**4. Discussion**

The performances of GWR and GTWR models are relatively similar at the annual and monthly scales according to the reductions in average MAE, MRE, and RMSE. However the application of the GTWR model is not effective with respect to all the indicators, which reflects the extreme uncertainties of rainfall in time.

It is preferable to use a daily time scale to calculate spatial weights. Precipitation is formed by precipitation processes, and the duration of each process is usually several days. Therefore, as the timescale goes from yearly to daily the fine temporal details of precipitation gradually become prominent. Figure 11 shows the precipitation process in August 2008. The average monthly precipitation stands for the rainfall of the whole month. Consequently, the performance of the GTWR(D) model with a daily time scale is much better than the GTWR(M) and GTWR(Y) models' at the monthly (Table 2) and annual (Table 3) scales.

The periodicity of precipitation has some impact on improving the accuracy of interpolation. As shown in Figure 6 shows the GTWR(C) model performs better than GTWR(M). However, the frequency and amplitude of each precipitation cycle are very different, as shown in Figures 3 and 11. The calculation of the periodic function needs further study.

The introduction of kriging is reasonable in the improvement of interpolation accuracy. Table 4 shows the normality test of residuals between the results of the GTWR model and the actual precipitation. The residuals of most months fitted normal distribution (significance > 0.05), while other months, such as May, June, September, and December, fitted approximate normal distribution (kurtosis < 10 and skewness < 3). Compared with the GTWR model, the GTWRK model performed well in terms of average MAE, MRE, and RMSE (Tables 2 and 3). The GTWRK model produces a more accurate spatio-temporal precipitation. The GTWRK model's performance varies performs through the seasons. This work represent summer as June, July, and August. It seen in Table 4, the residuals in these months all fitted the approximately normal distribution, which affects the accuracy of the GTWRK model to some extent.

**Table 4.** The normality test of residuals between GTWR and the actual precipitation.


\* This is a lower bound of the true significance; a. Lilliefors significance correction.
