  A method is presented for computing approximate expressions for the
stochastic force term $\xi_{ab}$ which appears in the Einstein-Langevin
equation of stochastic gravity. Within this framework, $\xi_{ab}$ is a
stochastic tensor field whose probability distribution mimics the probability
distribution of the fluctuations of the quantum stress tensor operator; it is
defined to be a random tensor field of zero mean whose correlation function is
given by the expectation value of the symmetrized two point function of the
stress energy fluctuation operator, called the noise kernel. Approximate
expressions are obtained by means of a truncated Karhunen-Loeve transform
defined on a random lattice of spacetime points. Due to the singular nature of
the noise kernel, a coarse graining procedure is used to regulate divergences;
as a result, the expressions obtained for $\xi_{ab}$ approximate values which
might be seen by a probe measuring fluctuations in the stress energy using a
sampling profile of finite width. Two realizations of $\xi_{ab}$ in Minkowski
spacetime for the conformally invariant quantum scalar field in the Minkowski
vacuum state are presented.
