Effect of increasing number of neurons using artificial neural network to estimate geoid heights

Abstract

Nowadays the GPS measurements are one of the most frequently used technique in geodesy. With this technique ellipsoidal height can be reckoned. However in the engineering practice orthometric heights (height above sea level) are used. The orthometric heights are determined by levelling. Transforming the GPS-derived ellipsoidal heights to orthometric heights it is important to know the distance between the ellipsoidal and the geoid surface, called the geoid height or geoid undulation. GPS levelling method is easy to determine geoid height of related region. Geoid height calculated by soft computing methods such as fuzzy logic and neural networks has gained more popularity recently. In this study, it examined effect of increasing number of neurons in neural networks to determine geoid height. The neural network approach used in this study is based on a back propagation neural network learning the functional relationship between geographic position and geoid undulation. Thus, inputs to the neural network are geographic position (latitude and longitude), and the output from the network is the predicted geoid undulation.

Full text

International Journal of the Physical Sciences Vol. 6(3), 529-533, 4 February, 2011
Available online at http://www.academicjournals.org/IJPS
DOI: 10.5897/IJPS10.626
ISSN 1992 - 1950 ©2011 Academic Journals
Full Length Research Paper
Effect of increasing number of neurons using artificial
neural network to estimate geoid heights
Mehmet Yilmaz1* and Ersoy Arslan2
1Department of Geodesy and Photogrammetry Engineering, Faculty of Engineering, Harran University,
Osmanbey Campus, Sanliurfa Turkey.
2Department of Geodesy and Photogrammetry Engineering, Istanbul Technical University, Istanbul, Turkey.
Accepted 10 January, 2011
Nowadays the GPS measurements are one of the most frequently used technique in geodesy.
With this technique ellipsoidal height can be reckoned. However in the engineering practice orthometric
heights (height above sea level) are used. The orthometric heights are determined by levelling.
Transforming the GPS-derived ellipsoidal heights to orthometric heights it is important to know the
distance between the ellipsoidal and the geoid surface, called the geoid height or geoid undulation.
GPS levelling method is easy to determine geoid height of related region. Geoid height calculated by
soft computing methods such as fuzzy logic and neural networks has gained more popularity recently.
In this study, it examined effect of increasing number of neurons in neural networks to determine geoid
height. The neural network approach used in this study is based on a back propagation neural
network learning the functional relationship between geographic position and geoid undulation.
Thus, inputs to the neural network are geographic position (latitude and longitude), and the output from
the network is the predicted geoid undulation.
Key words: Geoid height, GPS, Neural networks, neuron.
INTRODUCTION
There are two types of heights used in geodesy. These
are: orthometric height which is reckoned from geoid and,
the second one is ellipsoidal height reckoned from
ellipsoid. Orthometric height is a physical height on the
other hand; ellipsoidal height is a mathematical height.
These two height systems cannot be coincided with each
other. In most engineering and surveying project,
orthometric heights are required because orthometric
height reflects the topography better than ellipsoidal
height. The difference between the two height systems
are called geoid height (undulation) and can be obtained
in the following simple equation:
N=h-H (1)
Where, N denotes the geoid height, h and H are the
*Corresponding author. E-mail: mehmetyilmaz40@gmail.com,
yilmazmeh@harran.edu.tr. Tel: (90) 414 3183811, 90 505
2647499. Fax: (90) 414 344 00 31.
ellipsoidal and orthometric heights, respectively. The
importance of accurately obtaining the geoid height has
increased in recent years with the advance of quantity
and quality of satellite positioning systems such as
(GPS). GPS provides height information relative to a best
fitting earth ellipsoid rather than the geoid. To convert
ellipsoidal heights derived from GPS to conventional
orthometric heights the relationship between ellipsoid and
geoid mentioned Equation 1 must be known. Orthometric
heights can be readily computed from (1) if the geoid and
ellipsoidal height are known. Ellipsoidal heights, or
ellipsoidal height differences, can be derived from GPS
more economically than orthometric heights.
Determination of the latter requires time-consuming
leveling. More details can be found in Wellenhof and
Moritz (2006), Featherstone (2001), Engelis et al. (1985),
Torge (2001),Yilmaz and Arslan (2008).
GPS/levelling geoid is easy to calculate and geoid
heights obtained by GPS/levelling can be used to
modelling the geoid in the region of interest using
polynomial coefficients (interpolation) (Yanalak and
Baykal, 2001) and soft computing model such as fuzzy

530 Int. J. Phys. Sci.
logic (Yilmaz, 2005) and Akyilmaz (2005).
In this study, we deal with Artificial Neural Network
(ANN) method alternative to other geoid determination
methods such as surface polynomials, fuzzy logic for the
interpolation of geoid heights.
Some of these studies are Seager at al. (1999), Kuhar
et al. (2001), Veronez et al. (2005), Kutoglu (2006),
Kavzoglu and Saka (2005), Palencz and Volgyesi (2003).
Although many studies have been performed by using
ANN to determine geoid height, effect of increasing
number of hidden nodes in the layer has not been studied
exactly in these studies. Kavzoglu and Saka have just
examined two different numbers of hidden nodes (8 and
10) in the layer. Another studies have been performed
with ANN versus surface polynomial by Kutoglu (2006)),
Veronez et al. (2005); Weiwei and Xiudung (2010).
METHODOLOGY
Artificial neural networks (ANNs)
Artificial neural networks (ANNs), or shortly, neural networks
(NN) have been used for the structure and functionality of
biological natural of human brain. Therefore, ANN is found to be
more flexible and suitable than other modeling methods (Zhang et
al., 1997). ANN is based on the neural architectures of the
human brain (Haykin, 1994), and described as group of simple
processing units, known as neurons (nodes), that are arranged
in parallel layers that are connected to each other by weighted
connections. By virtue of hidden layers of neurons that lie
between the input and output layers of the network, and the
nonlinear activation functions that are used to translate nodal
input to output, ANN provides linear and nonlinear modeling
without the requirement of preliminary information and
assumption as to the relationship between input and output
variables. This provides ANN an advantage over other statistical
and conventional prediction methods such as logistic regression
and numerical methods, in which nonlinear interactions
between variables must be modeled in explicit functional form (Tu,
1996). ANN trained with feed-forward back-propagation algorithm
has been studied extensively and applied successfully to various
areas, such as automotives (Majors et al., 2002), banking (Arzum
and Yalcin, 2007), electronics (Bor-ren and Hof, 2003), finance
(Xiaotian et al., 2008), industry (Cheginia et al., 2008), oil and
gas(Peranbur and Preechayasomboon, 2002), and robotics (Huang
et al., 2008) as well as others. The most ANNs contain three
layers: input, output and hidden layer. Generally, there are various
types of ANN techniques for example feed forward network, radial
basis network, generalized regression network and recurrent neural
network.
Feed-forward back-propagation and radial basis ANN the
most often used of networks type. They have been utilized to
solve a number of real problems, although they gained a wide
use, however the challenge remains to select the best of them.
In other words, there are no perfectly clear methods to determine
the best network type. In this paper, geoid heights are calculated
using feed-forward back-propagation.
A Feed forward neural network
Multi-layer feed-forward network was first established by Rumelhart
(Rumelhart et al., 1986). Among the existing several neural
networks such as recurrent networks, Hopfield networks, etc.
the feed-forward is most popular, primarily due to their
simplicity from the viewpoint of structure and ease of
mathematical analysis, good representational capabilities. Feed
forward network has been applied successfully to various
application domains, such as prediction, controlling, system
modeling and identification, signal processing and patter
classification (Bilski, 2005). Overall, feed-forward architecture as
shown in Figure 1 demonstrates an arrangement of
interconnected nodes called neurons by sets of connections
weight organized into three groups called layers, that is, input,
hidden, and output layers (Abdalla et al., 2010).
In feed forward network each neuron in a layer receives
weighted inputs from a previous layer and transmits its output
to neurons in the next layer. The sums of weighted inputs are
computed by Equation (2) and this sums is transferred by an
activation function shown in Equation (3). The output values of
network are compared with the actual output and the error of
network is computed with Equation (4). The training process
continues until this error met acceptable value.
(2)
(3)
(4)
where is input neuron, is weight coefficient of each input
neuron, is bias, is the summation of weighted inputs,
is the response of system, is the nonlinear
activation function, is the observed output value, E is the
error between output observed value and network result.
Furthermore, in the Figure 1, N is the number of input patter and M
is number of neurons in hidden layer (Abdalla et al., 2010).
The back-propagation algorithm for training of feed-forward
network was inspired by Rumelhart 1986. The training process
adjusts the connection weight and bias of network in order to
minimize the error function (that is, instantaneous sum squared
error) defined in Equation 4.
The adjustment of connection weight are conducted by back-
propagating the errors to the network. To achieve this, the
connection weight is adjusted by an amount proportional to the
gradient of error with respect to the weight, shown as follows:
(5)
where is the learning rate parameter which is used to
controlling the convergent speed of the training algorithm and
the local gradient of .
The BP algorithm presents a better performance with a
second-order term referred to as the momentum coefficient ,
which introduces the old connection weight change as a
parameter for the calculation of the new connection weight
change.
(6)
Data
In this study 1005 points whose latitude, longitude, ellipsoidal height

Yilmaz and Arslan 531
Figure 1. General structure of feed forward network.
Figure 2. Distribution of model and test points in Istanbul (black dot shows model and red dot show test points).
and orthometric height are known were used to construct neural
network models in the region. The points are homogenously
distributed and randomly selected in Istanbul; the point density is
nearly one point in 10 km2. The data covers the region between 41°
30´2.79″ > ϕ > 40° 48´13.75″ and 29° 54´ 24.24″ > λ > 27° 59´
3.05″. The standard deviation of the ellipsoidal heights after the
adjustment of the network has been found to be ± 2.56 cm (Ayan et
al., 2006). To construct the neural network models latitude and
longitude are taken as inputs and geoid heights of the points are
taken as outputs. To check for the calculation, randomly selected
178 points which had not been included in the preparation of the
neural network models are used. The distribution of the 1005 model
points and the 178 test points used in the ANN can be seen in
Figure 2.

532 Int. J. Phys. Sci.
Table 1. Summary of results at model and test points obtained by neural networks using different numbers of neuron.
Number of
neuron
Model points Test points
Maximum
Error (cm)
Minimum
Error (cm) RMSE (cm) Maximum
Error (cm)
Minimum
Error (cm) RMSE (cm)
5 20.49 -15.71 4.769 10.25 -10.34 4.403
10 12.44 -13.78 4.083 12.57 -3.96 3.814
15 11.89 -12.60 3.523 8.55 -5.49 3.496
20 11.61 -12.80 3.517 7.88 -7.13 3.508
25 13.80 -12.05 3.444 8.30 -5.11 3.419
30 10.52 -12.05 3.310 10.11 -5.27 3.259
35 11.39 -13.60 3.238 9.39 -4.62 3.300
40 11.76 -11.83 3.206 9.11 -6.06 3.303
Table 2. Number of points that error values greater than +7 cm and lower than -7 cm at both model and test
points.
Number of
neuron
Number of points error values
greater than +7 cm
Number of points error values lower
than -7 cm
Model points Test points Model points Test points
5 56 8 75 9
10 63 4 39 7
15 34 5 27 5
20 29 6 22 6
25 22 3 17 5
30 20 1 19 3
35 21 3 13 4
40 18 5 18 5
RESULTS AND DISCUSSION
In this study, geoid height is calculated by neural network
with taking eight different numbers of neurons. Models
are constructed by starting at neuron number 5 and each
time neuron number is increased in 5 and last models are
finished neuron number reached at 40. Therefore, eight
different neural network models are set up. It is aimed to
show both neural network method can be used in geoid
height calculations and effect of increasing number of
neurons. Summary of obtained results are shown in
Table 1.
When Table 1 is examined, the highest RMSE value is
4.769 cm obtained using 5 neurons, the lowest RMSE
value is 3.206 cm obtained using 40 neurons in neural
network calculations at model points. On the other hand,
the highest RMSE value is 4.403 cm obtained using 5
neurons, the lowest RMSE value is 3.259 cm obtained
using 30 neurons in neural network calculations at test
points. It can be seen that RMSE values are decreasing
as the neuron numbers are increased until number of
neuron is 30 at both model and test points. After neuron
number is 30, RMSE value is still decreasing at model
points, however, it cannot be said same thing at test
points. Because After neuron number is 30 RMSE values
is getting increasing at test points. This indicates that if
neuron number is selected larger than 30, neural network
model is over fitting.
Maximum and minimum error values are varied
between -15.71 and +20.49 cm at 5 neurons and
maximum and minimum error values are varied between
-11.83 and +11.76 cm at 40 neurons at model points.
Number of points that error values greater than +7 cm
and lower than -7 cm are examined at both model and
test points and results about this are given in Table 2. If
these points are carefully searched, it is seen that some
points have large error values at all neural models. Points
numbered at 735, 747, 858, 898, 179 and 874 have
generally large minimum error values and at 730, 1033,
736, 567 and 132 huge maximum error values at model
points. Because these points have large errors at all
neural models, it can be inferred that either height of
these points are defective or incompatible. Therefore,
heights of these points must be checked or throw away
from neural model to get better results. Same thing has
done at test points and points numbered at 458, 683, 840

have generally huge minimum error values and at 130,
873, 819, 91 have big maximum error values.
Conclusion
This study shows that neural network can be used as
calculation method in geoid height determination.
Changing number of neuron numbers affects the geoid
height results. To find suitable number of neuron is a trial
and error task. If appropriate number of neuron is not
selected, model can be overfitting and these leads to
wrong results. This means neural network model gives
better result at model points but it also gives worse
results at test points. RMSE value is used to validate
neural network model. If RMSE values are close at both
model and test points, constructed neural model can be
used (as it happen neuron number from 5 - 30),
otherwise neural model cannot validate (as it happen in
neuron number 35 and 40).
It is also important to keep in mind the number of points
used in neural network. If selected points are
represented the study area well enough, precision of
neural network and other calculation methods of geoid
height will be high. And lastly, quality of points also effect
precision of geoid height results.
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