Fuzzy set theory based methodology for the

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Fuzzy set theory based methodology for the analysis of measurement uncertainties in river discharge and stage Rajesh R. Shrestha and Slobodan P. Simonovic

Abstract: The discharge and stage measurements in a river system are characterized by a number of sources of uncertainty, which affects the accuracy of a rating curve established from measurements. This paper presents a fuzzy set theory based methodology for consideration of different sources of uncertainty in the stage and discharge measurements and their aggregation into a combined uncertainty. The uncertainty in individual measurements of stage and discharge is represented using triangular fuzzy numbers, and their spread is determined according to the International Organization for Standardization (ISO) standard 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of various uncertainties into overall stage–discharge measurement uncertainty. In addition, a fuzzified form of ISO 748 formulation is used for the calculation of combined uncertainty and comparison with the fuzzy aggregation method. The methodology developed in this paper is illustrated with a case study of the Thompson River near Spences Bridge in British Columbia, Canada. The results of the case study show that the selection of number of velocity measurement points on a vertical is the largest source of uncertainty in discharge measurement. An increase in the number of velocity measurement points provides the most effective reduction in the overall uncertainty. The next most important source of uncertainty for the case study location is the number of verticals used for velocity measurements. The study also shows that fuzzy set theory provides a suitable methodology for the uncertainty analysis of stage–discharge measurements. Key words: discharge calculation, fuzzy arithmetic, measurement uncertainty, uncertainty aggregation, velocity–area method. Re´sume´ : Les mesures de de´bit et de niveau dans un re´seau hydrographique sont caracte´riseés par plusieurs sources d’incertitudes, ce qui affecte la prećision d’une courbe des de´bits jauge´s e´tablis a` partir des mesures prises. Cet article pre´sente une theórie des ensembles flous pour tenir compte des diverses sources d’incertitude dans les mesures de niveau et de de´bit ainsi que leur regroupement dans une incertitude combineé. L’incertitude des mesures individuelles du niveau et du de´bit est repre´senteé en utilisant des nombres flous triangulaires et leur dispersion est de´termineé en utilisant les lignes directrices ISO 748. L’arithme´tique floue baseé sur le principe d’extension est utiliseé pour le regroupement des diverses incertitudes dans l’incertitude globale de la mesure du de´bit et du niveau. De plus, une forme floue de la formulation ISO 748 est utiliseé pour calculer l’incertitude combineé et la comparaison a` la me´thode floue de regroupement. La me´thode de´veloppeé dans cet article est illustreé par une e´tude de cas de la rivie`re Thompson pre`s du pont Spences en ColombieBritannique, au Canada. Les re´sultats de cette e´tude de cas montrent que le choix du nombre de points de mesure de la vitesse sur une verticale est la plus grande source d’incertitude de la mesure du de´bit. Une augmentation du nombre de points de mesure repre´sente la re´duction la plus efficace de l’incertitude globale. La seconde source en importance d’incertitude pour l’emplacement de l’e´tude de cas est le nombre de verticales utiliseés pour les mesures de vitesse. L’e´tude montre e´galement que la theórie des ensembles flous est approprieé pour l’analyse des incertitudes dans des mesures de de´bit et de niveau. Mots-cle´s : calcul du de´bit, arithme´tique floue, incertitude de mesure, regroupement d’incertitudes, me´thode de la zone de vitesse. [Traduit par la Re´daction]

Received 13 May 2008. Revision accepted 25 September 2009. Published on the NRC Research Press Web site at cjce.nrc.ca on 4 March 2010. R.R. Shrestha.1 Environment Canada, Water and Climate Impacts Research Centre (W-CIRC), University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada. S.P. Simonovic. Department of Civil and Environmental Engineering, The University of Western Ontario, London, ON N6A 5B8, Canada. Written discussion of this article is welcomed and will be received by the Editor until 31 July 2010. 1Corresponding

author (e-mail: [email protected]).

Can. J. Civ. Eng. 37: 429–439 (2010)

doi:10.1139/L09-151

Published by NRC Research Press

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Introduction River stage and discharge records are essential for hydrologic and hydraulic analyses. While stage is measured directly, discharge is calculated using the velocity–area method from measurements of flow velocity, depth, and channel cross section. The result of a measurement is only an estimate of the true value of the measurement and is therefore only complete when accompanied by a statement of its uncertainty (Herschy 2002). The measurements are affected by random and systematic measurement errors and other inaccuracies, such as approximation of velocity distribution and channel geometry with a finite number of measurements. Such errors lead to the uncertainty in both the stage and discharge values, which propagates into the rating curve established from the measurements. The measurement uncertainty further propagates into the discharge value derived from the rating curve. Therefore, the measurements obtained from gauging stations should not always be readily accepted without the understanding and quantification of different sources of uncertainty that may affect them (Whalley et al. 2001). Several guidelines by the International Organization for Standardization (ISO 1997, 1998), Environment Canada (Terzi 1981), and the US Geological Survey (Rantz 1982) have outlined different sources of uncertainty in the measurement of discharge and stage. An extensive literature review of the measurement uncertainty is available in Pelletier (1988). The velocity–area method based on current meter measurements of the velocity is the widely accepted method for discharge determination (Herschy 1999; Whalley et al. 2001) and is the standard in Canada (Terzi 1981; Pelletier 1988). In this method, flow velocity, water depth, and cross-section width are measured at a number of points distributed over a number of verticals covering the channel cross section. Point measurements are then aggregated over a cross section, and the total discharge in the cross section is determined. For the quantification of stage–discharge uncertainty, the ISO standard 748 suggests the range of values at a 95% confidence level for different sources of uncertainty. This recommendation is based on investigations carried out since 1968. The ISO standard 748 recommends independent determination of uncertainty in each measurement for the application to a particular case study. However, in most cases, an independent value of the confidence interval in the measurement is not available, which limits the applicability of statistical quantification of the uncertainties. It is to be noted too that randomness is not the only source of uncertainty in discharge measurements, as they can also be affected by systematic uncertainty, human error, and other subjective uncertainties that cannot be treated in a statistical framework. For example, the evaluation of individual current meter discharge measurement on the basis of hydrographer observation can be subjective, as each measurement can receive different ratings based on the perception of the hydrographer (Clemmens and Wahlin 2006). The ISO (1993) guide for expression of uncertainties has recognized these limitations by distinguishing two different categories of uncertainties according to the method used to estimate their numerical values: type A, a method of evaluation of uncertainty by the statistical analysis of series of observations; and type B, evaluation of uncertainty by means other than the statistical analysis of series of observations.

Can. J. Civ. Eng. Vol. 37, 2010

The ISO standard 748 also provides a statistical framework for aggregation of confidence levels of measurement uncertainties. The combined uncertainty is expressed as the ratio between the sum of percentage errors in the segment discharges and the sum of the segment discharges (Herschy 1995). However, such an aggregation method only provides a means of combining the confidence levels and cannot provide a confidence interval of the output unless the probability distribution function that characterizes its dispersion is known (Ferrero and Salicone 2003). Therefore, there are a number of limitations in the application of the statistical methodology in the aggregation of the overall uncertainties in discharge and stage measurements. The fuzzy set theory based approach is explored in this paper as an alternative way of analyzing various uncertainties associated with discharge measurements. The fuzzy approach provides a nonprobabilistic framework for representation of uncertainties using vaguely defined boundaries of fuzzy sets. El-Baroudy and Simonovic (2006) and Guyonnet et al. (2003) used fuzzy sets to treat uncertainties due to lack of knowledge and scarcity of data, respectively. In recent years, fuzzy sets have been used for the expression of uncertainty in measurement by a number of researchers (Xia et al. 2000; Mauris et al. 2001). The study by Xia et al. (2000) considered the application of fuzzy sets for the estimation of uncertainty when the number of measurements is very small and the probability distribution is unknown. Mauris et al. (2001) used fuzzy sets for the representation of vertical interpretation of the probability distribution and nested stacks of intervals as horizontal interpretation of the distribution function for representation of measurement uncertainty. The study also showed that fuzzy representation of measurement uncertainty in terms of the possibility distribution is compatible with the ISO (1993) guide for expression of uncertainties, as it can characterize dispersion of observed data and provide a confidence interval that contains an important proportion of the observed values. Another approach for the consideration of measurement uncertainties uses random-fuzzy variables (Ferrero and Salicone 2003, 2004; Urbanski and Wasowski 2003) to define random properties of uncertainties in terms of the probability distribution and systematic components in terms of the possibility function. However, in the absence of information on the random uncertainties, a purely fuzzy treatment can still be used. Fuzzy sets also provide a suitable methodology for the aggregation of uncertainties expressed in terms of fuzzy numbers of different measurement quantities. A number of different aggregation methods such as t-norm, t-conorm, mean operator, Yager’s operator, and g-operators (Zimmerman 2001) can be used for operations on fuzzy sets such as union and intersection. In the work presented here, the aggregation of measurement quantities expressed in terms of fuzzy numbers requires mathematical operations like addition and multiplication. The extension principle (Zadeh 1965) fuzzy arithmetic provides a suitable methodology to generalize these crisp mathematical operations to fuzzy sets. Therefore, fuzzy arithmetic is used for the aggregation of uncertainties in different measurement quantities. Previous studies by Pappenberger et al. (2006) and Shrestha et al. (2007) used a fuzzy set based approach for the repPublished by NRC Research Press

Shrestha and Simonovic

resentation of uncertainty in the stage–discharge relationship and the analysis of flood inundation. Consideration of the stage and discharge measurement uncertainty in a fuzzy framework will lead to treatment of uncertainties from the source. The main objective of this paper is to use the fuzzy set based approach to represent all uncertainties associated with implementation of the velocity–area method for the development of stage–discharge relationships. The methodology focuses on the fuzzy representation of uncertainties in the measurement of flow velocity and water depth and propagation of measurement uncertainties into calculation of discharge and stage. The paper starts with a discussion of different sources of uncertainties in the stage and discharge measurements. The second section of the paper discusses two different methods of uncertainty propagation considered in this study: (i) aggregation of uncertainties in measurements of velocity, width, depth, and stage based on fuzzy arithmetic; and (ii) use of a fuzzified form of ISO standard 748 formulation for the calculation of overall uncertainty. The third section of the paper illustrates the methodologies with a case study from the Thompson River near Spences Bridge, British Columbia. The last section of the paper discusses the dominant sources of measurement uncertainties for the case study and provides recommendations for their reduction.

Sources of uncertainty in discharge and stage measurements Uncertainty in discharge measurement The standard method for discharge calculation in a cross section is the velocity–area method, where velocities, widths, and depths are measured in a finite number of verticals in a cross section. A major source of uncertainty according to ISO standard 748 is in the approximation of bed profile and the velocity distribution using a limited number of verticals. In general, selection of too few verticals may lead to a considerable error in discharge. ISO standard 748 recommends that the interval shall not be greater than 1/15 of the width in the case of regular bed profiles and 1/20 of the width in the case of irregular bed profiles. ISO standard 748 also suggests the uncertainty for number of verticals at a 95% confidence level to be 15% for five verticals and up to 2% for 45 verticals. The velocity measurement involves three types of uncertainty: (i) number of limited points on a vertical, (ii) exposure time of velocity measurement, and (iii) current meter measurement. The first type of uncertainty is due to approximation of the velocity distribution on a vertical using a limited number of sampling points. Common methods of determination of the mean velocity are usually based on the one- or two-point method, which involves measurement of velocity at 0.6 of the depth (0.6D) or at 0.2 and 0.6 of the depth (0.2D and 0.8D), respectively. According to ISO standard 748 suggestions, uncertainty at the 95% confidence level may be as high as 15% for one-point measurement and up to 7% for two-point measurement. The second uncertainty arises due to limited exposure time of local point velocity on the vertical, with an assumption of the steady flow condition. An instantaneous measurement of the velocity at a point could be considerably different from the mean

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velocity at that point. The mean flow velocity determined from measurement during a finite measuring time will therefore be an approximation of the true mean flow velocity at that point (Sauer and Meyer 1992). By observing the velocity for a longer period of time, the pulsation differences are averaged and the mean velocity during exposure approaches the true velocity. According to the ISO standard 748 suggestions, uncertainty at the 95% confidence level due to exposure times of 0.5, 1.0, 2.0, and 3.0 min is 7%, 6%, 5%, and 4%, respectively. The third type of uncertainty arises in the current meter measurement of velocities, which can consist of errors due to several sources like the effect of suspension (rod or cable suspension), a boundary effect (like near rough boundaries and shallow depth), the effect of oblique flow, vertical motion (wave action or rocking of the boat), and the effect of turbulence (Pelletier 1988). However, the range of uncertainty for these sources is not available, and uncertainty in the current meter measurement is usually only considered in terms of current meter rating (individual or group rating). The random uncertainty value is generally small, i.e., a magnitude of ±1% at the 95% confidence level for an individually rated current meter (Herschy 1995). A relatively small uncertainty also arises in the measurements of water depth and channel width. Under most discharge measurement conditions, measurements of the overall width and distances between verticals can be made with reasonable precision, and ISO standard 748 suggests a relative error of ±0.3% for the width measurement between 0 and 100 m. The instrumental error in the measurement of depth depends to a large extent on the composition of the river bed. ISO standard 748 suggests an uncertainty of up to ±3% for depths up to 0.3 m and up to ±1% for depths greater than 0.3 m. For depth measurements between 0.4 and 6.0 m using a sounding rod, a relative error of ±0.7% is suggested. Uncertainty in stage measurement The uncertainty in stage measurement depends on the characteristics of the gauging station and the water surface elevation. Since the stage can be measured directly, it is reasonable to assume that errors in the measurement of stage are small compared with those in the measurement of discharge (Clarke 1999). However, displacement of measured values from the reference point caused by processes such as turbulent fluctuations, wind, and stationary waves can lead to error in the measured stage (Schmidt 2002). Different measurement instruments have different accuracies, and the stage measurement uncertainty in a float-operated autographic recorder is in the order of ±10 mm (Herschy 1995). A source of uncertainty often neglected in stage measurement is the determination of mean reference gauge height corresponding to the measured discharge. According to Rantz (1982), if the change in stage is uniform or no greater than 0.05 m, the mean stage can be obtained by averaging the stage at the beginning and end of the measurement. In the case of a non-uniform stage, mean stage can be obtained by weighting each stage by partial discharge. There is no suggested uncertainty range available for the determination of mean stage. Published by NRC Research Press

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Uncertainty propagation in the measurement of stage and discharge Introduction of fuzzy arithmetic approach The fuzzy set theory based analysis of uncertainty in the stage and discharge measurements involves aggregation of different fuzzy quantities. When the measurements of velocity, width, and discharge are expressed as fuzzy numbers, total uncertainty can be calculated using simple mathematical operations like addition and multiplication. The extension principle (Zadeh 1965) based fuzzy arithmetic provides a means to generalize crisp mathematical operations to fuzzy sets. A simple tool for the fuzzy arithmetic operation is the alpha-level cut (a-cut), which can be used for resolving fuzzy numbers into crisp numbers so that crisp mathematical operations such as addition, subtraction, division, square, and square root can be performed (Simonovic 2008). An example of a fuzzy number with a-cut and its support is shown in Fig. 1. Let an a-cut intersect the membership function of a fuzzy number at two points a1 and a2 (a1, a2 [ A). Then, the subset Aa contains all possible values of the fuzzy variable A, including and between a1 and a2, which are referred to as the lower and upper bounds of the

Fig. 1. Fuzzy number and a-level cut.

a-cut. The subset Aa also contains a set of elements, which have at least a membership value greater than or equal to a, as given by ½1

Aa ¼ fa 2 A; mA ðAÞ ag

where çA(A) is the grade of membership of a in the fuzzy subset A, two fuzzy numbers A and B are considered at the a level: Aa = [a1, a2] and Ba = [b1, b2].

The individual arithmetic operations on the a-cut of A and B can be defined in terms of the following equations (Klir 1997; Simonovic 2008): ½2

½a1 ; a2 þ ½b1 ; b2 ¼ ½a1 þ b1 ; a2 þ b2

½3

½a1 ; a2 ½b1 ; b2 ¼ ½a1 b2 ; a2 b1

½4

½a1 ; a2 ½b1 ; b2 ¼ ½min ða1 b1 ; a1 b2 ; a2 b1 ; a2 b2 Þ; max ða1 b1 ; a1 b2 ; a2 b1 ; a2 b2 Þ

½5

a1 a1 a2 a2 a1 a1 a2 a2 ; max ; for 0=2½b1 ; b2 ½a1 ; a2 =½b1 ; b2 ¼ min ; ; ; ; ; ; b1 b2 b1 b2 b1 b2 b1 b2

For calculation of the fuzzy square root, 8 pffiffiffi pffiffiffi > > ½ a; a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi < pffiffiffiffiffiffiffi pffiffiffi ½6 ½a1 ; a2 ¼ ½ a; a > pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi > : ½ a1 ; a2

Similarly, for calculation of ing equation can be used: 8 2 2 ½a ; a > > < 1 2 2 2 ½7 ½a1 ; a2 2 ¼ ½a2 ; a1 > > : ½a21 ; a22

Salicone (2007) proposed the following relation: when 0 a1 a2 when a1 < 0; a2 > 0 when a1 a2 0

the fuzzy square, the followwhen 0 a1 a2 when a1 < 0; a2 > 0 when a1 a2 0

Aggregation of uncertainties using fuzzy arithmetic For consideration of uncertainties in measurement of velocity, depth, and width, each of the measurement quantities is expressed as a symmetrical triangular fuzzy number, with the spread given by percentage fraction between –xi and +xi and the central value at 0 as shown in Fig. 2.

The uncertainty in the measurement of velocity consists of three different sources: (i) uncertainty in the number of b p , (ii) current meter rating X b c , and (iii) points on a vertical X b time of exposure X e . Each of these uncertainties is independent of one another, so it is unlikely that these uncertainties will occur at the same time. Therefore, the total uncertainty can be considered to be less than the arithmetic sum of the individual uncertainties. A method based on ISO standard 748 is employed, which uses the square root of the sum of squares of individual uncertainties instead of the arithmetic sum, for the calculation of reduced total uncertainty. As the individual uncertainties are expressed in terms of percentage fraction between –xi and +xi and the central value at 0, the combined fuzzy uncertainty in the mean vePublished by NRC Research Press


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Fig. 2. Expression of measurement uncertainty in terms of fuzzy number.

½11

n1 X biþ1 bi1 Q¼ v i di 2 i¼2

Using the fuzzified values of velocity, depth, and discharge from eqs. [8]–[10], the total discharge is calculated b as a fuzzy number Q: ! n1 X b b b b iþ1 i1 b b¼ b ½12 Q v i di 2 i¼2 To take into consideration the uncertainty due to a limited b m , expressed as a fraction, the total number of verticals, X uncertainty in the discharge measurement is calculated using the following relationship: ½13

locity bv i is calculated as the sum of total uncertainties plus unity multiplied by the crisp mean value of velocity measurement v i : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b p Þ2 þ ðX b c Þ2 þ ðX b e Þ2 b ½8 v i ¼ v i ½1 þ ðX Similarly, the fuzzy number of the width measurement b bi and depth measurement b di considering the measurement uncertainties is expressed as ½9 ½10

In the case of stage measurement, two different sources of uncertainty are considered: (i) error in the measuring instrub ins ; and (ii) error in the determination of mean referment, X ence gauge height corresponding to the measured discharge, b ref . Since uncertainty ii is dependent on uncertainty i, the X uncertainty in one measurement will affect the other measurement. Therefore, an arithmetic sum of individual uncertainties is used for the calculation of the sum of uncertainties i and ii. Therefore the aggregated uncertainty in stage measurement can be expressed as ½14

b b ins þ X b ref Þ h ¼ h½1 þ ðX

where h is the measured stage, and b h is the fuzzified stage.

b bbÞ bi ¼ bi ð1 þ X b bd Þ di ¼ di ð1 þ X

b b and X b d are the fuzzy numbers of uncertainties in where X width bi and depth di measurements, respectively. The computation of discharge using the midsection velocity area method is the standard in Canada (Pelletier 1988). For the width measurement from a common reference point, the discharge Q measurement using the midsection method may be expressed as

Aggregation of uncertainties using fuzzified ISO method For comparison with the previous method, the aggregation of uncertainties using fuzzy variables with a conventional treatment of measurement, uncertainties according to ISO standard 748 are used. In this case, instead of aggregation of confidence level of uncertainty, each of the uncertainties is fuzzified and aggregated using fuzzy arithmetic. According to the ISO standard 748 suggested formulation, a combination of confidence level of uncertainties is expressed as

12 3 1=2 m X 2 2 2 2 2 1=2 B fðbi di v i Þ½ðXb Þ þ ðXd Þ þ ðXp Þ þ ðXc Þ þ ðXe Þ gC 7 6 B i¼1 C 7 6 2 C 7 6 XQ ¼ 6ðXm Þ þ B m B C 7 X @ A 5 4 ðb d v Þ 2

½15

b tot ¼ Qð1 b þX bmÞ Q

0

i i i

i¼1

It is to be noted that eq. [15] is different from the equation given in ISO standard 748. The derivation of the equations for the aggregation of uncertainties according to ISO 748 is documented in Herschy (1995), where two different forms of equations are listed. The equation without squaring of the terms on the right-hand side is used in this study be-

cause it confirms with the definition of total uncertainty as a ratio of the sum of the percentage errors in the segment discharges to the sum of the segment discharges. Hence, the original form of the equation by Herschy is used for the aggregation of uncertainties. Expressing each uncertainty in terms of fuzzy numbers leads to Published by NRC Research Press

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2

½16

0

12 31=2 m X 2 2 2 2 2 1=2 b b b b b B fðbi di v i Þ½ðX b Þ þ ðX d Þ þ ðX p Þ þ ðX c Þ þ ðX e Þ gC 7 6 B i¼1 6 C 7 2 6 C 7 b b X Q ¼ 6 ðX m Þ þ B m B C 7 X @ 4 A 5 ðb d v Þ i i i

i¼1

The total uncertainty in discharge measurement is expressed as ½17

b tot ¼ Qð1 þ X bQÞ Q

Case study Stage and discharge measurements from an Environment Canada hydrometric station (08LF051) at Thompson River near Spences Bridge are used for the fuzzy analyses of measurement uncertainties. The measurement data between the years 1970 and 2000 are used for the analysis. The Thompson River is a major tributary of the Fraser River in British Columbia, Canada, and has a gross drainage area of 54 900 km2 at the gauging station. The station is located in a narrow gorge with well-defined banks. The analysis of the river cross sections from 1970 to 2000 showed very little change in channel geometry, indicating that the cross section is very stable. Environment Canada used individually rated current meters for discharge measurement at the Spences Bridge gauging station. The measurement data consisted of 20–30 verticals in a cross section with a single point velocity measurement on each vertical. Based on these measurements, the midsection method was used to obtain the total discharge. The available information from the Spences Bridge gauge and general practice of discharge measurement in Canada (Terzi 1981; Pelletier 1988) are used in the analyses. The ISO standard 748 suggested random uncertainty values (at the 95% confidence level) are used as a reference for the expression of each uncertainty in terms of triangular fuzzy number. As already outlined, ISO 748 recommends determination of the values independently for application to a particular case. To account for the lack of information on random uncertainties and possible systematic uncertainties, the spread of each fuzzy number, in this case, is increased by 50%. Therefore, the fuzzy number of each of the uncertainty sources is viewed as a combination of both the random and systematic uncertainties. The increase in the spread of the fuzzy number by 50% is subjective in this particular case. An appropriate value of spreads can be chosen objectively at a particular gauging station by determining individually the random and systematic uncertainty sources for each measurement. The left or right spread (one half of the support) of the symmetrical triangular fuzzy number taken for each of the errors is given in Table 1. In the case of uncertainty in the stage measurement, half of the support of the fuzzy number of errors considered is summarized in Table 2. Results and discussion Fuzzy arithmetic aggregation method The aggregation of uncertainties using fuzzy variables

leads to nonlinear fuzzy numbers of discharge values. The membership functions of the largest measured discharge and corresponding stage are shown in Figs. 3 and 4, respectively. The results show high uncertainty in the discharge due to measurement uncertainties characterized by the wide support of the discharge fuzzy number. The central value of the discharge and stage fuzzy numbers represents the values without consideration of uncertainty. The left and right spreads represent the total uncertainties in the measurement. The fuzzy numbers can be interpreted in terms of membership levels, with 0 as the highest uncertainty, i.e., the extreme possible measurement value. The closer the membership level is to 1, the lower is the uncertainty. The independent (non-interactive) measurements of discharge and stage are combined to form a joint fuzzy number of the corresponding measurements. This leads to a tridimensional representation of the fuzzy number as shown in Fig. 5. The joint membership function of the stage and discharge values provides a visualization of the uncertainties in stage and discharge at any membership level. The joint membership function of the fuzzy numbers mQh is given by ½18

mQh ¼ min ðmQ mh Þ

where mQ is the discharge membership level, and mh is the stage membership level. Figure 6 shows a ‘‘top view’’ of the uncertainty in observed stage and discharge values represented by spread of the joint membership functions of the fuzzy numbers. Each rectangle in Fig. 6 represents four points a, b, c, and d with membership level 0 as shown in Fig. 5. It is evident from Fig. 6 that the spread of the discharge fuzzy numbers increases with the higher discharge while the spread of the stage fuzzy numbers remains constant. This is due to the fact that uncertainties in each of the elements of discharge measurement (velocity, depth, and width) are expressed in terms of percentage values, whereas constant uncertainty is used for all stage values. Fuzzified ISO method The result of the fuzzified ISO method also shows the large spread of the fuzzy number of largest measured discharge as shown in Fig. 7. In this case also, there is an increase in the spread of discharge fuzzy numbers with the higher discharges as shown in Fig. 8. A comparison of the left and right spread of the fuzzy numbers for the minimum, mean, and maximum discharge is given in Table 3, which shows that the spreads are higher in the case of the fuzzy aggregation method than in the case of the ISO method. This is due to the fact that aggregation of uncertainty using the fuzzy arithmetic method uses a direct combination of fuzzy numbers of different uncertain quantities, and there is no reduction of uncertainty. In the case of the ISO method, the fuzzified form of the ISO equation (eq. [15]) is used, Published by NRC Research Press


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Table 1. Half of the support of the fuzzy number of errors in discharge measurement due to different uncertainty sources. Uncertainty source Limited number of verticals (20–30) Limited number of points in a vertical (single point) Limited exposure time (one minute) Current meter rating (individual rating for velocity>0.5 m/s) Depth measurement (0.4–6.0 m) Width measurement (0–100 m)

Half of the support of fuzzy number (%) 7.5–4.5 22.5 9.0 1.5 1.05 0.45

Table 2. Half of the support of the fuzzy number of errors in stage measurement due to different uncertainty sources. Uncertainty source Measurement instrument Determination of mean reference gauge height

Half of the support of fuzzy number (mm) 10 10

Fig. 3. Membership function of fuzzy number of maximum observed discharge using fuzzy arithmetic aggregation method. 1

Fig. 5. Joint membership function of maximum observed stage and discharge in terms of tri-dimensional representation of fuzzy number (fuzzy arithmetic aggregation method).

0.9 0.8

1 0.8 Membership level

Membership level

0.7 0.6 0.5 0.4

0.6 0.4

b

0.2

0.3 0

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c

8.9 8.92

0.1 0 2500

8.94

d

8.96

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Fig. 6. Uncertainty in observed stage and discharge represented by spread of the joint membership functions using fuzzy arithmetic aggregation method. 6000

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Fig. 4. Membership function of fuzzy number of maximum observed stage using fuzzy arithmetic aggregation method.

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Recommendations for reduction of uncertainties in the case study The possibilities of reducing the total uncertainty in discharge measurement are analyzed by considering different spreads of the membership functions of the uncertainty sources. Three parameters with the highest range of uncertainty values are chosen for the analyses, which include approximation due to the limited number of verticals, velocity uncertainties due to the limited numbers of points on a vertical, and the measurement exposure time. Different values of uncertainties used for the analyses are summarized in Table 4. Figures 9 and 10 show the effects of the uncertainty in the number of verticals for the fuzzy aggregation and fuzzified ISO methods, respectively. For the reduction of uncertainty from 6.0% to 1.5%, the reduction in support of the fuzzy number in the fuzzy aggregation and the fuzzified ISO methods is 13.4% and 2.5%, respectively. This shows that the uncertainty due to a limited number of verticals has a more significant effect with the application of the fuzzy aggregation method than with the application of the fuzzified ISO method. This difference is due to the fact that the ISO method combines the uncertainties as a square root of the sum of squares of all the uncertainties, and the highest value of uncertainty dominates. This leads to a lower effect of elements with the low uncertainty level. In the case of uncertainty due to a limited number of points on a vertical, a higher reduction of uncertainties is observed in both methods, as shown in Figs. 11 and 12. The reduction of support of the fuzzy numbers of 47% and 55% is obtained in the fuzzy aggregation and fuzzified ISO methods, respectively, when the uncertainties are reduced from 22.5% to 2.5%. The uncertainty due to a limited number of points on a vertical has the highest range of values of all the uncertainties considered, which therefore provides a high possibility for the reduction of the uncertainties, regardless of the method used for aggregation. However, it is interesting to note that the effect of the uncertainty is less dominating in the fuzzy aggregation method, which uses the direct combination, compared to the fuzzified ISO method, which uses the square root of the sum of squares of all the uncertainties. The effect of uncertainties due to exposure time is similar in both methods, as shown in Figs. 13 and 14. For the reduction of uncertainty from 9.0% to 3.0%, reduction in the support of the fuzzy number in the fuzzy aggregation and fuzzified ISO methods is 4.8% and 5.9%, respectively. This shows that there is a limited possibility for the reduction of uncertainties for the exposure time. Based on the effects of different sources of uncertainties in this particular case study, it can be seen that the number

Fig. 7. Membership function of fuzzy number of maximum observed discharge using fuzzified ISO method.

0.6 0.5 0.4 0.3 0.2 0.1 0 2500

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Fig. 8. Uncertainty in the observed stage and discharge represented by spread of the joint membership functions using fuzzified ISO method 6000

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which combines the fuzzy numbers of uncertainties as a square root of the sum of squares of all the uncertainties. Therefore, there is reduction of uncertainties in the ISO method. It is to be noted also that the ISO method leads to a linear fuzzy number of discharge, and the fuzzy aggregation leads to a nonlinear fuzzy number. The right spread of the fuzzy numbers for minimum, mean, and maximum are higher than the left spread in the case of the fuzzy aggregation method, whereas the left and right spreads are equal in the case of the fuzzified ISO method.


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1

2

3

4 5 Stage (m)

6

7

8

9

of points on a vertical is the greatest source of uncertainty in the case of the Spences Bridge gauge. Therefore, reduction in the uncertainty in this source provides the most significant reduction in the overall discharge uncertainty. According to the ISO standard 748 suggestions, the uncertainty due to the number of verticals is 15% and 7% at the 95% confidence level for one- and two-point measurement of discharge, respectively. It can be seen from the analysis that the reduction of uncertainty in the fuzzy number from 22.5% to 10.5% (with 1.5 multiplication factor) results in a reduction in the overall uncertainty in terms of support of the fuzzy number of 33% in the fuzzy aggregation method. The other two parameters considered in the analysis show very limited impact on the reduction of the uncertainty. For example, reduction of uncertainty in the number of verticals in a cross section from 6.0% to 3.0% results in an 8.6% reduction of total uncertainty with the application of the aggregation method. This corresponds to the increase in Published by NRC Research Press


437

Table 3. Comparison of the left and right spreads of the fuzzy numbers of discharge using fuzzy aggregation and fuzzified ISO method. Left spread (m3/s) Min 45 38

Fuzzy arithmetic aggregation method Fuzzified ISO method

Mean 406 338

Table 4. Values of uncertainties used for the uncertainty reduction analyses. Uncertainty values (%) 6.0 4.5 22.5 15.0 9.0

7.5

1.5 2.5

0.9

Membership level

Max 1361 1031

6.0

3.0

0.7

Uncertainty=22.5% Uncertainty=15.0% Uncertainty=10.5% Uncertainty=2.5%

0.8


0.6 0.5 0.4 0.3 0.2

0.7

0.1

0.6

0 2500

3000

3500

0.5


5000

5500

6000

0.4

Fig. 12. Reduction in uncertainties due to the number of points in a vertical (fuzzified ISO method)

0.3

1

0.2

0.8 3000

3500


5000

5500

6000 0.7

Fig. 10. Reduction in uncertainties in the number of verticals (fuzzified ISO method). 1 Uncertainty=6.0% Uncertainty=4.5% Uncertainty=3.0% Uncertainty=1.5%

0.9 0.8 0.7

Membership level

0 2500


0.9

0.1

Membership level

Mean 459 338

Fig. 11. Reduction in uncertainties due to the number of points in a vertical (fuzzy arithmetic aggregation method).

3.0 10.5

1

0.8

Min 49 38

1

Fig. 9. Reduction in uncertainties in the number of verticals (fuzzy arithmetic aggregation method). 0.9

Max 1211 1013

Membership level

Criteria No. of verticals No. of points in a vertical Exposure time

Right spread (m3/s)

0.6 0.5 0.4 0.3 0.2 0.1

0.6

0 2500

0.5

3000

3500


5000

5500

6000

0.4 0.3 0.2 0.1 0 2500

3000

3500


5000

5500

6000

number of verticals from 25 to 35. On the other hand, a reduction in uncertainty in the number of verticals will lead to a significant increase in the overall uncertainty (Fig. 11). An increase in the uncertainty from 6.0% to 13.5% (corresponding to a reduction in the number of verticals from 25 to 10) will lead to a 25% increase in the support of the discharge fuzzy number. Similarly, an increase in the exposure time from 1 min to 3 min results in a 3.0% reduction in the support of the membership function with the use of the fuzzy aggregation method. Therefore, the increase in the Published by NRC Research Press

438


Fig. 13. Reduction in uncertainty due to exposure time (fuzzy arithmetic aggregation method). 1 Uncertainty=9.0% Uncertainty=7.5% Uncertainty=6.0% Uncertainty=3.0%

0.9 0.8

Membership level

0.7

Conclusions

0.6 0.5 0.4 0.3 0.2 0.1 0 2500

3000

3500


5000

5500

6000

Fig. 14. Reduction in uncertainty due to exposure time (fuzzified ISO method). 1 Uncertainty=9.0% Uncertainty=7.5% Uncertainty=6.0% Uncertainty=3.0%

0.9 0.8

Membership level

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2500

fuzzy number by 50% is subjective in this particular case. An appropriate value of spreads can be chosen by determining individually the random and systematic uncertainty sources for each measurement. This will lead to a more objective determination of total uncertainties at a gauging station.

3000

3500


5000

5500

6000

number of point measurements on a vertical provides the most significant opportunity for reduction of the discharge measurement uncertainty. In this regard, the use of an acoustic Doppler current profiler (ADCP) can lead to a significant reduction in discharge measurement uncertainties. The method measures ensembles of velocities throughout the sample window in a measurement section, leading to better accuracy in discharge measurement (Simpson 2001). Our preliminary analysis (Shrestha and Simonovic 2009) suggests that the total uncertainty in discharge measurement using the ACDP is far smaller than that using a current meter . It should be noted that the combined uncertainties and conclusions discussed here may only apply to the case study under consideration. For example, gauging stations with unstable cross sections will have higher uncertainties in depth and width measurements, which may increase the contribution of the particular component to the overall discharge measurement uncertainty. The increase in the spread of the

The understanding and quantification of different sources of uncertainty in the measurement of discharge and stage are of great importance for hydrologic and hydraulic analyses of river systems. The paper presents an original fuzzy set theory based approach for the consideration of different sources of uncertainty in measurement of stage and discharge and their aggregation into a combined uncertainty. Each of the measurement quantities is represented as a triangular fuzzy number with the spread determined on the basis of the International Organization for Standardization (ISO) standard 748 guidelines. The extension principle based fuzzy arithmetic is used for the aggregation of different uncertainties and calculation of the total measurement uncertainty. The paper explores the methods of combining the independent and dependent sources of uncertainties. As the individual uncertainties in the velocity measurement are independent of one another, it is unlikely that these uncertainties will occur at the same time. Therefore, the total uncertainty is considered as the square root of the sum of squares of individual uncertainties. As the uncertainties in stage measurement are dependent, the uncertainty in one source will affect the uncertainty in the other source. Therefore, an arithmetic sum of individual uncertainties is used for the calculation of total uncertainty. The results of the study are compared with the fuzzified form of ISO standard 748 formulation for the calculation of combined measurement uncertainty. The results for the Spences Bridge location on the Thompson River in British Columbia, Canada, show high uncertainty in the measurement of discharge (expressed by the wide support of the discharge fuzzy number). The analysis of different uncertainty sources shows that the number of points on a vertical for the measurement of velocity is the largest source of uncertainty in the discharge measurement. Therefore, the increase in the number of points on a vertical results in the largest reduction in the measurement uncertainty. The number of verticals in a cross section is another important source of uncertainty in discharge measurement. Although there is a limited reduction in uncertainty when the number of verticals is increased beyond 25, there will be a considerable increase in uncertainty when the number of verticals is reduced below 10. These results can be used as a basis for improvement in the measurement methods and subsequent reduction in the stage–discharge measurement uncertainties. It can also be demonstrated that the fuzzy set theory based approach is an effective alternative for treating measurement uncertainty in a nonprobabilistic framework. The method considers uncertainties using vaguely defined boundaries of fuzzy sets and can incorporate different sources of uncertainty arising from random and systematic errors. This approach is very significant for most measurement locations where confidence levels of uncertainty sources in measurePublished by NRC Research Press


ment are unknown. The next step in this research will be to propagate uncertainties in the discharge and stage measurement into development of the stage–discharge relationship.

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