## 3 Inherent Optical Properties (IOPs)
### Snow and Ice IOPs
#### 3.5.2 Fast, yet Accurate Parameterization of Snow/Ice IOPs  
Building on previous work [50; 51; 52], [PERSON] et al. [53] created a generic tool for computing snow/ice IOPs (\(\tau\), \(\omega\), and \(g\)). This tool can be used to generate wavelength-dependent ice/snow IOPs from ice/snow physical parameters: real and imaginary parts of the ice/snow refractive index, brine pocket concentration and effective size (sea ice), air bubble concentration and effective size (sea ice), volume fraction and absorption coefficient of sea ice impurities, asymmetry factors for scattering by snow grains, brine pockets, and air bubbles, and sea ice thickness. We can compute \(Q^{\prime}_{\alpha}(r)\), \(Q^{\prime}_{\beta}(r)\), and \(p_{\rm P}(\lambda,\Theta,r)\) using a Mie code, but evaluation of Equations (58)-(60) requires knowledge of the particle size distribution \(n(r)\), which is usually unknown. Equations (58)-(60) can be considerably simplified by making the following assumptions [53]:  
* The particle distribution is characterized by an effective radius [Equation (70)], which obviates the need for an integration over \(r\). * The particles are weakly absorbing, so that [51] \[Q^{\prime}_{\alpha}(r)\equiv Q^{\prime}_{\alpha}\approx\frac{16\pi r_{\rm eff} \,\bar{m}_{i,{\rm P}}}{3\lambda}\frac{1}{m_{\rm rel}}[m_{\rm rel}^{3}-(m_{\rm rel }^{2}-1)^{3/2}]\] (83) where \(\bar{m}_{i,{\rm P}}\) is the imaginary part of the refractive index of the particle, \(\lambda\) is the wavelength in vacuum, and \(m_{\rm rel}=\bar{m}_{r,{\rm P}}/\bar{m}_{r,{\rm med}}\) is the ratio of the real part of the refractive index of the particle (\(\bar{m}_{r,{\rm P}}\)) to that of the surrounding medium (\(\bar{m}_{r,{\rm med}}\)). * The particles are large compared to the wavelength (\(2\pi r/\lambda\gg 1\)) which implies \[Q^{\prime}_{\beta}(r)\equiv Q^{\prime}_{\beta}=2.\] (84) The scattering phase function may be represented by the one-parameter Henyey-Greenstein scattering phase function [see Equation (10)], which depends only on the asymmetry factor defined in Equation (7). With these assumptions, Equations (58)-(59) become: \[\alpha_{\rm P}(\lambda) =\alpha(\lambda)\frac{1}{m_{\rm rel}}[1-(m_{\rm rel}^{2}-1)^{3/2 }]f_{\rm V}\] (85) \[\beta_{\rm P}(\lambda) =\frac{3}{2}\frac{f_{\rm V}}{r_{\rm eff}}.\] (86) Here \(\alpha(\lambda)=4\pi\bar{m}_{i,{\rm P}}/\lambda\) is the absorption coefficient of the material of which the particle is composed, and \(f_{\rm V}=\frac{4\pi}{3}\int n(r)r^{3}dr\approx\frac{4}{3}\pi r_{\rm eff}^{3} n_{\rm e}\), where \(n_{\rm e}=\) number of particles per unit volume with radius \(r_{\rm eff}\). Since Equation (86) is identical to Equation (82), it is clear that \(f_{\rm V}\) represents the volume fraction of the particles as defined in Equation (80). For wavelengths \(\lambda\leq 1.2\)\(\upmu\)m, the absorption and scattering efficiency for snow grains, brine inclusions in sea ice, and air bubbles in ice may be parameterized by Equations (83) and (84), and the asymmetry factor \(g\) can be held constant with wavelength and set equal to 0.85, 0.89, and 0.997 for air bubbles, snow grains, and brine pockets, respectively. To extend the validity to NIR wavelengths, we may use the following modified parameterizations [53]:  
\[Q_{\alpha}=0.94[1-\exp(-Q_{\alpha}^{\prime}/0.94)];\qquad Q_{\beta}=2-Q_{ \alpha};\qquad g=g_{0}^{(1-Q_{\alpha})^{0.6}} \tag{87}\]  
where \(Q_{\alpha}^{\prime}\) is given by Equation (83). Here \(g\) is the asymmetry factor of the scattering phase function, and \(g_{0}\) is the asymmetry factor for non-absorbing particles. For large particles (\(r>\sim 50\)\(\upmu\)m) \(g_{0}\) depends only on the real part of the refractive index. For a medium consisting of several absorbing and scattering constituents the total absorption and scattering efficiencies are just the sum of those due to the separate constituents.