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     

A 1D microphysical cloud model for Earth,and Earth-like exoplanets: Liquid water and water ice clouds in the convective troposphere

One significant difference between the atmospheres of stars and exoplanets is the presence of condensed particles (clouds or hazes) in the atmosphere of the latter. In current 1D models clouds and hazes are treated in an approximate way by raising the surface albedo, or adopting measured Earth cloud properties. The former method introduces errors to the modeled spectra of the exoplanet, as clouds shield the lower atmosphere and thus modify the spectral features. The latter method works only for an exact Earth-analog, but it is challenging to extend to other planets.The main goal of this paper is to develop a self-consistent microphysical cloud model for 1D atmospheric codes, which can reproduce some observed properties of Earth, such as the average albedo, surface temperature, and global energy budget. The cloud model is designed to be computationally efficient, simple to implement, and applicable for a wide range of atmospheric parameters for planets in the habitable zone.We use a 1D, cloud-free, radiative–convective, and photochemical equilibrium code originally developed by Kasting, Pavlov, Segura, and collaborators as basis for our cloudy atmosphere model. The cloud model is based on models used by the meteorology community for Earth’s clouds. The free parameters of the model are the relative humidity and number density of condensation nuclei, and the precipitation efficiency. In a 1D model, the cloud coverage cannot be self-consistently determined, thus we treat it as a free parameter.We apply this model to Earth (aerosol number density 100 cm^?3, relative humidity 77%, liquid cloud fraction 40%, and ice cloud fraction 25%) and find that a precipitation efficiency of 0.8 is needed to reproduce the albedo, average surface temperature and global energy budget of Earth. We perform simulations to determine how the albedo and the climate of a planet is influenced by the free parameters of the cloud model. We find that the planetary climate is most sensitive to changes in the liquid water cloud fraction and precipitation efficiency.The advantage of our cloud model is that the cloud height and the droplet sizes are self-consistently calculated, both of which influence the climate and albedo of exoplanets.     

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.     

Use of Systematic, Palaeoflood and Historical Data for the Improvement of Flood Risk Estimation. Review of Scientific Methods

The catastrophic floods recently occurring in Europe warn of the critical need forhydrologic data on floods over long-time scales. Palaeoflood techniques provideinformation on hydrologic variability and extreme floods over long-time intervals(100 to 10,000 yr) and may be used in combination with historical flood data (last1,000 yr) and the gauge record (last 30–50 yr). In this paper, advantages anduncertainties related to the reconstruction of palaeofloods in different geomorphologicalsettings and historical floods using different documentary sources are described.Systematic and non-systematic data can be combined in the flood frequency analysisusing different methods for the adjustment of distribution functions. Technical toolsintegrating multidisciplinary approaches (geologic, historical, hydraulic and statistical)on extreme flood risk assessment are discussed. A discussion on the potential theoreticalbases for solving the problem of dealing with non-systematic and non-stationary data ispresented. This methodology is being developed using new methodological approachesapplied to European countries as a part of a European Commission funded project (SPHERE).     

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.     

Predicting evaporation from mountain streams

Evaporation can be an important control on stream temperature, particularly in summer when it acts to limit daily maximum stream temperature. Evaporation from streams is usually modelled with the use of a wind function that includes empirically derived coefficients. A small number of studies derived wind functions for individual streams; the fitted parameters varied substantially among sites. In this study, stream evaporation and above-stream meteorological conditions (at 0.5 and 1.5 m above the water surface) were measured at nine mountain streams in southwestern British Columbia, Canada, covering a range of stream widths, temperatures, and riparian vegetation. Evaporation was measured on 20 site-days in total, at approximately hourly intervals, using nine floating evaporation pans distributed across the channels. The wind function was fit using mixed-effects models to account for among-stream variability in the parameters. The fixed-effects parameters were tested using leave-one-site-out cross-validation. The model based on 0.5 m measurements provided improved model performance compared to that based on 1.5 m values, with RMSE of 0.0162 and 0.0187 mm h⁻¹, respectively, relative to a mean evaporation rate of 0.06 mm h⁻¹. Inclusion of atmospheric stability and canopy openness as predictors improved model performance when using the 1.5 m meteorological measurements, with minimal improvement when based on 0.5 m measurements. Of the wind functions reported in the literature, two performed reasonably while five others exhibited substantial bias.