## 3 Inherent Optical Properties (IOPs)
### Snow and Ice IOPs
#### 3.5.3 Impurities, Air Bubbles, Brine Pockets, and Snow  
If the volume fraction of impurities within a snow grain or brine pocket is not too large, which is the case for typical situations occurring in nature, scattering by impurities can be ignored, so that their effects can be included by simply adding the imaginary part \(\tilde{m}_{i,\text{imp}}\) of the refractive index for impurities to \(\tilde{m}_{i,\text{p}}\) in Equation (83). For typical impurities in snow and ice, the wavelength dependence of \(\tilde{m}_{i,\text{imp}}\) can be parameterized as [53]  
\[\tilde{m}_{i,\text{imp}}(\lambda)=\tilde{m}_{i,\text{imp}}(\lambda_{0})\,( \lambda_{0}/\lambda)^{\eta} \tag{89}\]  
where \(\eta\) would be close to zero for black carbon, but larger for other impurities, and \(\tilde{m}_{i,\text{imp}}(\lambda_{0}=440\,\text{nm})\) has values that depend on the type of impurity. Equation (89) is based on the observation that the absorption coefficient \(\alpha\) of non-algal impurities tend to have a smooth increase towards shorter wavelengths [54; 55; 56; 57], and \(\alpha\) is connected to the imaginary part of the refractive index through \(\alpha=4\pi\,\tilde{m}_{i,\text{imp}}/\lambda\). For snow, the number of snow grain particles per unit volume is \(N=\frac{1}{3}\pi r_{\text{eff}}^{2}\frac{\rho_{\text{s}}}{\rho_{\text{i}}}\), where \(r_{\text{eff}}\) is the effective particle radius, while \(\rho_{\text{s}}\) and \(\rho_{\text{i}}\) are the mass densities of snow and pure ice, respectively. The optical thickness and the single-scattering albedo can be calculated from Equations (87) and (88), using the refractive indices of pure ice [49] and impurities [Equation (89)]. We assume that sea ice consists of pure ice with embedded brine pockets, air bubbles, and impurities. To include the effects of the embedded components, we first calculate the absorption coefficient \(\alpha\) for sea ice [53]  
\[\alpha=\pi r_{\text{br}}^{2}N_{\text{br}}Q_{\alpha,\text{br}}+\left[1-\frac{4} {3}\pi r_{\text{br}}^{3}N_{\text{br}}-\frac{4}{3}\pi r_{\text{bu}}^{3}N_{\text {bu}}\right]\frac{4\pi(\tilde{m}_{i,\text{p}}+f_{\text{imp}}\tilde{m}_{i, \text{imp}})}{\lambda} \tag{90}\]  
where \(f_{\text{imp}}\) is the volume fraction of impurities, \(N_{\text{br}}\) and \(N_{\text{bu}}\) are the number concentrations of brine pockets and air bubbles, respectively, \(r_{\text{br}}\) and \(r_{\text{bu}}\) are the corresponding effective radii, and \(Q_{\alpha,\text{br}}\) is the absorption efficiency for brine pockets. The two terms on the right side of Equation (90) represent the absorption coefficients of brine pockets and surrounding ice (including impurities), respectively. In Equation (90), we have used the general relation \(\alpha=4\pi\,\tilde{m}_{i,\mathrm{p}}/\lambda\), where \(\lambda\) is the wavelength in vacuum, and the expression inside the square brackets is the volume fraction of the ice surrounding all brine pockets and bubbles. The air bubbles were assumed to be non-absorbing (\(Q_{a,\mathrm{bu}}=0\)), and the impurities were assumed to be uniformly distributed in the ice with \(\tilde{m}_{i,\mathrm{p}}\) and \(\tilde{m}_{i,\mathrm{imp}}\) being the imaginary parts of the refractive indices for pure ice and impurities, respectively. For brine pockets, which are in the liquid phase, the refractive index of sea water was used. The volume fraction \(f_{\mathrm{imp}}\) of impurities typically lies in the range between \(1\times 10^{-7}\) and \(1\times 10^{-5}\). The scattering coefficient \(\beta\) of sea ice is given by [53]  
\[\beta=\beta_{\mathrm{br}}+\beta_{\mathrm{bu}};\ \ \ \beta_{\mathrm{br}}=\pi r_{ \mathrm{br}}^{2}N_{\mathrm{br}}Q_{\beta,\mathrm{br}};\ \ \ \beta_{\mathrm{bu}}=\pi r_{ \mathrm{bu}}^{2}N_{\mathrm{bu}}Q_{\beta,\mathrm{bu}} \tag{91}\]  
where \(\beta_{\mathrm{br}}\) and \(\beta_{\mathrm{bu}}\) are the scattering coefficients for brine pockets and air bubbles, respectively, and \(Q_{\beta,\mathrm{br}}\) and \(Q_{\beta,\mathrm{bu}}\) are the corresponding scattering efficiencies.