| Abstract
Stochastic methods of space-time interpolation and conditional simulation
have been used in statistical downscaling approaches to increase the resolution
of measured fields. One of the popular interpolation methods in geostatistics
is kriging, also known as optimal interpolation in data assimilation. Kriging
is a stochastic, linear interpolator which incorporates time/space variability by
means of the variogram function. However, estimation of the variogram from
data involves various assumptions and simplifications. At the same time, the high
numerical complexity of kriging makes it difficult to use for very large data sets.
We present a different approach based on the so-called Spartan Spatial Random
Fields (SSRFs). SSRFs were motivated from classical field theories of statistical
physics [1]. The SSRFs provide a different approach of parametrizing spatial
dependence based on “effective interactions,” which can be formulated based
on general statistical principles or even incorporate physical constraints. This
framework leads to a broad family of covariance functions [2], and it provides new
perspectives in covariance parameter estimation and interpolation [3]. A significant
advantage offered by SSRFs is reduced numerical complexity, which can lead to
much faster codes for spatial interpolation and conditional simulation. In addition,
on grids composed of rectangular cells, the SSRF representation leads to an explicit
expression for the precision matrix (the inverse covariance). Therefore SSRFs
could provide useful models of error covariance for data assimilation methods. We
use simulated and real data to demonstrate SSRF properties and downscaled fields.
keywords: interpolation, conditional simulation, precision matrix
References
[1]Â Â Â Hristopulos, D.T., 2003. Spartan Gibbs random field models for geostatistical
applications, SIAM Journal in Scientific Computation, 24, 2125-2162.
[2]Â Â Â Hristopulos, D.T., Elogne, S. N. 2007. Analytic properties and covariance
functions of a new class of generalized Gibbs random fields, IEEE Transactions
on Information Theory, 53(12), 4667-4679.
[3]Â Â Â Hristopulos, D.T., Elogne, S. N. 2009. Computationally efficient spatial
interpolators based on Spartan Spatial random fields, IEEE Transactions on Signal
Processing, 57(9), 3475-3487. |