  When using optimal linear prediction to interpolate point observations of a
mean square continuous stationary spatial process, one often finds that the
interpolant mostly depends on those observations located nearest to the
predictand. This phenomenon is called the screening effect. However, there are
situations in which a screening effect does not hold in a reasonable asymptotic
sense, and theoretical support for the screening effect is limited to some
rather specialized settings for the observation locations. This paper explores
conditions on the observation locations and the process model under which an
asymptotic screening effect holds. A series of examples shows the difficulty in
formulating a general result, especially for processes with different degrees
of smoothness in different directions, which can naturally occur for
spatial-temporal processes. These examples lead to a general conjecture and two
special cases of this conjecture are proven. The key condition on the process
is that its spectral density should change slowly at high frequencies. Models
not satisfying this condition of slow high-frequency change should be used with
caution.
