## 4 Results and Discussion
### Comparison of the Interpolation Methods  
The analysis of deterministic and geostatistical methods allowed to identify strengths and weaknesses of the applicability of the different algorithms to monthly rainfall in New Zealand. Since IDW considers only the distance between estimated and sampled points and leaves out the pattern of spatial dependence of rainfall observation, as expected, substantially different results have been obtained by applying the IDW and the geostatistical methods.  
In Table 3 results of the cross-validation are shown in terms of MAE and RMSE values, providing clear indications of the effectiveness of kriging in the spatial interpolation of monthly rainfall observation in New Zealand. In fact, the IDW technique has been clearly outperformed by the geostatistical approaches, thus confirming past studies (e.g., [12, 63]) which found IDW to be less accurate than geostatistical interpolators in the estimation of monthly and annual rainfall.  
Figure 5: Mean monthly precipitation interpolated using ordinary kriging (OK) method.  
Figure 6: Mean monthly precipitation interpolated using ordinary cokriging (COK) method.  
Specifically, considering the RMSE, the COK has been identified as the most accurate interpolator in 11 months with the exception of April when the KED resulted more precise. However, taking into account the MAE, the KED has been identified as the most accurate interpolator in April, May, June and July, and the COK in the remaining months. The COK can be thus indicated as the best method for the spatial interpolation of rainfall in New Zealand for almost all months. The IDW and OK methods were not the best approaches for interpolating rainfall in any analysed month. Overall, the COK evidenced the best least bias and scatter in the cross-validation test, with the lowest errors. Finally, interesting results can be obtained from the visual comparison among the maps obtained from the application of the different geostatistical interpolation algorithms. In fact, similar rainfall spatial distributions, have been produced with the geostatistististical interpolators, and all these distributions seem to be consistent with natural precipitation. In particular, as demonstrated in previous studies [66; 67; 12; 68; 53; 14], adding elevation and geographical coordinates as secondary variables in the application of the KED and the COK, did not lead to a significant improvement of the results, that is explainable considering the weak correlation identified between precipitation and the secondary variables. In fact, as pointed out by [PERSON] [12], better results of the COK with respect to the OK can be observed with an increase in the correlation between precipitation and altitude. Moreover, [PERSON] [26] evidenced that the impact of the auxiliary data on the COK interpolations generally depends on the correlation between primary and secondary variables but also on their spatial continuity patterns. Perhaps, as suggested by [PERSON] et al. [66], using monthly time series of covariates (e.g., monthly wind speed or humidity) as additional information in the KED and the COK instead of variables fixed in time (e.g., topography and geographical coordinates) can lead to a much better cross-validation than those obtained by the OK. Unfortunately, these variables are available in New Zealand only for short observation periods and in very few stations.