process with minimum variance. Theoretically, the variance 
of the prediction errors decreases with increasing orders p 
and q. Hence this críterion alone is not sufficient. There- 
fore, Akaike /3 / introduced his information criterion, 
which determines the order by minimizing the function 
AICrerine,? + 2(p+q) / N 
A further criterion to be considered is the significance of 
the parameters a and b. The maximum order p and q is 
determined by the number of significant parameters. 
6. Modelling Digital Terrain Profiles 
The statistical properties of  ARIMA processes and their 
relations to autocorrelation functions and to spectral 
analysis justify to consider ARIMA processes as an adequate 
model for digital terrain profiles. Here the process 
orders, the statistical properties of the prediction errors 
and the variations of the process parameters within the 
same type of terrain are of main interest. To achieve this 
we studied terrain profiles from different regions. The 
experiments comprised 52 profiles out of 9 regions. Most 
profiles were measured on the occasion of the ISPRS-Test of 
Comm..III/3.-(:.Torlegárd /15/ ) and sare described in 
Frederiksen /7/ and Jacobi and Kubik /9/. Table 1 gives a 
survey of the terrain profiles analysed. 
For process identification every profile was modeled by 
ARIMA processes of increasing order. Stationarity could be 
reached with the first derivatives (i.e. the slopes of the 
profile) on flat terrain. The second derivatives (i.e. the 
curvatures) are sufficiently stationary in all other 
profiles. 
With regard to the process order, the empirical investiga- 
tions established that the profiles of a local region 
(similar type of terrain) can be modeled by the same 
process order. Table 2 records process orders established 
for the different types of terrain. It is a remarkable 
result that the process orders are very low. Consequently a 
very limited number only of parameters are required to 
describe the profiles. A maximum order of p+d+q = 6 is 
sufficient for all cases considered. 
Besides, table 2 includes the statistical properties of the 
prediction errors: the mean values of the standard devia- 
tions, the expectations and the correlation coefficients. 
Within a given region the standard deviations of the pre- 
diction errors are very homogeneous. The expectations and 
the correlation coefficients are not significant in any 
case. Therefore the prediction errors may be described by a 
white noise process. Consequently terrain profiles satisfy 
a condition of ARIMA processes. 
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