  The problem of estimating the parameters of a linear regression model
$Z(s,t)=m_1g_1(s,t)+ \cdots + m_pg_p(s,t)+U(s,t)$ based on observations of $Z$
on a spatial domain $G$ of special shape is considered, where the driving
process $U$ is a Gaussian random field and $g_1, \ldots, g_p$ are known
functions. Explicit forms of the maximum likelihood estimators of the
parameters are derived in the cases when $U$ is either a Wiener or a stationary
or nonstationary Ornstein-Uhlenbeck sheet. Simulation results are also
presented, where the driving random sheets are simulated with the help of their
Karhunen-Lo\`eve expansions.
