Statistical Analysis of Extreme Dry Events in the Tunisian Sub-Humid Region Based on Probability Laws ^†

Abstract

Dry event duration analysis from series of daily rainfall observations in a Ghézala Dam catchment in Northern Tunisia, where the average rainfall is about 600 mm, was executed. The dry events are a series of dry days framed by a rainfall event. Rainfall events are defined as an uninterrupted series of rainfall days lasting at least a day having received a precipitation superior or equal to a threshold value of 4 mm d⁻¹. The rainfall events are characterized by depth and duration. A correlation exists between these two parameters. Therefore, the rainfall depth per event is conditioned by the event duration. Respectively, the negative binomial and the geometric distribution best fits the dry event duration and the rainfall event duration. For the length of the climatical cycle, the incomplete gamma was found to be the appropriate distribution.

1. Introduction

Precipitation is a very important element of climate that affects both the natural environment and human society. Whether short or long, droughts or high intensities of rainfall have harmful consequences on society and the environment [1].

It is recognized in the literature that the wet–dry spell model is used to simulate wet and dry spells separately by fitting their durations to an appropriate probability distribution [2,3].

In this contribution, we are interested in the intensive statistical analysis of the uninterrupted sequences of dry and rainfall days, and, consequently, their alternation during the rainy season. We use discrete and continuous probability distribution functions.

The case study is a dam watershed with a Mediterranean climate located in the north of Tunisia.

This concept of event-based investigation takes a step ahead of continuous data generation methods. Synthetic data can be used in irrigation management studies or analysis of the effects of climate change.

2. Main Data

The case study is the watershed of the Ghézala dam (lat. 37°02′75″ N, long. 9°32′07″ E) Northern Tunisia (Figure 1).

The rainy season lasts from September to April, where the average annual rainfall is 680 mm. The dry season lasts almost from May to August. The climate is sub-humid, where the greatest percentage of rain is recorded during the three winter months (December, January, and February).

Here, we use daily rainfall records from the rain gauge installed at the Ghézala dam site for the period from 1968 to 2007.

3. Method

The methodology used here is based on the characterization of rainfall and dry events following the principle in Figure 2. We first define a negligible rainfall threshold. Here, a threshold of 4 mm d⁻¹ was chosen corresponding to the average daily evapotranspiration in the watershed. Thus, according to Figure 2, a rainfall event is an uninterrupted sequence of rainy days including at least one day having received precipitation greater than or equal to the threshold value. The dry event is, then, the time interval between two successive rainfall events.

The process is therefore specified by the probability laws of the duration of wet periods and the duration of dry periods (Figure 2).

To index the climatic parameters in the time marker, the rainy season takes the letter n and the event the letter m, as follows:

$$n=1,2,\dots ,n$$

(1)

$$m=1,2,\dots ,m$$

(2)

According to Figure 2, during any rainy season, the duration of any rainfall event will be denoted D_n,m and that of a dry event Z_n,m. The cumulative rainfall amounts corresponding to this rainfall event will be calculated as follows (Equation (3)):

$$H_{n,m}={\displaystyle \sum _{i=1}^{Dn,m}{h}_j}$$

(3)

where h_j is the daily rainfall amount recorded in mm; note that h_j is positive.

The varying duration of the events requires that the duration of the event should condition the cumulative rainfall amounts corresponding to each event. The identifying and fitting of conditional probability distributions to rainfall amounts can be a problem, particularly in the case of short records and extreme (long) duration events [4].

In the same way as shown in Figure 2, we define the length L_n of the rainy season as the time interval between the start of the first and the end of the last event. The climate cycle C_n is determined as the time between the start of two subsequent rainy seasons.

4. Results and Discussion

In order to study the dependence relationship between the different parameters extracted from the wet–dry period analysis, a regression analysis was carried out. The results of the maximum determination coefficients obtained are shown in

Table 1. Coefficients of determination for the parameters of the analysis by events.

Seasonal Characteristics	Nn	Zn,m Max (Day)	Cn (Day)
Ht,n (mm)	0.49	0.5	0.02
Ln (day)	0.33	0.21	0.62
Cn (day)	0.22	0.23	1
Event-related characteristics	Hn,m (mm)	Dn,m (day)
Dn,m (day)	0.64	1
Zn,m (day)	0.02	0.02

Legend: Z_n,m max: longest dry event of the rainy season (day).

.

As expected, Table 1 shows that there is a relationship between the duration of the rainfall event and its corresponding rainfall depth, while no significant pairwise correlation between Z_n,m and the duration D_n,m and cumulative rainfall amounts H_n,m could be recorded (Table 1). Therefore, the assumption made that precipitation during a rainy season is part of an independent random process appears to be justified.

Table 1 shows that the number of events per season N_n is independent of other parameters except, as expected, the total precipitation amount H_t,n. In fact, this amount of rain is, among other things, the sum of the cumulative precipitation amounts H_n,m.

The length of the climatic cycle C_n is independent of the number of events per rainy season N_n; however, it shows a dependence on the length of the rainy season L_n.

Fitting Probability Distribution Functions to Event Based Analysis Parameters

Table 2. Probability distribution functions, pdf, fitted to the different parameters extracted from the event-based analysis. The Kolmogorov–Smirnov test was used to assess the goodness-of-fit at the 95% significance level.

Parameter of the Event Based Analysis	Probability Distribution Function Retained	Formula
Number of events per rainy season	Poisson	$f\left(N,\lambda \right)={\displaystyle \frac{e^{-\lambda }{\lambda }^N}{N!}}$ N = 0, 1, 2, … N = the number of events during a rainy season. λ = average number of events per rainy season.
Duration of rainfall events	Geometric	$f(m)=p{q}^{m-1}$ m = duration of the rainfall event in days. $p=1/\stackrel{\_}{m}$, q = 1 − p.
Rainfall depth per event (events lasting at least 6 days)	Gamma	$f\left(x,\propto ,\beta \right)={\displaystyle \frac{{\beta }^{\alpha }{x}^{\alpha -1}{e}^{-\beta x}}{(\propto )}}$ α = shape parameter and β = scale parameter.
Duration of dry events	Negative binomial	$f\left(n\right)={\displaystyle \frac{\left(r+n-1\right)}{n!\left(r-1\right)!}}.p^r.q^n$ where n = 0, 1, 2, … , the variance of the dry event duration and r et p are estimated. by: $m=\overline{n}-1$$;r={\displaystyle \frac{m.p}{1-p}}$$;p={\displaystyle \frac{m}{{\sigma }_n^2}}$$;q=1-p$
Advent of the rainfall event	Law of the leaks	$f\left(x\right)=\left\{\begin{array}{l}{e}^{-\lambda }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{f}x=0\\ {\lambda e}^{-\lambda -{\displaystyle \frac{x}{s}}}{\displaystyle \frac{I_1\left(2\sqrt{\lambda \frac{x}{s}}\right)}{\sqrt{\lambda \frac{x}{s}}}}\hspace{1em}\hspace{1em}\hspace{1em}ifx>0\end{array}\right.$ where λ is the positive shape parameter; s the positive scale parameter; I1 the modified Bessel function of order 1. Its descriptive characteristics are the mean and variance estimated by the maximum likelihood method: $\widehat{\mu }=\lambda s$ ${\widehat{\sigma }}^2=2\lambda s^2$

gives the probability distribution functions, pdf, fitted, and Figure 3 shows some adjustments of this pdf to the different parameters extracted from the event based analysis. It appears that the Poisson density function adequately describes the distribution of the number of events per season (Figure 3a). The arithmetic mean and the standard deviation are, respectively, 22.54 and 4.67. The Kolmogorov–Smirnov test assessed the goodness-of-fit at the 95% significance level using the SIMFIT software version 5.5 [5].

Analysis of the results obtained shows that approximately 33% of rainfall events last at most one day. The maximum duration observed is 13 days. However, the frequency of these long-lasting events decreases rapidly with increasing duration. The arithmetic mean is 2.79 days and the standard deviation is 1.87. The geometric pdf proved to be the most adequate for the fitting (Figure 3b).

Since there is a relationship between rainfall depth H_n,m and duration D_n,m (Table 1), we used conditional pdfs of the rainfall depth for different values of event duration. The H_n,m recorded during the events are grouped into 4 mm wide classes, starting with the 4–8 mm class. For example, for events lasting at least 6 days, the gamma distribution provided the best fit (Figure 3c). The fit was assessed by the Kolmogorov–Smirnov test. The two statistical parameters arithmetic mean and standard deviation of this class are 69.81 mm and 38.02 mm, respectively.

From the results in Table 1 the dry event can be assumed to be independent of the rainfall event and the rainfall depth per event. Therefore, the distribution of the dry event, which can only take integer values (Figure 2), follows an unconditional probability distribution function.

Therefore, the negative binomial pdf was found to be the best form for the fitting (Figure 3d).

The interpretation of the results found highlights that dry periods of up to 30 days, or even more, can be recorded (a maximum of 56 days is recorded). The arithmetic mean and standard deviation for the dry event are 7.3 days and 7.9, respectively; i.e., 30.2 days and 3.6 for extreme events.

The best fit of the X-axis of the first rainfall event in the time axis (see Figure 2) is a law of the leaks (Figure 3e), using the HYFRAN software version 1.1 [6]. On average, the first rainfall event occurs in mid-September, while the probability of exceeding this value is 0.52 for a biennial return period. In the extreme case, the hydrological year begins around the first decade of October.

5. Conclusions

The dry spell phenomenon using rainfall records from the Ghézala dam can be well described by fitting a pdf to the length of the interevent period. An excellent fit is provided by the negative binomial pdf for prolonged dry periods between subsequent rainfall events. Conceptually, in a true Poisson process, the time “without event” should follow the exponential pdf or, in a discrete case, the geometric pdf [7]. Notably, this “flaw” could be eliminated by defining the interevent time as the dry event. Consequently, the present role of the interevent time would be taken over by the rainfall events duration. The theoretical requirements of the fitted geometric pdf are satisfied (Figure 3b).

Event-based analysis was used to generate synthetic rainfall event time series; coupling this with a rainfall–runoff model, one can obtain synthetic streamflow series to be used for reservoir simulation studies and the design of flood estimations.