  Measurement on sets with a specific geometric shape can be of interest for
many important applications (e.g. measurement along the isotherms in structural
engineering). In the present paper the properties of optimal designs for
estimating the parameters of shifted Ornstein-Uhlenbeck sheets, that is
Gaussian two-variable random fields with exponential correlation structures,
are investigated when the processes are observed on monotonic sets. Substantial
differences are demonstrated between the cases when one is interested only in
trend parameters and when the whole parameter set is of interest. The
theoretical results are illustrated by computer experiments and simulated
examples from the field of structure engineering. From the design point of view
the most interesting finding of the paper is the loss of efficiency of the
regular grid design compared to the optimal monotonic design.
