**3. Research methodology**

*Recent Advances in Flood Risk Management*

**2. Background of the study**

relief operations [3].

stated that:

reduction and mitigation efforts in the basin.

The combination of these two approaches is hoped to go a long way towards disaster

Hydrological extreme events such as floods and droughts have accompanied mankind throughout its entire history and these events are cyclic in nature. The twenty-first century, however, has been marked by an unusual number of natural disasters worldwide, among these events are the recent hurricane Matthew that devastated the Caribbean Islands of Haiti, parts of Jamaica and United States of America [1], the recent Nepal 2015 giant earthquake that killed more than 8000 people and injured more than 19,000 people [2], flooding and landslides in Brazil in 2011 and flooding in Mozambique and other parts of Southern Africa in 2000. Natural disasters such as floods often pose an intolerable threat to society, hence a holistic approach is needed to understand such phenomena, predict such catastrophic events and mitigate the impact of these natural disasters. The lower Limpopo River in Mozambique has a history of worst floods and droughts than all other national and international rivers in Mozambique. The most catastrophic and expensive of these reported natural disasters in Mozambique were the year 2000 floods which killed a total of more than 700 people and caused economic damages estimated at US\$500 million. It is argued by the International Federation of Red Cross (IFRC) that aid money can buy more than seven times as much humanitarian impact if spent before a disaster rather than on post-disaster

The chairman of the 2014 International Disaster and Risk Conference (IDRC) held in Davos, Switzerland, August 2014, Dr. Walter J. Ammann pointed out that the frequency and intensity of natural hazards such as floods and earthquakes are on the rise in these recent years [3]. In a separate study, a unique survey of 139 national meteorological and hydrological services carried out by the World Meteorological Organisation (WMO) in 2013 revealed that floods were the most frequently experienced extreme events worldwide over the course of the decade 2001–2010 [4]. Some studies have also shown that floods and droughts account for 90% of all the people that are affected by natural disasters [5]. According to Munich Re [6] the statistics of natural disasters for the year 2013 was dominated by floods that caused several billions of United States of America dollars in losses. Irina Bokova, the Director-General of UNESCO [7]

*Every year, more than 200 million people are affected by natural hazards, and the risks are increasing – especially in developing countries, where a single major disaster can set back healthy economic growth for years. As a result, approximately one trillion dollars have been lost in the last decade alone. This is why disaster risk reduction is so essential. Mitigating disasters requires training, capacity building at all levels, and it calls for a change of thinking to shift from post-disaster reaction to* 

The present study considers floods in the lower Limpopo River basin of Mozambique. The lower Limpopo River basin is characterised by extreme natural climatic conditions alternating between extreme floods and severe droughts. Droughts affect the country on an average of 7–8-year cycle and are usually associated with the El Nino phenomenon which affects Southern Africa [8]. The provinces of Gaza and Inhambane, which house the lower Limpopo River basin,

*pre-disaster action –this is UNESCO's position.*

**56**

In this section we present the data source and fundamental principles of extreme value theory [9], as well as a brief discussion of some goodness-of-fit tests.

#### **3.1 Study sites, data and block maxima moving sums**

Hydrometric data has been collected in Mozambique since the early 1930s. However, due to war and other external factors there were periods in which no data were collected at some stations. For this study we obtained hydrometric data for the lower Limpopo River for the sites Chokwe (1951–2010) and Sicacate (1952–2010) from the Mozambique National Directorate of Water (DNA), the authority responsible for water management in Mozambique. The data obtained were daily flood heights (in metres) and were time series in nature.

In statistics of extremes there are two fundamental realisations used in flood frequency analysis namely block maxima and partial duration series commonly known as peaks-over-threshold (POT) [9]. The approach used in this study is block maxima. In hydrological studies, when sample sizes are large it is natural to block observations by years [9, 10].

Since in flood frequency analysis the years are natural blocks, the flood heights data in this study were blocked by years. Sequential steps were taken to obtain annual maxima data from the daily flood heights data series. Further sequential steps were taken to obtain the annual moving sums of 2 days (AM2), 5 days (AM5) and 10 days (AM10). A generalised extreme value (GEV) distribution

was then fitted to the annual daily (AM1) flood heights and their corresponding moving sums.

#### **3.2 Generalised extreme value model**

The GEV distribution is a well-known distribution in statistics of extremes. Comprehensive details of probability framework of block maxima and the practical reasons for using block maxima over POT are given in [9, 11]. Dombry [12] proved the consistency of maximum likelihood (ML) estimators when using block maxima approach.

The GEV cumulative distribution function, *G*, is given in Eq.(1) as:

$$\text{The GEV cumulative distribution function, } G \text{, is given in Eq.(1) as:}$$

$$G(\mu, \sigma, \xi; \mathbf{x}) = \begin{cases} \exp\left(-\left(1 + \xi \frac{\mathbf{x} - \mu}{\sigma}\right)^{-1/\xi}\right), 1 + \xi \frac{\mathbf{x} - \mu}{\sigma} > 0, \xi \downarrow \ \mathbf{0}, \\\exp\left(-\exp\left(-\frac{\mathbf{x} - \mu}{\sigma}\right)\right), \mathbf{x} \in \mathfrak{R}, \xi = \mathbf{0}, \end{cases} \tag{1}$$

where *μ*,σ and ξ are the location, scale and shape parameters, respectively. The parameters of the GEV in (1) are estimated by the ML method [11].

The log-likelihood function for the GEV in (1) is given in (2):

$$\mathcal{U}(\mu, \sigma, \xi; x) = -k \log \sigma - \left(\mathbb{1}/\xi + 1\right) \sum\_{t=1}^{k} \log \left[1 + \xi \left(\frac{x - \mu}{\sigma}\right)\right]\_{+} - \sum\_{t=1}^{k} \left[1 + \xi \left(\frac{x - \mu}{\sigma}\right)\right]\_{+}^{-1/\xi}, \tag{2}$$

where *k* is the number of blocks (years) and annual maxima flood height *x* = (*x*1,*x*2,…,*xk*).

#### **3.3 Anderson-Darling and Kolmogorov-Smirnov tests**

The goodness-of-fit of the GEV model to the annual maxima flood heights moving sums time series models was verified using Anderson-Darling (A-D) and Kolmogorov–Smirnov (K-S) tests. The A-D test is sensitive to the tails of the distribution, while the K-S test is sensitive to the centre of the distribution [13]. The moving sums time series models were ranked from 1 to 4, with 1 being the best according to the particular test. A model that attains the lowest value of the total rank (sum of A-D rank and K-S rank) satisfies the criteria for the best annual maxima moving sums time series model. Where there is a tie in the total ranks for two or more models, then the rank value of the A-D test is used as a tie-breaker (with smaller value being best) since the main emphasis in extreme value theory is in fitting the tails.

### **4. Results and conclusion**

This section presents the results of the study. **Tables 1** and **3** present the ML estimates of the parameters of the GEV distribution for Chokwe and Sicacate, respectively, for the models AM1, AM2, AM5 and AM10. **Tables 2** and **4** present results for the goodness-of-fit of the GEV distribution to the annual maxima moving sums time series models for Chokwe and Sicacate, respectively. **Table 5** presents the flood frequency tables of the return periods and their corresponding return levels for Chokwe and Sicacate based on the best fitting models.

**59**

*Fitting a Generalised Extreme Value Distribution to Four Candidate Annual Maximum Flood…*

**Model K-S Cv R A-D Cv R Total** AM1 0.054 0.172 3 0.237 2.50 4 7 AM2 0.051 0.172 1 0.234 2.50 3 4 AM5 0.052 0.172 2 0.185 2.50 1 3 AM10 0.062 0.172 4 0.218 2.50 2 6

*Key: Cv stands for critical value, R stands for rank, K-S is Kolmogorov-Smirnov, A-D is Anderson-Darling.*

**Model μ σ ξ** AM1 6.190 3.4587 −0.49005 AM2 12.267 6.8603 −0.48351 AM5 29.440 16.466 −0.45154 AM10 54.217 29.875 −0.38701

**Model K-S Cv R A-D Cv R Total** AM1 0.093 1.737 3 0.394 2.50 4 7 AM2 0.094 1.737 4 0.370 2.50 3 7 AM5 0.089 1.737 2 0.324 2.50 2 4 AM10 0.053 1.737 1 0.217 2.50 1 2

*Key: Cv stands for critical value, R stands for rank, K-S is Kolmogorov–Smirnov, A-D is Anderson-Darling.*

**Return period 20 50 100 200 500** Chokwe (AM5) 32.89 36.12 38.54 40.96 44.15 Sicacate (AM10) 103.58 115.66 124.72 133.75 145.66

**Model μ σ ξ** AM1 4.2671 1.8242 −0.11031 AM2 8.4538 3.4794 −0.09947 AM5 19.671 7.9968 −0.07243 AM10 35.889 14.8720 −0.05874

*DOI: http://dx.doi.org/10.5772/intechopen.82140*

*ML estimates of the GEV distribution parameters for Chokwe (1951–2010).*

*Goodness-of-fit tests of the GEV distribution for Chokwe (1951–2010).*

*ML estimates of the GEV distribution parameters for Sicacate (1952–2010).*

*Goodness-of-fit tests of the GEV distribution for Sicacate (1952–2010).*

*Return periods (years) and their corresponding return levels (m) for the two sites.*

**Table 1.**

**Table 2.**

**Table 3.**

**Table 4.**

**Table 5.**

*Fitting a Generalised Extreme Value Distribution to Four Candidate Annual Maximum Flood… DOI: http://dx.doi.org/10.5772/intechopen.82140*


**Table 1.**

*Recent Advances in Flood Risk Management*

**3.2 Generalised extreme value model**

*G*(*μ*,σ, ξ;*x*) =

*l*(*μ*,σ, ξ;*x*) = −*k* logσ − (1/ξ + 1)∑

*x* = (*x*1,*x*2,…,*xk*).

in fitting the tails.

**4. Results and conclusion**

moving sums.

approach.

was then fitted to the annual daily (AM1) flood heights and their corresponding

The GEV distribution is a well-known distribution in statistics of extremes. Comprehensive details of probability framework of block maxima and the practical reasons for using block maxima over POT are given in [9, 11]. Dombry [12] proved the consistency of maximum likelihood (ML) estimators when using block maxima

−1/ξ

*<sup>x</sup>* <sup>−</sup> *<sup>μ</sup>* \_\_\_\_ <sup>σ</sup> )]<sup>+</sup> <sup>−</sup> <sup>∑</sup>

*i*=1 *k*

[1 + ξ(

*<sup>x</sup>* <sup>−</sup> *<sup>μ</sup>* \_\_\_\_ <sup>σ</sup> )]<sup>+</sup>

−1/ξ

, (2)

exp(−exp(−*<sup>x</sup>* <sup>−</sup> *<sup>μ</sup>* \_\_\_\_ <sup>σ</sup> )),*<sup>x</sup>* <sup>∈</sup> <sup>ℜ</sup>, <sup>ξ</sup> <sup>=</sup> 0,

where *μ*,σ and ξ are the location, scale and shape parameters, respectively. The

log[1 + ξ(

where *k* is the number of blocks (years) and annual maxima flood height

The goodness-of-fit of the GEV model to the annual maxima flood heights moving sums time series models was verified using Anderson-Darling (A-D) and Kolmogorov–Smirnov (K-S) tests. The A-D test is sensitive to the tails of the distribution, while the K-S test is sensitive to the centre of the distribution [13]. The moving sums time series models were ranked from 1 to 4, with 1 being the best according to the particular test. A model that attains the lowest value of the total rank (sum of A-D rank and K-S rank) satisfies the criteria for the best annual maxima moving sums time series model. Where there is a tie in the total ranks for two or more models, then the rank value of the A-D test is used as a tie-breaker (with smaller value being best) since the main emphasis in extreme value theory is

This section presents the results of the study. **Tables 1** and **3** present the ML estimates of the parameters of the GEV distribution for Chokwe and Sicacate, respectively, for the models AM1, AM2, AM5 and AM10. **Tables 2** and **4** present results for the goodness-of-fit of the GEV distribution to the annual maxima moving sums time series models for Chokwe and Sicacate, respectively. **Table 5** presents the flood frequency tables of the return periods and their corresponding return levels for

), <sup>1</sup> <sup>+</sup> <sup>ξ</sup> *<sup>x</sup>* <sup>−</sup> *<sup>μ</sup>* \_\_\_\_ <sup>σ</sup> <sup>&</sup>gt; 0, <sup>ξ</sup> <sup>≠</sup> 0,

(1)

The GEV cumulative distribution function, *G*, is given in Eq.(1) as:

exp(−(<sup>1</sup> <sup>+</sup> <sup>ξ</sup> *<sup>x</sup>* <sup>−</sup> *<sup>μ</sup>* \_\_\_\_ <sup>σ</sup> )

parameters of the GEV in (1) are estimated by the ML method [11]. The log-likelihood function for the GEV in (1) is given in (2):

> *i*=1 *k*

⎧ ⎪ ⎨ ⎪ ⎩

**3.3 Anderson-Darling and Kolmogorov-Smirnov tests**

Chokwe and Sicacate based on the best fitting models.

**58**

*ML estimates of the GEV distribution parameters for Chokwe (1951–2010).*


*Key: Cv stands for critical value, R stands for rank, K-S is Kolmogorov-Smirnov, A-D is Anderson-Darling.*

#### **Table 2.**

*Goodness-of-fit tests of the GEV distribution for Chokwe (1951–2010).*


#### **Table 3.**

*ML estimates of the GEV distribution parameters for Sicacate (1952–2010).*


#### **Table 4.**

*Goodness-of-fit tests of the GEV distribution for Sicacate (1952–2010).*


#### **Table 5.**

*Return periods (years) and their corresponding return levels (m) for the two sites.*
