Exact optimisation of spatiotemporal monitoring networks by p‐splines with applications in groundwater assessment

This paper develops methods to optimise the sampling strategy for monitoring networks which have fixed locations with regular sampling but with only a proportion of these locations to be used on each sampling occasion. This creates the need for a dynamic spatiotemporal sampling design which makes optimal choices of the locations to be sampled on each occasion. This is a commonly occurring scenario in many environmental settings where there is an existing network of monitoring stations and sampling can be expensive. The particular context of optimisation of an existing groundwater monitoring network is discussed in the paper. The standard design criteria of integrated variance (IV) and variance of the integral (VI) are adapted to the spatiotemporal setting. p‐spline models are shown to allow exact computation of IV and VI, in the case of the additive errors, and a very good approximation of IV in the case of multiplicative errors. The speed of these exact computations allows the globally optimal sampling design to be identified efficiently. In the standard case of additive errors, the design criteria are able to exploit information across time. The only information needed is the location of sampling points, not the values sampled. This contrasts with the case of multiplicative errors where the design criteria are also influenced by the observed response data. Simulated and real examples are used to illustrate the results throughout.