Spatial sampling design based on convex design ideas and using external drift variables for a rainfall monitoring network in Pakistan

Spatial sampling design is concerned with the optimal allocation of samples to spatial coordinates in order to improve in a well-defined sense the estimation and prediction of spatial random fields. Unfortunately, objective functions in spatial sampling design seem to be so complicated so far that most often stochastic search algorithms are used to get these design criteria optimized. Our intention is to show that the minimization of the average kriging variance design criterion shows a mathematically tractable structure when considering the random field as a linear regression model with infinitely many random coefficients. Either the Karhunen–Loeve expansion or the polar spectral representation of the random field may be used to get such a favourable representation. Well-known convex experimental design theory may be applied then to this high dimensional cosine-sine-Bessel surface harmonics random coefficients regression model to calculate spatial sampling designs. We study a monitoring network for rainfall during the monsoon in Pakistan and consider both the optimal deletion and subsequent addition of monitoring stations from/to this network. Only deterministic optimization algorithms and no stochastic search algorithms are used for the task of network optimization. As external drift variables determining the rainfall trend wind, humidity and elevation are considered.