Second-order scattering approximation of pulse vector radiative transfer equation

The problem of polarized light propagating through scattering media can be explained using the vector radiative transfer equation. This equation is an integro-differential equation and is well-known to be unsolvable analytically. One of the approximate solutions is discrete ordinates method which is based on the discretization of the Stokes parameters and the Mueller matrix. Although it produces accurate results, it requires a lot of computational resources. In addition, there are limitations on the calculation for angles that are very close to the optical axis. The solutions at these angles are necessary for some applications such as atmospheric imaging. First-order scattering approximation has been applied to mitigate the computational resource situation. It can also be used to calculate the solution at the angles that are very close to optical axis. However, it lacks information about the cross-polarization and it is inaccurate when light encounters more scattering events. Second-order scattering approximation provides more accurate solutions and offers some information about cross-polarization. We develop the first-order and second-order scattering approximations and their solutions for the pulse wave case. We investigate the second-order approximation solutions and compare them to the solution from the complete vector radiative transfer equation in several cases.