AVERAGE VOLCANIC -632.8 nm SYNTHETIC AVERAGE VOLCANIC SCATTERING MATRIX. F11_au is normalized so that its average over all directions equals unity. DATA ARE FREELY AVAILABLE UNDER THE REQUEST OF CITATION OF: Muñoz et al. JGR, 109, D16201, doi:10.1029/2004JD004684, 2004, AND Munoz et al. JQSRT, 113, 565-574, 2012. The Average Volcanic Scattering Matrix was obtained as follows. First, the average volcanic ash phase function, F11 was determined by averaging the 7 measured phase functions for volcanic ash particles presented in Munoz et al. JGR, 109, D16201, doi:10.1029/2004JD004684, 2004, and the two phase functions of Lokon, and Pinatubo volcanic ash at 632.8 nm presented by Volten et al., 2001, and Munoz et al., 2002. Since no scattering cross sections are available, the phase functions were averaged giving them equal weights. The normalization is the same as for the indivual phase function. Second, each measured element ratio was multiplied with its corresponding normalized phase function. Third, for each pair of indices (i,j) the elements Fi,j of the average scattering matrix were obtained by averaging the 9 corresponding elements. Finally, division by the average phase function yielded the element ratios of the average aerosol scattering matrix. The resulting average matrix satisfies the Cloude test at each measured scattering angle. To facilitate the use of the average scattering matrix for multiple scattering calculations we have constructed a Synthetic Average Volcanic Scattering Matrix based on the Average Volcanic Scattering Matrix. The Synthetic Average Volcanic Scattering Matrix is defined in the full range from 0 to 180 degrees. For the extension of the phase function, we have followed the procedure described by Liu et al.2003, that is based on the assumption that the diffraction forward scattering peak for small randomly oriented particles with moderate aspect ratios mainly depends on the size of the particles and is largely independent of their shapes and refractive indices. For the relative scattering matrix elements Fij/F11 a cubic spline extrapolation is used for both the forward and the backscattering direction.Values at exact forward and backward scattering were determined so that they satisfy the conditions given by Hovenier and van der Mee, 2000. In addition, the element ratios must run as smoothly as possible toward 0 and 180 degrees. In order to achieve this, a fit algorithm was applied that takes into account the experimental error bars in such a way that if the error is small, the measured value has a strong influence on the final solution. The resulting Synthetic Average Volcanic Scattering Matrix satisfies the Cloude (coherency matrix) test at all scattering angles. For the Synthetic Average Volcanic Scattering Matrix F11_au is normalized so that its average over all directions equals unity. Angle F11_au -F12/F11 F22/F11 F33/F11 F34/F11 F44/F11 0.0 8.77E+03 -0.00E+00 9.69E-01 9.69E-01 0.00E+00 9.48E-01 0.1 4.48E+03 -6.83E-05 9.69E-01 9.69E-01 -3.70E-04 9.48E-01 0.2 1.88E+03 -1.40E-04 9.69E-01 9.69E-01 -7.48E-04 9.48E-01 0.3 1.19E+03 -2.14E-04 9.68E-01 9.68E-01 -1.13E-03 9.48E-01 0.4 8.60E+02 -2.91E-04 9.68E-01 9.68E-01 -1.52E-03 9.48E-01 0.5 6.66E+02 -3.69E-04 9.68E-01 9.68E-01 -1.92E-03 9.49E-01 0.6 5.35E+02 -4.50E-04 9.68E-01 9.68E-01 -2.31E-03 9.49E-01 0.7 4.43E+02 -5.32E-04 9.68E-01 9.67E-01 -2.71E-03 9.49E-01 0.8 3.73E+02 -6.15E-04 9.68E-01 9.67E-01 -3.12E-03 9.49E-01 0.9 3.17E+02 -6.98E-04 9.67E-01 9.67E-01 -3.52E-03 9.49E-01 1.0 2.75E+02 -7.82E-04 9.67E-01 9.67E-01 -3.93E-03 9.49E-01 1.1 2.40E+02 -8.66E-04 9.67E-01 9.66E-01 -4.33E-03 9.50E-01 1.2 2.13E+02 -9.50E-04 9.67E-01 9.66E-01 -4.74E-03 9.50E-01 1.3 1.90E+02 -1.03E-03 9.67E-01 9.66E-01 -5.14E-03 9.50E-01 1.4 1.71E+02 -1.11E-03 9.66E-01 9.65E-01 -5.54E-03 9.50E-01 1.5 1.55E+02 -1.20E-03 9.66E-01 9.65E-01 -5.94E-03 9.50E-01 1.6 1.42E+02 -1.27E-03 9.66E-01 9.65E-01 -6.34E-03 9.50E-01 1.7 1.30E+02 -1.35E-03 9.66E-01 9.65E-01 -6.72E-03 9.51E-01 1.8 1.20E+02 -1.42E-03 9.66E-01 9.64E-01 -7.11E-03 9.51E-01 1.9 1.11E+02 -1.50E-03 9.66E-01 9.64E-01 -7.49E-03 9.51E-01 2.0 1.03E+02 -1.56E-03 9.65E-01 9.64E-01 -7.86E-03 9.51E-01 2.1 9.64E+01 -1.63E-03 9.65E-01 9.64E-01 -8.22E-03 9.51E-01 2.2 9.06E+01 -1.69E-03 9.65E-01 9.64E-01 -8.57E-03 9.51E-01 2.3 8.52E+01 -1.75E-03 9.65E-01 9.63E-01 -8.92E-03 9.51E-01 2.4 8.03E+01 -1.80E-03 9.64E-01 9.63E-01 -9.25E-03 9.51E-01 2.5 7.57E+01 -1.85E-03 9.64E-01 9.63E-01 -9.58E-03 9.51E-01 2.6 7.17E+01 -1.89E-03 9.64E-01 9.63E-01 -9.89E-03 9.51E-01 2.7 6.81E+01 -1.93E-03 9.64E-01 9.63E-01 -1.02E-02 9.51E-01 2.8 6.47E+01 -1.96E-03 9.64E-01 9.63E-01 -1.05E-02 9.51E-01 2.9 6.15E+01 -1.98E-03 9.63E-01 9.63E-01 -1.07E-02 9.51E-01 3.0 5.87E+01 -2.00E-03 9.63E-01 9.63E-01 -1.10E-02 9.51E-01 4.0 6.72E+01 -2.00E-03 9.61E-01 9.64E-01 -1.30E-02 9.52E-01 5.0 4.42E+01 -2.00E-03 9.63E-01 9.64E-01 -1.50E-02 9.57E-01 6.0 3.21E+01 -1.00E-03 9.65E-01 9.62E-01 -1.70E-02 9.42E-01 7.0 2.41E+01 1.00E-03 9.66E-01 9.61E-01 -1.70E-02 9.38E-01 8.0 1.87E+01 3.00E-03 9.64E-01 9.57E-01 -1.70E-02 9.30E-01 9.0 1.52E+01 4.00E-03 9.61E-01 9.54E-01 -1.80E-02 9.25E-01 10.0 1.28E+01 4.00E-03 9.58E-01 9.51E-01 -1.60E-02 9.26E-01 15.0 6.36E+00 7.00E-03 9.42E-01 9.30E-01 -2.10E-02 8.99E-01 20.0 3.97E+00 7.00E-03 9.24E-01 9.08E-01 -1.80E-02 8.67E-01 25.0 2.76E+00 9.00E-03 9.06E-01 8.94E-01 -1.60E-02 8.46E-01 30.0 2.04E+00 1.30E-02 8.88E-01 8.68E-01 -9.00E-03 8.19E-01 35.0 1.56E+00 1.50E-02 8.64E-01 8.45E-01 5.00E-03 7.89E-01 40.0 1.23E+00 2.00E-02 8.40E-01 8.11E-01 1.10E-02 7.56E-01 45.0 9.91E-01 3.00E-02 8.11E-01 7.74E-01 2.30E-02 7.27E-01 50.0 8.24E-01 3.70E-02 7.77E-01 7.32E-01 4.20E-02 6.93E-01 55.0 6.96E-01 4.90E-02 7.45E-01 6.91E-01 6.10E-02 6.61E-01 60.0 6.02E-01 6.40E-02 7.10E-01 6.34E-01 7.50E-02 6.21E-01 65.0 5.30E-01 7.40E-02 6.70E-01 5.75E-01 9.20E-02 5.83E-01 70.0 4.73E-01 8.20E-02 6.32E-01 5.19E-01 1.06E-01 5.45E-01 75.0 4.28E-01 9.10E-02 5.96E-01 4.54E-01 1.18E-01 5.06E-01 80.0 3.94E-01 1.01E-01 5.56E-01 3.94E-01 1.26E-01 4.66E-01 85.0 3.65E-01 1.08E-01 5.19E-01 3.30E-01 1.38E-01 4.32E-01 90.0 3.43E-01 1.13E-01 4.88E-01 2.61E-01 1.45E-01 3.95E-01 95.0 3.22E-01 1.13E-01 4.52E-01 2.01E-01 1.41E-01 3.60E-01 100.0 3.06E-01 1.11E-01 4.23E-01 1.47E-01 1.43E-01 3.19E-01 105.0 2.92E-01 1.12E-01 3.96E-01 7.60E-02 1.44E-01 2.76E-01 110.0 2.79E-01 1.05E-01 3.72E-01 1.80E-02 1.29E-01 2.50E-01 115.0 2.69E-01 9.80E-02 3.55E-01 -3.40E-02 1.27E-01 2.09E-01 120.0 2.59E-01 8.90E-02 3.38E-01 -9.10E-02 1.20E-01 1.74E-01 125.0 2.53E-01 8.70E-02 3.38E-01 -1.34E-01 1.09E-01 1.34E-01 130.0 2.49E-01 7.30E-02 3.24E-01 -1.92E-01 9.80E-02 9.80E-02 135.0 2.47E-01 6.00E-02 3.21E-01 -2.16E-01 7.90E-02 6.70E-02 140.0 2.45E-01 4.50E-02 3.26E-01 -2.48E-01 7.60E-02 3.40E-02 145.0 2.45E-01 2.90E-02 3.37E-01 -2.98E-01 6.20E-02 8.00E-03 150.0 2.47E-01 1.80E-02 3.53E-01 -3.11E-01 4.80E-02 -2.00E-02 155.0 2.47E-01 8.00E-03 3.62E-01 -3.49E-01 2.90E-02 -5.20E-02 160.0 2.53E-01 -7.00E-03 3.84E-01 -3.58E-01 2.00E-02 -8.20E-02 165.0 2.59E-01 -2.20E-02 4.09E-01 -3.83E-01 5.00E-03 -1.06E-01 170.0 2.71E-01 -1.90E-02 4.20E-01 -3.72E-01 -1.10E-02 -1.20E-01 171.0 2.77E-01 -1.80E-02 4.24E-01 -4.15E-01 -1.70E-02 -1.20E-01 172.0 2.88E-01 -1.70E-02 4.28E-01 -4.07E-01 -1.60E-02 -1.18E-01 173.0 2.92E-01 -2.00E-03 4.42E-01 -4.10E-01 -2.10E-02 -1.21E-01 173.1 2.92E-01 -4.33E-04 4.44E-01 -4.11E-01 -2.15E-02 -1.21E-01 173.2 2.92E-01 1.05E-03 4.45E-01 -4.12E-01 -2.20E-02 -1.21E-01 173.3 2.92E-01 2.44E-03 4.47E-01 -4.13E-01 -2.25E-02 -1.22E-01 173.4 2.93E-01 3.75E-03 4.48E-01 -4.15E-01 -2.29E-02 -1.22E-01 173.5 2.93E-01 4.97E-03 4.50E-01 -4.16E-01 -2.32E-02 -1.22E-01 173.6 2.93E-01 6.11E-03 4.52E-01 -4.17E-01 -2.35E-02 -1.22E-01 173.7 2.94E-01 7.18E-03 4.53E-01 -4.19E-01 -2.38E-02 -1.22E-01 173.8 2.94E-01 8.17E-03 4.55E-01 -4.20E-01 -2.40E-02 -1.22E-01 173.9 2.94E-01 9.08E-03 4.56E-01 -4.22E-01 -2.42E-02 -1.22E-01 174.0 2.95E-01 9.92E-03 4.58E-01 -4.23E-01 -2.44E-02 -1.22E-01 174.1 2.95E-01 1.07E-02 4.59E-01 -4.24E-01 -2.45E-02 -1.21E-01 174.2 2.95E-01 1.14E-02 4.61E-01 -4.26E-01 -2.45E-02 -1.21E-01 174.3 2.96E-01 1.20E-02 4.62E-01 -4.28E-01 -2.46E-02 -1.21E-01 174.4 2.96E-01 1.26E-02 4.64E-01 -4.29E-01 -2.46E-02 -1.21E-01 174.5 2.96E-01 1.31E-02 4.65E-01 -4.31E-01 -2.45E-02 -1.20E-01 174.6 2.97E-01 1.35E-02 4.67E-01 -4.32E-01 -2.45E-02 -1.20E-01 174.7 2.97E-01 1.39E-02 4.68E-01 -4.34E-01 -2.43E-02 -1.20E-01 174.8 2.97E-01 1.42E-02 4.69E-01 -4.36E-01 -2.42E-02 -1.19E-01 174.9 2.98E-01 1.45E-02 4.71E-01 -4.38E-01 -2.40E-02 -1.19E-01 175.0 2.98E-01 1.47E-02 4.72E-01 -4.39E-01 -2.38E-02 -1.18E-01 175.1 2.99E-01 1.49E-02 4.74E-01 -4.41E-01 -2.36E-02 -1.18E-01 175.2 2.99E-01 1.50E-02 4.75E-01 -4.43E-01 -2.34E-02 -1.17E-01 175.3 2.99E-01 1.51E-02 4.77E-01 -4.45E-01 -2.31E-02 -1.17E-01 175.4 3.00E-01 1.51E-02 4.78E-01 -4.47E-01 -2.28E-02 -1.16E-01 175.5 3.00E-01 1.51E-02 4.79E-01 -4.49E-01 -2.25E-02 -1.16E-01 175.6 3.01E-01 1.50E-02 4.81E-01 -4.50E-01 -2.21E-02 -1.15E-01 175.7 3.01E-01 1.49E-02 4.82E-01 -4.52E-01 -2.17E-02 -1.14E-01 175.8 3.01E-01 1.48E-02 4.84E-01 -4.54E-01 -2.13E-02 -1.14E-01 175.9 3.02E-01 1.46E-02 4.85E-01 -4.56E-01 -2.09E-02 -1.13E-01 176.0 3.02E-01 1.44E-02 4.86E-01 -4.58E-01 -2.05E-02 -1.12E-01 176.1 3.03E-01 1.41E-02 4.88E-01 -4.60E-01 -2.00E-02 -1.11E-01 176.2 3.03E-01 1.38E-02 4.89E-01 -4.62E-01 -1.96E-02 -1.11E-01 176.3 3.04E-01 1.35E-02 4.90E-01 -4.64E-01 -1.91E-02 -1.10E-01 176.4 3.04E-01 1.32E-02 4.92E-01 -4.66E-01 -1.86E-02 -1.09E-01 176.5 3.05E-01 1.29E-02 4.93E-01 -4.68E-01 -1.81E-02 -1.08E-01 176.6 3.05E-01 1.25E-02 4.94E-01 -4.70E-01 -1.76E-02 -1.07E-01 176.7 3.05E-01 1.21E-02 4.96E-01 -4.72E-01 -1.70E-02 -1.07E-01 176.8 3.06E-01 1.17E-02 4.97E-01 -4.75E-01 -1.65E-02 -1.06E-01 176.9 3.06E-01 1.12E-02 4.98E-01 -4.77E-01 -1.59E-02 -1.05E-01 177.0 3.07E-01 1.08E-02 5.00E-01 -4.79E-01 -1.54E-02 -1.04E-01 177.1 3.07E-01 1.03E-02 5.01E-01 -4.81E-01 -1.48E-02 -1.03E-01 177.2 3.08E-01 9.84E-03 5.02E-01 -4.83E-01 -1.42E-02 -1.02E-01 177.3 3.08E-01 9.36E-03 5.04E-01 -4.85E-01 -1.36E-02 -1.01E-01 177.4 3.09E-01 8.87E-03 5.05E-01 -4.87E-01 -1.31E-02 -1.01E-01 177.5 3.09E-01 8.37E-03 5.06E-01 -4.89E-01 -1.25E-02 -9.97E-02 177.6 3.10E-01 7.88E-03 5.08E-01 -4.91E-01 -1.19E-02 -9.88E-02 177.7 3.10E-01 7.38E-03 5.09E-01 -4.93E-01 -1.13E-02 -9.80E-02 177.8 3.10E-01 6.88E-03 5.10E-01 -4.96E-01 -1.07E-02 -9.71E-02 177.9 3.11E-01 6.38E-03 5.12E-01 -4.98E-01 -1.01E-02 -9.62E-02 178.0 3.11E-01 5.89E-03 5.13E-01 -5.00E-01 -9.54E-03 -9.53E-02 178.1 3.12E-01 5.41E-03 5.14E-01 -5.02E-01 -8.96E-03 -9.45E-02 178.2 3.12E-01 4.93E-03 5.16E-01 -5.04E-01 -8.38E-03 -9.36E-02 178.3 3.13E-01 4.46E-03 5.17E-01 -5.06E-01 -7.81E-03 -9.27E-02 178.4 3.13E-01 4.01E-03 5.18E-01 -5.08E-01 -7.24E-03 -9.19E-02 178.5 3.14E-01 3.57E-03 5.20E-01 -5.10E-01 -6.69E-03 -9.10E-02 178.6 3.14E-01 3.15E-03 5.21E-01 -5.12E-01 -6.14E-03 -9.02E-02 178.7 3.15E-01 2.74E-03 5.22E-01 -5.14E-01 -5.60E-03 -8.94E-02 178.8 3.15E-01 2.36E-03 5.24E-01 -5.16E-01 -5.08E-03 -8.85E-02 178.9 3.16E-01 1.99E-03 5.25E-01 -5.18E-01 -4.56E-03 -8.77E-02 179.0 3.16E-01 1.65E-03 5.26E-01 -5.21E-01 -4.06E-03 -8.69E-02 179.1 3.16E-01 1.34E-03 5.28E-01 -5.23E-01 -3.57E-03 -8.62E-02 179.2 3.17E-01 1.05E-03 5.29E-01 -5.25E-01 -3.10E-03 -8.54E-02 179.3 3.17E-01 7.97E-04 5.30E-01 -5.27E-01 -2.64E-03 -8.47E-02 179.4 3.18E-01 5.72E-04 5.32E-01 -5.29E-01 -2.21E-03 -8.39E-02 179.5 3.18E-01 3.82E-04 5.33E-01 -5.30E-01 -1.79E-03 -8.32E-02 179.6 3.19E-01 2.26E-04 5.34E-01 -5.32E-01 -1.39E-03 -8.25E-02 179.7 3.19E-01 1.09E-04 5.36E-01 -5.34E-01 -1.01E-03 -8.19E-02 179.8 3.19E-01 3.05E-05 5.37E-01 -5.36E-01 -6.47E-04 -8.12E-02 179.9 3.20E-01 -6.34E-06 5.39E-01 -5.38E-01 -3.11E-04 -8.06E-02 180.0 3.20E-01 1.22E-07 5.40E-01 -5.40E-01 1.28E-06 -8.00E-02