  The mapping of dark matter clustering from real space to redshift space
introduces the anisotropic property to the measured density power spectrum in
redshift space, known as the redshift space distortion effect. The mapping
formula is intrinsically non-linear, which is complicated by the higher order
polynomials due to indefinite cross correlations between the density and
velocity fields, and the Finger-of-God effect due to the randomness of the
peculiar velocity field. Whilst the full higher order polynomials remain
unknown, the other systematics can be controlled consistently within the same
order truncation in the expansion of the mapping formula, as shown in this
paper. The systematic due to the unknown non-linear density and velocity fields
is removed by separately measuring all terms in the expansion directly using
simulations. The uncertainty caused by the velocity randomness is controlled by
splitting the FoG term into two pieces, 1) the "one-point" FoG term being
independent of the separation vector between two different points, and 2) the
"correlated" FoG term appearing as an indefinite polynomials which is expanded
in the same order as all other perturbative polynomials. Using 100 realizations
of simulations, we find that the Gaussian FoG function with only one
scale-independent free parameter works quite well, and that our new mapping
formulation accurately reproduces the observed 2-dimensional density power
spectrum in redshift space at the smallest scales by far, up to $k\sim
0.2h$Mpc, considering the resolution of future experiments.
