**3. Results and discussions**

## **3.1 Trend in annual maximum precipitation**

In this study, the Mann-Kendall non parametric test is computed to characterize the time course of annual maximum precipitation at the national scale. The trend is considered significant if the value of the probability (*p*-value) is greater than 0.05 (95%). The **Figure 3** shows the distribution of the significance trend (red color) and non-significant trend (blue color). From the results, we can observe that 130 (15%)

**Figure 3.** *The spatial distribution of the annual maximum precipitation trend in Algeria country.*


#### **Table 2.**

*Statistical descriptive of AMPmax.*

stations have a significant trends and most of them are positive (97% of the total stations) at the national scale.

The maximum value of the annual maximum precipitation (AMPmax) for each station is calculated and presented in **Table 2**. From the **Table 2**, we can see that the mean value of AMPmax is 22.46 mm and 47% of the total stations with values greater than the mean of AMPmax. The coefficient of variation is 61%, indicating the significant spatial distribution of AMPmax at the national scale. Therefore, we applied a Kriging and Co-kriging approaches to better understanding the spatial distribution of the annual maximum precipitation in Algeria country.

In this study, we have 865 selected grids and 9496 predicted grids have locations where spaced every 20 m in the East and North grid directions and covered the irregularly shaped of the country (**Figure 4**). Due to the large numerical range of AMPmax values and to allow easy interpretation of the results, we worked with the logarithmic transformation of the variable. In this application, we chosen a base 10 logarithms (log10) for the data and we randomly selected control and test datasets. In this study, 30% of the total grids were excluded for testing (assessment).

### **3.2 Choosing the variogram model**

We start by plotting the experimental variogram before adjusting the latter with the different models. The sum of the square errors (SSErr) and the regression coefficient (R<sup>2</sup> ) provided an accurate measure of the fit of the model to the variogram data, with a lower SSErr and a higher R<sup>2</sup> indicating better fit of the model.

**Figure 4.** *The spatial distribution of the prediction grid.*

*Evaluation of the Spatial Distribution of the Annual Extreme Precipitation Using Kriging… DOI: http://dx.doi.org/10.5772/intechopen.101563*


#### **Table 3.**

*The parameters of each model.*

The values of the parameters of the different fitted models are presented in **Table 3**.

Theoretical and empirical semi-variogram were prepared for the AMPmax as shown in **Figure 5**. From the results, we can see that the spherical model has been found to be the most accurate model for annual maximum precipitation.

The spatial dependence is generally accessible in terms of the ratio between the nugget (C0) and the sill (C0 + C) expressed as a percentage. The AMPmax is considered to be a strong spatial dependence when the ratio value is less than 25%, moderate spatial dependence when this value is between 25% and 75%, and low spatial dependence when the value is greater than 75%. From **Table 3**, we can clearly see that the spatial dependence of AMPmax for the best-fitting semivariogram model is strong and with a ratio of 24.56%.

#### **3.3 AMPmax interpolation**

The Spherical model is used to interpolate the AMPmax for both Kriging and co-Kriging methods at the national scale. In the first step, we compared graphically the forecasted and estimated GEV parameters (μ, σ and ξ). From **Figure 6**, we can see that a very clear spatial pattern for the estimates of the location and scale parameters however an absence of the spatial pattern for the shape parameter. The northern region is very marked compared to the rest of the regions with a significantly

**Figure 5.** *Empirical semivariance and its fitted model.*

**Figure 6.**

*The spatial distribution of the forecasts of the GEV parameters for the two methods.*

higher value of the location and scale parameters. On the other hand, the co-Kriging method clearly provided new regions where the values are high. Generally, the high values could be observed in northern Algeria.

In order to compare the two methods Kriging and Co-Kriging, we used Cross Validation method and some statistical indicators such as Mean Errors (ME), Root Mean Square Errors (RMSE) and Squared Deviation Ratio (MSDR) **Table 4**.

**Figure 7** display a bubble plots of the cross-evaluation error of the two methods, where positive values are drawn in green and negative values are drawn in red, and the size of the bubble is proportional to the distance from zero.

From **Table 4** and **Figure 7**, we can clearly notice that the Kriging forecast error map with the three parameters of the GEV distribution shows low errors expected near the stations, while co-Kriging gives the lowest errors on average at the national scale, especially for the shape parameter.

After the validation of the two methods, Co-kriging method used to estimate the return levels (RLs) of the annual maximum of daily precipitations for the different stations using Eq. (6).

The return periods are shown in **Table 5** for the 20, 50, and 100-year. The results show that the maximum annual maximum precipitation observed in 1982,

*Evaluation of the Spatial Distribution of the Annual Extreme Precipitation Using Kriging… DOI: http://dx.doi.org/10.5772/intechopen.101563*


#### **Table 4.**

*Cross-evaluation errors for the two methods.*

#### **Figure 7.**

*The spatial distribution of cross-evaluation errors for the three parameters of the GEV law by the two methods.*

1992, 1994, 2001, 2006, and 2007 exceeds the 20-year regression level. AMP exceeding the maximum AMP during the observation period (123.59) begins to appear in the confidence interval of 50 year.


**Table 5.**

*Statistical descriptive of the return periods.*

**Figure 8.** *Spatial representation of return periods (co-kriging method).*

**Figure 8** shows the results for the three cases RLs considered, in the first and second cases (20 years and 50 years), we notice roughly the same results, although there are some remarks that need to be made. In 50-year RLs, there is an increase in the eastern part, far south and in the center of Algeria. Otherwise in 100-year RLs, we noticed a great difference, especially in the eastern region, in the far southern the state of Tamanrasset, the western region in the state of Tindouf, and in The Middle of the desert the Adrar region.
