  We study parameter estimation for a linear parabolic second-order stochastic
partial differential equation (SPDE) in two space dimensions with a small
dispersion parameter using high frequency data with respect to time and space.
We set two types of $Q$-Wiener processes as a driving noise. We provide minimum
contrast estimators of the coefficient parameters of the SPDE appearing in the
coordinate process of the SPDE based on the thinned data in space, and
approximate the coordinate process based on the thinned data in time. Moreover,
we propose an estimator of the drift parameter using the fact that the
coordinate process is the Ornstein-Uhlenbeck process and statistical inference
for diffusion processes with a small noise.
