Box and Jenkins /4/ consider the noise process to be an
additional moving average process whereas Förstner /6/
applies Wiener filter technique to obtain estimates for the
process x(t). For applying the Wiener filter technique the
variances of the innovation process 0.2 and of the
observation process a, * must be known. To determine these
variances Fórstner /6/ uses variance component estimation
technique.
Our investigations on terrain profiles presented in chapter
6 do not take into account additional observation errors.
Hence the estimates refer actually to the combined process
y(t). This is permissible as long as x(t). > n(t).
8. Conclusions
The presented concept introduces ARIMA processes for model-
ling digital terrain profiles. From a theoretical point of
view, the generality of the approach with his various rela-
tions to other concepts is of main interest. These instruc-
tive relations present our concept to be open to further
developments. Hence ARIMA processes may be considered to be
a superior model to review other concepts for modelling
terrain. The extension to twodimensional processes for
modelling terrain is possible. Image analysis techniques
are already applying such processes.
Experiments have confirmed, that ARIMA processes are suited
for modelling digital terrain profiles. Main advantages are
high accuracy - as demonstrated by the statistical proper-
ties of the prediction errors - and parsimonity “4% only a
few parameters are required. Complications resulting from
the necessity of the determination of the process order for
different terrain types can practically be avoided by
selecting processes of a general, high order (s 6).
9. References
/1/ Ackermann, F.: The accuracy of digital height models,
Proceedings of the 37th Photogrammetric Week,
Stuttgart, 1980
/2/ » Akaike,:H.: Maximum likelihood identification of
Gaussian ARMA models, Biometrika 60,2, pp. 255 -
265, 1973
/3/ Akaike, H.: A new look at the statistical model iden-
tification, IEEE Trans. AC-19, pp. 716. - 723, 1974
^/ Box, Q.E.P.; Jenkins, G.M.: Time Series Analysis:
Forecasting and Control, Holden-Day, San
Francisco, 1976
/5/ Burg, J. P.: A new analysis technique for time series
data, Adv. Study Inst. Sign.Proc., Enschede, 1968
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