| We examine sampling designs that will be used to interpolate a spatial random field. We compare existing sampling methods under various spatial models, as well as develop a Bayesian sampling method when the field is not fully specified.; We investigate the fields that are well modelled, with determined parameters, with the objective of determining a best sampling design for future use. We develop designs for fields that assume either stationary variance function or a step variance function of the location. Sampling designs are selected with the objective of minimizing the average estimation error or maximizing the entropy of the sampling set. Designs selected by sequential and annealing algorithms are compared. We present circumstances in which it might be beneficial to choose one method over another, especially in relation to the size of the sampling set. An example of predicting maximum temperatures in Mojave desert is presented.; For a field that has not been extensively studied and is not well characterized, a different approach may be appropriate. We develop the Bayesian sequential sampling algorithm and contrast it to the prior sequential method, for fields that have step variance function. With the newly developed Bayesian method, parameters of the field are updated with each new sampling location, and used in the further selection of sampling locations. We compare several updating methods, including MLE, prior mean or mean of likelihood, and posterior mean, given the sampled data. Designs selected with this sequential method, using posterior mean updating of the field parameters, have the least estimation error. The predicted error also most accurately reflects the realized estimation error.; We further derive designs for fields with a more complex variance function. We also present a discussion of how much improvement in the error prediction one can really expect depending on what is known about the field parameters. The robustness of the selection algorithms is examined when the variance function is mis-specified. We apply these methods to predict the maximum daily temperatures in the "Four Corners" US states of Utah, Arizona, New Mexico and Colorado. |
