Implementation of a Gauss convoluted Pandel PDF for track reconstruction in neutrino telescopes

A probability distribution function is presented which provides a realistic description of the detection of scattered photons. The resulting probabilities can be described analytically by means of a superposition of several special functions. These exact expressions can be evaluated numerically only for small distances and limited time residuals, due to computer accuracy limitations. In this report, we provide approximations for the exact expressions in different regions of the distance–time residual space, defined by the detector geometry and the space–time scale of an event. These approximations can be evaluated numerically with a relative error with respect to the exact expression at the boundaries of less than 10−3.