## 4 Results and Discussion  
In this section, a comparison of the results obtained through deterministic and geostatistical algorithms is presented. In Table 1, the results of the exploratory statistical analysis of monthly rainfall are shown. Mean monthly rainfall ranges between 83.2 mm (February) and 117.4 mm (July). The maximum and the minimum monthly values have been detected in January (632.6 mm) and in July (20.4 mm), respectively. The coefficients of variation evaluated for the monthly rainfall values range between 49% and 66%, thus evidencing a low spatial heterogeneity of rainfall in New Zealand (Table 1). The exploratory statistical analysis of the precipitation observation was mainly performed to evaluate whether the monthly values fit a normal distribution. Indeed normality is a desirable property for kriging, that will generate the best absolute estimate only if the random function fits a normal distribution ([PERSON], 1970). As a result, appreciably different monthly mean and median values and high values of skewness coefficients were observed (Table 1). Therefore, a priori logarithmic transformation of the data was performed after which normality was obtained (similar mean and median values and close to zero values of the coefficients of skewness). Due to the station density, 1 station per 913 km\({}^{2}\), the spatial variability has been considered to be equal in the different directions, and thus an omnidirectional semi-variograms, evaluated for the monthly rainfall observation, has been considered to carried out the spatial dependence analysis. To represent the empirical semivariogram by fitting a theoretical model, it was found that using a combination of theoretical models results in a more accurate fit onto the empirical semivariance than using a single model. Therefore, the semivariance of three models, precisely Gaussian, spherical, and circular, were used to fit the empirical semivariograms of monthly rainfall observation and for avoidance of negative interpolations. For all cases, two models combined showed the best fittings. The results of variograms parameters are different for each month (not shown). An example of experimental semi-variograms and their fitted models for the three geostatistical methods, computed for June, is shown in Figure 2.  
In order to match the interpolation performance of the adopted algorithms, the cross-validation has been applied. The cross-validation results (scatter plots of predicted versus measured values) for June are shown for the four methods in Figure 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Month** & **Mean** & **Median** & **Max** & **Min** & **CV** & **SK** & **KU** \\ \hline January & 90.0 & 77.3 & 632.6 & 36.9 & 0.66 & 5.21 & 35.06 \\ February & 83.2 & 73.9 & 512.0 & 27.9 & 0.58 & 4.52 & 29.42 \\ March & 94.4 & 80.9 & 609.7 & 40.8 & 0.60 & 4.81 & 33.29 \\ April & 98.2 & 88.4 & 547.5 & 35.9 & 0.57 & 3.74 & 20.67 \\ May & 113.0 & 102.9 & 553.5 & 31.8 & 0.54 & 3.09 & 14.98 \\ June & 113.8 & 106.0 & 439.5 & 25.8 & 0.53 & 1.90 & 6.31 \\ July & 117.4 & 111.8 & 380.0 & 20.4 & 0.49 & 1.41 & 3.42 \\ August & 110.9 & 101.7 & 433.5 & 21.4 & 0.52 & 1.92 & 6.03 \\ September & 98.9 & 85.6 & 542.3 & 26.5 & 0.64 & 3.42 & 16.52 \\ October & 106.2 & 91.3 & 612.0 & 37.4 & 0.66 & 3.96 & 20.34 \\ November & 94.5 & 79.7 & 573.3 & 34.5 & 0.66 & 4.27 & 23.48 \\ December & 101.8 & 88.7 & 595.5 & 37.2 & 0.61 & 4.26 & 23.93 \\ \hline \hline \end{tabular} CV = Coefficients of variation; SK = Coefficient of skewness; KU = Kurtosis.  
\end{table} 
Table 1: Descriptive statistics of the monthly precipitation observation for the 294 rainfall series. Figure 2: Scatterplots between observed and estimated monthly precipitation in June for (**a**) inverse distance weighting (IDW), (**c**) ordinary kriging (OK), (**e**) kriging with an external drift (KED), and (**g**) ordinary cokriging (COK). Related semi-variograms in June for (**b**) OK, (**d**) KED, and (**f**) cross-variograms in June for COK. Distance is expressed in meters.