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