Designing environmental monitoring networks to measure extremes

Abstract

This paper discusses challenges arising in the design of networks for monitoring extreme values over the domain of a random environmental space-time field {X _ij } i = 1, . . . , I denoting site and j = 1, . . . denoting time (e.g. hour). The field of extremes for time span r over site domain i = 1, . . . ,I is given by \(\{Y_{i(r+1)}=\max_{j=k}^{k+n-1} X_{ij}\}\) for k = 1 + rn, r = 0, . . . ,. Such networks must not only measure extremes at the monitored sites but also enable their prediction at the non-monitored ones. Designing such a network poses special challenges that do not seem to have been generally recognized. One of these problems is the loss of spatial dependence between site responses in going from the environmental process to the field of extremes it generates. In particular we show empirically that the intersite covariance Cov(Y _i(r+1),Y _i′(r+1)) can generally decline toward zero as r increases, for site pairs i ≠ i′. Thus the measured extreme values may not predict the unmeasured ones very precisely. Consequently high levels of pollution exposure of a sensitive group (e.g. school children) located between monitored sites may be overlooked. This potential deficiency raises concerns about the adequacy of air pollution monitoring networks whose primary role is the detection of noncompliance with air quality standards based on extremes designed to protect human health. The need to monitor for noncompliance and thereby protect human health, points to other issues. How well do networks designed to monitor the field monitor their fields of extremes? What criterion should be used to select prospective monitoring sites when setting up or adding to a network? As the paper demonstrates by assessing an existing network, the answer to the first question is not well, at least in the case considered. To the second, the paper suggests a variety of plausible answers but shows through a simulation study, that they can lead to different optimum designs. The paper offers an approach that circumvents the dilemma posed by the answer to the second question. That approach models the field of extremes (suitably transformed) by a multivariate Gaussian-Inverse Wishart hierarchical Bayesian distribution. The adequacy of this model is empirically assessed in an application by finding the relative coverage frequency of the predictive credibility ellipsoid implied by its posterior distribution. The favorable results obtained suggest this posterior adequately describes that (transformed) field. Hence it can form the basis for designing an appropriate network. Its use is demonstrated by a hypothetical extension of an existing monitoring network. That foundation in turn enables a network to be designed of sufficient density (relative to cost) to serve its regulatory purpose.