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 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.