soot (Soot)</center> |
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| 231 | </td> |
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| 232 | |
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| 233 | <td VALIGN=TOP WIDTH="17%"> |
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| 234 | <center>70 |
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| 235 | <p>29 |
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| 236 | <p>1</center> |
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| 237 | </td> |
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| 238 | |
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| 239 | <td VALIGN=TOP WIDTH="16%"> |
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| 240 | <center>2.26278E-06 |
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| 241 | <p>9.37437E-01 |
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| 242 | <p>6.25607E-02</center> |
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| 243 | </td> |
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| 244 | </tr> |
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| 245 | |
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| 246 | <tr> |
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| 247 | <td VALIGN=TOP WIDTH="34%"> |
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| 248 | <center>Urban</center> |
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| 249 | </td> |
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| 250 | |
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| 251 | <td VALIGN=TOP WIDTH="34%"> |
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| 252 | <center>water-soluble (W-S) |
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| 253 | <p>soot (Soot) |
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| 254 | <p>dust (Dust-Like)</center> |
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| 255 | </td> |
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| 256 | |
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| 257 | <td VALIGN=TOP WIDTH="17%"> |
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| 258 | <center>61 |
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| 259 | <p>22 |
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| 260 | <p>17</center> |
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| 261 | </td> |
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| 262 | |
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| 263 | <td VALIGN=TOP WIDTH="16%"> |
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| 264 | <center>5.88931E-01 |
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| 265 | <p>4.11069E-01 |
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| 266 | <p>1.64128E-07</center> |
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| 267 | </td> |
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| 268 | </tr> |
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| 269 | |
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| 270 | <tr> |
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| 271 | <td VALIGN=TOP WIDTH="34%"> |
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| 272 | <center>Maritime</center> |
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| 273 | </td> |
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| 274 | |
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| 275 | <td VALIGN=TOP WIDTH="34%"> |
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| 276 | <center>oceanic (Ocean) |
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| 277 | <p>water-soluble (W-S)</center> |
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| 278 | </td> |
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| 279 | |
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| 280 | <td VALIGN=TOP WIDTH="17%"> |
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| 281 | <center>95 |
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| 282 | <p>5</center> |
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| 283 | </td> |
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| 284 | |
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| 285 | <td VALIGN=TOP WIDTH="16%"> |
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| 286 | <center>4.29942E-04 |
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| 287 | <p>9.99573E-01</center> |
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| 288 | </td> |
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| 289 | </tr> |
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| 290 | |
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| 291 | <tr> |
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| 292 | <td VALIGN=TOP WIDTH="34%"> |
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| 293 | <center>Stratospheric</center> |
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| 294 | </td> |
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| 295 | |
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| 296 | <td VALIGN=TOP WIDTH="34%"> |
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| 297 | <center>sulfuric acid (75% H<sub>2</sub>SO<sub>4</sub>)</center> |
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| 298 | </td> |
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| 299 | |
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| 300 | <td VALIGN=TOP WIDTH="17%"> |
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| 301 | <center>100</center> |
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| 302 | </td> |
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| 303 | |
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| 304 | <td VALIGN=TOP WIDTH="16%"> |
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| 305 | <center>1.0</center> |
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| 306 | </td> |
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| 307 | </tr> |
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| 308 | |
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| 309 | <tr> |
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| 310 | <td VALIGN=TOP WIDTH="34%"> |
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| 311 | <center>Volcanic</center> |
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| 312 | </td> |
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| 313 | |
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| 314 | <td VALIGN=TOP WIDTH="34%"> |
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| 315 | <center>volcanic ash (V-Ash)</center> |
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| 316 | </td> |
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| 317 | |
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| 318 | <td VALIGN=TOP WIDTH="17%"> |
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| 319 | <center>100</center> |
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| 320 | </td> |
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| 321 | |
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| 322 | <td VALIGN=TOP WIDTH="16%"> |
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| 323 | <center>1.0</center> |
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| 324 | </td> |
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| 325 | </tr> |
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| 326 | |
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| 327 | <tr> |
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| 328 | <td VALIGN=TOP WIDTH="34%"> |
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| 329 | <center>Upper Atmosphere</center> |
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| 330 | </td> |
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| 331 | |
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| 332 | <td VALIGN=TOP WIDTH="34%"> |
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| 333 | <center>sulfuric acid (75% H<sub>2</sub>SO<sub>4</sub>)</center> |
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| 334 | </td> |
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| 335 | |
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| 336 | <td VALIGN=TOP WIDTH="17%"> |
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| 337 | <center>100</center> |
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| 338 | </td> |
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| 339 | |
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| 340 | <td VALIGN=TOP WIDTH="16%"> |
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| 341 | <center>1.0</center> |
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| 342 | </td> |
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| 343 | </tr> |
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| 344 | |
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| 345 | <tr> |
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| 346 | <td VALIGN=TOP WIDTH="34%"> |
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| 347 | <center>Cloudy </center> |
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| 348 | </td> |
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| 349 | |
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| 350 | <td VALIGN=TOP WIDTH="34%"> |
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| 351 | <center>water</center> |
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| 352 | </td> |
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| 353 | |
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| 354 | <td VALIGN=TOP WIDTH="17%"> |
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| 355 | <center>100</center> |
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| 356 | </td> |
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| 357 | |
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| 358 | <td VALIGN=TOP WIDTH="16%"> |
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| 359 | <center>1.0</center> |
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| 360 | </td> |
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| 361 | </tr> |
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| 362 | </table></center> |
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| 363 | |
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| 364 | <dir> |
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| 365 | <dir><sup>*)</sup> N<sub>i</sub>- number of particles of i-component; N- |
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| 366 | total number of particles in an aerosol sample. |
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| 367 | <center> |
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| 368 | <p><b>Table 2. Integrated optical properties of the urban aerosol model |
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| 369 | (a non-cloudy model).</b></center> |
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| 370 | </dir> |
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| 371 | </dir> |
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| 372 | |
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| 373 | <center><table BORDER COLS=6 WIDTH="80%" > |
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| 374 | <tr> |
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| 375 | <td ALIGN=CENTER VALIGN=CENTER WIDTH="20%"> |
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| 376 | <center><i> Num</i></center> |
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| 377 | </td> |
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| 378 | |
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| 379 | <td> |
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| 380 | <center> <font face="Symbol">l(m</font>m)</center> |
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| 381 | </td> |
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| 382 | |
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| 383 | <td> |
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| 384 | <center><font face="Symbol">s</font><i><sub>ex</sub></i>(km<sup>-1</sup>) </center> |
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| 385 | </td> |
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| 386 | |
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| 387 | <td> |
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| 388 | <center><font face="Symbol">w</font></center> |
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| 389 | </td> |
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| 390 | |
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| 391 | <td> |
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| 392 | <center><font face="Symbol">m</font></center> |
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| 393 | </td> |
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| 394 | |
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| 395 | <td> |
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| 396 | <center><i>K<sub>n</sub></i></center> |
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| 397 | </td> |
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| 398 | </tr> |
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| 399 | |
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| 400 | <tr> |
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| 401 | <td> |
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| 402 | <center>1 |
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| 403 | <br>2 |
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| 404 | <br>3 |
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| 405 | <br>4</center> |
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| 406 | </td> |
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| 407 | |
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| 408 | <td ALIGN=CENTER VALIGN=TOP> |
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| 409 | <center>0.200 |
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| 410 | <br>0.250 |
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| 411 | <br>0.300 |
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| 412 | <br>0.337</center> |
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| 413 | </td> |
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| 414 | |
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| 415 | <td ALIGN=CENTER VALIGN=TOP> |
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| 416 | <center> 0.13889E-05 |
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| 417 | <br>0.12610E-05 |
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| 418 | <br>0.11042E-05 |
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| 419 | <br>0.98538E-06</center> |
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| 420 | </td> |
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| 421 | |
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| 422 | <td ALIGN=CENTER VALIGN=TOP> |
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| 423 | <center>0.53439E+00 |
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| 424 | <br>0.59215E+00 |
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| 425 | <br>0.65632E+00 |
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| 426 | <br>0.66404E+00</center> |
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| 427 | </td> |
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| 428 | |
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| 429 | <td ALIGN=CENTER VALIGN=TOP> |
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| 430 | <center> 0.68971E+00 |
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| 431 | <br>0.64587E+00 |
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| 432 | <br>0.61527E+00 |
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| 433 | <br>0.60741E+00</center> |
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| 434 | </td> |
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| 435 | |
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| 436 | <td ALIGN=CENTER VALIGN=TOP> |
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| 437 | <center> 1.001 |
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| 438 | <br>1.000 |
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| 439 | <br>1.000 |
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| 440 | <br>1.000</center> |
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| 441 | </td> |
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| 442 | </tr> |
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| 443 | </table></center> |
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| 444 | |
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| 445 | <br> |
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| 446 | <dir> |
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| 447 | <dir><i>Num</i>- line number; |
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| 448 | <br><i><font face="Symbol">l</font></i> - wavelength in micrometers; |
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| 449 | <br><font face="Symbol">s</font><i><sub>ex </sub></i>- extinction coefficient |
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| 450 | in km<sup>-1</sup>; |
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| 451 | <br><i><font face="Symbol">w</font></i>- single scattering albedo; |
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| 452 | <br><i><font face="Symbol">m</font></i>- asymmetry factor; |
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| 453 | <br><i>K<sub>a</sub></i>- normalization factor. |
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| 454 | <br> |
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| 455 | <p><br> |
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| 456 | <center> |
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| 457 | </dir> |
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| 458 | </dir> |
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| 459 | </body> |
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| 460 | </html> |
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