Kriging is the most popular method for spatial interpolation in the Earth Sciences. It is a clever, constrained, parametric regressor that is sometimes called nonparametric. Yakowitz and Szidarovsky [1985] developed theoretical results to establish the consistency and convergence properties of Kriging. For Kriging to work, they showed that proper variogram selection was critical. They formalized a Nadaraya Watson kernel regression estimator for spatial regression, established its consistency and convergence rates, utility for estimating functionals (e.g., integrals or derivatives of the surface), and developed a NN estimator of the local mean square error of estimation. Their Monte Carlo analysis showed that the kernel regressor was competitive with Kriging where the Kriging assumptions were valid, and superior where the second order assumptions broke down. Stein [1990] has since shown that the Kriging estimate can be unbiased even if the variogram is mis-specified (it needs to be of the right class though). The same cannot be said for the Kriging estimate of the MSE. The Kriging vs Nonparametric regression debate continues in the statistics literature, with equivalences and differences being highlighted, and the winner depending on the modeling perspective (stochastic process vs deterministic function observed with noise) adopted.
Some hydrologic applications of nonparametric spatial analysis are
Owosina et al. [1992] compare two multivariate kernel regression
estimators, MARS, LOESS, TPSS and Kriging for reconstructing spatial surfaces
from a variety of irregularly sampled synthetic (with varying signal to noise
ratios) and ground water data sets. Model parameters were chosen automatically
using cross validatory measures in all cases. In terms of RMSE and Mean Absolute
Deviation, overall algorithm ordering (best to worst) across the data sets was
TPSS, LOESS, KERNEL, MARS, KRIGING. The differences between the best and worst
algorithm were dramatic in some cases. Methods for interpolating ground water
data irregularly sampled in space and time were also illustrated.
Lall et al. [1994a] develop weighted, local polynomial
estimators for spatial surfaces in the spirit of LOESS, with polynomial order
and number of nearest neighbors chosen by GCV. A local GCV function is used to
assess pointwise performance of the estimator. Global parameter selection using
averaged local GCV scores is shown to be superior to the classical GCV.
Applications to synthetic and ground water data show the superiority of the
method to the best Ordinary Kriging estimator.
Lall and Ali [1992] consider the recovery of subsurface
stratigraphy from drill log information. This information is encoded in a binary
function (1 for ``sand,'' 0 for ``clay'') and two models to interpret this data
are developed. The first considers the occurrence of sand in the vertical as a
nonhomogeneous Poisson process (NPP) at any point in the domain and uses a
3-dimensional kernel method to interpolate this rate from neighboring drill
logs. Bandwidth parameters are chosen by cross validation. The NPP model is
refined, and fully implemented with examples by Ali and Lall [1993a,b].
The second model considers the process of sand/clay occurrence in the vertical
as a 2 state non-homogeneous Markov process. The transition intensity of this
Markov process at any point in the aquifer is estimated using kernel methods,
with bandwidth chosen by cross validation.