Karhunen-Loeve expansions for axially symmetric Gaussian processes: modeling strategies and L~2 approximations

摘要

Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen-Loeve expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen-Loeve coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen-Loeve expansions are used to approximate axially symmetric processes. For such approximations, bounds for the L-2-error are provided. Numerical experiments are conducted to illustrate our findings.