Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity

J. Hydrol. Hydromech., 63, 2015, 4, 287–294 DOI: 10.1515/johh-2015-0036 287 Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity Daniel Marton*, Miloš Starý, Pavel Menšík Brno University of Technology, Faculty of Civil Engineering, Institute of Landscape Water Management, Veveří 331/95, 616 00, Brno, Czech republic. * Corresponding author. Tel.: +420 541 147 773. Fax: +420 541 147 771. E-mails: marton.[email protected], [email protected], [email protected] Abstract: The paper contains a sensitivity analysis of the influence of uncertainties in input hydrological, morphological and operating data required for a proposal for active reservoir conservation storage capacity and its achieved values. By introducing uncertainties into the considered inputs of the water management analysis of a reservoir, the subsequent analysed reservoir storage capacity is also affected with uncertainties. The values of water outflows from the reservoir and the hydrological reliabilities are affected with uncertainties as well. A simulation model of reservoir behaviour has been compiled with this kind of calculation as stated below. The model allows evaluation of the solution results, taking uncertainties into consideration, in contributing to a reduction in the occurrence of failure or lack of water during reservoir operation in low-water and dry periods. Keywords: Uncertainties; Reliability; Reservoir storage capacity; Monte Carlo method; Mean monthly flows; Evaporation; Elevation–volume curve; Elevation–area curve. INTRODUCTION The current knowledge in the field of climatology indicates a gradual change in hydroclimatic conditions all over the world. Climate changes are reflected in the changes in the hydrological cycle due to the redistribution of precipitation during the year and they contribute to more frequent occurrences of extremes in the form of floods and dry periods. Clear signs of climatic changes have appeared in the Czech Republic in recent years. It should be noted that, from the hydrological point of view, 2011 and 2012 were considered to be extremely dry (Zahradníček et al., 2014). The temperatures in the winter in 2014 were considerably above average. In that period, the water storage in snow cover was the lowest in the last twenty years. The consequences were extraordinary manipulations at some water reservoirs. It is apparent that the subject of advanced management and control of surface water resources is becoming more and more important. The manipulation rules of large open water reservoirs were approved in the period of construction of those waterworks and subsequently reviewed for the current hydrological conditions. It will be necessary to carry out a thorough review in the future in relation to their adaptability due to climate change and hydrological cyclic evolution. Therefore, the tasks of water management analysis of reservoirs for active reservoir conservation storage capacity will always be necessary and research in this field is valuable. In particular, the application of new optimization methods for water management analysis of reservoirs, new reservoir performance definitions and, last but not least, the introduction of analysis of uncertainties in these problems or combinations of the above mentioned applications and knowledge. Under the given conditions, it is necessary to introduce input data uncertainties into the proposal for control of reservoir storage capacity. The procedures which will be described below refer to “The influence of uncertainties in the calculation of mean monthly discharges on reservoir storage” (Marton et al., 2011). This paper describes, in detail, the introduction of uncertainties of measurement in determining below average monthly flows over the stage-discharge curve in a river and a number of measurements of hourly records of river stages in a hydrometric profile. For this sort of computation, the Monte Carlo method was used. One of the results was creating the random time series of mean monthly flows, which was affected by uncertainties of measurement in the hydrometric profile. The random series of mean monthly flows served as input data for the water management analysis of the reservoir storage capacity, when the spectrum of reservoir storage volumes for the maximum hydrological reliability of 100% were determined repeatedly using single-pass simulations of reservoir operation. The final spectrum of random reservoir storage capacities was evaluated statistically and the interval of possible values of reservoir storage volumes was found. The other work referred to in this paper is a paper by Marton et al. (2014) describing the calculation of reservoir storage capacity in the conditions of measurement uncertainties and extended to using the AR and ARMA models, generators of artificial streamflow series of mean monthly discharges. Random discharge series, in this case as data inputs for artificial streamflow series generators, were used, resulting in random samples of artificial streamflow series. These random series were evaluated using a reservoir simulation model. The calculation result was a spectrum of reservoir storage capacities for maximum reliability of 100 %, which was statistically evaluated. Both papers have indicated that the current water volumes in reservoirs can be underestimated and, in dry periods, may result in an unexpected failure in surface water supply. As regards current knowledge, uncertainty can be derived from set theory, but also uncertainty can be derived using statistics. From set theory, it is necessary to mention the application of uncertainty based on the theory of fuzzy sets (Zadeh, 1965) and the theory of possibilities (Klir, 2005). However, the first definition of uncertainty was by Knight (1921), nowadays known as Knightian uncertainty. The uncertainty concept is currently viewed from different aspects, with plenty of definitions and points of view, such as the uncertainty associated with the definition of risk, uncertainties applied in forecasting problems, and also the uncertainty of measurement. Uncertainties Unauthenticated Download Date | 8/10/17 12:51 PM Daniel Marton, Miloš Starý, Pavel Menšík 288 applied in hydrology were described, for example, by Beven and Binley (1992). They described in detail a method called GLUE – generalized likelihood uncertainty estimation. This was followed by numerous publications which deal with this issue, such as Beven (2007). Uncertainties in measurement were first formulated on the basis of the WECC (1990) agreement. A statistical approach using the concept of uncertainty of measurement, which clearly defined the introduction and calculation of measurement uncertainties, was introduced as the “Guide to Expression of Uncertainty in Measurement” (GUM 1993). The ISO GUIDE 99998 Standard (2004) deals with the distribution and propagation of uncertainties using Monte Carlo simulation. Hydrological applications, including propagation of uncertainties into hydrological inputs when measuring precipitation, water inflows into reservoirs and evaporation in the water balances of reservoirs, were dealt with by Winter (1981). LaBaugh and Winter (1984) examined the influence of uncertainties in the measurements of water inflow into reservoirs, water outflows from reservoirs and evaporation, and other hydrological and operating parameters on the volume and chemical analysis of water in reservoirs. The latest publications, for example Campos et al. (2014), examined the risks and uncertainties in, and influence on reservoir storage using Monte Carlo simulations. Kuria and Vogel (2014) carried out an analysis of uncertainties in reservoir storage using the water supply yield model. Coxon et al. (2015) publish paper which apply novel framework for discharge uncertainty to UK gauging stations. Another research area in the application of water management simulation and modelling is uncertainty associated with non-stationary processes in time series. This issue is too expensive for the purposes of this work and needs extensive independent research. The aim of the paper is to present another possible application of the Monte Carlo method for introducing uncertainties into the hydrological, morphological and operating input data required for reservoir storage capacity design, which is crucial in low-water periods. This is also connected with the calculation of reliability for water outflow from the reservoir in adaptive conditions. Input hydrological, morphological and operating data for the solution are considered, especially water inflow into the reservoir, water losses from the reservoir by evaporation from the water surface and by dam seepage, the reservoir elevation–volume curve and the reservoir elevation–area curve. A reservoir storage model was created for this purpose using a single-pass simulation method to determine reliability for water outflows from the reservoir, both considering water losses from the reservoir and ignoring such losses. By introducing uncertainty into the input data, the Monte Carlo method is used to determine, by repeated solutions, the spectrum of reliability of reservoir storage capacity. This is then evaluated and appropriately interpreted. As the uncertainties in the input data are unknown, the work focuses on compiling sensitivity analysis between the uncertainty in the input data of a solution for the storage capacity of the reservoir and the uncertainty of the achieved reliability of controlled water outflows from the reservoir. METHOD Monte Carlo method The general procedure for generating uncertainty affected hydrological, morphological and operating input data for the related water management analysis of a reservoir for its storage capacity is as follows. Uncertainties of input quantities are introduced into the calculations using the Monte Carlo method. Using the distribution curve F(X), a random position of values NXi within the interval of a given uncertainty are generated as input value Xi. Value Xi is considered random and independent of values Xi–1 and Xi+1. This presumption will allow the introduction of the normal probability distribution N( μ (X), σ (X)). Then, each input value Xi is considered as a mean value μ(X) and the amount of uncertainty is defined as the standard deviation σ (X). Subsequently, a cumulative distribution function Fi(X) of normal standardized probability distribution is generated for each mean value μ (Xi). The pseudorandom number generator generates a random number from an interval in which random quantity value NXi is generated. Fig. 1. Principle of generating uncertainties in input elements using the Monte Carlo method. Unauthenticated Download Date | 8/10/17 12:51 PM Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity 289 Fig. 2. Symbolic introduction of considered quantities affected with uncertainties. The basic principle of generating random positions of points in the two-dimensional coordinate system (NXi,NYi) is identical to the theory described above. The dissimilarity is given by plotting a point which requires plotting two Monte Carlo generators independent of each other. Each generator produces a random position of point NXi (e.g. water level elevation Nhi) and with it a random value NYi (water volume in reservoir NVi). The result on the reservoir elevation–volume curve is then a random point coordinate (NVi,Nhi) of the reservoir elevation–volume curve. See Fig. 1. As observed, the reservoir inflow, water surface evaporation, dam seepage and reservoir elevation–area and elevation– volume curves are considered to be hydrological and operating inputs. The principle of introducing uncertainties into the calculation of reservoir storage capacity is shown in Fig. 2. The generated random curves of water inflows into the reservoir, water evaporation from the water surface, seepages and random area and elevation–volume curves serve as input values for a simulation model which, using single-pass simulation, simulates the behaviour of the reservoir in the conditions of data affected with uncertainty. Reservoir simulation model and reservoir performance calculation The basis for a reservoir simulation model is an adjusted equation in the cumulative form converted to the following inequalities (1) Starý (2005). 0≤∑󰇛−󰇜∆+󰇛 −󰇜∆ ≤,   (1) where Oi is the reservoir outflow, Qi, the reservoir inflow for i = 1, …, n, and Δt is the time step of calculation (one month). Ok+1 is the outflow from the reservoir in the following time step, when in step i+1 the value Oi+1 is first replaced with the value of the required outflow Op. The time course of the numbered sum simulates the course of emptying the reservoir storage by time steps i = 1, …, k. For i = 0 it is necessary to enter the starting solution condition after the sum value. Inequality (1) is limited from both the left and the right. From the left it is limited by value 0 (full storage capacity) and from the right by value Vz,max (empty storage capacity) characterizing the reservoir storage capacity available for the reservoir. By calculating the value of the expression, the current emptying of the storage volume , 󰆒 is obtained and it is then tested as to whether it lies in a particular interval 〈0, VZ,max〉. If not, it is necessary to find value Oi+1 (for the sum of the expression to be equal to zero, idle discharge will occur, or if equal to Vz,max – a failure will occur). The general definition of reliability was successively described by (Hashimoto et al., 1982; Klemeš, 1967; Kritskiy and Menkel, 1952). The classification of a failure in the reservoir storage capacity for the following calculation of reliability is as follows (2). , = , =1,  ≥  , =0,  <  (2) Zt,i = 1 describes the reservoir storage capacity in a no failure situation (satisfactory state). Zt,i = 0 describes the reservoir storage capacity in a failure situation (unsatisfactory state). The required reliability can be further calculated from values Zt,i. Generally, reliability is calculated by time-based reliability as temporal reliability and occurrence reliability, and volumetric reliability is calculated separately. The paper uses the formula for the calculation of temporal reliability PT (3). = ∑,   (3) where k is the number of months in the period being solved. PRACTICAL APPLICATION The model was applied in practice to the existing reservoir, Vír I, which is situated in the Vysočina Region, Czech Republic. This is a multi-use reservoir serving mainly as flood protection and surface water accumulation for water supply and hydroelectric purposes. The reservoir is built in the Svratka River basin and has been in operation since 1957. The Svratka is the main inflow into the reservoir. The mean long-term inflow into the reservoir Qa is 3.34 m3 s–1. Input values for the calculation were made up of a time series of mean monthly flows over 60 years with the measurement period from 1950 to 2010. The mean annual evaporation from the water surface EANNUAL = 613 mm. The monthly evaporation values from water surface were derived in a simplified manner according to the percentage distribution of evaporation according to the ČSN 75 2405 Standard (2004) and from the mean annual evaporation values for Vír I reservoir, see Table 1. Unauthenticated Download Date | 8/10/17 12:51 PM Daniel Marton, Miloš Starý, Pavel Menšík 290 Table 1. Monthly distribution of evaporation amount during the calendar year. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Em [%] 6 9 12 14 16 15 11 7 5 2 1 2 Em [mm] 36.78 55.17 73.56 85.82 98.08 91.95 67.43 42.91 30.65 12.26 6.13 12.26 Table 2. Calculation without considering water losses from the reservoir. Measurement uncertainties are applied for water inflow into the reservoir. Input data uncertainty Ua = 3 σ ±3% ±6% ±9% ±15% O p [m3 s–1] μ (P T ) U a(P T ) μ (P T ) U a(P T ) μ (P T ) U a(P T ) μ (P T ) U a(P T ) 2.2 100.000 0.000 100.000 0.000 100.000 0.000 99.999 0.033 2.3 99.590 0.001 99.591 0.033 99.598 0.095 99.611 0.200 2.5 98.907 0.001 98.906 0.024 98.896 0.109 98.852 0.277 2.7 96.668 0.354 96.688 0.453 96.701 0.540 96.672 0.753 2.9 93.165 0.302 93.230 0.479 93.250 0.621 93.250 0.879 3.0 90.849 0.338 90.923 0.573 90.984 0.749 91.084 0.987 Fig. 3. Relation between required outflow Op and temporal reliability PT without considering. The total reservoir volume is VTOTAL 56.193 x 106 m3, active storage volume VZ,max is 44.056 x 10 6 m3 and flood reservoir volume VFLOOD is 8.337 x 106 m3. The total dam height is 67.3 m. The ecological flow from the reservoir QECO is 0.53 m3 s–1. The value of seepage through the dam was derived from empirical observation and for the gravity concrete dam it is 0.15 l s–1 per 1000 m2. The calculation of temporal reliability PT for an increased outflow from the reservoir was analysed with and without considering water losses from the reservoir. When water losses were considered in the calculations of temporal reliability, the described procedures for generating uncertainty-affected hydrological, morphological and operating inputs were applied. The analysis was carried out for the values of increased required outflow Op lying in the interval Op∈〈2.1; 3.0〉 m3 s–1. The selected number of repetitions using the Monte Carlo method was 300. Input uncertainties for the analysis ranged in intervals ±3, ±6, ±9, and ±15%. The algorithm simulating the behaviour of the reservoir then calculated random discharges NOi of water from the reservoir and temporal reliability of NPT. Then, random courses of monthly filling and emptying of the reservoir storage capacity were calculated. For a better presentation of the results, these values were evaluated statistically. The mean value μ (X) for each random set is considered to be the resultant value and the standard deviation σ (X) is considered to be the standard uncertainty related to a particular result. The total, extended uncertainty, type “Ua” covering almost 100% or specifically 99.97% of occurrences of the monitored quantity, corresponded to value μ (X)±3 σ . Sensitivity analysis was carried out for the calculation without considering water losses from the reservoir, when only Unauthenticated Download Date | 8/10/17 12:51 PM Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity 291 inflow into the reservoir was burdened with uncertainty, see Table 2 and Fig. 3. Calculations were also made while considering water losses from the reservoir. First, only the evaporation values, elevation–volume curve, elevation–area curve and seepage through the dam were affected with uncertainty, see Table 3 and Fig. 4. Then, reservoir inflow, evaporation, elevation– volume curve, elevation-area curve and seepage through the dam were affected with uncertainty, see Table 4 and Fig. 5. The shape of the curves in Figs. 3, 4, and 5 in the range from Op = 2.3 m3 s–1 to Op = 2.5 m3 s–1 is caused by a large time step in the calculations (1 month), and also by a step increase in the number of failure months, which is a small number in the given Table 3. Calculation with considering water losses from reservoir. Uncertainties considered for evaporation, elevation-volume (area) curves and seepage through dam body combinations. Input data uncertainty Ua = 3 σ ±6% ±9% ±15% O p [m3 s–1] μ (P T ) U a(P T ) μ (P T ) U a(P T ) μ (P T ) U a(P T ) 2.2 100.000 0.000 100.000 0.000 100.000 0.000 2.3 99.590 0.001 99.590 0.001 99.590 0.001 2.5 98.906 0.024 98.899 0.092 98.889 0.140 2.7 96.311 0.001 96.311 0.001 96.311 0.001 2.9 92.896 0.001 92.896 0.001 92.896 0.001 3.0 90.516 0.203 90.514 0.204 90.522 0.224 Fig. 4. Relation between required outflow Op and temporal reliability PT, with considering water losses from reservoir for input uncertainties ±6, ±9, ±15, ±30% and evaporation, elevation-volume(area) curves, dam seepage combinations. Table 4. Calculation with considering water losses from reservoir. Uncertainties considered for all inflow, evaporation, elevation-volume (area) curves, seepage through dam body combinations. Input data uncertainty Ua = 3 σ ±3% ±6% ±9% ±15% O p [m3 s–1] μ (P T ) U a(P T ) μ (P T ) U a(P T ) μ (P T ) U a(P T ) μ (P T ) U a(P T ) 2.2 100.000 0.000 100.000 0.000 99.996 0.070 99.978 0.191 2.3 99.587 0.057 99.566 0.155 99.555 0.179 99.539 0.204 2.5 98.859 0.196 98.843 0.215 98.811 0.293 98.721 0.451 2.7 96.316 0.084 96.348 0.338 96.350 0.510 96.300 0.797 2.9 92.939 0.290 92.957 0.454 92.943 0.584 92.896 0.838 3.0 90.556 0.402 90.575 0.591 90.589 0.734 90.608 1.038 Unauthenticated Download Date | 8/10/17 12:51 PM Daniel Marton, Miloš Starý, Pavel Menšík 292 Fig. 5. Relation between required outflow Op and temporal reliability PT, with considering water losses from reservoir for input uncertainties ±3, ±6, ±9 a ±15% and an inflow, evaporation, elevation-volume(area) curves and dam seepage combination. Fig. 6. The course of filling reservoir storage capacity in the conditions of entered input data uncertainties Ua = ±3% a ±15% for the selected low water period. Unauthenticated Download Date | 8/10/17 12:51 PM Analysis of the influence of input data uncertainties on determining the reliability of reservoir storage capacity 293 range of required outflow from the reservoir Op. For values Op = 2.6 m3 s –1 and higher, there is an apparent increase in the failure months, due to which the curves are smooth. The more significant interval of spacing in the curves in Fig. 5, unlike Figs. 3 and 4, in the field of required outflow from the reservoir Op = 2.2 m 3 s –1 is caused by the number of failure months occurring in the evaluated set. The process contingency applied when uncertainties are introduced into all input data of the solution results in a significant increase in the number of failure months compared to the solution in which only uncertainties of reservoir inflow or uncertainties for evaporation, elevation– volume (–area) curves and seepage through the dam combination are applied. The analysis also included values of filling the reservoir storage capacity. Fig. 6 then shows the course of filling the reservoir for a particular number of repetitions and for the selected low water period. SUMMARY The final comparison is from selected required outflow from the reservoir Op = 2.5 and Op = 3.0 m3 s–1, where the influence is clearly graded. In the variant without applying water losses from the reservoir, the temporal reliability for Op = 2.5 m3 s–1 is in interval PT∈〈98.906%; 98,908%〉, i.e. PT = 98.907% ±0.001% for input uncertainty ±3% and in interval PT∈〈98.575%; 99,129%〉 PT = 98.852% ±0.277% for input uncertainty ±15%. The interval of temporal reliability for the Op = 3.0 m3 s–1 range in interval PT∈〈90.511%; 91,187%〉 is PT = 90.849% ±0.338% for input uncertainty 3% and PT∈〈90.097%; 92,071%〉 PT = 91.084% ±0.987% for input uncertainty 15%. In the variant without applying all combinations, i.e. considering uncertainties in both inflow and water losses from the reservoir, the interval of temporal reliability was in PT∈〈98.663%; 99,055%〉 PT = 98.859% ±0.196% for input uncertainty ±3% and for uncertainty ±15% in interval PT∈〈98.27%; 99,172%〉 PT = 98.721% ±0.451%. For Op = 3.0 m3 s –1 the interval of temporal reliability acquired values PT∈〈90.154%; 90,958%〉 PT = 90.556% ±0.402% for input uncertainty 3% and PT∈〈89.57%; 91,646%〉 PT = 90.608% ±1.038% for input uncertainty 15%. The above mentioned results show a logical conclusion that with increasing input data uncertainty, the uncertainty in temporal reliability also increases. Converted to the number of failure months, increased discharge Op = 2.5 m3 s–1 in the solution without considering uncertainties and with considering reservoir water losses, corresponds to eight failure months, and for Op=3.0 m3 s –1 to 70 months. When considering input uncertainties ±3%, the number of failure months is from 8 to 9 months for Op = 2.5 m3 s–1 and 66 to 72 months for Op = 3.0 m3 s–1. For the input data uncertainty of ±15%, the number of possible failure months is 8 to 12 for Op = 2.5 m3 s–1 and 60 to 76 months for Op = 3.0 m3 s–1. The presented results show how uncertainties can influence the increase in failure months and which intervals the temporal reliability can then acquire. CONCLUSIONS In the manipulation rules for the Vír I reservoir, the stated temporal reliability for the hydrological period 1931 to 1991 is PT = 99.59% for a required reservoir outflow Op = 2.5 m3 s–1. This means that the current state is underestimated by approximately 1% compared to the calculations which were undertaken for that reservoir. Underestimation can be explained by the length of the input streamflow series introduced to the calculations, which were not updated until 2010. In addition, there were few low water years in the first half of the 1990s. For example, in the presented analysis, there is the apparent effect of underestimation of temporal reliability for the value of the required outflow from the reservoir Op = 2.3 m3 s–1, which is, with the amount of temporal reliability, nearest to temporal reliability according to the manipulation rules of the reservoir for the solution including considering input data uncertainties, and thus the reservoir is classified inappropriately in the significant class (A – PT ≥ 99.5%, B – PT ≥ 98.5%, C – PT ≥ 97.5%, D – PT ≥ 95%) according to ČSN 75 2405. For the uncertainty of ±6%, the mean value of temporal reliability was PT = 99.566%, which corresponds to the significance of the reservoir A – PT ≥ 99.5%. In considering uncertainty, the lower interval of temporal reliability corresponded to value PT = 99.411% and thus also the reservoir significance would fall to a lower significant class corresponding to class B – PT ≥ 98.5%. From this point of view, there is space for future reviews of the manipulation rules of reservoirs and possible amendment to the ČSN 75 2405 standard which should take input data uncertainties into consideration. In that case, it will be necessary to take the value of temporal reliability PT as the lower limit of the resultant interval and thus to incline more to the safe side in the solution. Currently, the results cannot be generalized, but the computational algorithm is written in general terms and it can be applied also to other reservoirs. When carrying out the sensitivity analysis, the same value of uncertainties was always counted for all input data. Under these conditions it was shown that water inflow was the most significant source of uncertainties. However, other input measurement uncertainties also have an influence on the result, which must be taken into account. At the present time, the authors are not aware of which values the elevation–volume (–area) curves can acquire, when their stated actual course is affected by sedimentation of the reservoir and other effects. Here it is possible to assume that higher uncertainties of elevation–volume (–area) curves can affect the results more. From this point of view, the results may be different and thus also the intervals describing the occurrence of the calculated temporal reliability PT may be different. Finally, it must be stated that the presented sensitivity analysis was only carried out for one reservoir and the results cannot be generalized. It can be assumed that different results will be obtained for other reservoirs and all reservoir systems with different sizes of reservoir storage capacity, various sources of inflows and flood altitude. In this respect, there is space for further research, for example using software based on the paper by Menšík et al. (2015), as well as cooperation with waterworks administrators. Acknowledgement. This paper was supported by the Brno University of Technology’s project “CZ.1.07/2.3.00/30.0039 Excellent young researchers at Brno University of Technology” and the specific research project FAST-S-15-2694 “Uncertainty propagation in the hydrological and water management applications for mitigation of drought on the open water reservoir.” REFERENCES Beven, K., 2007. Towards integrated environmental models of everywhere: Uncertainty, data and modelling as a learning process. Hydrol. Earth Syst. Sci., 11, 460–467. Beven, K.J., Binley, A.M., 1992. The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6, 279–298. Unauthenticated Download Date | 8/10/17 12:51 PM Daniel Marton, Miloš Starý, Pavel Menšík 294 Campos, J.N.B., Souza Filho, F.A., Lima, H.V.C., 2014. 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JAWRA Journal of the American Water Resources Association, 17, 82–115. doi: 10.1111/j.17521688.1981.tb02593.x. Zadeh, L.A., 1965. Fuzzy sets. Information and Control, 8, 3, 338–353. doi: 10.1016/s0019-9958(65)90241-x. Zahradníček, P., Trnka, M., Brázdil, R., Možný, M., Štěpánek, P., Hlavinka, P., Žalud, Z., Malý, A., Semerádová, D., Dobrovolný, P., Dubrovský, M., Řezníčková, L., 2014. The extreme drought episode of August 2011–May 2012 in the Czech Republic. Int. J. Climatol., doi: 10.1002/joc.4211. Received 20 March 2015 Accepted 2 June 2015 Unauthenticated Download Date | 8/10/17 12:51 PM