Stochastic modelling of rainfall occurrences in ...

Hydrological Sciences - Journal - des Sciences Hydrologiques, 33,5, 10/1988

Stochastic modelling of rainfall occurrences in continuous time

G. TSAKERIS National Technical University of Athens, Department of Civil Engineering, Division of Water Resources, 5 Iroon Polytechniou Str., 157 73 Athens, Greece

Abstract The paper is concerned with the modelling of rainfall occurrence in continuous time. The Alternating Renewal Process is employed for the evaluation of probability distribution functions for total wet and dry periods over a homogeneous time interval (0, r). The derived general solution is simplified by assuming that the individual wet and dry intervals are random variables following an Erlang distribution, in particular an exponential distribution. Data on a continuous time scale from the Mikra Station in Greece are used to illustrate the proposed methodology. Modélisation stochastique d'occurrences de pluie en temps continu Résumé Ce rapport examine la modélisation d'occurrences de pluie en temps continu. LAlternating Renewal Process est employé pour l'évaluation des fonctions de distribution de probabilité de périodes sèches totales dans un intervalle de temps homogène (0, r). La solution générale dérivée est simplifiée en faisant l'hypothèse que les intervalles secs et humides sont des variables aléatoires qui suivent une distribution d'Erlang et plus particulièrement une distribution exponentielle. Les donnés en temps continu de la station de Mikra en Grèce sont utilisées pour expliquer par un exemple la méthodologie proposée. INTRODUCTION Rainfall modelling has for a long time been the subject of study by various reseachers from various disciplines including climatology, meteorology and hydrology. In view of the large number of aspects of rainfall modelling, it would be an almost impossible task to investigate rainfall modelling as a whole in one short article. One aspect of rainfall modelling with which many hydrologists have been concerned over the last 20 years is the temporal structure of rainfall. Generally, two groups of random variables enter into the description of the temporal structure of point rainfall. The first sequence is concerned with the frequency of occurrence of rainfall events whereas the second deals with the rainfall depths associated within the rainfall events. Numerous models have been proposed for modelling rainfall occurrence and rainfall amount. A closer look at the literature on the subject reveals Open for discussion until 1 Aprill989.

437

G. Tsakiris

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that there is no unified mathematical model incorporating the modelling of both frequency and amount of rainfall with a universal acceptability. This is mainly due to the variability of the temporal structure of rainfall as it is encountered in different parts of the world. The definition of a rainfall event depends on the time scale over which the rainfall process is to be described. Therefore both discrete and continuous time scales have been used in the literature. Focussing on rainfall occurrence modelling at short time scale (e.g. hours, or at the most, days) several counting processes have been used. For a comprehensive review of these processes the reader is referred to the reviews on the subject prepared by Waymire & Gupta (1981) and to individual papers by Todorovic & Yevjevich (1969), Todorovic & Woolhiser (1976), Buishand (1978), Carey & Haan (1978), Roldan & Woolhiser (1982), Srikanthan & McMahon (1983), Tsakiris et al. (1984). The most popular rainfall occurrence models are the well-known Poisson process for a continuous time scale, and the Markov chain models for a discrete time scale. Applications of the Poisson process to temporal point rainfall are given by Todorovic & Yevjevich (1969), Todorovic & Woolhiser (1976) and Eagleson (1978). On the other hand, from the existing numerous applications of Markov chains one can refer to Allen & Haan (1975), Carey & Haan (1978), Caskey (1963), Chin (1977), Gabriel & Newmann (1962), Haan et al. (1976), Mimikou (1983a, b). Apart from the above types of models, the Alternating Renewal Process has been used in discrete time rainfall modelling (Buishand, 1977). Various other models, incorporating nonhomogeneous Poisson and Markov processes and some variants of the Polya process (Neyman-Scott cluster process in Kavvas & Delleur (1981)) have also been used. Discussion of all these models is beyond the scope of this paper. The objective of this paper is to present and use the Alternating Renewal Process in continuous time scale for modelling rainfall occurrence. Before entering into more specific definitions of the rainfall process let us consider a schematic representation of this process as it is presented in Fig. 1. In this figure the sequence of dry (X) and wet (Y) intervals appear on the horizontal axis of time whereas the cumulative rainfall amount is recorded on

i

\-

CL LU Q

Ho _J

Î H?

< 0

(3)

A better description of this evolution may be realized by assuming that the process starts with a dry state (X(t) = 0) at time / = 0, and that the sequence Xv Yv X2, Y2, ... of dry and wet periods, respectively, consists of independent, non-negative, continuous random variables. A schematic representation of an alternating sequence of wet and dry intervals is presented in Fig. 2(a). In Fig. 2(b) a sample function of the process Wit) is illustrated. Assume that all the dry periods (Z.) and all the wet periods (Y) are identically distributed as X and Y according to: P(X « x) = F(x)

(4)

P(Y « y) = Giy)

(5)

The distribution of I " Xi is denoted by F (r) which is the «-fold convolution of F, namely:

440

G. Tsakiris Y

Y

1

Y

2

3 Wet

V—

0. An analogous definition holds for Gn(r), being the n-fold convolution of G. In order to apply the Renewal Theory it is assumed that the wet intervals are of zero durations. Then the sequence Xv X2, Xy ... can be considered as a renewal process as illustrated in Fig. 3.

-Nr TIME

Fig. 3 A schematic representation of a Renewal Process of dry intervals (wet intervals are assumed of zero duration). A sequence of independent identically distributed random variables forms a Renewal Process (Ross, 1970). According to Renewal Theory, the points in time at which the process is switched from a wet to a dry interval are called "renewals". In accordance with this theory the distribution of the total dry period in the interval (0, t) may be found (Barlow & Hunter, 1961; Muth, 1968; Takacs, 1957):

441

Stochastic modelling of rainfall occurrences

P{D(t) i r} = Z^o Fn{r)[Gn(t - r) - Gn+1(t - r)]

for t > r

(7)

in which the term inside the brackets on the right-hand side of equation (7) represents the probability of having n renewals in the time period (0, t), while F (r) represents the cumulative density function of the sum of the n separate dry intervals. The simplification adopted for the derivation of equation (7) is presented in Fig. 3. Similarly the distribution of the total wet period in ((Q, t) is obtained (Tsakiris et al, 1984): P{W(t) « r} = ll_0 Gn(r)[Fn(t - r) - Fn+1(t - r)]

for t > r

(8)

COMPUTATION OF THE DISTRIBUTION D(t) AND W{t) The solution of equations (7) and (8) depends mainly on the form of F(x) and G(x). The main difficulty encountered is the evaluation of convolutions. In the general case equations (7) and (8) may be solved using numerical techniques (McConalogue, 1978). According to Li (1971), the infinite sum in equations (7) and (8) may be approximated by a finite sum up to a number N by adding an error term e(N), as follows: P{D(t) « r} = X£Q Fn(r)[Gn{t - r) - Gn+1(t - /•)] + e(N)

for t > r (9)

Li (1971) showed that the error e(N) is bounded and therefore satisfactory approximations may be obtained. There are special cases for which the distribution of D(t) can be calculated using equation (9) with the aid of a computer. The computations may be significantly simplified if F(r) and G(r) are members of the family of special Erlang distributions. Erlang distributions exhibit two major advantages. First they are relatively simple involving only factors such as e~' and powers of t. Secondly, they are reproductive: if F(r) is an Erlang distribution F (r) is Erlang also. The simplest Erlang distributions have the one-parameter exponential distribution of the form: F(r) = 1 - e ' X r

(10)

G(r) = 1 - c^r

(11)

with rates X and Grayman & fitting the dry proposed the use for the sequence

ii, respectively, and X, ii > 0. Eagleson (1971) used a Weibull density function for and wet periods. More recently Eagleson (1978) of exponential density functions with different parameters of wet and dry spells, respectively. To support the idea

442

G. Tsakiris

of exponentially distributed wet and dry spells, rainfall data from Boston (Massachusetts) were presented. As can be seen in the following section on applications the same assumption seems realistic for the data of the Mikra station (Greece). From equations (10) and (11) the following equations may be derived: Gn(t - r) - Gn+1(t - r) = e*M r

Hn(t -r)n

(12)

\nji-ie~\y

F„(r) = f — ty J "w o (n - 1)!

/

for n = 1, 2, ...

(13)

where FQ(r) = 1. Inserting equations (12) and (13) in equation (7), the latter becomes:

P{D{t) « r} = Z^o

(t r)

^'

lin(t

-r)n

I J

—

ày

(14)

o in - 1)!

or finally, according to Li (1971):

P{D{t) « r} = e " ^ ^

1 + 7 [\n(t - r)] f elyy-l% {2 J [\n(t - r)y]}dy (15)

where all the symbols retain the meanings attributed to them previously and Ix(.) is the modified Bessel function of the first kind. Similar expression may be obtained for the total wet period W(t). ANALYSIS OF RAINFALL OCCURRENCE DATA AND DISCUSSION In order to apply the methodology presented above, rainfall data in continuous time were obtained from various meteorological stations in Greece. The data were provided by the National Meteorological Service of Greece. The common drawback of these data is that they are of short length covering not more than six consecutive years. By analysing the rainfall data of several meteorological stations, some similarities in the behaviour of rainfall characteristics and in connection with the presented methodology were observed. Therefore the rainfall data of only one station, Mikra (Thessaloniki), are analysed in this paper. Mikra meteorological station is located in the northern part of the country, on the outskirts of Thessaloniki (longitude: 22°58 ' ; latitude: 40°31 ' ; altitude: 3 m). The data cover the irrigation season of 120 days duration starting on 1 May each year. This period is divided into 15 equal intervals of eight-days duration.

443

Stochastic modelling of rainfall occurrences

Such intervals are convenient as decision stage intervals for irrigation water distribution on a rotational basis. This is the most common distribution method in the large gravity irrigation systems in Greece. From the rainfall characteristics the following two are of major importance in relation to the presented methodology: the duration of rainfall (wet interval) and time between rainfalls (dry interval). Further, the irrigation season is divided into two sub-seasons (1, from the first to the seventh decision stage interval, and 2, from the eighth to the fifteenth) which were found to be homogeneous with respect to the main rainfall characteristics. As a result, all the computations were made separately for these two subseasons. In the analysis no seasonality effect was taken into account. To distinguish between independent rainfall observations, a two-hour intervening dry interval was found suitable, as was proposed previously by Grayman & Eagleson (1971) and Eagleson (1978). One-parameter exponential distributions were fitted to the wet and dry intervals of both subseasons. In Fig. 4 the derived probability density function of dry intervals is plotted together with the available data in histogram form. The rates of the exponential distributions, calculated by the method of maximum likelihood, are

(a)i SUBSEA SOU 1

\

f(X) - 0.0136

exp

(-0.0136

t)

0.1

SL

0.0 103

200

DRY INTERVAL

(b)

400

0.3 SUBSEA SOU

o

300 (hours)

2

0.2 fit)

= 0.0057

exp

(-0.0057

t)

co 0.1