  This article focuses on the estimation and design aspects of a bivariate
collocated cokriging experiment. For a large class of covariance matrices, a
linear dependency criterion is identified, which allows the best linear
unbiased estimator of the primary variable in a bivariate collocated cokriging
setup to reduce to a univariate kriging estimator. Exact optimal designs for
efficient prediction for such simple and ordinary reduced cokriging models with
one-dimensional inputs are determined. Designs are found by minimizing the
maximum and the integrated prediction variance, where the primary variable is
an Ornstein-Uhlenbeck process. For simple and ordinary cokriging models with
known covariance parameters, the equispaced design is shown to be optimal for
both criterion functions. The more realistic scenario of unknown covariance
parameters is addressed by assuming prior distributions on the parameter
vector, thus adopting a Bayesian approach to the design problem. The equispaced
design is proved to be the Bayesian optimal design for both criteria. The work
is motivated by designing an optimal water monitoring system for an Indian
river.
