  In this paper we perform a comparative study of the forward and backward
Liouville mapping applied to the modeling of ring-shaped and non-gyrotropic
velocity distribution functions of particles injected in a sheared
electromagnetic field. The test-kinetic method is used to compute the velocity
distribution function in various areas of a proton cloud moving in the vicinity
of a region with a sharp transition of the magnetic field and a non-uniform
electric field. In the forward approach the velocity distribution function is
computed for a two-dimensional spatial bin, while in the backward approach the
distribution function is averaged over a spatial bin with the same size as for
the forward method and using a two-dimensional trapezoidal integration scheme.
It is shown that the two approaches lead to similar results for spatial bins
where the velocity distribution function varies smoothly. On the other hand,
with bins covering regions of configuration space characterized by sharp
spatial gradients of the velocity distribution function, the forward and
backward approaches will generally provide different results.
