  We study the effective reduction for a nonlocal stochastic partial
differential equation with oscillating coefficients. The nonlocal operator in
this stochastic partial differential equation is the generator of non-Gaussian
L\'{e}vy processes, with either \textbf{integrable} or \textbf{non-integrable}
jump kernels. We examine the limiting behavior of this equation as a scaling
parameter tends to zero, and derive a reduced (local or nonlocal) effective
equation. In particular, this work leads to an effective reduction for a data
assimilation system with L\'{e}vy noise, by examining the corresponding
nonlocal Zakai stochastic partial differential equation. We show that the
probability density for the reduced data assimilation system approximates that
for the original system.
