chapter  4
26 Pages

Geostatistical Space-Time Models, Stationarity, Separability, and Full Symmetry

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.1 Introduction

Environmental and geophysical processes such as atmospheric pollutant concentrations, precipitation ﬁelds and surface winds are characterized by spatial and temporal variability. In view of the prohibitive costs of spatially and temporally dense monitoring networks, one often aims to develop a statistical

model in continuous space and time, based on observations at a limited number of monitoring stations. Examples include environmental monitoring and model assessment for surface ozone levels (Guttorp et al., 1994; Carroll et al., 1997; Meiring et al., 1998; Huang and Hsu, 2004), precipitation forecasts (Amani and Lebel, 1997) and the assessment of wind energy resources (Haslett and Raftery, 1989). Geostatistical approaches model the observations as a partial realization of a spatio-temporal, typically Gaussian random function

Z(s, t), (s, t) ∈ Rd×R, which is indexed in space by s ∈ Rd and in time by t ∈ R. Henceforth, we assume that second moments for the random function exist and are ﬁnite. Optimal least-squares prediction, or kriging, then relies on the appropriate speciﬁcation of the space-time covariance structure. Generally, the covariance between Z(s1, t1) and Z(s2, t2) depends on the space-time coordinates (s1, t1) and (s2, t2), and no further structure may exist. In practice, however, estimation and modeling call for simplifying assumptions, such as stationarity, separability, and full symmetry.