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Measurement on sets with a specific geometric shape can be of interest formany important applications (e.g. measurement along the isotherms in structuralengineering). In the present paper the properties of optimal designs forestimating the parameters of shifted Ornstein-Uhlenbeck sheets, that isGaussian two-variable random fields with exponential correlation structures,are investigated when the processes are observed on monotonic sets. Substantialdifferences are demonstrated between the cases when one is interested only intrend parameters and when the whole parameter set is of interest. Thetheoretical results are illustrated by computer experiments and simulatedexamples from the field of structure engineering. From the design point of viewthe most interesting finding of the paper is the loss of efficiency of theregular grid design compared to the optimal monotonic design.

Optimal designs for parameters of shifted Ornstein-Uhlenbeck sheets measured on monotonic sets