AOMC: Radiative Transfer Theory

This document provides a summary of the theoretical basis for the AOMC model.

Theory of Radiative Transfer

The propagation of light through an aquatic medium is governed by the Radiative Transfer Equation (RTE). The RTE accounts for the various ways light can be absorbed and scattered as it interacts with water and its dissolved and particulate constituents. The key to solving the RTE lies in characterizing the Inherent Optical Properties (IOPs) of the medium. IOPs are properties that depend only on the substances in the medium, not on the ambient light field.

Inherent Optical Properties (IOPs)

The primary IOPs that determine the fate of light are:

Absorption and Scattering

• Absorption Coefficient (\(a\)): The fraction of radiant energy absorbed by the medium per unit path length (units: m⁻¹).
• Scattering Coefficient (\(b\)): The fraction of radiant energy scattered by the medium per unit path length (units: m⁻¹).

These two properties combine to form the beam attenuation coefficient (\(c\)), which represents the total loss of energy from a light beam due to both absorption and scattering: \[ c(\lambda) = a(\lambda) + b(\lambda) \]

The single-scattering albedo (\(\omega_0\)) describes the probability that a photon will be scattered rather than absorbed upon interaction: \[ \omega_0(\lambda) = \frac{b(\lambda)}{c(\lambda)} \]

Additivity of IOPs

For a medium containing multiple constituents (e.g., pure water, phytoplankton, sediments), the bulk IOPs are the sum of the contributions from each constituent. The contribution of each constituent is its concentration (\(C_i\)) multiplied by its specific absorption or scattering coefficient (\(a_i^*\) or \(b_i^*\)).

The bulk absorption coefficient is given by: \[ a(\lambda) = a_w(\lambda) + \sum_{i=1}^{N} a_i^*(\lambda)C_i \] The bulk scattering coefficient is given by: \[ b(\lambda) = b_w(\lambda) + \sum_{i=1}^{N} b_i^*(\lambda)C_i \] where the subscript \(w\) refers to pure water and \(i\) refers to the i-th constituent.

The Volume Scattering Function (VSF)

While the scattering coefficient \(b\) describes the total amount of scattering, the Volume Scattering Function (VSF), \(\beta(\psi, \lambda)\), describes the angular distribution of the scattered light. It is defined as the radiant intensity scattered from a small volume element in a given direction \(\psi\) per unit of incident irradiance (units: m⁻¹ sr⁻¹).

The scattering coefficient \(b\) is the integral of the VSF over all solid angles: \[ b(\lambda) = 2\pi \int_{0}^{\pi} \beta(\psi, \lambda) \sin(\psi) d\psi \] A normalized version of the VSF, known as the scattering phase function (\(\tilde{\beta}\)), gives the probability distribution of a scattering event at a given angle: \[ \tilde{\beta}(\psi, \lambda) = \frac{\beta(\psi, \lambda)}{b(\lambda)} \]