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.     

Local temperature estimation under climate change

Summary A methodology to estimate the space-time distribution of daily mean temperature under climate change is developed and applied to a central Nebraska case study. The approach is based on the analysis of the Markov properties of atmospheric circulation pattern (CP) types, and a stochastic linkage between daily (here 500hPa) CP types and daily mean temperatures. Historical data and general circulation model (GCM) output of daily CP corresponding to 1 × CO₂ and 2 × CO₂ scenarios are considered. The relationship between spatially averaged geopotential height of the 500 hPa surface — within each CP type — and daily mean temperature is described by a nonparametric regression technique. Time series of daily mean temperatures corresponding to each of these cases are simulated and their statistical properties are compared. Under the climate of central Nebraska, the space-time response of daily mean temperature to global climate change is variable. In general, a warmer climate appears to cause about 5°C increase in the winter months, a smaller increase in other months with no change in July and August. The sensitivity of the results to the GCM utilized should be considered.On leave from the Department of Meteorology, Eötvós Loránd University, Budapest, Hungary.With 14 Figures     

Kriging with imprecise (fuzzy) variograms. II: Application

The geostatistical analysis of soil liner permeability is based on 20 measurements and imprecise prior information on nugget effect, sill, and range of the unknown variogram. Using this information, membership functions for variogram parameters are assessed and the fuzzy variogram is constructed. Both kriging estimates and estimation variances are calculated as fuzzy numbers from the fuzzy variogram and data points. Contour maps are presented, indicating values of the kriged permeability and the estimation variance corresponding to selected membership values called levels.     

An event-based stochastic model of phosphorus loading into a lake

In most lakes, phosphorus (P) is the nutrient controlling the trophic state. Thus, for effective control of eutrophication, the uncertainty in P-loading should be encoded as a probability density function (pdf). Specifically, the pdf of P-loading Y from non-point agricultural sources is sought by means of an event-based stochastic model.P-loading events are triggered by precipitation events (X₁, X₂, T), in which X₁ is the rainfall amount, X₂ the duration, and T the interarrival time between events. (X₁, X₂) are dependent random variables, while T is assumed to be exponentially distributed. The precipitation event causes runoff, which carries dissolved P into the lake with a concentration C₁ and sediment yield, Z, which carries fixed or sorbed P into the lake in a fraction C₂ of Z. Seasonal loading of P is calculated by adding random numbers of random variables. The model accounts separately for dissolved P and sorbed P. Explicit expressions are given for the mean and variance of each type of P-loadings. The case study of a sub-watershed of Lake Balaton, Hungary, is used to illustrate the methodology. Precipitation data, empirical rainfall-runoff-sediment yield relationships and a small number of observations of events are used to calibrate the model and estimate the means and variances of loading per event and per season. Then a simulation method is used to estimate complete pdf of these random variables. Use of the model for alternative methods of controlling P-loading is briefly discussed, as well as the economics of control.     

Fuzzy rule-based downscaling of precipitation

Summary A fuzzy rule-based methodology for downscaling local hydrological variables from large-scale atmospheric circulation is presented. The method is used to estimate the frequency distribution of daily precipitation conditioned on daily geopotential fields. The task is accomplished in two steps. First, the exceedence probabilities corresponding to selected precipitation thresholds are estimated by fuzzy rules defined between geopotential fields (premises) and exceedence events (response). Then a continuous probability distribution is constructed from the discrete exceedence probabilities and the observed behaviour of precipitation. The methodology is applied to precipitation measured at Essen, a location in the Ruhr catchment, Germany. Ten years of precipitation data (1970–1979) were used for training and another ten years (1980–1989) for validation. The 700 hPa geopotential fields are used to characterise large-scale circulation. The application example demonstrates that this direct downscaling method is able to capture the relationship between premises and the response; namely both the estimated exceedence probabilities and the frequency distribution reproduce the empirical data observed in the validation period.