**1. Introduction**

Natural disasters cause loss of human life and damage to infrastructure every year throughout the world. In Algeria, extreme rains are the source of flooding which can cause catastrophic damage both in inhabited areas and in the countryside.

One of the basic problems encountered in meteorology is the need to assess the meteorology risk caused by extreme precipitation in order to avoid human and material losses. Thus, the location and severity of floods can be determined.

In the twenties to the middle of the last century, the theory of extreme values has witnessed a remarkable development [1–3] most studies focused on the monthly or yearly mean values and we find their application in many fields like; rainfall in Algeria [4, 5], extreme precipitations in Argentina [6] Mapping snow depth return levels [7], precipitation and temperature [8, 9].

On the other hand, a lot of studies say there is a great spatial difference in rainfall [10] for this reason it was used interpolation methods, the kriging method is considered as the most used for spatial interpolation of rainfall [11–13], the kriging method has a special feature which is complementing the sparsely sampled primary variable, in the case of secondary variable there is another method called cokriging that outperforms the kriging method [14].

According to [15] In order to examine the spatiotemporal variations of meteorological variables there is a statistical method that allows to apply multiple strategies by cluster analysis to pinpoint the similar places, local and universal meteorology techniques which has been raising lately.

Our goal in this chapter is to compare the two methods kriging and co-kriging using the GEV and determine the location and severity of floods in all regions of Algeria.

### **2. Methodology and data**

#### **2.1 Generalized extreme value distribution**

The cumulative distribution function is proposed by [16].

$$F(\mathbf{x}) = \begin{cases} \exp\left\{-\left[\mathbf{1} + \xi \frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma}\right]^{-\frac{1}{\xi}}\right\} \xi \neq \mathbf{0} \\\\ \exp\left(-\exp\left(-\frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma}\right)\right) \xi = \mathbf{0} \end{cases} \tag{1}$$

By deriving the Eq. (1) we get the density function

$$f(\mathbf{x}) = \begin{cases} \frac{1}{\sigma} \left[ 1 + \xi \left( \frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma} \right) \right]^{-\frac{1-\xi}{\sigma}} \exp\left\{ - \left[ 1 + \xi \left( \frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma} \right) \right]^{-\frac{1}{\xi}} \right\} \xi \neq \mathbf{0} \\\\ \frac{1}{\sigma} \exp\left( - \left( \frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma} + \exp\left( - \frac{\mathbf{x} - \boldsymbol{\mu}}{\sigma} \right) \right) \right) \xi = \mathbf{0} \end{cases} \tag{2}$$

The logarithm of the likelihood function is given by: For *ξ* 6¼ 0

$$d(\xi, \mu, \sigma, Y) = -n \ln \sigma - \left(\mathbf{1} + \frac{\mathbf{1}}{\xi}\right) \sum\_{n=1}^{n} \ln \left(\mathbf{1} + \xi \left(\frac{\mathbf{x}\_{i} - \mu}{\sigma}\right)\right) - \sum\_{i=1}^{n} \left[\mathbf{1} + \xi \left(\frac{\mathbf{x}\_{i} - \mu}{\sigma}\right)\right]^{\frac{-1}{\xi}} \tag{3}$$

For *ξ* ¼ 0

$$d(\xi, \mu, \sigma, Y) = -n \ln \sigma - \sum\_{n=1}^{n} \exp \left( -\frac{\mathbf{x}\_i - \mu}{\sigma} \right) - \sum\_{i=1}^{n} \left( \frac{\mathbf{x}\_i - \mu}{\sigma} \right) \tag{4}$$

3 and 4 with differentiating the two parameters:

$$\begin{cases} n - \sum\_{n=1}^{n} \exp\left(-\frac{\mathbf{x}\_{i} - \mu}{\sigma}\right) = \mathbf{0} \\\\ n + \sum\_{n=1}^{n} \frac{\mathbf{x}\_{i} - \mu}{\sigma} \left[ \exp\left(-\frac{\mathbf{x}\_{i} - \mu}{\sigma}\right) - \mathbf{1} \right] = \mathbf{0} \end{cases} \tag{5}$$

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

## **2.2 Return period**

The return period, also known as a recurrence interval is the estimated average time between events such as earthquakes, floods, landslides, or river floods. From Eq. (1) we can write the return level as following:

$$Z\_T = \begin{cases} \mu + \frac{\sigma}{\xi} \left\{ 1 - \ln \left[ 1 - \frac{1}{T} \right]^{\xi} \right\} \xi \neq \mathbf{0} \\\ \mu - \sigma \ln \left( -\ln \left( 1 - \frac{1}{T} \right) \right) \xi = \mathbf{0} \end{cases} \tag{6}$$

### **2.3 Variogram model**

Various parameter variogram models have been used in the literature. Here is some of the most popular content.

#### **Spherical model.**

The Spherical model has linear behavior at small separation distances near the origin, but flattens at large distances, which means that it shows a gradual decrease in spatial dependence until a certain distance beyond which the spatial dependence tends to smooth.

$$\gamma(h) = c\_0 + c\_1 \left(\frac{3}{2}\frac{h}{a} - \frac{1}{2}\left(\frac{h}{a}\right)^3\right) \text{ for } 0 < h \le a$$

$$\gamma(h) = c\_0 + c\_1 \text{ for } h \ge a$$

Where c0 is the nugget effect. The sill is c0 + c1. The range for the spherical model can be computed by setting g(h) = 0.95(c0 + c1).

#### **Gaussian model.**

The Gaussian model is used when the data exhibits strong continuity at short lag distances which means the spatial correlation is very high between two neighboring points.

*<sup>γ</sup>*ð Þ¼ *<sup>h</sup> <sup>c</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>*<sup>1</sup> <sup>1</sup> � *<sup>e</sup>*�ð Þ *<sup>h</sup>=<sup>α</sup>* <sup>2</sup> � �where *<sup>c</sup>*<sup>0</sup> is the nugget effect. *<sup>c</sup>*<sup>0</sup> <sup>+</sup> *<sup>c</sup>*<sup>1</sup> is the sill. The range is 3*α*. This model describes a random field that is considered to be too smooth and possesses the peculiar property that *Z*(*s*) can be predicted without error for any *s* on the plane.

#### **2.4 Data description**

The precipitation data used in this study are for the National centers for environmental information NOAA of USA, this data used especially in cases where surface data are difficult to obtain or insufficient. Our data represented by the annual daily maximal of rainfall from1979 to 2012 calculated in the 856 Algerian stations (**Figure 1**).

The preliminary analysis of the annual maximum precipitation data during the analysis period (1979–2012) included descriptive statistical calculations (**Table 1** and **Figure 2**). More precisely, we calculated the minimum (Min), the maximum (Max), the mean (Mean), the standard deviation (std.dev) and the coefficient of variation (coef.var). **Table 1** presents the values of the descriptive statistics for the annual time series of maximum precipitation for all stations (from 1979 to 2012). The results show that the maximum values is observed in the years 1982, 1992,

**Figure 1.** *A map of Algeria showing rainfall stations.*


#### **Table 1.**

*Statistical descriptive of all years.*

1994, 2001, 2006, and 2007, while the mean and highest values are observed in 1982 (**Figure 2**). The lowest value of coef.var. is for the year 1996 (68%), and the highest for 1984 (14%). On this basis, the observed data showed that all years had a coef. var. greater than 68%, highlighting the high variability of annual maximum precipitation over Algeria.

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

**Figure 2.** *Boxplots of annual extreme precipitation.*
