Pardo-Iguzquiza, E. and Atkinson, P.M.
Modelling the semivariograms and cross-semivariograms
required in downscaling cokriging by numerical
Computers & Geosciences, 33, (10), . (doi:10.1016/j.cageo.2007.05.004).
Full text not available from this repository.
A practical problem of interest in remote sensing is to increase the spatial resolution of a coarse spatial resolution image
by fusing the information of that image with another fine spatial resolution image (from the same sensor or from sensors
on different satellites). Thus, the problem is how to introduce spatial ‘detail’ into a coarse spatial resolution image
(decrease the pixel size) such that it is coherent with the spectral information of the image. Cokriging provides a
geostatistical solution to the problem and has several interesting advantages: it is a sound statistical method by being
unbiased and minimizing a prediction variance (c.f. ad hoc procedures), it takes into account the effect of pixel size, and
also autocorrelation in each image as well as the cross-correlation between images, it may be extended to incorporate extra
information from other sources and it provides an estimation of the uncertainty of the final predictions. When formulating
the cokriging system, semivariograms and cross-semivariograms (or covariances and cross-covariances) appear, some of
which cannot be estimated from data directly. Cross-variograms between different variables as well as crosssemivariograms
between different supports for the same variable are required. The problem is solved by using linear
systems theory in which any variable for any pixel size is seen as the output of a linear system when the input is the same
variable on a point support. In remote-sensing applications, the linear system is specified by the point-spread function (or
impulse response) of the sensor. Linear systems theory provides the theoretical relations between the different
semivariograms and cross-semivariograms. Overall, one must ensure that the whole set of covariances and crosscovariances
is positive-definite and models must be estimated for non-observed semivariograms and cross-semivariograms.
The models must also be realistic, taking into account, for example, the parabolic behaviour close to the origin presented in
regularized semivariograms and cross-semivariograms. The solution proposed is to find by numerical deconvolution a
positive-definite set of point covariances and cross-covariances and then any required model may be obtained by numerical
convolution of the corresponding point model. The first step implies several numerical deconvolutions where some
model parameters are fixed, while others are estimated using the available experimental semivariograms and crosssemivariograms,
and some goodness-of-fit measure. The details of the proposed procedure are presented and illustrated
with an example from remote sensing.
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