Experiential information

Measurements or estimates of physical and meteorological variables at the point or plot scale are associated with inherent errors or uncertainties. The up-scaling procedures used to scale up from the point or plot scale to the catchment scale introduce additional uncertainties, governed by the adequacy of the spatial extrapolation procedure used in the model. The interpretation or processing of point observations for obtaining the corresponding estimates at the catchment scale often rely on the experiences and intuitive feel of experts about the general catchment characteristics and climate, about the small-scale variability of catchment properties, and meteorological variables.

But the expert information is not always given in quantitative terms, and instead may be expressed in heuristic terms, such as ''large'' or ''small''. For example, referring to a catchment's soil depth, an expert may express the catchment's mean soil depth as ''somewhat deeper than 0.5 m''. How can we map these expressions of human thinking, expressed verbally as above, in a mathematically useful manner without losing the character of imprecision? It turns out that methods based on fuzzy logic do indeed allow us to do this. They can be used to incorporate both the uncertainties inherent in traditional estimates of catchment properties, as well as the imprecision of heuristic statements and estimations made by experts. These methods are discussed in the next section where the basic principles of fuzzy theory are outlined in the context of the water balance modeling to be presented later on in this paper.

10.3 FUZZY MEMBERSHIP FUNCTIONS

Zadeh (1965) first defined a fuzzy set by generalizing the mathematical concept of an ordinary set. From those early days, applications of this new concept of uncertainty have been successful in a wide range of topics. In this section, we briefly introduce the generalities of the relevant fuzzy theory while the associated basic arithmetic rules are presented in the Appendix. In particular, we discuss the concept and formulation of membership functions, which form the basis for the definition of fuzzy model parameters and variables. However, instead of repeating much of the fuzzy mathematical concepts presented elsewhere (Kaufmann and Gupta 1991; Ozelkan and Duckstein 2001), we quickly turn the focus to aspects of its application.

If experts are asked about the mean profile depth of lithosol, the common soil type of the high topographic elevation zones in the headwaters of the Upper Enns catchment, the answer may be given in the following terms: Soils in this regions are ''shallow, approximately around 20cm". Another expert's opinion could be that the mean soil depth is ''most likely to be greater than 100mm but very unlikely to be greater than 300 mm". Others may intuitively think of the soils having a depth of a' 'few decimeters'. Experts base their estimates not only on the interpretation of a few plot scale measurements but also on their heuristic experiences within the region of interest, including extraneous or surrogate information, such as geological maps. The combination of these verbal statements and point measurements and their transcription into a crisp estimate of the mean regional soil depth at the catchment scale is somewhat difficult. If one is forced to cope with limited information for model parameterizations, it may be more practical to treat uncertainty about these estimates, with whatever information is available, and to locate the unknown true value inside some closed interval. The first statement mentioned 20 cm as the soil depth, but it did not mean precisely 20 cm. Other information such as the one included in the second statement is much more conservative, being aware that point measurements of soil depth fluctuate between 100 mm and 300 mm. The specification of the mean soil depth can thus be expressed in intervals of confidence, such that the unknown true value is located inside the closed interval [dl,dr ]where d1 and dr stand for the left and right interval boundaries - in fuzzy logic this is defined as the interval of confidence [100mm, 300mm].

A fuzzy number may be considered as an extension of the concept of the interval of confidence. Instead of defining just one interval of confidence, the latter is considered at several levels, which may be called levels of presumption From the lowest (i = 0) to the highest level of presumption (i = 1), multiple levels of presumption may be defined. Continuing with the example of soil depth, in general, we can say that the larger we define the interval for the unknown mean soil depth, the smaller is our level of presumption in making that statement. Conversely, with a smaller interval of confidence the level of presumption is higher, the highest level of presumption being attained when the interval of confidence shrinks to a single crisp value, say, 200 mm [d1 = dr = dc = 200 mm] For example, the mean catchment soil depth may be expressed using a triangular function, relating interval of confidence to level of presumption. The interval of confidence at the lowest levels of presumption (i = 0) can be defined as [100 mm, 300 mm], this being a very conservative estimate. On the other hand, say, we estimate the mean profile depth as 200 mm at the highest level of presumption (i = 1). We can thus relate the interval of confidence to multiple levels of presumption that lie in the range [0, 1] - this is called the fuzzy membership function. For a continuous transition of the level of presumption ¡i in the range [0, 1], any finite number of characteristic values, or a continuous function, can be defined. An example of the resulting fuzzy membership function of the mean catchment soil depth is presented in Figure 10.1. Note that the theory of fuzzy sets should not be confused with the theory of probability, and a fuzzy set is not a random variable. A fuzzy set is merely an extrapolation of the concept of the interval of confidence to multiple levels of presumption in the range [0, 1].

Defining formally, D is called a fuzzy set of a referential set, for example, R, if the set consists of ordered pairs such that

Again referring to the example cited above d may be the suspected soil depth at multiple levels of presumption within [0, 1], whereas D is the membership function of the estimated mean soil depth. On the basis of the definition of a fuzzy set and the coupled concepts of the

Non Convex Fuzzy Sets

Figure 10.1 A binary set, a non-normal and non-convex fuzzy set, and a triangular fuzzy number

Figure 10.1 A binary set, a non-normal and non-convex fuzzy set, and a triangular fuzzy number level of presumption and interval of confidence, a fuzzy number is defined as a fuzzy subset.

A fuzzy number D in R is a fuzzy subset in R that is convex and normal (Kaufmann and Gupta 1991) (Figure 10.1). Normality in the context of fuzzy sets means that there exists at least one value of d e D such that ixD(d) = 1. A fuzzy set is (quasi) convex if the membership function of D, laid) e [0, 1], does not show a local extreme and the membership function of D is always nondecreasing on the left of the single peak, and nonincreasing on the right of the single peak.

A triangular fuzzy number is a special type of a fuzzy number with two linear functions on either sides of the peak. Left-right symmetry is not a necessary condition for a triangular fuzzy number. A simple method of defining a triangular fuzzy number is by assessing the symmetric or semi-symmetric membership function with three points (Dubois and Prade 1980), as generally used in this study: D = (dl,dc,dr) = (100mm, 200mm, 300 mm). Of course, any crisp number can be defined as a triangular fuzzy number with dl = dc = dr : D = (200mm, 200 mm, 200mm).

10.4 CASE STUDY CATCHMENT: LOCATION, CLIMATE AND HYDROLOGY

The study catchment for the application presented in this paper is the Upper Enns catchment with its outlet at Liezen. This catchment is located in the Austrian Central Alpine region north of the main Alpine ridge (see Figure 10.2), and stretches almost linearly from West to East. The Enns river and its tributaries drain parts of the Niedere Tauern, the Dachstein and the Totes Gebirge (Table 10.1). The low elevation zones are mainly extensively used grasslands or arable lands free from wooded areas, whereas the adjacent mountain slopes are typically covered with coniferous forests.

The Upper Enns catchment is characterized by an Alpine climate. The meso-climate within the basin is highly variable because of the shielding effects of the Salzburger Alpen in the north, the Hohen Tauern in the west and the Julischen Alpen (Julijske Alpe in Slovakia) in the south of the main Enns valley, and due to the strong changes in elevation of the valley-ridge system of the mountains. Thus, the Upper Enns catchment belongs to a moderately dry mountain region of the Alps. Regional mean monthly precipitation shows strong seasonal variations, with the first maximum in July (160 mm) due to summer storms with high intensities, a second but small peak in December (100 mm), interrupted by the minima in February (60 mm) and October (70 mm). Estimates of mean monthly evapotranspiration for the Upper Enns catchment vary seasonally, with a maximum of around 60 mm in July and a minimum of almost zero in the winter months of November to March, corresponding to seasonal fluctuations of mean air temperatures of +12°C in August to — 1.5°C in January.

For part of the year, the precipitation falls in the form of snow, during times when temperatures remain below freezing, leading to the temporary accumulation on the ground as snowpack. Once the temperatures warm up during the spring, the snowpack begins to melt, and contributes to both soil moisture storage and snowmelt runoff. Snow accumulation during winter and melt during spring introduce elements of carry over of storage and time delays to the hydrologic system. They cause a reduction of discharge in winter, which is then mainly fed from groundwater storage that is gradually depleted. In spring, the melting snowpack increases the discharge to its maximum level and also leads to the recharge of

Figure 10.2 The topography of the Upper Enns catchment with river network, catchment boundary, and hydro-meteorological monitoring stations. Reproduced by permission of John Wiley & Sons Ltd.

Table 10.1 Table of basin characteristics

Name of the basin/area Mountain Range

Elevation range of entire catchment [m a.s.l]

Latitude and longitude

Area in km2

Geology

% glacierized

Vegetation type (dominantly)

Mean (1972-1993) Q at catchment outlet [mm] Mean (l972-1993) P [mm] Mean (1972-1993) E (if determined) [mm] Mean (l972-1993) T [°C]

Upper Enns catchment Alps

700-2995

Dominantly limestone

Grasslands (700-800 m a.s.l), coniferous forests (800-1800 m a.s.l), alpine pasture (1800-3000 m a.s.l) 940 1200 270 4

the groundwater table. Hydrological characteristics of the Upper Enns catchment have been previously described in detail by Nachtnebel etal. (1993) and Eder etal. (2001,2003).

10.5 BASIC MODEL CONSTRUCT

The water balance model we use here is a lumped conceptual model, based on a daily time step, developed and tested previously by Eder et al. (2003) for the Upper Enns catchment. The model incorporates the processes of runoff generation by the mechanisms of saturation overland flow (whenever soil moisture storage capacity is exceeded), interflow (shallow subsurface flow) whenever soil moisture storage exceeds the limited storage capacity corresponding to the soil's field capacity, and deep groundwater flow (or baseflow). Evapotranspiration is simply assumed to be equal to the potential evapotranspiration estimated by the Thornthwaite method, except during precipitation events when fluxes of evapotranspiration are neglected. Precipitation is partitioned into rainfall or snowfall based on a single threshold air temperature, and snowfall accumulates into a snowpack during the winter months. The model applies a temperature - index algorithm for simulating snow processes. The rate and timing of the snowmelt process are estimated on the basis of the same threshold air temperature, and a fixed snowmelt factor.

Fuzzy input (t)

Figure 10.3 Water balance model concept accounting for snowmelt runoff (QN), saturation excess runoff (Qse), inter flow ( Qin) and base flow ( Qbf)

Figure 10.4 The fuzzy water balance model approach: fuzzy input data and fuzzy parameters result in fuzzy outputs. Carry-over of fuzzy system states from time t to t + 1

Figure 10.3 Water balance model concept accounting for snowmelt runoff (QN), saturation excess runoff (Qse), inter flow ( Qin) and base flow ( Qbf)

Fuzzy output (t)

Figure 10.4 The fuzzy water balance model approach: fuzzy input data and fuzzy parameters result in fuzzy outputs. Carry-over of fuzzy system states from time t to t + 1

The model structure is presented schematically in Figure 10.3. The water balance dynamics of the catchment is characterized by the following two coupled equations, involving two state variables representing soil moisture storage and snow water storage, respectively (refer to the list of symbols and abbreviations and their brief descriptions):

The structure of the model was arrived at through a systematic, data-driven procedure known as the downward approach (Klemes 1983). This version of the model uses eight parameters, all of which were estimated a priori for the Upper Enns catchment; the physical meanings of these parameters and the details of their estimation are provided in Eder et al. (2003). The input data required for the running of the model are P and Ep. The model uses the parameters Cfc {a function of (Dtp,6fc, 6pwp)},Ctp {a function of (Dtp, 6>pwp)}, T^mf, tc-in and tc-bf.

In this paper, this model is recast from its formerly deterministic form into a new, fuzzyfied form, based on the types of fuzzy membership functions mentioned above and associated rules of fuzzy arithmetic briefly summarized in the Appendix. With the change to fuzzy form, the model now uses fuzzy input data and parameter values (Figure 10.3), and in turn produces various time series of fuzzy model outputs (fluxes) and system states (Figure 10.4). These are: Pr, Qs, Q.N, sn, Ea, S, QSe, Qin and Qbf.

The justification for the choice of membership functions for the various fuzzy parameters and input data is described next.

10.6 ESTIMATION OF FUZZY PARAMETERS AND INPUTS

Because of uncertainties in the observation of various point data, and because of insufficient knowledge about the spatial distribution of catchment physiographic properties and climate inputs, the estimation of mean catchment properties and climatic variables is difficult, leading to uncertainty in the specification of model parameters and climatic inputs. In this paper, such uncertainties are expressed by means of fuzzy membership functions for each of the model parameter values and climatic inputs.

We have chosen to use triangular membership functions to describe the fuzziness of all parameters and climate inputs used in the model. Triangular membership functions are the simplest to use, although any other shapes of membership functions could be used if proved to be more appropriate for any physical reason. Mathematical operations with triangular fuzzy numbers always result in fuzzy numbers, which are not necessarily triangular any more, but still retain a unique maximum at the highest level of presumption. Computational results obtained from the fuzzy water balance model for the highest level of presumption are therefore identical to results of a conventional (deterministic) water balance model of the same structure.

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