## 1 Introduction
### Notation  
Radiative transfer practitioners in the atmosphere, liquid water (ocean, lakes, rivers) and cryosphere (snow/ice) communities use different nomenclatures and terminologies. This situation can be confusing and frustrating to students and researchers addressing interdisciplinary problems in environmental optics. In this paper, we will adopt the notation of [3]. Letting \(z\) be the vertical position in the plane-parallel medium under consideration, and letting \(\Theta\) be the scattering angle, we will denote the:  
1. the absorption coefficient [m\({}^{-1}\)] by the letter \(\alpha(z)\);
2. the scattering coefficient [m\({}^{-1}\)] by the letter \(\beta(z)\);
3. the extinction coefficient [m\({}^{-1}\)] by the letter \(\gamma(z)=\alpha(z)+\beta(z)\);
4. the single-scattering albedo by \(\varpi(z)=\beta(z)/(\alpha(z)+\beta(z))\);
5. the volume scattering function [m\({}^{-1}\)sr\({}^{-1}\)] by vsf\((z,\cos\Theta,\phi)\) and the related scattering phase function (dimensionless) by \(p(z,\cos\Theta,\phi)\);
6. the scattering phase matrix (dimensionless) by \(\mathbf{P}_{S}(\Theta)\) in the Stokes vector representation \(\mathbf{I}_{S}=[I,Q,U,V]^{T}\) and by \(\mathbf{P}(\Theta)\) in the Stokes vector representation \(\mathbf{I}=[I_{\parallel},I_{\perp},U,V]^{T}\).  
The corresponding notation used in the Ocean Optics community is \(a\) instead of \(\alpha\), \(b\) instead of \(\beta\), and \(c\) instead of \(\gamma\). Since \(\alpha,\beta\), and \(\gamma\) are the three first letters in the Greek alphabet it should be easy to recall the connection with \(a\), \(b\), and \(c\).  
This tutorial review is organized as follows. In Section 2 the input parameters needed to describe the coupled system are specified. Then in Section 3 we describe the inherent optical properties (IOPs) of the two adjacent coupled slabs, consisting of absorption and scattering coefficients as well as scattering phase functions. In addition, one needs the scattering phase matrix for polarized radiation. For unpolarized radiation only one element of this matrix, namely the scattering phase function, is required. These IOPs appear in the radiative transfer equation (RTE) described in Section 4, where we also review how to formulate and solve the RTE for unpolarized as well as polarized radiation. In Section 5 we provide several examples of how the solution of the forward problem discussed in Section 4 can be used to solve the corresponding inverse problem. In the forward problem, the IOPs are assumed to be known, so that the solution of the RTE provides the total (and polarized) radiances. To solve the inverse problem we ask: given the measured total (and/or polarized) radiances can we determine the IOPs? The inverse problem is generally much more difficult to solve than the forward problem. It can be formulated as a classical, nonlinear minimization problem, which can be solved in an iterative manner. We will also demonstrate how neural networks can be used to help tackle the inverse problem in a reliable and efficient manner. Finally, in Section 6 we briefly discuss some remaining problems in Ocean Optics, while in Section 7 we provide a brief summary.