## Model performance with fuzzy parameters and climatic inputs

The fuzzy water balance model described in the previous section is implemented using fuzzy estimates of daily precipitation, mean daily temperature and physically based model parameters from the Upper Enns catchment over a period of 21 years. The resulting time series of streamflows from the model are processed to generate daily, monthly and annual water balances. In this paper, for the sake of brevity, the focus is on annual and monthly model water balances, whereas only statistical measures of the model performance are presented at the daily time step.

In Figure 10.5(a), monthly time series of fuzzy precipitation and fuzzy potential evapotranspiration (model input), as well as simulated fuzzy estimates of actual evapotranspiration, are presented for the 1975-1979 period. In comparison to time series of crisp values usually presented as a continuous graph, in this case the intervals of confidences of fuzzy values at a certain level of presumption are joined together and are visualized as continuous ''ribbons''. In Figure 10.5(a), and in every following figure, the level of presumption (u) chosen for presentation is 0.8.

The parameters governing the water holding capacity of the soil, that is, Ctp and Cfc, appear in Figure 10.5(b) as constant values over time. The time variation of the fuzzy state variables of soil moisture, S(t), and snow water equivalent, SN(t), are set in relation to Ctp and Cfc. The results from the continuous simulation of the water balance model show, in the case of the seasonal accumulation and depletion of snow, that the higher the snow water equivalent the higher also its fuzziness. The same is true ofthe soil moisture storage.

In order to gain insights into the generation of calculated fuzzy water balance components presented in Figure 10.5 (at u of 0.8), the results of one particular day, that is, May 1, 1997, are presented in Figure 10.6. On that day in spring 1997, potential evapotranspiration is low as temperature is still low at that time of the year (Figure 10.6(a)). Precipitation is much higher than potential evapotranspiration on this particular day. The fuzzy numbers of the state variables concerning the soil moisture and the snow storage as well as water holding

(a) Monthly precipitation, potential and actual évapotranspiration 20

(a) Monthly precipitation, potential and actual évapotranspiration 20

22 32 42

1975-1979 time clip (month)

22 32 42

1975-1979 time clip (month)

(b) Monthly soil moisture storage and snow water equivalent in the snowpack

50 40 30

(b) Monthly soil moisture storage and snow water equivalent in the snowpack

A> , {SN> p] [P] |
■ |
20 10 | |

L A A |
Cfc | ||

1975-1979 time clip (month) Fuzzy observations and results: intervals of confidence for m = 0.8 22 32 42 52 1975-1979 time clip (month) Fuzzy observations and results: intervals of confidence for m = 0.8 Figure 10.5 (a) Fuzzy climatic input variables precipitation (P) and potential évapotranspiration (Ep) serve as input for the basic water balance model that results in simulated actual evapotranspiration (Ea), (b) fuzzy snow cover (water equivalent) (SN ), and soil moisture (S) presented in relation to fuzzy water holding capacity of the total soil profile (Ctp) and until field capacity (Cfc) tp capacity of the soil profile are presented for the same date in Figure 10.6(b). In particular, it may be noted that at a high level of presumption, say, 0.8, the soil moisture is clearly higher than field capacity, whereas at low levels of presumption the uncertainty of soil moisture increases such that soil moisture could be larger or smaller than field capacity. The concept of fuzzy membership functions, which is based on an infinite number of intervals of confidences between the lowest and the highest levels of presumption (0 and 1), does not allow one to judge if it is more likely that soil moisture is higher or smaller than field capacity, say, for the level of presumption of 0.2. This is because the interval of confidence for field capacity is located within the interval of confidence for soil moisture. Another interesting point to note is that the fuzzy numbers for potential evapotranspiration, and storage measures, soil moisture and snow water equivalent, are not triangular fuzzy numbers any more, even though they have been generated on the basis of triangular fuzzy numbers of input and parameter values. Generally, May is a month with high snowmelt, which is difficult to predict precisely, that is, because of fuzzy estimates for temperature and melt factors. Consequently, calculated storage values of the snowpack are also fuzzy. The use of 0.8 as the level of presumption (Figure 10.6) results in specific magnitudes for the intervals of confidence of different fuzzy numbers on any given day. The absolute magnitudes of the intervals of confidences change over time, and these may also change seasonally. The performance of the fuzzy water balance model is presented in terms of characteristic signature plots and hydrographs, which are compared to corresponding graphs based on observed streamflows (Figure 10.7). The (a) Precipitation and potential évapotranspiration (time clip: 01.05.1976) 1 |

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