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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:
[email protected]
_______________________________________________________________________
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 MSAccess aiming in overcoming the limitations of the existing methods. An integrated reliable approach towards torrential risk estimation, requires consideration of a longperiod’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 TNorms in order to produce Unified Indices. Each watershed is assigned various Unified Risk Indices (depending on the TNorm used) each one representing its overall torrential risk, seen from a different angle. TORRISDESSYS was applied successfully for the watersheds of KoroneiaVolvi 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 torrentialhistory of each watershed over time. Finally, the method of KronfellnerKraus 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 twodimensional 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 TNorms 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 NorthEastern 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 KronfellnerKraus 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 (TNorms). Various TNorms were applied. The following Table 2 presents the TNorms that were applied for this purpose. Table 2 Description of the TNorms 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 TNorms. This means that the problem is studied under different perspectives. The Drastic Product TNorm 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 TNorm. 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 TNorms, 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 TriangularTrapezoidal MF and the cluster of the SigmoidNormal 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 TNorm also has a high compatibility of 65% to the G2 model. This proves that the TRAPMF combined with any type of TNorm except for the Drastic Product, operates quite similarly to the G2 method.
Table 6
TNORM 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 TNorm 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 TNorm 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 modelDSS 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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