  The paper covers the design and analysis of experiments to discriminate
between two Gaussian process models, such as those widely used in computer
experiments, kriging, sensor location and machine learning. Two frameworks are
considered. First, we study sequential constructions, where successive design
(observation) points are selected, either as additional points to an existing
design or from the beginning of observation. The selection relies on the
maximisation of the difference between the symmetric Kullback Leibler
divergences for the two models, which depends on the observations, or on the
mean squared error of both models, which does not. Then, we consider static
criteria, such as the familiar log-likelihood ratios and the Fr\'echet distance
between the covariance functions of the two models. Other distance-based
criteria, simpler to compute than previous ones, are also introduced, for
which, considering the framework of approximate design, a necessary condition
for the optimality of a design measure is provided. The paper includes a study
of the mathematical links between different criteria and numerical
illustrations are provided.
