Nonstationary covariance models for global data

Abstract

With the widespread availability of satellite-based instruments, many geophysical processes are measured on a global scale and they often show strong nonstationarity in the covariance structure. In this paper we present a flexible class of parametric covariance models that can capture the nonstationarity in global data, especially strong dependency of covariance structure on latitudes. We apply the Discrete Fourier Transform to data on regular grids, which enables us to calculate the exact likelihood for large data sets. Our covariance model is applied to global total column ozone level data on a given day. We discuss how our covariance model compares with some existing models.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS183 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Long-term time-dependent stochastic modelling of extreme waves

“…For example, processes may be approximately stationary with respect to longitude but with highly dependent covariance structures with respect to latitude. In order to capture the nonstationarity in such global data, with a spherical spatial domain, a class of parametric covariance models are proposed in Jun and Stein (2008). These assume that processes are axially symmetric, i.e.…”

“…A non-negative constant C corresponds to including homogeneous models for the case A(L) = B(L) = 0. In order to apply this model to real applications, the A and B functions need to be estimated, and it is suggested to use linear combinations of Legendre polynomials (Jun and Stein 2008). The covariance model is applied to global column ozone level data and it is shown that the strong nonstationarity with respect to latitudes as well as the local variation of the process can be well captured with only a modest number of parameters.…”

Spatial mapping of ground-based observations of total ozone

“…This is why Jun and Stein (2007) assume that the spatial process driving the TCO data is an axially symmetric process whose first two moments are invariant to rotations about the Earth's axis, and constructed space-time covariance functions on the sphere × time that are weakly stationary with respect to longitude and time for fixed values of latitude. Jun and Stein (2008) further used linear combinations of Legendre polynomials to represent the coefficients of partial differential operators in the covariance functions. These covariance functions produce covariance matrices that are neither of low rank nor sparse for irregularly distributed observations, as it is the case with ground-based stations.…”

“…As pointed out in Jun and Stein (2008), the spatial mean structure on a sphere can be modeled using a regression basis of spherical harmonics; however, since the data set only contains measurements from one specific event, it is not possible to identify which part of the variation in the data come from a varying mean and which part can be explained by the variance-covariance structure of the latent field. To obtain basic identifiability, the parameters κ(s) and τ (s) are taken where P k are Legendre polynomials,…”

Validation of CMIP5 multimodel ensembles through the smoothness of climate variables

“…Because surface temperatures are closely related to altitude, we estimate m ijl by regressing on the altitude from the sea level in addition to spherical harmonics for n 012, for each climate ensemble realisation in CMIP5 and the NCEP/NCAR reanalysis. The choice of n 012 is made following the literature dealing with similar data sets (Jun and Stein, 2008;Stein, 2008;Jun, 2011Jun, , 2014.…”