International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
observed and predicted changes in height on the platform
0.2
0.1
= Ka
E o ;
= {
® H i
0.1 {
|
02
0 50 100 150 200 250
epoch number
02
E 01
5 + i
5 it m
s 9 i i
3
3 01
a
02
0 50 100 150 200 250
epoch number
Figure 8. Actual (black) and predicted (grey) height changes
(top) and corresponding prediction errors in July 9 2001,
between 10:15-11:15
Note that the epoch numbers given in both days at 10:15 —
11:15 interval are less than those at 7:20 — 8:20 intervals. These
data gaps are due to the inconvenient satellite constellation
resulting with unsuccessful integer ambiguity solution for the
relevant epochs of kinematic GPS observations.
The quality measures of the predictions for the relevant time
spans in July 9, 2001 are given in Table 2.
July 9, 2001
7:20 — 8:20 10:15 — 11:15
u (m) 0.000 0.000
r 0.844668 0.845583
Mean abs. error (m) 0.012 0.018
Standard deviation (m) 0.020 0.025
Max. error (m) 0.079 0.082
Min. error (m) -0.105 -0.080
Table 1. Quality measures of the prediction results for July 2,
2001.
The resulting standard deviations are obtained as the same
values with the mean square errors of the point heights derived
from the adjustment of GPS observations. Recalling the figures
5. 6, 7 and 8, there are some parts of time sequences very
precisely estimated whereas a very small part are slightly less
precisely predicted. This is due to the sampling rate of the input
values which were assumed to be either linearly varying or
constant values during each hourly period. However, in general
very good approximations were achieved.
In addition to the criterion given in Table 1 and Table 2, the
remaining residual sequences are investigated by using fast
fourier transform in order to examine the frequency content of
the residuals. Fig. 9 shows the frequency content of the
remaining residual series of the approximations.
July 2 2001, 7:20-8:20 July 2 2001, 10:15-11:15
2.5 5
|
2; | 4
1.5 i 3 {1a
05: 1
ES i LH i Ti
0. 0-— dE. US 3 à n
0 0 50 100 150
frequency [Hz] frequency [Hz]
July 9 2001, 7:20-8:20 July 9 2001, 10:15-11:15
4. . « 4 :
i
3 3: i
H
2 |; 2 | i
i |
i | |
id i la
uy ly | ir
1 M DS 1 i | |
Bi ult iy Ap ! i
mo A W “il
0 LAN 0I gi. ^
0 50 100 0 50 100 150
frequency [Hz] frequency [Hz]
Figure 9. The frequency content of each neural approximation
residuals. Note the chaotic form of the residual frequencies for
all four models.
The frequency content in the remaining residual reflects a wide
spectrum of frequencies exist which approve that the resulting
prediction errors are highly normally distributed random errors
with zero mean and variances equal to the mean variances of
adjusted heights from GPS observations (See Table 1, Table 2).
6. CONCLUSIONS
The use of artificial neural networks for modelling deformation
process of engineering structures as well as natural hazards such
as landslides offers geodesists a good alternative for the
description of resulting deformations as a function of causing
effects which are generally more or less non-geodetic
observations. In case of neural modelling, the determined
parameters, i.e. the weights between consecutive neurons
implicitly describe the mapping between the inputs and outputs,
but cannot be used in any other way as representing a typical
mathematical function for deformation process.
One has to note that the results from the neural network are
particularly depend on the selection of inputs and outputs, and
the architecture of the network to be used as they are capable of
learning anything. One disadvantage of neural network
applications is that there is no single similar solution to any
given input-output data set as the estimated parameters of the
network depends on various settings, which are especially
considered during learning process. In most cases, these settings
are selected by personal human judgement. Therefore, the
solution of neural network is referred as sub-optimal solution.
This means that the obtained solution is just the one among
other solutions which provide the similar precision of
approximation and/or prediction.
In this study, Matlab Version 6.5 Neural Toolbox is used for
computations. During the network architecture and learning
process, the number of layers and the neurons in each layer as
well as the number of training run has been cared to be kept as
minimum as possible in order to avoid overfitting and
overtraining.
The results given in Table 1 and Table 2 show that a very good
approximation can be succeeded even if the input data sampling
rate is very low, i.e. every one hour, a measurement of input
data has been used to generate the input matrix of training and
testing data sets. The more frequently the input data is
measured the better approximation can be obtained by using
neural network methods. On the other hand, the more accurate
706
on
Eise
Heu
Defi
(Hai
fach
208,
Miit
mod
mon
Miir
appr
Kutt
Stati
Zuri
Pfeu
exan
Com
G.W
W. V
Mod
Wels
ficati
Symp
27-3(
Wels
Class
