Analytical solution of beam spread function for ocean light radiative transfer

We develop a new method to analytically obtain the beam spread function (BSF) for light radiative transfer in oceanic environments. The BSF, which is defined as the lateral distribution of the (scalar) irradiance with increasing depth in response to a uni-directional beam emanating from a point source in an infinite ocean, must in general be obtained by solving the three-dimensional (3D) radiative transfer equation (RTE). By taking advantage of the highly forward-peaked scattering property of the ocean particles, we assume, for a narrow beam source, the dependence of radiance on polar angle and azimuthal angle is deliberately separated; only single scattering takes place in the azimuthal direction while multiple scattering still occurs in the polar direction. This assumption enables us to reduce the five-variable 3D RTE to a three-variable two-dimensional (2D) RTE. With this simplification, we apply Fourier spectral method to both spatial and angular variables so that we are able to analytically solve the 2D RTE and obtain the 2D BSF accordingly. Using the relations between 2D and 3D solutions acquired during the process of simplification, we are able to obtain the 3D BSF in explicit form. The 2D and 3D analytical solutions are validated by comparing with Monte Carlo radiative transfer simulations. The 2D analytical BSF agrees excellently with the Monte Carlo result. Despite assumptions of axial symmetry and spike-like azimuthal profile of the radiance in deriving the 3D BSF, the comparisons to numerical simulations are very satisfactory especially for limited optical depths (< O(5)) for single scattering albedo values typical in the ocean. The explicit form of the analytical BSF and the significant gain in computational efficiency (several orders higher) relative to RTE simulations make many forward and inverse problems in ocean optics practical for routine applications.