An innovative Decision Support System using Fuzzy Reasoning for the Estimation of
Mountainous Watersheds Torrential Risk: The case of Lakes Koroneia and Volvi
L. Iliadis1, F. Μaris1 and T.Tsataltzinos1
1
Democritus University of Thrace, Department of Forestry & Management of the
Environment & Natural Resources, 68200, Orestiada, Greece. email:
liliadis@fmenr.duth.gr
_______________________________________________________________________
Abstract
This manuscript describes the design, development and testing of the
TORRISDESSYS Fuzzy Logic Computer System that performs Torrential Risk
estimation of mountainous watersheds. The System was developed in MS-Access
aiming in overcoming the limitations of the existing methods. An integrated reliable
approach towards torrential risk estimation, requires consideration of a long-period’s
torrential history and input of as many involved parameters as possible (e.g. the
percentage of compact geological forms, forest cover). TORRISDESSYS considers
actual values of a wide range of independent variables for as many years as data exist.
It produces Partial Risk Indices using Sigmoid, Normal Distribution, Trapezoidal and
Triangular Fuzzy Set Membership functions. Each watershed is assigned many Partial
Risk Indices (each one corresponding to a specific risk factor) which are unified (under
different perspectives) using Fuzzy T-Norms in order to produce Unified Indices. Each
watershed is assigned various Unified Risk Indices (depending on the T-Norm used)
each one representing its overall torrential risk, seen from a different angle.
TORRISDESSYS was applied successfully for the watersheds of Koroneia-Volvi
lacustrian area.
_______________________________________________________________________
Introduction
Torrential Risk estimation is a major issue not only in Greece but in many parts of the world.
This project deals with the construction of an innovative fuzzy model that is applied by a
modern Decision Support System (DSS) in order to evaluate the Degree of Torrential Risk
(DTR) for every single mountainous watershed of an area under study.
Existing methods of DTR calculation appear to have certain limitations. More specifically the
function of Gavrilovic considers mainly the average annual values of a limited number of
factors (sediments, meteorological and topographic data, vegetation and erosion) and it does
not take into account the torrential-history of each watershed over time. Finally, the method
of Kronfellner-Kraus determines the DTR based on the estimation of the maximum possible
load of sediments (Kotoulas 1997). None of these methods succeeds in considering the risk
coming from each involved factor independently, they fail in assigning weights to the few
parameters that they evaluate and the most important is that they do not produce an overall
risk index.
The developed System applies Fuzzy Algebra and Fuzzy Set theory concepts. Fuzzy Logic
was introduced by Zadeh in 1965 to relax the harsh constraint that everything that can be
said about anything is either absolutely true or absolutely false (Kandel 1992). In real life
situations this is rarely the case. Zadeh suggested that it is possible to understand a
statement as being 0.70 true. The use of Fuzzy numbers and Fuzzy Sets offered the
scientists powerful tools for performing classification and ranking tasks (Kandel 1992).
This study works towards two complementary parallel directions. The first direction involves
the determination of the main n Risk factors affecting the specific Risk problem.
Consequently, a number of n Fuzzy Sets are formed, each Fuzzy Set corresponding to a
Risk factor.
Five different and major fuzzy membership functions, the Trapezoidal (TRAPMF), the
Triangular (TRIAMF) the Sigmoid (SIGMF) and the Normal Distribution (NODIMF), are used
to estimate the degree of membership of the m watersheds under examination, to each one
of the n corresponding Fuzzy Sets. In this way each watershed is assigned n real numbers
(ranging from 0 to 1) that show its degree of Risk for each of the n Risk factors respectively.
Consequently the following two-dimensional Risk matrix 1 (n × m) is formed containing the
produced Risk indices.
Table 1
General format of the Risk Degree Array.
µ12 ...... µ1m ⎤
µ 22 ...... µ 2 m ⎥⎥
µ( Χ i j ) =
where i = 1 to n and j = 1 to m
⎢...... ...... ...... ...... ⎥
⎥
⎢
⎣ µ n1 µ n 2 ...... µ nm ⎦
⎡ µ11
⎢µ
⎢ 21
In this way, the System assigns Partial Risk Indices (PRI) to each watershed, associated
with each parameter. The number of PRI of each area equals to the number of the involved
parameters. So if the system evaluates the impact of five different parameters to the problem
of torrential risk, it produces five different PRI for each watershed. So each area can be very
risky due to one factor and not risky at all due to another. After the first task is over we can
have an idea of the reasons that make each area risky or not. But we do not have a clear
view of its torrential risk. The problem is still the lack of the one and only Unified Risk Index
(URI) that will produce an overall index concerning the risk of the watershed.
Fuzzy Logic and Fuzzy Algebra offer powerful tools towards this direction. These tools and
methods can provide aid towards the construction of successful models. This is performed
by applying five different types of Fuzzy Relations which are called T-Norms and they
operate conjunction between the Partial Risk Indices under different perspectives.
This approach and its application for the Torrential Risk estimation of the main watersheds of
North-Eastern Greece have already been presented thoroughly in another paper of our
research team (Iliadis et. al. 2004). The original contribution of this work is that it applies for
the first time, the Sigmoid Membership function and the Normal Distribution function for the
estimation of the DTR. Fuzzy Logic is widely used in various fields of engineering, decision
support and data analysis applications. A survey done in 1994 identified a total of 684 such
applications in Europe (Kecman, 2001).
The second main direction of this research is the application of the System and the
development of a comparative study between the output of three existing methods to the one
proposed here. The results of the two methods of Gavrilovic (G1) and (G2) (Kotoulas 1997)
and the results of the method of Kronfellner-Kraus are compared to the output of
TORRISDESSYS for the lakes Koroneia and Volvi, located in North Greece, close to the city
of Thessaloniki.
Materials and Methods
The Partial Risk Estimation model
Classification and ranking problems (like the one of risk estimation) can be faced using two
different Algebraic Modeling approaches. The one uses Crisp Sets (Leondes, 1998) and it
has the great disadvantage of using specific boundaries. According to the crisp approach a
watershed with 20mm of rain height may be considered as high risky, but another with
19.999 mm of rain height may be considered as having average risk. Areas that belong to
the boundaries are always a problem when crisp Sets are applied. The following function 1
defines a Crisp Set.
⎧1, if X ∈ S
µ s ( Χ) = ⎨
⎩0, if X ∉ S
Function 1 Characteristic function of a Crisp Set
In Crisp sets a function of this type is also called characteristic function.
On the other hand the approach proposed in this paper and in Iliadis et. al. (2004) uses
Fuzzy Sets.
Fuzzy sets can be used to produce the rational and sensible clustering
(Kandel, 1992). For Fuzzy sets there exists a degree of membership (DOM) µs(Χ) that is
mapped on [0,1] and every area belongs to the “Torrential Risky Area” Fuzzy Set with a
different degree of membership (Kandel, 1992). Functions 2 and 3 represent the Triangular
and the Trapezoidal Membership functions (MF) respectively that determine the Risk indices
for each examined area (Kecman 2001).
⎧0 if X < a
⎪ (X - a)/(c - a) if X ∈[a, c]
⎪
µ s (Χ) = ⎨
⎪(b - X)/(b - c) if X ∈[c, b]
⎪⎩0 if X > b
Function 2 Triangular Membership function
if X ≤ a
⎧0,
⎪(X - a)/(m - a), if X ∈ (a, m)
⎪⎪
µ s (X) = ⎨1,
if X ∈[m, n]
⎪(b - X)/(b - n), if X ∈ (n, b)
⎪
⎪⎩0,
if X ≥ b
Function 3 Trapezoidal Membership function
The following functions 4 and 5 correspond to parametric forms of the Sigmoid and of the
Normal Distribution (ND) MF respectively. The above MF are applied for the first time
towards the estimation of the DTR. (Tzionas et al., 2004).
f(x, a, c) =
1
− a ( x −c )
1+ e
Function 4 Sigmoid Membership function
It should be clarified that in the Sigmoid MF the parameters a, c determine its shape and
position respectively. Here we have chosen randomly to use a=1 and c=2. Also a parametric
form of the ND was applied as a MF. In the case of the ND the parameter c declares the
position of the curve and σ is the standard deviation.
− ( x −c ) 2
f n (x, σ, c) = e
2σ 2
Function 5 Normal Distribution Membership function
The Risk Unification model
The determination of the PRI is the first important step. The major step towards the final
target is the unification of the various degrees of Membership (PRI) in order to produce the
URI. Thus, the conjunction operation had to be performed using all types of unification
operations of Fuzzy Algebra (T-Norms). Various T-Norms were applied. The following Table
2 presents the T-Norms that were applied for this purpose.
Table 2
Description of the T-Norms used in this project.
Minimum Approach URI = MIN(µΑ (Χ), µ Β ( Χ))
Algebraic Product URI = µ Α (Χ) * µ Β ( Χ)
Drastic Product
URI = MIN(µ Α (Χ), µ Β ( Χ))..if ..MAX (µ Α (Χ), µ Β ( Χ)) = 1 otherwise URI = 0
Einstein Product
URI = µ Α (Χ) * µ Β ( Χ ) / (2-( µ Α (Χ) + µ Β (Χ) − µ Α (Χ) * µ Β (Χ)))
Hamacher Product
URI = µ Α (Χ) * µ Β ( Χ )/ (µ Α (Χ) + µ Β ( Χ) − µ Α (Χ) * µ Β ( Χ))
Their nature proves that each one of them produces a Unified Risk Index under a different
perspective. This can be very useful because each Unified Risk Index expresses a valid
Degree of Torrential Risk (DTR) under a different perspective. A comparison between the
results of the TORRISDESSYS and the output of other existing methods offers very
interesting Rankings and Classifications of the areas under examination.
The Software
A Decision Support System (DSS) was developed using the relational database MS Access.
The Access environment was used due to the fact that the reasoning of the System is quite
straightforward and there are no complicated or special Rules involved in it, whereas the
data are the main key concept leading the System to the goal. Consequently it is a data
driven DSS.
All data were stored in properly designed Tables that follow the principles of the first three
Normal forms (Date, 1990). The functions of the Mathematical model were performed using
single and multiple Queries written in the Structured Query Language (SQL). The system
uses a very friendly Graphical user interface and its results are obtained in a straightforward
manner. The input and the output is performed using Access forms and Command Buttons.
The potential user does not need any special experience in Computers to run the System.
The Software will be available (to be used) from the Laboratory of Forest Informatics
(Department of Forestry and Management of the Environment and Natural Resources) of the
Democritus University of Thrace, after December 2005. Requests can be forwarded to Dr. L.
S. Iliadis in the email mentioned in the first page of this paper. A user guide will also be
available in the following web page of the department http://www/fmenr/duth.gr, after
January 2006. The following screenshot is a sample of the System’s interface.
Screenshot 1 The System’s Interface
Area of Study
The torrential streams and the watersheds of the lacustrian area of Koroneia and Volvi were
determined and the whole analysis and application of the System was performed on them.
The following Table 3 presents the main torrential streams with their code names.
Table 3
Code
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
B11
B12
B13
B14
B15
B16
B17
B18
B19
B20
B21
B22
B23
The watersheds and torrential streams of the Vovli area
Torrent
Torrent
Torrent
Code
Code
Watershed
Watershed
Watershed
Sxolari
B24
Anonymous
B32
Nea Maditos
Anonymous
B25
Anonymous
B33
Apollonias
Anonymous
B26
Anonymous
B34
Neas Apollonias
Tris Petres
B27
Kavakolakkos
B35
Anonymous
Kastrirema
B28.1
Vambakias
B36
Anonymous
Anonymous
B28.2
Arethousas
B37
Anonymous
Anonymous
B28.3
Stefanina
B38
Spitakia
Anonymous
B28.4
Anonymous
B39
Stibos
Anonymous
B28.5
Anonymous
B40
Anonymous
Anonymous
B28.6
Anonymous
B41
Anonymous
Saint Ioannis
B28.7
Manas Lakkos
B42
Anonymous
Anonymous
B28.8
Mauroudas
B43.1
Sarakinas
Megalis Volvis
B28.9
Vagena
B43.2
Platanorema
Arapi
B28.10
Palati
B43.3
Anonymous
Anonymous
B28.11
Filadelfio
B43.4
Anonymous
Anonymous
B28.12
Ksiropotamos
B43.5
Arapi
Anonymous
B28.13
Lefkouda
B43.6
Adam
Tselepi
B28.14
Anonymous
B43.7
Kouliarotiko
Zesto Nero
B28.15
Anonymous
B43.8
Anonymous
Gazolakkos
B28.16
Anonymous
B43.9
Terani
Anonymous
B29
Modi
B43.10
Anonymous
Anonymous
B30
Anonymous
Siant Nikolaos
B31
Anonymous
Torrential Risk results
The research area where the DSS was applied for the estimation of the torrential risk is
located in the Northern part of Greece and more specifically in the lacustrian area of
Koroneia and Volvi, near the city of Thessaloniki. Data was gathered from Greek public
services who are responsible for meteorological and map data. Also, our research team
gathered important data. Initially the limits of the research areas were estimated. Maps of the
Geographical Army Service (GAS) with a scale of 1:250.000 were used for this purpose. The
upper and lower limits of the watershed areas are 300 και 2 km2 respectively (Kotoulas
1997). For every research area and for each torrential stream the morphometric
characteristics were specified.
The morphometric characteristics were produced after the process of maps (scale
1:100.000) of the GAS and the accuracy of the data was confirmed by visits of our research
teams in the research areas. The results were stored in descending order in matrices. For
each research area a matrix was used. According to the bibliography as it is included in the
book of “Mountainous Hydronomy” of professor Kotoulas (1969, 1973, 1979, 1987, 1997)
and it contains papers of various researchers on a global scale, the most important
morphometric characteristics of the watersheds that influence the torrential risk of an area
are the following: The area, the perimeter, the shape of the watershed, the degree of the
round shape of the watershed, the maximum altitude, the minimum altitude, the average
altitude, the average slope of the watershed, and its maximum altitude. The following Table
4, presents the four most risky watersheds of the Volvi area, based on four different MF and
on five different T-Norms. This means that the problem is studied under different
perspectives. The Drastic Product T-Norm assigns a high Degree of Risk in the areas that
have extreme values for one or more parameters. The Algebraic Product case estimates an
overall Degree of Risk and the Minimun approach characterizes as risky, the areas that have
very low values for some of the parameters that influence the problem.
Table 4
The four most Torrential risky streams in the Volvi area for each MF.
AREA OF VOLVI
Normal Distribution Membership Function
Sigmoid Membership Function
Algebraic
Drastic
Product MIN
Product Einstein Ham
B28.3
B29
B26
B28.3
B28.3
B28.15
B28.2
B19
B28.15 B28.15
B29
B28.3
B20
B29
B29
B28.16
B43.5
B21
B28.16
B28.2
Trapezoidal Membership Function
Algebraic
Drastic
Product MIN
Product Einstein Ham
B28.3
B43.1
B1
B28.3
B8
B28.2
B27
B2
B28.2
B3
B28.6
B29
B3
B28.6
B2
B28.9
B28.2
B4
B28.9
B7
Triangular Membership Function
Algebraic
Product
MIN
Algebraic
Product
B16
B28.4
B15
B28.11
B43.1
B16
B28.4
B15
Drastic
Product
Einstein
B16
B28.4
B15
B42
B28.11
B15
B28.4
B16
Ham
B16
B28.4
B15
B28.11
B28.11
B28.4
B43.1
B43.2
MIN
B43.1
B28.11
B43.2
B17
Drastic
Product
Einstein
B28.11
B17
B16
B40
B28.11
B28.4
B15
B17
Ham
B28.1
1
B28.4
B43.1
B43.2
Obviously the stream B28.3 appears to be the most risky one according to the Normal
Distribution (ND) and Sigmoid (SIG) MF, regardless the T-Norm. Other areas that are
characterized as risky based on the ND and on the SIG are B28.2, B28.15, B28.9, B28.6 and
B28.15. Of course the Drastic Product operates again in its own way, giving distinct areas as
risky, due to its nature that characterizes as risky the areas that have extreme values for one
or more parameters. Though there seems to be a disagreement between the Drastic and the
others, it is absolutely normal, due to the different perspective of the Drastic Product. The
following Table 5 presents the four most risky watersheds of the Koroneia area, based on
four different MF and on five different T-Norms, each one having a different view on the
problem.
Table 5
The four most Torrential risky streams in the Koroneia area for each MF.
AREA OF KORONEIA
Normal Distribution Membership Function
Sigmoid Membership Function
Algebraic
Drastic
Algebraic
Drastic
Product MIN
Product Einstein Ham
Product MIN
Product Einstein
Κ4
Κ2
Κ15
Κ4
Κ4
Κ4
Κ2
Κ1
Κ4
Κ2
Κ4
Κ2
Κ2
Κ2
Κ2
Κ24
Κ2
Κ2
Κ9
Κ14
Κ3
Κ9
Κ9
Κ15
Κ10
Κ3
Κ15
Κ8
Κ5
Κ4
Κ8
Κ13
Κ17
Κ1
Κ4
Κ17
Trapezoidal Membership Function
Triangular Membership Function
Algebraic
Drastic
Algebraic
Drastic
Product MIN
Product Einstein Ham
Product MIN
Product Einstein
Κ19
Κ19
Κ19
Κ19
Κ19
Κ19
Κ21
Κ19
Κ19
Κ18
Κ16
Κ23
Κ18
Κ23
Κ16
Κ18
Κ23
Κ16
Κ18
Κ16
Κ23
Κ18
Κ16
Κ23
Κ18
Κ17
Κ21
Κ19
Κ22
Κ17
Κ17
Κ18
Κ23
Κ18
Κ17
Κ21
Ham
Κ4
Κ2
Κ15
Κ17
Ham
Κ19
Κ18
Κ17
Κ21
The following Table 6 presents the percentage of compatibility between the G1, the G2, the
Kronfellner – Kraus method and the developed system. The equation of Gavrilovic considers
the average annual production of sediments, the average annual temperature, the average
annual rain height of the watershed, its area, the kind of its geodeposition, the vegetation,
the erosion of the watershed, and the calculation of the special degradation is really
important. The main difference between the G1 and the G2 methods is that G1 refers to the
total area under examination and G2 refers to the average per Km2. The maximum
compatibility has proven to be as high as 70% and the lowest 10%.
Conclusions
The System characterizes the same specific areas as the most torrential risky ones with
unimportant differences in their order of ranking, in the cases of the Trapezoidal and
Triangular MF and the same streams in the cases of Normal Distribution and Sigmoid MF.
So there seem to be do different clusters of Risky areas. There exists the cluster of the
Triangular-Trapezoidal MF and the cluster of the Sigmoid-Normal Distribution MF. The
determination of the most suitable and accurate cluster will become through the years as our
research goes on, comparing to the actual risky streams.
The average compatibility between the G1 and the TORRISDESSYS is 34%, between the
G2 and the TORRISDESSYS is 42.5% and between the Kronfellner – Kraus method and
TORRISDESSYS is 29.5%. The average compatibility is low in all three cases, but in the
best case it equals to 70% and in the worst it is 10%. So there is a lot of deviation.
On the other hand the Trapezoidal MF combined with the Algebraic Product, with the
Einstein Product and with the Hamacher product seems to have Compatibility as high as
70% to the G2 method. It should be mentioned that the Trapezoidal MF combined with the
MIN T-Norm also has a high compatibility of 65% to the G2 model. This proves that the
TRAPMF combined with any type of T-Norm except for the Drastic Product, operates quite
similarly to the G2 method.
Table 6
T-NORM
applied
Algebraic
Product
MIN
Drastic
Product
Compatibility test between the System and other existing methods.
Compatibility Test
Function
Function
G2
Function
Kronfellner
– Kraus
Triangular
45%
Triangular
55%
Triangular
40%
Trapezoidal
30%
Trapezoidal
70%
Trapezoidal
25%
40%
35%
Sigmoid
Sigmoid
Sigmoid
Normal
Normal
Normal
Distribution 30% Distribution 15% Distribution
30%
35%
Triangular
50%
Triangular
45%
Triangular
30%
Trapezoidal
30%
Trapezoidal
65%
Trapezoidal
25%
40%
15%
Sigmoid
Sigmoid
Sigmoid
Normal
Normal
Normal
Distribution 45% Distribution 15% Distribution
40%
40%
Triangular
25%
Triangular
45%
Triangular
15%
Trapezoidal
20%
Trapezoidal
50%
Trapezoidal
10%
Sigmoid
Sigmoid
Sigmoid
Normal
Normal
Normal
Distribution
Distribution
Distribution
Triangular 50% Triangular 55% Triangular
25%
40%
25%
40%
35%
Sigmoid
Sigmoid
Sigmoid
Normal
Normal
Normal
Distribution 30% Distribution 15% Distribution
25%
Trapezoidal
70%
-
Trapezoidal
Trapezoidal
Einstein
G1
35%
Triangular
50%
Triangular
55%
Triangular
40%
Trapezoidal
30%
Trapezoidal
70%
Trapezoidal
25%
10%
40%
Sigmoid
Sigmoid
Sigmoid
Normal
Normal
Normal
Hamacher Distribution 30% Distribution 15% Distribution
10%
25%
This does not mean that the Drastic Product approach (that has a 50% compatibility to the
G2 method) is not effective. The Drastic Product approach always assigns high risk indices
to the watersheds having extreme values for some of their parameters. For example if an
area has 0% Compact Geological forms this means that it has extreme value for this variable
and thus the output of the Drastic product will give an extreme value for the URI. Obviously
the low compatibility of the Drastic Product in most of the vases is due to the fact that it has a
completely different perspective, which does not make it less efficient.
The compatibility between the Triangular and the G2 model varies from 45% to 55% which is
not so bad considering the fact that the TRIAMF does not have a tolerance interval for the
high membership values, but it offers only one peak instead.
The comparison between the G1 and the TORRISDESSYS results shows that the
agreement is the highest (50%) in the case of the TRIAMF regardless the T-Norm used.
The comparison between the Kronfellner – Kraus model and the system proves that the
Triangular MF is the closest one in all of the cases except of the case when it is combined
with the Drastic Product. In the case of the MIN T-Norm the Sigmoid MF and the Normal
Distribution function are the closest ones to the Kronfellner – Kraus model with compatibility
as high as 40%.
The percentage of agreement of the TORRISDESSYS to the other methods is interesting but
it is not a measure of the TORRISDESSYS’s validity. Also it is a fact that the existing and
used methods so far do not agree to each other, but they are useful. The evaluation of the
system will continue for several years, comparing the watersheds characterized as risky to
the actual ones. It is a system that offers many different approaches. The optimal MF and TNorm for each area should be determined after years of testing and experience.
More research is required for the SIG MF. Various shapes of SIG functions should be tried
by changing the values of a and c, in order to determine which one gives better results. This
could have been the result of future study.
The final conclusion is that a new very promising model-DSS has been developed and
applied, but it is going to take several years of further research to prove the degree of its
output accuracy.
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