Order-of-scattering point spread and modulation transfer functions for natural waters

The point spread and modulation transfer functions for natural waters are derived using the small-angle approximation to radiative transfer theory. The functional forms are expanded into a summation of terms that represent each order-of-scattering contribution to the total. The beam spread function is shown to be a product of an angle function that depends only on the phase function of the medium and a weighting factor that depends only on the optical properties and depth. The modulation transfer function is similarly shown as a product of a function depending only on the spatial frequency and a weighting function. These results are compared with Monte Carlo calculations using two different phase functions, with excellent agreement. The results suggest the small-angle approximation to be valid over a much larger angular range than previously thought.