  The collision of a spherical grain with a granular bed is commonly
parametrized by the splash function, which provides the velocity of the
rebounding grain and the velocity distribution and number of ejected grains.
Starting from elementary geometric considerations and physical principles, like
momentum conservation and energy dissipation in inelastic pair collisions, we
derive a rebound parametrization for the collision of a spherical grain with a
granular bed. Combined with a recently proposed energy-splitting model [Ho
${\it et\ al.}$, Phys. Rev. E 85, 052301 (2012)] that predicts how the impact
energy is distributed among the bed grains, this yields a coarse-grained but
complete characterization of the splash as a function of the impact velocity
and the impactor-bed grain-size ratio. The predicted mean values of the rebound
angle, total and vertical restitution, ejection speed, and number of ejected
grains are in excellent agreement with experimental literature data and with
our own discrete-element computer simulations. We extract a set of analytical
asymptotic relations for shallow impact geometries, which can readily be used
in coarse-grained analytical modeling or computer simulations of geophysical
particle-laden flows.
