  We present numerical simulations on propagation of Ultra-High Energy Cosmic
Rays (UHECRs) above $10^{19}$ eV in a structured extragalactic magnetic field
(EGMF) and simulate their arrival distributions at the earth. We use the IRAS
PSCz catalogue in order to construct a model of the EGMF and source models of
UHECRs, both of which reproduce the local structures observed around the Milky
Way. We also consider modifications of UHECR arrival directions by the galactic
magnetic field. We follow an inverse process of their propagation from the
earth and record the trajectories. This enables us to calculate only
trajectories of UHECRs arriving at the earth, which saves the CPU time. From
these trajectories and our source models, we construct arrival distributions of
UHECRs and calculate the harmonic amplitudes and the two point correlation
functions of them. We estimate number density of sources which reproduces the
Akeno Ground Air Shower Array (AGASA) observation best. As a result, we find
that the most appropriate number density of the sources is $\sim 5 \times
10^{-6}$ Mpc$^{-3}$. This constrains the source candidates of UHECRs. We also
demonstrate skymaps of their arrival distribution with the event number
expected by future experiments and examine how the EGMF affects their arrival
distribution. A main result is diffusion of clustering events which are
obtained from calculations in the absence of the EGMF. This tendency allows us
to reproduce the observed two point correlation function better.
