Spatial Interpolation of Ewert’s Index of Continentality in Poland
Abstract
The article presents methodological considerations on the spatial interpolation of Ewert’s index of continentality for Poland. The primary objective was to perform spatial interpolation and generate maps of the index combined with selection of an optimal interpolation method and validation of the use of the decision tree proposed by Szymanowski et al. (Meteorol Z 22:577–585, 2013). The analysis involved four selected years and a multiyear average of the period 1981–2010 and was based on data from 111 meteorological stations. Three regression models: multiple linear regression (MLR), geographically weighted regression (GWR), and mixed geographically weighted regression were used in the analysis as well as extensions of two of them to the residual kriging form. The regression models were compared demonstrating a better fit of the local model and, hence, the nonstationarity of the spatial process. However, the decisive role in the selection of the interpolator was assigned to the possibility of extension of the regression model to residual kriging. A key element here is the autocorrelation of the regression residuals, which proved to be significant for MLR and irrelevant for GWR. This resulted in exclusion of geographically weighted regression kriging from further analysis. The multiple linear regression kriging was found as the optimal interpolator. This was confirmed by cross validation combined with an analysis of improvement of the model in accordance with the criterion of the mean absolute error (MAE). The results obtained facilitate modification of the scheme of selection of an optimal interpolator and development of guidelines for automation of interpolation of Ewert’s index of continentality for Poland.
Keywords
Ewert’s index of continentality spatial interpolation regression kriging geographically weighted regression Poland1 Introduction
The concept of continentality in climatology refers to all characteristics of climate influenced by continents and is regarded as a notion opposed to climate oceanity. The continental impact is most frequently considered in the context of its effect on air temperature (thermal continentality) and precipitation (pluvial or hygric continentality). The continental climate is characterised by higher diurnal and annual amplitude of air temperature, hot summers, and cold winters. As the distance from the ocean increases, the cloudiness and precipitation rates decline and the annual distribution of precipitation changes and exhibits a distinct summer maximum. The oceanic climate is characterised by high air humidity and high rates of precipitation distributed evenly throughout the year, with low annual air temperature amplitudes, cool summers, and mild winters.
In a macroscale, it can be assumed that climate features of any place on the globe comprise a signal coming from the continents and a supposedly opposite signal from the oceans. To quantify these interactions, a number of indices, primarily describing the “strength” of land impacts, have been developed. Given the disproportion between the land and ocean cover (ca. 30 %—lands and 70 %—oceans), they can be regarded as a specific modification of the prevailing oceanic climate on the planet. As a rule, indicators of continentality express the relation between continental and oceanic features of climate and, hence, underline the relative nature of continentality. The continentality has been assigned particular importance in characterisation of areas with the socalled transitional climate combining continental and oceanic impacts, which exhibit high yeartoyear variability depending on macrocirculation features. For Poland, such analyses have been performed by Kożuchowski and Marciniak (1986, 1992), Kożuchowski and Wibig (1988), and Kożuchowski (2003). Recently, continentality indices have been applied in environmental analyses, particularly as a variable supporting the analysis of variability and spatial distribution of, e.g., air temperature (Hogewind and Bissolli 2011), evapotranspiration (Marti and Gasque 2010) and bioclimatic changes (Torregrosa et al. 2013). The analytical usefulness of continentality indices has also been corroborated in the investigations of glacier mass balance (Holmlund and Schneider 1997), plant species range and the treeline (Caccianiga et al. 2008), and plant pollen in northern Europe (Salonen et al. 2012).
Given the recent climate changes, it is expected that continentality indices bring important signs of these changes, particularly when researchbased thereon will address the dynamic yeartoyear changes, multiyear trends, and predictions supplemented with detailed analysis of extreme values and circulation relationships. On the one hand, maps of continentality indices can provide data concerning a number of environmental elements, thereby being an important variable in multidimensional spatial modelling of these elements, as mentioned above. Methodology for accurate spatialization of continentality indices based on discrete point observations is a prerequisite for generation of such maps. This paper is focused on spatial interpolation of the thermal continentality index for Poland. It has a methodological character allowing indication of a preferable spatialization algorithm in accordance with the environmental characteristics of the study area and the nature of the input data set.
The annual temperature amplitude, i.e., the difference between the mean temperature of the warmest and coldest month, is the simplest absolute measure of thermal continentality. The drawback of this indicator lies in its dependence on seasonal changes in the quantity of incoming solar energy, which results in an increase in the annual amplitude with latitude. The effect is compensated for in the thermal continentality indices by dividing the temperature amplitude by the sine of latitude with concurrent introduction of empirical parameters facilitating rescaling of the index into the assumed interval in accordance with theoretical assumptions (Conrad 1946).
Various continentality indices used in climatology are based on the average annual temperature amplitude scaled with the sine of latitude. However, Driscoll and Yee Fong (1992) suggest that there is no conclusive evidence for the validity of the use of this divisor with respect to amplitude changes determined by the incoming solar energy. This group comprises indicators specified by Gorczyński (1920), Johansson (1926), Raunio (1948), Conrad (Conrad and Pollak 1950), Hela (1953), Ivanov (1953), Khromov (1957), Ewert (1963), and Hogewind and Bissolli (2011). Formulas extending this type of thermal continentality indices with other environmental parameters include indicators developed by Spitaler (1922), Ringleb and Johansson (after Szreffel 1961), and Ivanov (1959). Berg (1944), Bailey (1968), and Oliver (1970) proposed formulas based on other assumptions than annual temperature amplitude. These ideas have been put forward by Driscoll and Yee Fong (1992) and Mikolášková (2009).
Classic approach to present continentality indices uses handdrawn isolines (Swoboda 1922; Kożuchowski and Marciniak 1992), with generalisation typical of this technique, especially in small scales. In recent studies, index maps are generated in the GIS environment, taking advantage of raster maps and showing more details (Mikolášková 2009; Torregrosa et al. 2013). The spatialization approaches applied are relatively simple, onedimensional, and disregard selection of an optimal interpolation method and accuracy of the results.
Currently, environmental research, including climatology, employs various interpolation techniques (Hengl 2007; Li and Heap 2008; Szymanowski et al. 2012), which often leads to difficulties in choosing a method that is appropriate in a given case. This problem has been addressed, e.g., in the COST 719 research project “The Use of GIS in Climatology and Meteorology” (Dobesch et al. 2007; COST Action 719 Final Report 2008), but the investigations did not yield a conclusive solution. The spectrum of methods employed for spatial interpolation of climate elements is very wide and comprises deterministic and geostatistical techniques and their combinations. The best results are usually obtained with multivariate methods considering the role of environmental factors, in particular the elevation and coordinates (Szymanowski et al. 2012). This group includes, among others, the multiple linear regression (MLR) and residual kriging (MLRK, regression kriging) methods. Analyses of the spatial properties of climate elements, particularly nonstationarity, indicate that the local geographically weighted regression (GWR) model with residual kriging (geographically weighted regression kriging, GWRK) is more suitable for modelling the spatial variability of these elements (Szymanowski and Kryza 2011, 2012). Methodological research aiming at interpolation of the temperature field for Poland resulted in construction of a decision tree for selection of an optimal multivariate interpolation method considering the potential nonstationarity of the spatial process (Szymanowski et al. 2012, 2013).
The main objective of this study is to perform spatial interpolation and generate maps of Ewert’s index of continentality (1972) for selected cases in Poland representing longterm annual mean values and years with extreme values or characteristics of the spatial distribution of this indicator. This methodological paper emphasises the choice of an optimal interpolation method and indication of guidelines for automated or semiautomated interpolation of the indicator for a large data set (multiyear series) to identify the trend in climate change. Second aim is to validate the usefulness of the scheme of selection of the optimal interpolation method developed by Szymanowski et al. (2012, 2013) for interpolation of the continentality indices.
2 Study Area
Poland is located in the temperate transitional climate zone with clear continental and oceanic impacts. Westerly winds and polar air masses are predominant. The spatial distribution of the lowest values of the mean annual air temperature is determined by elevation, with the top parts of the mountains (−0.7 °C), and continentality, with the northeastern part of the country being the second coolest region (<7 °C). The mean annual temperature increases from the northeast to the southwest, where it exceeds 8.5 °C (Woś 2010). July, with temperatures ranging from 17.3 to 18.8 °C (except for the mountains), is the warmest month, and January, with temperatures in the range between −3.4 and −1.3 °C, is the coldest (Kożuchowski 2011). In January, longitudinal distribution of isotherms dominates, with temperatures decreasing eastwards. In July, the course of isotherms exhibits latitudinal distribution with temperatures decreasing from the central regions northwards, towards the Baltic Sea, and southwards along the increasing elevation in the mountains. The mean annual temperature amplitude varies between ca. 15 °C in high mountain areas and 22 °C in the eastern part of the country. The longitudinal course of isoamplitudes typical for the east of the country is deformed in the west. The longitudinal distribution is only observed there in the central part, whereas in the north and south, the arrangement has a latitudinal direction similar to the course of the coast and mountain ranges (Woś 2010). A similar distribution is characteristic for the multiyear continentality indices. The areal average value of Ewert’s index of continentality is 44.3 % (Kożuchowski 2011). In decades dominated by oceanic influences, it ranges from below 38 % at the Baltic Sea to over 48 % in the east of the country. In decades dominated by continental influences, the values of Ewert’s index range from below 44 % and over 56 %, respectively (Kożuchowski and Marciniak 1992).
3 Data and Methods
3.1 Ewert’s Index of Continentality
Summary of statistics of Ewert’s index of continentality (K) in Poland for the four selected years and the 1981–2010 average
Statistics  K _{1989}  K _{1990}  K _{2002}  K _{2006}  K _{1981–2010} 

Mean  36.19  38.52  58.71  67.52  50.26 
Minimum  22.79  31.26  35.36  49.25  39.54 
Maximum  40.88  42.78  70.10  75.92  55.52 
Range  18.09  11.51  34.74  26.67  15.97 
Standard deviation  3.43  2.72  7.01  6.03  4.18 

the mean value of the index (K _{1981–2010});

the years 1989 and 2006 with the lowest and highest areal means for Poland, respectively (K _{1989}, K _{2006}); and

the years 1990 and 2002 with the smallest and largest index ranges, respectively (K _{1990}, K _{2002}).
The study was based on measurement data provided by 111 synoptic meteorological stations, including 53 Polish stations and 58 stations located outside Polish borders (Fig. 1). The localization of each station was carefully verified by comparing station metadata with ortophotomaps and digital elevation model. The values of the annual amplitude of air temperature were calculated based on information contained in global summary of the day (GSOD) database and provided by German weather service—Deutscher Wetterdienst (Klimadaten für Deutschland—online—frei; http://www.dwd.de).
3.2 Environmental Variables

variables of the overall spatial trend comprising layers of raster cell coordinates (X, Y) and the sea distance index (SDI);

terrain elevation represented by the digital elevation model (DEM) and derivative layers: the concavity/convexity index (CCI), the foehn index (FI), and insolation (IT); and

land cover and its derivatives: the percentage of natural surfaces (NS) and artificial surfaces (AS) in the neighbourhood of the data point.
Variables X and Y denote coordinate values in the local coordinate system PUWG92. The influence of the Baltic Sea is illustrated by the SDI index. To consider the decreasing impact of the Baltic Sea along with distance, the index was constructed as a square root of the smallest Euclidean distance (expressed by the number of cells) of a raster cell from the coastline. SRTM3 elevation data (http://www2.jpl.nasa.gov/srtm) were used as a digital elevation model (DEM). To achieve the goal of this study, the data were transformed into the PUWG92 system and resampled into 250m resolution. This resolution is a compromise between the computational costs and details of information introduced to the models and expressed on the maps. High resolution is of special importance when the interpolated variable is potentially strongly dependant on local factors (including changes in altitude). The choice of spatial resolution applied is supported by earlier studies of Szymanowski et al. (2012, 2013) and Szymanowski and Kryza (2015). Based on the 250m DEM, the concavity/convexity index (CCI) expressing the cool air accumulation effects of concave terrain forms and the foehn index (FI) illustrating the thermal impact of the foehn wind were calculated (Szymanowski et al. 2007). Insolation (IT), expressed by sums of energy of potential total radiation incoming to the terrain surface, is a variable illustrating the role of energy factors. The calculations were performed with the use of the r.sun program implemented in the GIS—GRASS software (GRASS Development Team 2011). The r.sun program is a functionally bestdeveloped module for calculation of radiation in GIS, which can be successfully applied to large areas (Šuri and Hofierka 2004) and works with highresolution terrain models (Kryza et al. 2010). Variables describing the percentage of the surface area arbitrarily called “artificial” (AS) and “natural” (NS) in the surroundings with a radius of 2500 m around each point were prepared on the basis of CLC2000—CORINE Land Cover 2000 database (2004) for European Union countries and the USGS Land Cover database (2011) for Ukraine, Belarus, and Russia. The values of all environmental variables were extracted from raster layers at the coordinates of localization of each meteorological station and were used to specify regression models as described below.
3.3 Statistical and Spatial Analysis Methods
In practice, the decision concerning the choice of the regression model for approximation of the deterministic component of the universal model can be taken only on the basis of the degree of the goodnessoffit of the model to the observation data. It was based on such measures of the fit as the adjusted coefficient of determination (R _{adj.} ^{2} ), standard error of estimation (STE), corrected Akaike information criterion (AICc), and analysis of variance (ANOVA) for regression model residuals. ANOVA was used to check whether the improvement of the model fit, expressed as a decrease in the sum of squared residuals of the model, was statistically significant (Szymanowski and Kryza 2012).
The justification for the use of the GWR model instead of MLR was tested using two tests of the spatial variability of local geographically weighted regression coefficients, i.e., Monte Carlo implemented in the GWR3.0 software (Fotheringham et al. 2002) and the geographical variability (GV) test from the GWR4.0 software (Nakaya 2016). In the case of stationarity of any of the explanatory variables, it would be justified to apply a mixed GWR (MGWR) model instead of a fully local approach (Nakaya et al. 2005). The mixed models allow to mix in one model the explanatory variables which are spatially nonstationary (like for ordinary GWR) and the stationary predictors (like used in MLR). Calibration of the semiparametric GWR model was performed for comparison using the GWR4.0 software.
The other decision in the presented scheme of selection of an optimal interpolation method concerns the possibility of extending the model with a stochastic component, i.e., interpolation of regression model residuals with ordinary kriging. This procedure is employed when there is a significant spatial autocorrelation of regression residuals, which is a basis for modelling a variogram that is different from the pure nugget effect. If the pure nugget effect was the only variogram possible to fit to the experimental variogram of regression residuals, the deterministic model would be corrected at each studied point by an average of the residuals, which equals zero, in accordance with the assumptions of MLR as a best linear unbiased predictor (BLUP) model. In the GWR model, which does not meet the criterion of unbiasedness, the average of the residuals varies, although it is sufficiently close to zero (Fotheringham et al. 2002) to assume that the modification of prediction by residual kriging is negligible in the absence of autocorrelation (Fotheringham et al. 2002). Therefore, the absence of autocorrelation of the residuals excludes the extension of the MLR or GWR models into MLRK or GWRK, respectively (Szymanowski et al. 2012).
Spatial autocorrelation is a mathematical expression of spatial relationships described by Tobler’s first law of geography (Tobler 1970), i.e., decreasing similarity of features of geographical objects along the increasing distance between them. It describes the degree of correlation of the variable value in one location with the value of the same variable in a different location, which implies that the values of the analysed variable determine and, concurrently, are determined by realisation of the variable in different locations. These relationships result in spatial clustering of similar values, which is referred to as positive autocorrelation. The value of the autocorrelation and its statistical significance was determined by calculating Moran’s I statistics (Moran 1950).
Due to the limited 111element input data set, the quantification of the modelling results was performed using the leaveoneout crossvalidation (CV) technique. It yielded a 111element set of CV errors, which were used in the validation of the model in two ways. The values of the summary diagnostic measures were calculated and analysis of the spatial distribution of the CV errors was carried out, particularly in terms of systematic local and regional trends.
Three synthetic measures, i.e., mean error (ME), mean absolute error (MAE), and RMSE, were used in the analysis of the CV errors. The relationship between the sizes of the aforementioned errors can be defined as ME ≤ MAE ≤ RMSE, with the two latter measures having only nonnegative values with the predicted zero value. The use of the square of the CV errors in the RMSE index makes the measure substantially biased even by an inconsiderable number of large errors, although the other errors may be small and acceptable. According to some researchers, MAE is regarded as the most natural diagnostic measure (Willmott and Matsuura 1995).
The quantitative analysis of the model was also accompanied by visual evaluation, which by definition serves identification of features of the model that cannot be shown by methods based on the actual values of the interpolated variable. In particular, it facilitates detection of such little realistic effects as spatial discontinuity, unusually large or small values of the modelled variable, strong directional or regional trends, and various artefacts illustrating the characteristics of the interpolation algorithm rather than those of the interpolated variable. Although it is based on the expert knowledge of the modelled element, such validation is subjective and only complementary to quantitative evaluation.
A number of various computer programs were used in the study, both commercial packages and free software. The spatial analyses primarily concerning the structure of the layers of the explanatory variables and spatial interpolation as well as the final maps were generated in the ArcGIS and GIS GRASS software. The analysis of stepwise regression and the global model was carried out in the STATISTICA program, and the complementary analyses of geographically weighted regression (ANOVA, nonstationarity tests, calibration o semiparametric GWR model) were performed using the GWR3.0 (Fotheringham et al. 2002) and GWR4.0 (Nakaya 2016) packages. For cross validation, the R scripts were developed, using gstat and spgwr packages.
4 Results and Discussion
4.1 Ewert’s Index Spatial Predictors
Explanatory variables and standardized regression coefficients in multiple linear regression (MLR) models of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average (descriptions of variables in the text)
Model parameter (in the order of significance)  Standardized coefficients  P value 

K _{1989}  
SDI  0.753  0.000 
DEM  −0.612  0.000 
X  0.121  0.050 
K _{1990}  
Y  −0.956  0.000 
DEM  −0.359  0.000 
K _{2002}  
X  0.725  0.000 
DEM  −0.569  0.000 
SDI  0.101  0.014 
K _{2006}  
DEM  −0.525  0.000 
X  0.323  0.000 
K _{1981–2010}  
SDI  0.754  0.000 
DEM  −0.576  0.000 
X  0.363  0.000 
Each time, a maximum of three variables was introduced into one model, and only elevation was included in all the five models. This is related to the expected decline in the temperature amplitude together with elevation. Variable X describing the rate of change in the east–west orientation was introduced four times, except for the index calculated for 1990 (Table 2). Similarly, one of the variables characterising the zonal distribution of Ewert’s index, Y, or SDI was introduced in four cases. Importantly, in none of the models, do these variables appear simultaneously. In the geographical conditions of Poland with the Baltic Sea in the north of the country, and given arrangement of meteorological stations, variable Y (northing) and the sea distance index SDI are correlated, and hence, only one of them was introduced into the regression model even if both were significantly correlated with the continentality index (SDI—3 times, Y—1 time; Table 2). There was no significant correlation with one of the zonal variables Y or SDI only in 2006.
The values of standardized regression coefficients indicate a varying impact of the environmental variables on the individual cases, which concurrently imply significant differences in the distribution of the continentality index. The sign of the coefficients identifies the direction of the relationships: in general, Ewert’s index of continentality decreases together with the terrain elevation and northwards and increases along the distance from the sea and eastwards (Table 2).
4.2 Regression Models
Selected statistics of multiple linear regression (MLR), geographically weighted regression (GWR), and mixed geographically weighted regression (MGWR) models of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
Regression model  Auxiliary variables (in the order of significance)  Bandwidth size  Adjusted R ^{2}  Corrected Akaike information criterion (AICc)  Standard error of estimation (STE) 

K _{1989}  
MLR  SDI, DEM, X  –  0.66  586.36  3.33 
GWR  65  0.77  550.19  2.74  
MGWR  65  0.77  549.32  2.74  
K _{1990}  
MLR  Y, DEM  –  0.76  510.86  2.38 
GWR  48  0.84  472.13  1.92  
K _{2002}  
MLR  X, DEM, SDI  –  0.87  566.53  3.05 
GWR  65  0.91  538.4  2.59  
MGWR  65  0.91  537.07  2.59  
K _{2006}  
MLR  DEM, X  –  0.37  684.13  5.20 
GWR  32  0.75  596.74  3.27  
K _{1981–2010}  
MLR  SDI, DEM, X  –  0.90  427.45  1.63 
GWR  65  0.91  538.4  2.59 
ANOVA of multiple linear regression (MLR), geographically weighted regression (GWR), and mixed geographically weighted regression (MGWR) models of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
Source  K _{1989}  K _{1990}  K _{2002}  K _{2006}  K _{1981–2010} 

Sum of squares  
MLR residuals  1186.5  612.8  992.3  2918.9  283.5 
GWR residuals  733.3  353.1  659.6  945.9  195.0 
GWR improvement  453.2  259.7  332.7  1972.9  88.5 
MGWR residuals  748.4  –  670.2  –  – 
MGWR improvement  438.1  –  322.1  –  – 
Local parameter variability tests for geographically weighted regression (GWR) models of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
GWR model parameter  Monte Carlo test (GWR3.0 software) p value  Geographical variability test (GWR4 software) difference of criterion 

K _{1989}  
Intercept  0.000  −6.234 
DEM  0.020  0.874 
SDI  0.000  −29.289 
X  0.000  −10.570 
K _{1990}  
Intercept  0.000  −15.714 
DEM  0.000  −0.843 
Y  0.000  −24.766 
K _{2002}  
Intercept  0.000  −6.920 
DEM  0.140*  1.363* 
SDI  0.000  −28.086 
X  0.000  −16.900 
K _{2006}  
Intercept  0.000  −0.305 
DEM  0.000  −5.312 
X  0.000  −22.551 
K _{1981}–_{2010}  
Intercept  0.000  −12.387 
DEM  0.000  −0.286 
SDI  0.000  −14.413 
X  0.000  −11.086 
The mixed MGWR models for the two cases considered were characterised by a slightly lower residual sum of squares than the GWR models (Table 4). However, the coefficients of determination and estimation errors were similar to the GWR model, and AICc did not indicate a significant improvement in the fit to the observations (Table 3).
The analysis presented in this section shows a better fit of the local GWR regression model than that of the global MLR model. This suggests the nonstationarity of the spatial process in each analysed case of the spatial distribution of Ewert’s index of continentality. No significant difference between the GWR and mixed MGWR models was found either. Thus, accordingly to Occam’s razor principle, where simpler models are preferable to more complex ones, because they are better testable and falsifiable, simpler GWR model was used for further analysis.
4.3 Residual Kriging Models
Spatial autocorrelation of multiple linear regression (MLR) and geographically weighted regression (GWR) residuals of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
Model  Moran’s I statistics E(I) = −0.009  p value 

K _{1989}  
MLR  0.380  0.000 
GWR  0.083  0.145* 
K _{1990}  
MLR  0.357  0.000 
GWR  0.049  0.357* 
K _{2002}  
MLR  0.244  0.000 
GWR  −0.019  0.870* 
K _{2006}  
MLR  0.585  0.000 
GWR  0.098  0.095* 
K _{1981–2010}  
MLR  0.313  0.000 
GWR  0.078  0.165* 
Variogram fitting evaluation and parameters of spherical variograms of multiple linear regression (MLR) residuals of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
Variogram model fitting—RMSE  Spherical model parameters  

Spherical  Circular  Exponential  Partial sill  Range  Nugget 
K _{1989}  
2.61  2.62  2.66  5.3279  382,579  5.9659 
K _{1990}  
2.34  2.34  2.34  2.3679  870,121  4.0273 
K _{2002}  
2.69  2.70  2.72  3.0954  709,466  6.7452 
K _{2006}  
2.83  2.83  2.96  23.618  380,159  5.1770 
K _{1981–2010}  
1.43  1.44  1.44  0.29,026  500,120  2.3367 
4.4 CrossValidation Results
Summary statistics of crossvalidation (CV) errors for multiple linear regression (MLR), geographically weighted regression (GWR), and multiple linear regression—kriging (MLRK) models of Ewert’s index of continentality (K) for the four selected years and the 1981–2010 average
Statistics  MLR  GWR  MLRK 

K _{1989}  
ME  1.080  1.068  0.408 
MAE  2.180 (1.991÷2.369)  2.065 (1.893÷2.237)  1.109 (1.000÷1.218) 
RMSE  2.526  2.385  1.400 
MIN  −4.054  −3.254  −2.520 
MAX  5.504  5.117  3.690 
K _{1990}  
ME  1.010  0.862  0.434 
MAE  1.797 (1.652÷1.942)  1.739 (1.599÷1.879)  1.399 (1.275÷1.523) 
RMSE  2.111  2.018  1.660 
MIN  −3.452  −2.977  −2.631 
MAX  4.341  4.135  3.649 
K _{2002}  
ME  −0.479  −0.518  −0.111 
MAE  2.158 (1.938÷2.378)  2.071 (1.866÷2.276)  1.819 (1.621÷2.017) 
RMSE  2.982  2.848  2.690 
MIN  −5.582  −6.134  −7.992 
MAX  9.654  8.956  8.496 
K _{2006}  
ME  −2.989  −2.171  −0.442 
MAE  5.078 (4.727÷5.429)  4.317 (3.998÷4.636)  1.645 (1.500÷1.790) 
RMSE  5.748  5.044  2.107 
MIN  −9.414  −7.551  −5.536 
MAX  10.631  11.024  6.357 
K _{1981}−_{2010}  
ME  0.341  0.323  0.243 
MAE  1.331 (1.223÷1.439)  1.285 (1.182÷1.388)  1.118 (1.025÷1.211) 
RMSE  1.608  1.565  1.398 
MIN  −3.233  −2.975  −2.557 
MAX  3.353  3.598  3.664 
The better fit of the GWR models, comparing with MLR, was also confirmed by cross validation. In general, the GWR model was characterised by better CV summary statistics in each analysed case. This did not imply that the model was better in terms of all the five measures employed. Sometimes, the ME or one of the extreme errors was closer to zero in MLR than in GWR. However, the MAE and RMSE for GWR were always lower than for MLR. Importantly, GWR was characterised by a significantly lower MAE only in 2006 (Table 8).
In all the analysed cases, the MLRK models have smaller errors than both the regression models. Moreover, due to MAE criterion, MLRK was significantly different from MLR and GWR, so it can be assumed the optimal spatial interpolation algorithm. To confirm the choice, an additional visual assessment of maps of the continentality index was carried out.
4.5 Maps of Ewert’s Index of Continentality
To present the characteristics and differences between the maps generated with the three spatial models analysed, two cases were selected, i.e., the years 2002 and 2006. Due to the large range and considerable variability of the index (Table 1), these were the most demanding cases in terms of spatial interpolation of the analysed data set. This was also reflected in the largest crossvalidation errors (Table 8).
In the other analysed cases, MLRK were also regarded as optimal models. This supported the conclusions drawn from the analysis of the crossvalidation errors also confirming the necessity of introduction of a modified scheme of selection of an optimal predictor (Fig. 2).
5 Summary and Conclusions
The primary aim of the paper was to perform spatial interpolation and to generate maps of Ewert’s index of continentality in Poland for the selected years and an average from the period 1981–2010. The main emphasis was placed on the methodological side of spatial interpolation to develop guidelines for automation of the interpolation process.
Additional objective was to test, on the example of Ewert’s index, the validity of the scheme of selection of an optimal interpolation method developed by Szymanowski et al. (2012, 2013) for spatialization of air temperature. An optimal method was yielding the smallest errors and simultaneously allowing generation of an acceptable map on the basis of expert knowledge about the spatial characteristics of the interpolated variable.
The continentality index for the period 1981–2010 was calculated based on the daily data from 111 meteorological stations. Four years (1989, 1990, 2002, and 2006) characterised by extreme values of the areal average and index range were selected for the analysis. The set of analysed cases was complemented with the mean value of Ewert’s index for the multiyear period 1981–2010.
The set of potential predictors of the continentality index comprised nine environmental variables previously used for spatial interpolation of air temperature in Poland (Szymanowski et al. 2012, 2013). However, the stepwise regression analysis demonstrated that only four variables in the analysed cases exhibited a significant correlation with Ewert’s index, i.e., coordinates X and Y, elevation, and the distance from the sea. Regional or local variables were not significantly correlated with the index, which implies that the features of the continentality index field are mainly determined by macroscale and regional factors (distance from the Baltic Sea) with modification depending on terrain elevation. Noteworthy, the analysis was performed on the basis of the index calculated for the synoptic stations; the location of which, by definition, minimises the local impacts on climate elements. Data from lowerrank, climatological stations, where the impact may be more significant, were not included.
The correlation analysis performed to eliminate collinearity in the regression model additionally showed that variables Y (northing) and SDI were correlated in the geographical conditions of Poland and considering given set of meteorological station. Therefore, only one of them was included in the multiple linear regression MLR model (SDI—3 times, Y—1 time). The elevation variable was included in the models of each of the five cases, although this does not mean that its role was dominant. In fact, the role of the explanatory variables in the regression models varied between the cases, which indicates significant changes in the determinants and, hence, variable features of the Ewert’s index distribution. The general regularities of the distribution allow a conclusion that the continentality index for Poland decreases with elevation and northwards and increases along the distance from the sea and eastwards, which is also supported by the previous studies (Kożuchowski and Marciniak 1992).
The environmental correlation for each of the analysed cases was statistically significant, although for 2006, the MLR model explained solely 37 % of the Ewert’s index variation. In each case, the local GWR model was better fitted to the observations and it significantly improved the MLR regression results. The tests of the spatial variability of the GWR regression coefficients indicated their stationarity. However, compared with the GWR, the use of mixed local–global regression models (MGWR) did not significantly improve the fit.
The residuals of the MLR models were characterised by a positive spatial autocorrelation, which clearly justified extension of the model to the form of residual kriging, MLRK. In contrast, no statistically significant autocorrelation was found in any case of GWR, so the GWRK model was excluded from further considerations.
The cross validation was carried out for three models of each case of Ewert’s index. The local GWR model was characterised by smaller errors. In terms of the MAE criterion, MLR was found to be statistically significantly worse than GWR only for 2006. CV carried out for residual kriging models clearly indicated that the MLRK models produced smaller errors than both regression models. However, the specification of GWRK was not justified and gave grounds for modification of the scheme of optimal interpolator selection in the presence of the environmental correlation. The modified scheme of Szymanowski et al. (2013) now allows a situation where the absence of an autocorrelation of GWR residuals is accompanied by an autocorrelation of MLR residuals, which indicates that MLRK is an optimal method, even if GWR is better fitted to observations than MLR.
The visual assessment of the maps confirmed the results of the cross validation and the MAE criterion. The MLRK model is recommended to spatialize the Ewert’s index for Poland.

Regression models should be specified using four potential predictors: coordinates (X, Y), elevation (DEM), and distance from the sea (SDI).

The decision concerning selection of the interpolator should in each case be based on the proposed scheme (Fig. 2), including comparison of the fit of the regression models to observations by, e.g., comparing the coefficient of determination and autocorrelation regression residuals using Moran’s I statistics.

For autocorrelated residuals, the automatic fitting of the variogram can be done using the spherical model.
In the previous papers (Szymanowski and Kryza 2012; Szymanowski et al. 2012, 2013), it was assumed that the choice of an optimal interpolation method was determined by the stationarity or nonstationarity of the spatial process, which could be inferred from the better fit of one of the regression models, i.e., global or local. The results obtained in this study demonstrated greater importance of the geostatistical component in the universal model of spatial variation (Hengl 2007). Residual kriging is highly efficient if regression residues exhibit a strong positive autocorrelation contributing to the well fit of the theoretical variogram to the experimental one. This does not imply, however, a possibility of exclusion of the deterministic component if the environmental correlation is significant. Although this would lead to maintenance of the good fit of the model to observation in measurement points, the spatial distribution of the index would simultaneously “diverge” from environmental features beyond these points.
References
 Bailey, H. (1968). Hourly temperatures and annual range. Yearb. Assoc. Pac. Coast Geogr., 32, 25–40.CrossRefGoogle Scholar
 Berg, H. (1944). Zum Begriff der Kontinentalitat. Meteorologische Zeitschrift, 61, 283–284.Google Scholar
 Caccianiga, M., Andreis, C., Armiraglio, S., Leonelli, G., Pelfini, M., & Sala, D. (2008). Climate continentality and treeline species distribution in the Alps. Plant Biosyst., 142, 66–78.CrossRefGoogle Scholar
 Conrad, V. (1946). Usual formulas of continentality and their limits of validity, Eos T. American Geophysical Union, 27, 663–664.CrossRefGoogle Scholar
 Conrad, V., & Pollak, L. W. (1950). Methods in climatology (2nd ed.). Cambridge: Harvard University Press.CrossRefGoogle Scholar
 CORINE Land Cover 2000 in Poland. (2004). Final Report, Warsaw. http://www.igik.edu.pl/images/stories/sip/clc_final_report_pl.pdf. Accessed Aug 2012.
 COST Action 719 Final Report. (2008). In O. E. Tveito, M. Wegehenkel, F. van der Wel, & H. Dobesch (Eds.), The use of geographic information systems in climatology and meteorology. Luxembourg: Office for Official Publications of the European Communities.Google Scholar
 USGS Land Cover (2011). http://landcover.usgs.gov/usgslandcover.php. Accessed Mar 2011.
 Dobesch, H., Dumolard, P., & Dyras, I. (Eds.). (2007). Spatial interpolation for climate data: The use of GIS in climatology and meteorology. London: ISTE Ltd.Google Scholar
 Driscoll, D. M., & Yee Fong, J. M. (1992). Continentality: A basic parameter reexamined. International Journal of Climatology, 12, 185–192.CrossRefGoogle Scholar
 Ewert, A. (1963). Kontynentalizm termiczny klimatu. Przegl. Geofiz. XVI, 3, 143–150.Google Scholar
 Ewert, A. (1972). O obliczaniu kontynentalizmu termicznego klimatu. Przegl. Geogr. XLIV, 2, 273–286.Google Scholar
 Fotheringham, A. S., Brunsdon, C., & Charlton, M. (2002). Geographically weighted regression: The analysis of spatially varying relationships. Chichester: Wiley.Google Scholar
 Gorczyński, W. (1920). Sur le Calcul du Degré du Continentalisme et son Application dans la Climatologie. Geografiska Annaler, 2, 324–331.CrossRefGoogle Scholar
 Johansson, O. V. (1926). Über die Asymmetrie der Meteorologische Schwankungen, Societas Scientiarum Fennica Communications in Mathematical Physics, 3 I, Helsingfors, 124.Google Scholar
 Matheron, G. (1971). The theory of regionalised variables and its applications (Les Cahiers du Centre de Morphologie Mathematique de Fontainebleau, 5. Ecole Nationale Superieure’ des Mines de Paris).Google Scholar
 GRASS Development Team. (2011). Geographic resources analysis support system (GRASS) Software, Version 6.4.0. Open Source Geospatial Foundation. http://grass.osgeo.org.
 Hela, I. (1953). Regional distribution of the continentality in the climate of the oceans. Geophysica, 4, 41–47.Google Scholar
 Hengl, T. (2007). A practical guide to geostatistical mapping of environmental variables. Luxembourg: Office for Official Publications of the European Communities.Google Scholar
 Hogewind, F., & Bissolli, P. (2011). Operational maps of monthly mean temperature for WMO Region VI (Europe and Middle East). Idojaras, 115, 31–49.Google Scholar
 Holmlund, P., & Schneider, T. (1997). The effect of continentality on glacier response and mass balance. Annals of Glaciology, 24, 272–276.CrossRefGoogle Scholar
 Ivanov, N. (1953). Ob opredelenyi welichyny kontinentalnosti klimata, Izv. Vses. Geogr. Obshch, 85. Google Scholar
 Ivanov, N. (1959). Belts of continentality on the globe. Izv Vses Geogr Obshch, 91, 410–423.Google Scholar
 Kalarus, M., Schuh, H., Kosek, W., Akyilmaz, O., Bizouard, C., Gambis, D., et al. (2010). Achievements of the Earth orientation parameters prediction comparison campaign. Journal of Geodesy, 84, 587–596.CrossRefGoogle Scholar
 Khromov, S. P. (1957). K voprosu o kontinentalnosti klimata, Izv. Vses. Geogr. Obshch. 89. Google Scholar
 Kożuchowski, K. (2003). Cyrkulacyjne czynniki klimatu Polski. Czasopismo Geograficzne, LXXIV, 1–2, 93–105.Google Scholar
 Kożuchowski, K. (2011). Klimat Polski. Nowe Spojrzenie. Warszawa: Wydawnictwo Naukowe PWN.Google Scholar
 Kożuchowski, K., & Marciniak, K. (1986). Fluktuacje kontynentalizmu klimatu Polski na tle warunków cyrkulacyjnych i solarnych (1881–1980). Przegląd Geofizyczny, 31(39), 139–152.Google Scholar
 Kożuchowski, K., & Marciniak, K. (1992). Kontynentalizm termiczny klimatu na obszarze Polski, Wiad. IMGW XV(XXXVI), 4, 89–93.Google Scholar
 Kożuchowski, K., & Wibig, J. (1988). Kontynentalizm pluwialny w Polsce: Zróżnicowanie geograficzne i zmiany wieloletnie. Acta Geographica Lodziensia, 55, 102.Google Scholar
 Kryza, M., Szymanowski, M., Migała, K., & Pietras, M. (2010). Spatial information on total solar radiation: Application and evaluation of the r.sun model for the Wedel Jarlsberg Land, Svalbard, Pol. Polar Research, 31(1), 17–32.Google Scholar
 Li, J., & Heap, A. D. (2008). A review of spatial interpolation methods for environmental scientists. Canberra: Geoscience Australia.Google Scholar
 Marti, P., & Gasque, M. (2010). Ancillary data supply strategies for improvement of temperaturebased ET_{o} ANN models. Agricultural Water Management, 97, 939–955.CrossRefGoogle Scholar
 Mikolášková, K. (2009). A regression evaluation of thermal continentality, Geografie—Sbornik České Geografické Společnosti, 114(4), 350–362.Google Scholar
 Moran, P. A. P. (1950). Notes on continuous stochastic phenomena. Biometrika, 37(1), 17–23.CrossRefGoogle Scholar
 Nakaya, T. (2016). GWR4 User Manual. GWR 4 Windows Application for Geographically Weighted Regression Modelling, GWR4 Development Team. https://raw.githubusercontent.com/gwrtools/gwr4/master/GWR4manual_409.pdf.
 Nakaya, T., Fotheringham, A. S., Brunsdon, C., & Charlton, M. (2005). Geographically weighted Poisson regression for disease associative mapping. Statistics in Medicine, 24, 2695–2717.CrossRefGoogle Scholar
 Oliver, J. (1970). An air mass evaluation of the concept of continentality. The Professional Geographer, 22, 83–87.CrossRefGoogle Scholar
 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1988). Numerical recipes in C, the art of scientific computing. New York: Cambridge University Press.Google Scholar
 Raunio, N. (1948). The effect of local factors on meteorological observations at Tórshavn. Geophysica, 3, 173–179.Google Scholar
 Salonen, J. S., Seppä, H., Luoto, M., Bjune, A. E., & Birks, H. J. B. (2012). A North European pollen—climate calibration set: Analysing the climatic responses of a biological proxy using novel regression tree methods. Quaternary Science Reviews, 45, 95–110.CrossRefGoogle Scholar
 Spitaler, R. (1922). Klimatische Kontinentalität und Ozeanität. Petermann’s Geographische Mitteilungen, 68, 113.Google Scholar
 Šuri, M., & Hofierka, J. (2004). A new GISbased solar radiation model and its application to photovoltaic assessments. Transactions in GIS, 8, 175–190.CrossRefGoogle Scholar
 Swoboda, G. (1922). Linien gleicher Kontinentalität und Ozeanität 1. Weltkarte 2. Europa, Justus Perthes/Petermanns Mitteilungen, Gotha.Google Scholar
 Szreffel, C. (1961). Przegląd ważniejszych sposobów charakterystyki stopnia kontynentalizmu. Przegląd Geograficzny VI, 3, 191–199.Google Scholar
 Szymanowski, M., & Kryza, M. (2011). Application of geographically weighted regression for modelling the spatial structure of urban heat island in the city of Wroclaw (SW Poland). Procedia Environmental Sciences, 3, 87–92.CrossRefGoogle Scholar
 Szymanowski, M., & Kryza, M. (2012). Local regression models for spatial interpolation of urban heat island—an example from Wrocław, SW Poland. Theoretical and Applied Climatology, 108, 53–71.CrossRefGoogle Scholar
 Szymanowski, M., & Kryza, M. (2015). The role of auxiliary variables in deterministic and deterministicstochastic spatial models of air temperature in Poland. Pure and Applied Geophysics,. doi: 10.1007/s0002401511992.Google Scholar
 Szymanowski, M., Kryza, M., Smaza, M. (2007). A GIS approach to spatialize selected climatological parameters for winegrowing in Lower Silesia, Poland. In: Střelcová, K., Škvarenina, J., Blaženec, M (Ed.) Bioclimatology and natural hazards, International Conference, Poľana nad Detvou, Slovakia, 17–20 September 2007, CDROM, ISBN 9788022817608.Google Scholar
 Szymanowski, M., Kryza, M., & Spallek, W. (2012). Air temperature atlas for Poland: The methodical approach. Wrocław: Uniwersytet Wrocławski. (in Polish, English summary).Google Scholar
 Szymanowski, M., Kryza, M., & Spallek, W. (2013). Regressionbased air temperature spatial prediction models: An example from Poland. Meteorologische Zeitschrift, 22, 577–585.CrossRefGoogle Scholar
 Tobler, W. (1970). A computer movie simulating urban growth in the Detroit region. Econ. Geogr., 46, 234–240.CrossRefGoogle Scholar
 Torregrosa, A., Taylor, M. D., Flint, L. E., & Flint, A. L. (2013). Present, future, and novel bioclimates of the San Francisco, California Region. PLoS ONE, 8(3), e58450. doi: 10.1371/journal.pone.0058450.CrossRefGoogle Scholar
 Ustrnul, Z., & Czekierda, D. (2005). Application of GIS for the development of climatological air temperature maps: An example from Poland. Meteorological Application, 12, 43–50.CrossRefGoogle Scholar
 Ustrnul, Z., & Czekierda, D. (2009). Atlas of extreme meteorological phenomena and synoptic situations in Poland. Warszawa: IMGW.Google Scholar
 Willmott, C. J., & Matsuura, K. (1995). Smart interpolation of annually averaged air temperature in the United States. Journal of Applied Meteorology, 34, 2577–2586.CrossRefGoogle Scholar
 Woś, A. (2010). Klimat Polski w Drugiej Połowie XX Wieku. Poznań: Wydawnictwo Naukowe UAM.Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.