Istvan Bogardi

Numerical solute transport simulation using fuzzy sets approach

Abstract This paper applies fuzzy sets and fuzzy arithmetic to incorporate imprecise information into transport modeling of nonreactive solute materials in groundwater flow. The method is applied to both one- and two-dimensional uniform flow fields. Emphasis is on the solution methods of the fuzzy numerical model of solute transport, which is a function of fuzzy variables. The solution techniques, including the vertex method and the fuzzy-numerical simulation method (i.e. the single-value simulation method), are discussed in detail. The solute concentration outputs from the fuzzy finite-difference numerical models based on these two solution methods are compared with those from the fuzzy analytical models. The vertex method can avoid the widening of the fuzzy function value set, in this case, the fuzzy solute concentration function. This widening is due to multi-occurrence of variables in the function expression when using conventional interval analysis. However, in fuzzy finite-difference numerical simulation of solute transport, the vertex method may still overestimate the uncertainty in the concentration outputs since all the fuzzy variables in the fuzzy numerical model are taken to be independent. The fuzzy-numerical simulation method can control the growth of the imprecision in the solute concentration calculations by taking into account the interaction (dependence) of concentration variables in both space and time dimensions in the fuzzy finite-difference model of solute transport. It has the advantage of allowing the use of imprecise data for modeling and also processing the fuzzy information using generated crisp values of fuzzy variables. The adoption of fuzzy sets allows common-sense knowledge to be represented in defining values through the use of a membership function. This enables the subjective information to be incorporated in system modeling in a formal algorithm.

Steady State Groundwater Flow Simulation With Imprecise Parameters

A methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models. In this case, fuzzy numbers are used to represent parameter imprecision. As such, they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters. The imprecise input parameters may come from indirect measurements, subjective interpretation, and expert judgment of available information. In the methodology, a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model. This fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations. Two numerical solution methods are used to solve the fuzzy groundwater flow model: the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics. The iterative method is simple but may overestimate the uncertainty of hydraulic heads. The groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters. Sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results, and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads. A numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem. It is also applied to a simplified two-dimensional heterogeneous flow case to demonstrate the methodology.

Combination of fuzzy numbers representing expert opinions

Abstract Expert opinions or imprecise estimates of a physical variable are expressed as fuzzy numbers and five techniques for combining these numbers into a single fuzzy number estimate are developed. Seven characteristics of the combination technique are defined; namely, agreement preservation, order independence, transformation variance, possibility conservation, possibility interval conservation, relationship between uncertainty of individual estimates and overall uncertainty, and desirability of resultant estimate. The five techniques, listed in increasing order of preference, are (1) crisp weighting, (2) fuzzy weighting, (3) minimal fuzzy extension, (4) convex fuzzy extension and (5) mixed linear extension. An example of estimating nitrate concentration in ground water illustrates the approach. The cases of equally versus unequally reliable estimates are distinguished and guidelines for choice of combination technique are provided.

Application of a space-time stochastic model for daily precipitation using atmospheric circulation patterns

Space-time series of daily precipitation amount conditioned on daily circulation pattern (CP) types are calculated. A stochastic hydroclimatological model is used to define daily precipitation under the climate of eastern Nebraska. Principal component analysis and k means method result in nine CP types in west central United States on the basis of 40 years of data. Both the probability and the amount of daily precipitation are strongly related to CP types. The approach can be used to predict the regional or local hydrological effect of climate change.

Application of fuzzy rule-based modeling technique to regional drought

Fuzzy rule-based modeling is applied to the prediction of regional droughts (characterized by the modified Palmer index, PMDI) using two forcing inputs, El Nino/Southern Oscillation (ENSO) and large scale atmospheric circulation patterns (CPs) in a typical Great Plains state, Nebraska. Although, there is significant relationship between simultaneous monthly CP, lagged Southern Oscillation Index (SOI) and PMDI in Nebraska, the weakness of the correlations, the dependence between CP and SOI and the relatively short data set limit the applicability of statistical modeling for prediction. Due to the above difficulties, a fuzzy rule-based approach is presented to predict PMDI from monthly frequencies of daily CP types and lagged prior SOIs. The fuzzy rules are defined and calibrated using a subset called the learning set of the observed time series of premises and PMDI response. Then, another subset, the validation set is used to check how the application of fuzzy rules reproduces the observed PMDI. In all its eight climate divisions and Nebraska itself, the fuzzy rule-based technique using the joint forcing of CP and SOI, is able to learn the high variability and persistence of PMDI and results in almost perfect reproduction of the empirical frequency distributions. q 1999 Elsevier Science B.V. All rights reserved.

Fuzzy rule-based approach to describe solute transport in the unsaturated zone

Abstract A fuzzy rule-based model is developed for simulation of solute transport processes in the vadose zone. Underlying physical processes of solute transport described by appropriate differential equations are captured in fuzzy rules. These rules are derived from a training set obtained from different test runs of the SWMS_2D model which simulates water flow and solute transport in two-dimensional variably saturated media. Fuzzy rules operate between two adjacent cells at each time step. Solute concentration of the upper cell, and solute concentration difference between two adjacent cells are used as premises. For a given time step, the solute flux between the two cells is taken as the response, which is combined with the conservation of mass to update the new solute concentration for the new time step. The methodology is applied to solve the breakthrough curve of bromide movement in a soil column. It is also generalized to the same problem under different soil and boundary conditions. In both cases, the fuzzy solution appears to be similar to the measured bromide concentration as well as the SWMS_2D model results. The methodology provides a more efficient alternative to existing modeling methods.

A fuzzy rule‐based approach to drought assessment

A methodology for predicting regional droughts from atmospheric pressure patterns is presented. Drought characteristics are strongly related to general circulation patterns (CP). CPs are determined from daily atmospheric pressure data. The link between large-scale CPs and regional scale droughts is modeled using a fuzzy rule-based approach. A fuzzy rule-based model operates on an “if” → “then” principle, where “if” corresponds to a vector of fuzzy inputs and “then” corresponds to some fuzzy consequences. The rules are derived from a so-called training set which includes a daily time series of CP classes and a corresponding monthly sequence of Palmer Drought Severity Indices (PDSI). Split sampling of historical data available for a 35-year time period is used to derive and then to validate the rules. Then, these fuzzy rules may be applied to predict droughts in terms of atmospheric circulation patterns. The occurrence and persistence of CPs are expected to vary under global climate change. Thus the approach may also be useful in estimating the potential impact of climatic change (e.g., 2 × CO2 scenario) on droughts. The methodology is illustrated using drought index data from New Mexico and atmospheric pressure data over the western United States.

Kriging with imprecise (fuzzy) variograms. I: Theory

Imprecise variogram parameters are modeled with fuzzy set theory. The fit of a variogram model to experimental variograms is often subjective. The “accuracy” of the fit is modeled with imprecise variogram parameters. Measurement data often are insufficient to create “good” experimental variograms. In this case, prior knowledge and experience can contribute to determination of the variogram model parameters. A methodology for kriging with imprecise variogram parameters is developed. Both kriged values and estimation variances are calculated as fuzzy numbers and characterized by their membership functions. Besides estimation variance, the membership functions are used to create another uncertainty measure. This measure depends on both homogeneity and configuration of the data.

Imprecise (fuzzy) information in geostatistics

A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.

Linkage between the occurrence of daily atmospheric circulation patterns and floods: an Arizona case study

The daily occurrence of large-scale atmospheric circulation patterns (CPs) is linked with the partial duration series of floods, using a case study in Central Arizona (USA) to illustrate the approach. The probabilistic linkage is evaluated by means of two performance indices, relating flood occurrence, observed CP occurrence before floods and purely random count of CPs. Three seasons (summer, autumn and winter) are distinguished, and CPs are grouped into two flood-producing groups year round, and two more such groups in winter. Floods in five watersheds, whose areas range from 900 to 15 000 km2, are analyzed for the period 1940–1980. The number of days N to be considered before the flood day is investigated using the first performance index, yielding a value of 1–3 days. The two performance indices appear to measure the linkage in a suitable way, especially in the autumn and winter seasons. The first index, measuring the ratio of percentage of floods explained by the two (or four) groups of flood-producing CPs to the percentage of CPs in the population, is well above 1.5. The second index, measuring the probability of at least k days out of N with CP type i before the flood day, is considerably greater than the corresponding binomial or Bernoulli value. Results for summer, when precipitation stem from convective storms, are not as clear-cut. In any case, this method makes it possible to study the effect of non-stationarities in the time series of CPs, and results should improve when daily precipitation or at least daily flows are considered.