Vol. 115, No. 1-2 * Pages 1–122 * January - June 2011

The paper focuses on the basic mathematical and theoretical questions of spatial interpolation of meteorological elements. Nowadays, in meteorology the most often applied procedures for spatial interpolation are the geostatistical interpolation methods built also in GIS software. The mathematical basis of these methods is the geostatistics that is an exact but special part of the mathematical statistics. However, special meteorological spatial interpolation methods for climate elements also exist, such as Gandin optimum interpolation as well as the MISH method developed at the Hungarian Meteorological Service in the last few years. These meteorological interpolation methods are also based on the mathematical statistical theory. Therefore, the basic type of the interpolation formulas applied by the geostatistical and meteorological methods are similar. One of our intentions is to present some comparison of the various kriging formulas, such as ordinary, universal, regression, residual, detrended, etc., ones. In general, these formulas can be derived from the multiple linear regression formula by using the generalized-least-squares estimation for certain unknown parameters. But the main difference between the geostatistical and meteorological interpolation methods can be found in the amount of information used for modeling the necessary statistical parameters. In geostatistics, the usable information or the sample for modeling is only the system of predictors, which is a single realization in time, while in meteorology we have spatiotemporal data, namely the long data series which form a sample in time and space as well. The long data series is such a speciality of the meteorology that makes possible to model efficiently the statistical parameters in question.

Any transformation of a discrete variable into a continuous one is subject to uncertainty. Consequently, the identification and assessment of errors is essential for avoiding misinterpretations of models describing the spatial distribution of climatic parameters. Our study attempts to identify the main sources of errors affecting the statistical spatial models of climatic parameters and to assess their impact on the accuracy of these models. In particular, we focus on georeference errors, the representativeness of the stations network and the related extrapolation problem, the outliers problem, error propagation from simple to complex variables, the problems aroused by heterogeneous regions.

A spatial interpolation method for the construction of operational climate maps for the WMO RA VI Region (Europe and Middle East) is presented on the example of monthly mean temperature. The method is suitable for an in situ data base with relatively low data coverage in a relatively large and climatically heterogeneous area, and considers the classical geographical parameters latitude, longitude, and altitude by multi-dimensional linear regression, but improved by continentality, using a new continentality index. A comparison of several interpolation methods reveals that radial basis functions (subtype multiquadratic) seems to be the most appropriate approach. Separate regressions for land and sea areas further improve the results.

This study deals with modeling of spatial distribution of summer (JJA) precipitation over the Czech Republic. The aim is to analyze the time variation of the relationships between geographical factors and precipitation during summer. Various candidates geographical predictors are evaluated in the stepwise regression models for summer precipitation, namely: (1) a set of omnidirectional parameters of the elevation that characterize an area of 3 ´ 3 km around meteorological stations, (2) various cross products calculated on the basis of geographical coordinates and elevation or topographic parameters, (3) slope and four facets of slope aspect characterizing the orographic regimes in the Czech Republic, (4) land cover parameters describing an area of 10 × 10 km around meteorological stations, and (5) geographical coordinates. The orographic parameters are derived from the 1 km resolution digital elevation model (DEM); the land cover parameters are derived from the 1 km resolution CORINE (COoRdination of INformation on the Environment) land cover data. Daily precipitation data for the period 1971–2003 have been used. The precipitations were collected from 203 stations throughout the country. Stepwise regression models of summer precipitation are generated for each year, and each overlapping decade from 1971 to 2003. To ensure the stability of the regression equations and comparability of regression models in time, similar suitable and stable independent variables in time should be selected. Therefore, orthogonally rotated principal component analysis (PCA) and frequency of significant predictors entering models are used to select them. Multivariate regression precipitation models are generated using definitive (PCA or stepwise based) selected predictors. Ten independent geographical variables have been selected as the most important predictors for precipitation regression models. They consist of latitude, longitude, slope aspect of the grid westward from the central grid, slope aspect of the grid northward from the central grid, slope of the grid northeastward from the central grid, slope of the grid eastward from the central grid, slope of the grid northward from the central grid, maximum value of elevation (percentile 95%) of northwestern grid from the central grid, minimum value of elevation (percentile 5%) of the central grid, and vegetation. The relationships between these significant predictors and precipitation are stable in time. No significant trend in regression coefficients has been found during 1971–2003.

Air temperature and precipitation data organized within a 10 ´ 10 km grid covering the whole of Slovakia were subject to analysis. The source data are produced by the ALADIN Climate/CZ regional climatic model. The output of the global climatic model ARPEGE-Climat (Meteo-France) provided the driving data for the regional model. The IPCC A1B scenario provides the information on the future development of greenhouse gas emissions. Such scenario was developed within the 6th Framework Programme project CECILIA (Central and Eastern Europe Climate Change Impacts and Vulnerability Assessment).

Geostatistical prediction of annual mean air temperature and precipitation data was carried out for the reference (1961–1990) and distant future (2071–2100) climates. The experimental data were non-stationary and significantly correlated with elevation. Therefore, we used non-stationary multivariate geostatistical techniques allowing for the integration of such information. In particular, we used kriging of residuals, universal kriging with external drift, and external drift kriging in the scope of IRF-k (intrinsic random functions of order k). Prediction based on linear regression of elevation data was used as a complementary technique. Accuracy assessment was based upon the mean square errors produced by cross-validation in case of kriging-based predictions and upon the mean square residual in case of linear regression-based prediction.

We found that all kriging-based techniques outperformed the linear regression-based approach, yielding mean square error lower by 53–75%. External drift kriging in the scope of IRF-k produced slightly better results for most of the climate variables analyzed. The poorest results were achieved in the case of annual mean air temperature for the period 1961–1990, where the variogram of residuals was very erratic.

External drift kriging-based techniques were found to be very efficient for interpolating annual mean air temperature and annual precipitation data organized in regular grids. Accuracy assessment indicated that the three predictors used yielded almost identical results for a single variable, while significant differences in mean square error were observed in a between-variables comparison.

For various studies, it is necessary to work with a sufficiently long series of daily data that is processed in the same way for the whole area. National meteorological services have their own tools for data quality control; data are usually available non-homogenized (with respect to artificial changes in the series due to relocations, change of observers, etc.). In the case of areas across borders of individual countries, researchers from both sides of a frontier can obtain quite different results depending upon the data they use. This was one of the reasons for processing stations from the area along borders of four countries in the Central European region within the international CECILIA project (Central and Eastern Europe Climate Change Impact and Vulnerability Assessment, project of EC No. 037005). For the processing of the series, quality control has been carried out, gaps have been filled and, in the end, a series at a new position (grid points of RCM output) were calculated. An interpolation technique which is able to deal with all these tasks is described in this work and then applied to a series of various meteorological elements in Central Europe.

The calculation of extreme climate indices defined by several international projects requires homogeneous time series. To this effect, long term daily extreme temperatures and daily precipitation sums were homogenized, quality controlled, and further processed by the method MASH (Multiple Analysis of Series for Homogenization). After homogenization of station observation series, a gridding procedure was performed on the daily observations by the method MISH (Meteorological Interpolation based on Surface Homogenized Data Basis). The idea behind the MISH interpolation scheme stems from the following principles: gridded data can be created (interpolated) at higher quality with respect to certain climate statistical parameters; and these parameters can be modeled by using the long climate data series. In the MISH procedure, the modeling of the statistical parameters for a given location is based on the long term homogenized monthly data of neighboring stations.

In this paper, we present the computations of extreme temperature and precipitation climate indices for the period of 1901–2009 using datasets which were processed by the above homogenization and gridding algorithms. The obtained trends of several extreme indices as well as their spatial distributions are demonstrated on graphs and maps.

Recent developments in GIS techniques have produced a wide range of powerful methods for capturing, modeling, and displaying of climate data. The main aim of the study was to identify an optimal interpolation method to describe the spatial differentiation of the climatic water balance in Poland based on meteorological data (temperature, precipitation, solar radiation) collected at 15 weather stations from 1986 to 2006. A climatic water balance index (CWB) was created based on a simplified definition, where it is interpreted as the difference between the precipitation total (P) and potential evapotranspiration (PE). The latter was calculated using the so-called Turc Equation. Four different spatial interpolation methods were used: (1) inverse distance weighting (IDW), (2) local polynomial (LP), (3) radial basis function (RBF), and (4) ordinary kriging. A subjective visual analysis of maps, root mean square error values, and coefficients of correlation indicated that the best CWB interpolation methods are the radial basis function method and the ordinary kriging method. However, spatial interpolation results suggest that the problem is more complex. Calculations performed for selected points of reference suggest that local geographic factors play an important role in the shaping of CWB. Such results also confirm the need to perform spatial climatic water balance analysis with special attention being paid to local conditions. Further research is needed that takes into account different temporal and spatial scales and aims to test established methods in other regions in Europe.