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basis of these descriptions of the terrain. For instance, if 
the terrain is described by its power spectrum, this 
spectrum may be approximated by an exponential function 
(Frederiksen et.al, 1978). The parameters of this 
exponential function are then used as characteristic 
parameters for describing different terrain types. When at 
last the characteristic parameters are selected, they are to 
be used for terrain classification in the daily work. 
This paper addresses the problem of selecting the 
characteristic parameters as described above. That is, given 
a certain description of the terrain, for instance its power 
spectrum or its autocorrelation function, which parameters 
are needed to describe the terrain? The proper selection of 
these parameters is very essential for its practical use. 
The number of characteristic parameters should be small, 
they should be easy to estimate and they should be well 
suited to the purpose. 
Unless the underlying impellents of the elevations of the 
terrain are known, our selection of characteristic 
parameters has to be based on empirical studies. The 
Karhunen-Loeve expansion is an objective method for this 
purpose. The object of this paper is to describe the basic 
principles of Karhunen-Loeve expansion when used for terrain 
classification. The aim is not to establish a general model 
for terrain classification, but rather to pay attention to 
the possibilities and limitations of an empirical approach 
like the Karhunen-Loeve expansion. 
2. BASIC PRINCIPLE OF THE KARHUNEN-LOEVE EXPANSION 
The Karhunen-Loeve (K-L) expansion is a method for the 
transformation Of a set of functions into a set of 
orthonormal functions. The method is well known from 
different sciences. In Digital Image Processing, the method 
is used for data compression (Rosenfeld and Kak, 1982), 
while in hydrology it is used for the study of similarities 
and dissimilarities among hydrological data sets (Stokes, 
1974). In Rao (1965), the method is described theoretically 
in the section covering principal component analysis of 
random variables. 
For the completeness of this paper, a brief review of K-L 
expansion is given in this chapter. A more detailed 
description can be found in several textbooks, for instance 
in Rosenfeld and Kak (1982) or Rao (1965). 
Assume that we in one way or another can observe a variable 
t and its outcome e). Assume also that each function 
ey (t) can be expressed as a linear combination of a set of 
base functions a,(t), such that 
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