## 3 Inherent Optical Properties (IOPs)
### Scattering Phase Function for Particles  
Measurements have shown that the particle size distribution (PSD) function in oceanic water can be accurately described by an inverse power law (Junge distribution) \(F(r)=C_{r}/r^{\xi}\), where \(F(r)\) is the number of particles per unit volume per unit bin width, and \(r\) [\(\mu\)m] is the radius of the assumed spherical particle. \(C_{r}\) [cm\({}^{-3}\cdot\mu\)m\({}^{\xi-1}\)] is the Junge coefficient, and \(\xi\) is the PSD slope, which typically varies between 3.0 and 5.0 [65; 66]. By assuming an inverse power law (Junge distribution) for the PSD, [67] derived an analytic expression for the scattering phase function of oceanic water (hereafter referred to as the FF scattering phase function), given by [64]  
\[p_{\text{FF}}(\Theta) = \frac{1}{4\pi(1-\delta)^{2\delta^{\
u}}}\Bigg{\{}\
u(1-\delta)-(1 -\delta^{\
u})+\frac{4}{u^{2}}[\delta(1-\delta^{\
u})-\
u(1-\delta)]\Bigg{\}} \tag{102}\] \[+\ \frac{1-\delta^{\
u}_{180}}{16\pi(\delta_{180}-1)\delta^{\
u} _{180}}[3\cos^{2}\Theta-1]\]  
where \(\
u=0.5(3-\xi)\), \(u=2\sin(\Theta/2)\), \(\delta\equiv\delta(\Theta)=\frac{u^{2}}{3(h_{\text{r}}-1)^{2}}\), \(\delta_{180}=\delta(\Theta=180^{\circ})=\frac{4}{3(h_{\text{r}}-1)^{2}}\), \(\Theta\) is the scattering angle, and \(\tilde{m}_{\text{r}}\) is the real part of the refractive index.  
Integrating \(p_{\text{FF}}(\Theta)\) over the backward hemisphere (setting \(x=\cos\Theta\)), one obtains the backscattering ratio or backscatter fraction defined in Equation (8) [64]  
\[b_{\text{FF}} =\frac{1}{2}\int_{\pi/2}^{\pi}p_{\text{FF}}(\cos\Theta)\sin \Theta d\Theta=\frac{1}{2}\int_{0}^{1}p_{\text{FF}}(-x)dx\] \[=1-\frac{1-\delta^{\
u+1}_{90}-0.5(1-\delta^{\
u}_{90})}{(1- \delta_{90})\delta^{\
u}_{90}} \tag{103}\]  
where \(\delta_{90}=\delta(\Theta=90^{\circ})=\frac{4}{3(h_{\text{r}}-1)^{2}}\sin^{2} (45^{\circ})=\frac{2}{3(h_{\text{r}}-1)^{2}}\). Equation (103) can be solved for \(\
u\) in terms of \(b_{\text{FF}}\) and \(\delta_{90}\), implying that \(\
u\) and thus \(\xi\) can be determined if the real part of the refractive index \(\tilde{m}_{\text{r}}\) and the backscatter ratio \(b_{\text{FF}}\) are specified. As a consequence, the FF scattering phase function can be evaluated from a measured value of \(b_{\text{FF}}\) if the real part of the refractive index \(\tilde{m}_{\text{r}}\) is known.  
As already mentioned, in the CCRR bio-optical model, the Petzold scattering phase function with a backscattering ratio of 0.019 is used to represent mineral (non-algal) particles. These scattering phase functions are shown in Figure 2 together with the Rayleigh scattering phase function, which represents scattering by water molecules.  
A moment-fitting program [68] may be used to create Legendre expansion coefficients \(\chi_{\ell,\text{PET}}\) and \(\chi_{\ell,FF}\) for the Petzold and FF scattering phase functions. Hence, the total scattering phase function Legendre expansion coefficients are given by:  
\[\chi_{\ell}=\frac{\beta_{\text{Pig}}(\lambda)\chi_{\ell,\text{FF}}+\beta_{\text {MIN}}(\lambda)\chi_{\ell,\text{PET}}+\beta_{\text{w}}(\lambda)\chi_{\ell, \text{water}}}{\beta_{\text{pig}}(\lambda)+\beta_{\text{MIN}}(\lambda)+\beta_{ \text{w}}(\lambda)}. \tag{104}\]  
Thus, to use the CCRR bio-optical model one must specify the three input parameters CHL, MIN, and \(\alpha_{\text{CDOM}}(443)\).