  Ultra-high energy cosmic rays generate extensive air showers in Earth's
atmosphere. A standard approach to reconstruct the energy of an ultra-high
energy cosmic rays is to sample the lateral profile of the particle density on
the ground of the air shower with an array of surface detectors. For cosmic
rays with large inclinations, this reconstruction is based on a model of the
lateral profile of the muon density observed on the ground, which is fitted to
the observed muon densities in individual surface detectors. The best models
for this task are derived from detailed Monte-Carlo simulations of the air
shower development. We present a phenomenological parametrization scheme which
allows to derive a model of the average lateral profile of the muon density
directly from a fit to a set of individual Monte-Carlo simulated air showers.
The model reproduces the detailed simulations with a high precision. As an
example, we generate a muon density model which is valid in the energy range
1e18 eV < E < 1e20 eV and the zenith angle range 60 deg < theta < 90 deg. We
will further demonstrate a way to speed up the simulation of such muon profiles
by three orders of magnitude, if only the muons in the shower are of interest.
