## 3 Inherent Optical Properties (IOPs)  
### General Definitions  
An _inherent optical property_ (IOP) depends only on the medium itself, and is independent of the ambient light field within the medium [4]. An _apparent optical property_ (AOP) depends also on the illumination, i.e., on light propagating in particular directions inside and outside the medium (Apparent optical properties (1) depend both on the medium (the IOPs) and on the geometric (directional) structure of the polarized radiance distribution, and (2) display enough regular features and stability to be useful descriptors of a water body [4]. Hence, a radiance or an irradiance would satisfy only the first part of the definition, while a radiance reflectance or irradiance reflectance, obtained by division of the radiance or the upward irradiance by the downward irradiance, would satisfy also the second part of the definition.). Two important IOPs are the absorption coefficient \(\alpha(z)\) and the scattering coefficient \(\beta(z)\) defined as [3]  
\[\alpha(z)=\frac{1}{I^{i}}\left(\frac{dI^{\alpha}}{dz}\right),\qquad\beta(z)= \frac{1}{I^{i}}\left(\frac{dI^{\beta}}{dz}\right)\qquad[\text{m}^{-1}]. \tag{1}\]Here \(I^{i}\) is the incident radiance entering a volume element \(dV=dAdz\) of the medium of cross sectional area \(dA\) and thickness \(dz\), and \(dI^{\alpha}>0\) and \(dI^{\beta}>0\), respectively, are the radiances that are absorbed and scattered in all directions as the light propagates the distance \(dz\) along the direction of the incident light. The extinction coefficient is given by \(\gamma(z)=\alpha(z)+\beta(z)\), and the single-scattering albedo is defined as \(\omega(z)\equiv\beta(z)/\gamma(z)\). The angular distribution of the scattered light is given in terms of the _volume scattering function_ (vsf), which is defined as  
\[\text{vsf}(z,\mathbf{\hat{\Omega}}^{\prime},\mathbf{\hat{\Omega}})=\frac{1}{I ^{i}}\frac{d^{2}I^{\beta}}{dzd\omega}=\frac{1}{I^{i}}\frac{d}{dz}\left(\frac{ dI^{\beta}}{d\omega}\right)\qquad[\text{m}^{-1}\,\text{sr}^{-1}]. \tag{2}\]  
Here \(\mathbf{\hat{\Omega}}^{\prime}\) and \(\mathbf{\hat{\Omega}}\) are unit vectors, and \(d^{2}I^{\beta}\) is the radiance scattered from an incident direction \(\mathbf{\hat{\Omega}}^{\prime}\) into a cone of solid angle \(d\omega\) around the direction \(\mathbf{\hat{\Omega}}\) as the light propagates the distance \(dz\) along the direction \(\mathbf{\hat{\Omega}}^{\prime}\) (see Figure 1). The plane spanned by \(\mathbf{\hat{\Omega}}^{\prime}\) and \(\mathbf{\hat{\Omega}}\) is called the _scattering plane_, and the _scattering angle_\(\Theta\) is given by \(\cos\Theta=\mathbf{\hat{\Omega}}^{\prime}\cdot\mathbf{\hat{\Omega}}\). Integration of Equation (2) over all scattering directions yields  
\[\beta(z) =\frac{1}{I^{i}}\frac{d}{dz}\int_{4\pi}\left(\frac{dI^{\beta}}{d \omega}\right)d\omega=\frac{1}{I^{i}}\left(\frac{dI^{\beta}}{dz}\right)\] \[=\int_{4\pi}\text{vsf}(z,\mathbf{\hat{\Omega}}^{\prime},\mathbf{ \hat{\Omega}})d\omega=\int_{0}^{2\pi}\int_{0}^{\pi}\text{vsf}(z,\cos\Theta, \phi)\sin\Theta d\Theta d\phi\qquad[\text{m}^{-1}] \tag{3}\]  
where \(\Theta\) and \(\phi\) are respectively the polar angle and the azimuth angle in a spherical coordinate system in which the polar axis is along \(\mathbf{\hat{\Omega}}^{\prime}\). As indicated in Equation (3), the volume scattering function \([\text{vsf}(z,\cos\Theta,\phi)]\) is generally a function of both \(\Theta\) and \(\phi\), but for randomly oriented scatterers one may assume that the scattering potential is spherically symmetric implying that there is no dependence on azimuth, so that \(\text{vsf}=\text{vsf}(z,\cos\Theta)\).