Assume Z(Xx)to be a certain attribute, i.e., a regional variable,
then the experimental variogram y(h) (generally abbreviated to
variogram) may be obtained from a =I, 2, ..., N(h) pairs of
observations ( Z, (X5) , Z, (X +h) } defined on a support v
at locations {x, x+h} separated by a fixed lag h:
N(h)
Yo) = ZZ ba) Za WE)
where h = a vector of direction and distance.
If the study area is isotropic, h simply degrades to distance h. In
this paper, the support v represents a single pixel in a remotely
sensed image.
The kriging system is essentially a generalized least square
regression algorithm (Goovaerts, 2002), which is characterized
by the following formula:
n(x)
Z(x)- m(x)- * A, es. )- mi.) @)
where Z (x) = estimate of the regional variable Z(x)
z(x, ) = realization of Z at x,
Aq (Xx) = weight of z(x,)
m(x) ,m(x, ) = expected values of Z at xand x,
total
neighborhood of x .
n(x) = number of realizations in the
The purpose of kriging is to minimize the estimated
variance (x) =Var[Z(x)—Z(x)] under the unbiased
condition, i.e., E[Z(x)—Z(x)]=0.
The regional variable Z(x) is usually further decomposed to two
parts, formulated as follows:
Z(x) = R(x)+m(x) 3)
where R(x) = the residual component modelled as a stationary
random function with zero mean
m(x) = the trend component
2.1 Simple Kriging with Varying Local Mean
Traditionally, simple kriging considers the mean m(x) to be
known and constant, i.e., m(x) =m, through the study area
(Goovaerts, 2002). In other words, the mean m dose not depend
on location x but represents global information common to all
unsampled locations under the assumption of stationarity. Once
the trend component m(x) is known and varying with the
location x, Eq.2 turns to simple kriging with varying local mean.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
Given K land cover types of a certain study area, Z(x) in Eq.2
may be replaced by a Boolean variable 7,(x) to indicate
whether the unsampled location pertaining to the predefined kth
class. The indicator transform of 7, (x) is
I(x;k)-
1, if class is k
| f (4)
0, otherwise
Hence, Eq.2 may be rewritten as:
n(x)
Ii (x)= 2 a («Ji (sa )= Pr (<a)l+ pps), (5)
where i; (x, ) = indicator of a training sample at location x,
Pr (x), py (Xa) ^ the a posteriori probability (of hard
classification) or membership (of fuzzy classification),
obtained by a classifier, pertaining to unsampled
location x and training sample at location x,
I x (x) = estimate of the probability that the unsampled
location x pertains to the kth class.
Given K classes under consideration, Eq.5 will be repeated K
times. Then a normalization process will be applied to attain K
estimated probabilities, which finally constitute a K-class
indicator vector, i.e., {x (X).
Through information fusion of the indicator vectors of training
samples and the predicted probability vectors, Eq.5 aims at
revising the probability vectors. It incorporates both the known
categorical information of training samples and the predicted a
posteriori information of all the pixels throughout a remotely
sensed image. From the perspective of the information theory,
Eq.5 adequately excavates the information contained in the
input vectors (i.e., spectral features) which is partially wasted
by a classifier. It is I,(x,)— Pr (X,) that is the wasted
information, namely the aforementioned residual. Therefore,
Eq.5 utilizes the linear combinations of residuals pertaining to
the training samples to amend the posterior probabilities which
are directly predicted by a classifier (Zhang, 2009).
2.2 Cokriging
In the kriging paradigm, another algorithm to combine the
primary and secondary information is cokriging. Direct
measurements of the primary attribute of interest are often
supplemented by secondary information in order to improve the
estimation (Goovaerts, 1997).
Similarly, given K land cover types in a study area, assume
I(x) to be the primary variable and FP, (x) to be the
secondary variable, the cokriging estimate for /,(X) is