1-s2.0-002216949090149R-main

Journal of Hydrology, 120 (1990) 183-202 183 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

[i]

CRITICAL DROUGHT ANALYSIS BY SECOND-ORDER MARKOV CHAIN

ZEKAI SEN*

Civil Engineering Faculty, Department of Hydraulics, Technical University of Istanbul, Ayaza~a, Istanbul (Turkey)

(Received July 11, 1989; accepted after revision November 8, 1989)

ABSTRACT

Sen, Z., 1990. Critical drought analysis by second-order Markov chain. J. Hydrol., 120: 183-202.

Exact probability distribution functions (PDF) of critical droughts in stationary second-order Markov chains are derived for finite sample lengths on the basis of the enumeration technique. These PDF are useful in predicting the possible critical drought durations that may result from any hydrologic phenomenon during any future period provided that the second-order Markov chain is representative of the underlying probability generation mechanism. Necessary charts are provided for the expectation and variance of critical droughts. The application of the developed methodology is given for three representative annual flow series from different parts of the world. It is observed that their critical droughts confirm well with the second-order Markov chain.

INTRODUCTION

Drough t s are complex events which may impair social, economic, agricul- tu ra l and o ther act ivi t ies of a society. General ly, in water sciences they are defined as extended periods of wa te r deficits e i ther at a single site or in a region. Extended dry periods mus t be considered in the p lann ing of water resources development if shor tages are to be avoided. The d rough t concept varies with different c l imates and different wa te r sources and uses. For instance, d rough t s in many par ts of Saudi Arab ia are recognized after 2 or 3 years wi thou t s ignif icant rainfal l occur rences whereas in Bali any period of more t h a n 1 week wi thou t ra in is considered as a drought . On the o ther hand, for g roundwa te r hydrologis ts abnormal falls in g roundwa te r levels because of over-pumping are the s tar t of d rough t periods whereas for agr icu l tura l i s t s such a s ta r t is the day on which the avai lable soil mois ture is depleted to some small pe rcen tage of avai lable capac i ty or when the soil wa te r in the root zone is at or below the pe rmanen t wil t ing percentage. There have been var ious definit ions of droughts , s t a r t ing with the work of Russel (1896).

*Present address: Hydrogeology Department, Faculty of Earth Sciences, King Abdulaziz Uni- versity, P.O. Box 1744, Jeddah 21441, Saudi Arabia.

0022-1694/90/$03.50 © 1990 Elsevier Science Publishers B.V.

184 z SEN

However, a universal definition of droughts can be given on the basis of an observed sequence of the phenomenon concerned in addition to a t runcat ion level by which drought and flood properties can be identified. The definition adopted herein has been suggested by Yevjevich (1967). He defined a drought duration as an uninterrupted sequence of observations less than the t runcat ion level provided that this sequence is preceded and succeeded by at least one observation greater than the t runcat ion level. The truncation level may be either a constant value or a function of time. In practical studies, the selection of the t runcat ion level is not arbi t rary but is dependent on the water demand as well as on the type of water deficit being studied. Drought characteristics of any phenomenon are dependent on the underlying generating mechanism, and can be modelled by a suitable stochastic process such as independent, first- and second-order Markov processes. However, each of these processes yields different drought and flood durations. For instance, the annual surface water or precipitation occurrences can be approximated by independent or simple first-order Markov chains, which usually fail to preserve the critical drought durations properly. On the basis of hypothesis testing procedure. Chin (1976) presented definitive examples in which the simple first-order model was rejected and a second-order model was proposed for such occasions.

One of the most severe drought duration properties is the possible maximum duration likely to occur over the economic life of any water resources system. It will be referred to as the "crit ical drought durat ion" in this study. The exact probability distribution function (PDF) of the critical drought durat ion is difficult to obtain and most of the previous studies provided approximate asymptotic solutions for infinite samples only. Various probabilistic descrip- tions of this durat ion suitable for different hydrologic processes have been analyzed by Feller (1968), Salazar and Yevjevich (1975) and Sen (1980a-c), among many other researchers.

The main purpose of this paper is to find the exact PDF of the critical drought durations in any finite sample size that originates from a second-order s tat ionary Markov chain by the enumeration technique. A similar technique has been used recently by ~en (1990) in relation to runs theory with applications to drought prediction.

DROUGHT DEFINITION AND PROBABILISTIC DESCRIPTION

Let { Si }, (i = 1,2,3,...n) be an ordered set of any hydrologic variable observed along the time axis at n equally spaced instants. The underlying generating mechanism is assumed to be a second-order Markov chain, in which case the occurrence of an event at a specific instant i is dependent on the two preceding events at instances i - 1 and i - 2. This dependence can be measured by either the autocorrelat ion function (Yule, 1923) or autorun function based on the conditional probabilities (Sen, 1977). For the analysis of drought durations the lat ter measurement is more convenient, as it involves estimation of a set of transit ional probabilities from an observed sequence. Furthermore, these prob-

CRITICAL DROUGHT ANALYSIS 185

-X p . . c r i t i c a l

×o I V w w w xl:~ w w

Y i i

o

O : DRY

W : W E T

Fig. 1. T runca t ion of a given series: x0, t runca t ion level; s, surplus; d, deficit.

n t i m e

abilities have physical meanings, they reflect the internal persistence in the evolution of the phenomenon concerned and they are not dependent on the particular PDF of the hydrologic variable itself. Analysis of short-term records shows evidence of a tendency for dry years to cluster together. This indicates that the sequence of dry years is not completely random. In addition, because of the existence of long-term trends as well as persistence, it becomes important to be able to quantify drought duration and its severity.

Truncation of a hydrologic sequence along a meaningful level, x0, is shown schematically in Fig. 1. Such a t runcation yields two simple states of the phenomenon, namely, a surplus, s = {Si > x0}, and a deficit, d = {Si < x0}. The critical drought duration of length L with successive deficits, Sj < x0, Sj+I < Xo ...... Sj+L < Xo is the main concern herein. Description of any drought duration in the t runcat ion of sample function from a second-order Markov process requires definition of a set of transit ion probabilities as follows:

(1) The initial state probabilities as p = P ( S i > Xo) and complementarily q = 1 - p . These probabilities are independent of any other state and are sufficient to represent an independent process.

(2) Transition probabilities between two successive time instants. There are four complementary transition probability statements:

P ( s / s ) = P ( 8 i > x0, Si_l > x0)

P ( d / s ) = P(S~ < x0, S~_1 > x0) (1)

P ( s / d ) = P ( S i >x0, Si_l <x0)

P ( d / d ) = P ( S i < Xo, Si_l < Xo)

In fact, together with the basic probability statement in point (1) above, these transition probabilities describe a first-order Markov chain completely. The transitions from a given state to the two adjacent states are mutually exclusive

186 z. SEN

and the re fore P ( s / s ) + P ( d / s ) = 1 and P ( s / d ) + P ( d / d ) = 1. Because of these two p robab i l i ty s t a t emen t s it is neces sa ry to e s t ima te f rom a g iven sequence only two of the four t r ans i t i on probabi l i t i es in eqn. (1).

(3) T rans i t i on probabi l i t i es as a resu l t of surp lus and deficit s ta tes combina t ion a t th ree consecu t ive t ime ins t an t s descr ibe the second-order M a r k o v process wi th the o the r p robab i l i ty s t a t emen t s g iven in points (1) and (2), Fo r the s t a t i o n a r y second-order M a r k o v process the re a re e ight different combina t ions and, accordingly , t r ans i t i on probabi l i t ies :

P ( s / s s ) = P ( S i > Xo/8i_l > x0, Si .2 > Xo)

P ( d / s s ) = P ( S i < xo/Si 1 > Xo, Si ,2 > Xo)

P ( s / s d ) = P ( S i > xo/Si 1 > xo, Si 2 < Xo)

P ( d / s d ) = P ( S i < xo/Si 1> xo, Si ~ < Xo)

P ( s / d s ) = P ( S i > xo/Si_ l < Xo, Si 2 > Xo) (2)

P ( d / d s ) = P(S~ < xo/Si_~ < x0, Si_2 > x0)

P ( s / d d ) = P ( S i > xo/S i_ l < x0, Si_2 < x0)

P ( d / d d ) = P ( S i < xo/Si 1 < x0, Si 2 < Xo)

where P ( s / s s ) is the p robab i l i ty of occu r rence of a surp lus even t a t the cu r r en t t ime i n s t a n t g iven t h a t the events a t two p reced ing t ime in s t an t s are in surp lus s ta tes . Thus, o the r p robab i l i ty s t a t emen t s are se l f -explanatory. M u t u a l exclus- iveness of some t r ans i t ions implies t h a t P (s / s s ) + P ( d / s s ) = 1, P ( s / sd ) + P

( d / s d ) = 1, P ( s / d s ) + P ( d / d s ) = 1 a n d P ( s / d d ) + P ( d / d d ) = 1. Consequent- ly, only four of the probabi l i t i es in eqn. (2) need to be es t imated f rom an ava i l ab le obse rva t ion sequence.

CRITICAL DROUGHT DURATION

The s imples t case occurs for the s t a t i o n a r y second-order M a r k o v processes for which all the r e l evan t p robab i l i ty s t a t e m e n t s a re ident ica l and t ime independent . As an in i t ia l condi t ion the probabi l i t i es of the cr i t ica l d rough t dura t ion , L, in a sample of size i = 1 can be wr i t t en read i ly as

P , ( L = O) = p

P , ( L = 1) = q (3)

For sample size two the t r ans i t i ona l p robabi l i t i es in eqn. (1) become appl icable and lead to

P2(L - O/s) = PI (L = O ) P ( s / s )

P2(L = 1/s) = P I ( L = O ) P ( d / s )

P2(L = 1/d) = P~(L = 1) P ( s / d ) (4)

P2(L = 2/d) = PI (L = 1) P ( d / d )


where P2 (L = O/s) is the probability that the critical drought duration will be equal to zero given that the final state is at the surplus state. The transitional probabilities of the second-order Markov chain start to play a dominant role from a sample size of three onwards. For i = 3 the desired critical drought probability statements are

P3(L = O/ss) = P2(L = O/s) P ( s / s s )

P3(L = 1/ds) = P2(L = O/s) P ( d / s s ) + P2(L = 1/d) P ( d / s d )

P3(L = 1/ss) = P2(L = 1/d) P ( s / s d )

P3(L = 1/sd) = P2(L = 1/s) P ( s / d s ) (5)

P3(L = 2 / d 4 ) = P2(L = 1/s) P ( d / d s )

P3(L -- 2 / s4 ) = P ~ ( L - - 2 / d ) P ( d / s d )

P3(L -- 3 / 4 4 ) = P2(L = 2/d) P ( s / d d )

where P3(L = O/ss) is the probability that the critical drought duration will be equal to zero, given that the two previous instants are in surplus states. The probabilities of critical drought duration after sample size three are all dependent on the combinations of two previous sample sizes and hence a recursive formulation is obtained as follows. For instance, when i = 4 the probabilities are

P4(L = O/ss) =

P4(L = 1/ds) =

P4(L = 1/ss) =

P4(L = 1/ds) =

P4(L = 2 /dd) =

P4(L = 2 /ds ) =

P4(L = 2/ss) =

P4(L = 2 / sd ) =

P4(L = 3 /dd) =

P4(L = 3 / s4) =

P3(L = O/ss) P (s / ss )

Ps(L = O/ss) P (d / ss ) + P3(L = 1/ss) P (d /ss )

+ P3(L = 1/sd) P (d / sd )

Pa(L = 1/ss) P (s /ds) + P3(L = 1/sd) P ( s / sd)

P3(L = 1/ds) P (s /ds )

P3(L = 1/ds) P (4 /d s )

P3(L = 2 / sd) P (d / sd )

P3(L = 2 / sd) P ( s / sd)

P3(L = 2 /dd) P ( s /dd)

Pa(L = 2 /4d) P (d /dd )

P3(L = 3 /dd) P ( s /dd )

= 3 / d d ) P (d /dd)

drought probabilities for i = 5 gives

= O/ss) P (s / ss )

= O/ss) P (d / ss ) + P4(L = 1/ss) P (d / ss )

= 1/sd) P (d / sd )

P4(L = 4 /dd) = P3(L

Enumeration of critical

Ps(L = O/ss ) = P4(L

Ps(L = 1 /ds) = P , ( L

+ P4(L

(6)

188

Ps(L = 1/ss) = P4(L = 1 / sd) P ( s / s d ) + P4(L = 1/sd) P ( s / s d )

Ps(L = 1 / sd) = P4(L = l i d s ) P (s /ds)

Ps(L = 2 /dd) = P4(L = l i d s ) P ( d / d s ) + P4(L = 2/ds) P ( d / d s )

P~(L = 2 /ds ) = P4(L = 2/ss) P ( d / s s ) + P4(L = 2/sd) P (d / sd)

P~(L = 2/ss) = P4(L = 2/ss) P ( s / s s ) + P4(L = 2/sd) P ( s / s d )

P~(L = 2/sd) = P4(L = 2/ds) P ( d / d s ) + P4(L = 2 / d d ) P ( d / d d )

Ps(L = 3/dd) = P4(L = 2/ds) P ( d / d s )

P~(L = 3/ds) = P4(L = 3/sd) P ( d / s d )

Ps(L = 3/ss) = P4(L = 3/sd) P ( s / s d )

Ps(L = 3 /sd) = P4(L = 3/dd) P ( s / d s )

Ps(L = 4 /dd) = P4(L = 3 /dd) P (d /dd )

Ps(L = 4 /sd) = P4(L = 4 /dd) P ( s / d d )

P~(L = 5 /dd) = P4(L = 4/dd) P (d /dd )

and , f i na l ly , for i = 6 t h e p r o b a b i l i t i e s a r e

P6(L = O/ss) = Ps(L = O/ss) P ( s / s s )

P6(L = l i d s ) = Ps(L = O/ss) P ( d / s s ) + Ps(L = 1/sd) P ( d / s d )

+ Ps(L = 1 / s d ) P ( d / s d )

P~(L = l i d s ) = Ps(L = 1/sd) P ( s / s d ) + Ps(L = 1/sd) P ( s / s d )

P6(L = 1/sd) = Ps(L = 1/sd) P ( s / d s )

P6(L = 2 /dd) = Ps(L = l i d s ) P ( d / d s ) + Ps(L = 2 /ds) P ( d / d s )

P6(L = 2 /ds) = P~(L = 2/ss) P ( d / s s ) + Ps(L = 2 /sd) P ( d / s d )

P6(L = 2/ss) = Ps(L = 2/ss) P ( s / s s ) + Ps(L = 2 /sd) P ( s / s d )

P6(L = 2 /sd) = P~(L = 2/ds) P (d /ds ) + Ps(L = 2/ss) P ( d / s s )

P6(L = 3 /dd) = P~(L = 2/ds) P ( d / d s ) + Ps(L = 3 /ds) P ( d / d s )

PG(L = 3/ds) = P~(L = 3/ss) P ( d / s s ) + Ps(L = 3 /sd) P ( d / s d )

P6(L = 3/ss) = Ps(L = 3/ss) P ( s / s s ) + P~(L = 3 / s d ) P ( s / s d )

P6(L = 3 /sd) = Ps(L = 3/ds) P ( s / d s ) + P~(L = 3 /dd) P ( s / d d )

P6(L = 4 /dd) = Ps(L = 3 /ds) P ( d / d s )

P6(L = 4 /ds ) = Ps(L = 4 /sd) P ( d / s d )

P6(L = 4/ss) = P~(L = 4/sd) P ( s / s d )

Z. SEN

(7)

(8)

CRITICAL DROUGHT ANALYSIS

P6(L = 4 / sd ) = P~(L = 4 /dd) P ( s / d d )

P6(L = 5 /dd ) = Ps(L = 4[dd) P (d /dd )

P6(L = 5]sd) = P~(L = 5]dd) P ( s / d d )

P6(L = 5 /dd) = Ps(L = 5 /dd) P (d /dd )

189

A close inspection of eqns. (5)-(8) reveals the following pattern in the enumeration of the critical drought duration probabilities:

(1) the zero critical drought probabilities have one probability statement only, irrespective of the sample size;

(2) there are three probability statements for the critical drought to be equal to one in every sample length;

(3) two probability statements exist for critical drought durations equal to the sample size minus one;

(4) only one probability statement exists for critical drought durations equal to the sample size;

(5) all of the other critical drought durations have four probability statements. Hence, with these points in mind, the general probability statements for sample sizes n can be writ ten by mathematical induction method a s

P~L = O/ss) = P . _ , ( L

P~(L = l i d s ) = Pn- I (L

P . ( L = l i d s ) = Pn_I(L

Pn(L = 1/sd) = P . _ I ( L

P . ( L = j / d d ) = P . _ I ( L

P=(L = j / d s ) = Pn_I(L

Pn(L = j / s s ) = P~_I(L

P~(L = j / s d ) = P~_I(L

Pn(L = n - 2 / d d ) = P=_,(L

P~(L = n - 2 / d s ) = P n - I ( L

P . ( L = n - 2 / s s ) = P n - , ( L

Pn(L = n - 2 / s d ) = P~_I(L

Pn(L = n - 1 / d d ) = P~_I(L

P=(L = n - 1 / s d ) = P . _ I ( L

= O/ss) P (s /ss)

= O/ss) P (d /ss ) + Pn_I(L -~ 1]sd) P (d / sd )

+ Pn_I(L = l / s d ) P (d / sd )

= 1/sd) P (s]sd) + P~_, (L = 1]sd) P (s /sd)

= 1]sd) P (s / sd)

= j - l i d s ) P (d /ds ) + P~- I (L = j ]ds ) P (d]ds)

= j / s s ) P (d]ss) + P~_I(L = j / s d ) P (d / sd )

= j / s s ) P (s /ss) + P~- I (L = j / s d ) P (s / sd)

= j / d s ) P (d /ds ) + P . _ , ( L = j / s s ) P ( s / s s )

(where 2 < j < n - 2)

= n - 3) /ds) P (d /ds )

= n - 2 /dd) P (d / sd )

= n - 2]sd) P ( s / sd)

= n - 2]ds) P (s]ds)

= n - 2]dd) P (d /dd )

= n - 1]dd) P (s]dd)

(9)

190 z. SEN

P~(L = n / d d ) = Pn , (L = n - 1 / d d ) P ( d / d d )

These genera l equa t ions are valid for n > 3. The marg ina l probabi l i ty of any cr i t ical d rough t du ra t ion can be found as the enumera t ion of all the r e l evan t probabi l i ty s ta tements for this dura t ion in eqn. (9). In general , these margina l probabil i t ies are

P, (L - O ) = Pn(L = O/ss)

P . ( L = 1) = Pn(L = 1/ds) + Pn(L = 1/ss) + P~(L = 1/sd)

P . (L - j ) = Pn(L = j / d d ) + Pn(L = rids) + Pn(L - j / d s ) (10)

+ Pn(L = j / sd )

Pn(L = n - l ) = Pn(L = n - 1 / d d ) + Pn(L = n - 1 / s d )

P~(L - n) = Pn(L = n /ad )

As a check of the enumera t ion procedure , the summat ion of these marg ina l probabi l i t ies should be equal to one for any sample size. It is s ignif icant to note at this s tage tha t all of the a forement ioned probabi l i ty der iva t ions are val id for s t a t iona ry second-order Markov processes only, i r respect ive of the under ly ing probabi l i ty d is t r ibut ion func t ion of the hydrologic var iab le concerned.

NUMERICAL SOLUTION

The PDF derived in eqn. (9) can be solved numerically on digital computers for any truncation ]eve] corresponding to any set of conditional probabilities and sample size. The results are presented for a truncation level equal to the median of the underlying PDF of the hydrologic variable, i.e. forp = q -- 0.5. To assess the effect of the transitional probabilities of the second-order Markov process only, the first-order probabilities defined in eqn. (2) are also kept constant as 0.5. During the numerical calculation at each step the summation of the critical drought probabilities is checked against unity as follows:

Pn(L = i) = I (11) i 0

Figures 2-9 show the PDF of the critical drought duration for different sample sizes and transitional probabilities. These figures reveal the following signifi- cant points.

(I) All of the critical drought PDF are invariably positively skewed. However, the skewness decreases with increasing sample size.

(2) Figures 2-6 show the effect of the positive persistence parameters, P (s/ss) on the critical drought PDF. It is obvious that an increase in this parameter reduces the maximum frequency value at the expense of increasing the possibility of longer drought durations.

(3) Figures 7-9 indicate that the effects of other transitional probabilities are not as strong as P (s/ss).


.D 0

n

0.6

0.4

0.3

0.2

0.1

I I

I I

p = 0 .50

P(s lss) =0.20

0 / ' I I

0 10 5

Fig. 2. Critical drought duration distribution for different sample sizes at P (s/ss) = 0.20.

The expecta t ion , E(L), o f the cr i t ical drought durat ions and the ir var iance , V(L), are ca l cu la ted from the a f o r e m e n t i o n e d probabi l i ty s t a t e m e n t s as

E (L) = ~ iPn (L = i) (12) i=0

and

192

0.~

Z. SEN

X:I 0

D._

0.3

0.2

0.1

p=0.50

P(slss) =O.hO

0 5 10 15

Fig. 3. Cr i t i ca l d r o u g h t d u r a t i o n d i s t r i b u t i o n for d i f ferent sample sizes a t P (s/ss) := 0.40.

n V (L) - ~ i2p, (L = i) - E2(L) (13)

i = 0

respectively. The expectations of the critical drought duration are given in Figs. 10-12 for different transitional probabilities. Comparison of these figures shows that for the persistence P(s/ss) variations are ve ry sensitive in the expectation calculations especially at transitional probability values higher than 0.6, It is clear also that P ( d / d d ) is the l eas t sensitive to expectation variations. However, contrary to the effect observed for P(s/ss) increases in P(d /dd) cause decreases in the expected critical drought duration. The transitional probability P (s/ds) has a similar effect on the expected critical drought duration but to a lesser extent. The change of critical drought duration with


0.3 i i I

p = 0.50

0.2 I / ~ P(~ i~)=o.5o

° o ' " ; ~ ~ -~s ~ i

Fig. 4. Critical drought durat ion distr ibution for different sample sizes at P ( s / s s ) = 0.50.

02 P = 0.50

t "/ ~ / \ \

0 5 10 15

Fig. 5. Critical drought durat ion distr ibut ion for different sample sizes at P ( s / s s ) = 0.60.

194

0.2 i I l I i i

p :0 .50 ! {

>. P ( s l s s ) = 0 8 0

~ 0.1

a_

0 ~ I 0 5 10 15

J

Fig. 6. Critical drought durat ion distr ibution for different sample sizes at P ( s / s s ) = 0.80.

Z. SEN

0 . 4 -

Q3

25 ..El

o o.2 13_

0.1

p =0.50

P(sl d s ) =0 .20

I 0 5 10 15

J Fig. 7. Crit ical drought duration distribution for different sample sizes at P (s/ds) = 0.20.


04

03

c~

02 0..

t t i

01

p =0.50

P(slds) =0.40

0 5 10 15

J

195

Fig. 8. Critical drought duration for different sample sizes at P (s/ds) = 0.40.

sample length and the transitional probabilities are given in Figs. 13-15. In general, the transitional probabilities show a similar effect on the variance and expectation of critical drought.

APPLICATION

The methodology developed herein is applied to three annual stream flow records from various parts of the world as shown in Table 1. The characteristics of these records were given by Yevjevich (1963) in modular coefficient form, that is, as ratios of annual flows to the mean annual flow, and therefore, they are dimensionless.

Transitional probabilities of occurrences of surplus and deficit within each sequence on a t runcat ion level equal to the mean annual flow are given in Table 2. These elemental probability values are substituted into eqn. (9) and then eqn. (10) to find the critical drought probabilities, which are finally used

I I I

0.3 ~-

>.

i - i

o 0.2 a_

0.1 -

I I . . . . . . . . I

p =0.50

P ( s l d s ) : 0.60

0 0 5 I0 15

)

Fig. 9. Critical drought duration for different sample sizes at P ( s / d s ) = 0.60.

z. SEN

TABLE 1

Characteristics of rivers

River Location Sample size Mean discharge (years) (cfs)

Rhine Basle, Switzerland 150 36253 Danube Orshava, Romania 120 189455 Mississippi St. Louis, U.S.A. 96 175119

in e v a l u a t i n g the expec ted cr i t ica l d r o u g h t du ra t ions f rom eqn.(12). On the o the r hand, the a v e r a g e h i s to r ica l d rough t du ra t i on for al l possible sample sizes (5, 10, 15, 20 and 25) t h a t can be d r awn f rom the a v a i l a b l e records a re ca lcula ted . F igu re l6 shows bo th the h i s t o r i c a l and expe r imen ta l c r i t ica l


18

13

8 ILl

00

I ! I I ~

p= 0.50

I I I I 10 20 30 4 0 50

Samp le length

Fig. 10. Expectation of the critical drought duration for different sample sizes and P (s/ss).

I I I I I

p=050

P(dldd ) =020 ~. 2 ° ._=

0.8°\ \ \

5

0 I = = I = 10 20 30 & 0 5O

Fig. 11. Expectation of the critical drought duration for different sample sizes and P (d/dd).

198

1 5 I ! I i

!

l p : 0.50 i lO ~ . . . . . . - J (

/ - . . . .

0 V t M J L _J o 1o 20 30 40 50

Sample length

Fig. 12. Expectation of critical drought duration for different sample sizes and P (s/ds).

Z. S E N

3° l I 1 I / I /

/ /

p=0.50 ,~9~///,//

20 ~ "~

/

o i 0 10 20 30

Sample length

o/~0 I ~ - - . . . . . . . . . . . . . . . .

02.._gO I

40 50

Fig. 13. Variance of the critical drought duration for different sample sizes and P (s/ss).


1 0 -

I I 1 I

p=0.50

.J

P(dld d)= 0.20~

o~o\ ° 1 6 ° \ \ ~ _

I

,o 2o 3o :o ~o S a m p l e l e n g t h

Fig. 14. Variance of the critical drought duration for different sample sizes and P (d/dd).

15

10 -

.J v

I I I I

p=050

P(s( ds)=0.80~ o~o~

~ 1 i i i 10 2o 30 40 50

S a m p | e [ e n g t h

Fig. 15. Variance of the critical drought duration for different sample sizes and P (s/ds).

200

5

0 0

j -

i l I I I .1

lO

Mississippi I _ I 1 i I I

2 0 L

3O

Z, SEN

~ ' I 1 I I I I

I I J

Oanube l I I I

0 10 2 0 L

30

J

5 I - t I i I ~ ~ ~ '~ ~ k L.........~-~ i .

0 L/. I I I I I I I I I I I l 1 . ,.

0 l0 20 30

Fig. 16. Observed and experimental critical drought durations for Rhine, Danube and Mississippi rivers.

drought durations for the three rivers. The agreements seem to be very good, at least for small subsample sizes. In these figures the longest drought durations are obtained from historical records, with a modular t runcat ion level of one. A common feature in all t h r ee rivers is tha t there is an overestimation of E(L) when L > 20. This is mainly because of sampling error, which gives rise to biases in the longest drought durat ion estimations from historic sequences of finite length.

CONCLUSIONS

The critical drought durations in small samples have been analytically investigated on the basis of second-order Markov process and the enumerat ion


TABLE 2

Observed t r a n s i t i o n a l p r o b a b i l i t i e s

201

P r e c e d i n g yea r s

i - 2 i - 1

C u r r e n t y e a r

i

Rhine; Basle, Switzerland s d

0.49 0.51

s 0.25 0.24

d 0.23 0.28 s s 0.10 0.13

d s 0.14 0.11

s d 0.12 0.12

d d 0.12 0.17

Danube; Orshava, Romania 0.48 0.52

s 0.21 0.27 d 0.25 0.27

s s 0.09 0.14

d s 0.11 0.15 s d 0.13 0.12 d d 0.12 0.14

Mississippi; St. Louis, U.S.A. 0.45 0.55

s 0.25 0.21 d 0.20 0.34

s s 0.16 0.10

d s 0.10 0.11 s d 0.06 0.15

d d 0.14 0.19

technique. The PDFs of critical drought durations are expressed in terms of the elemental transitional probabilities as combinations of the present and two preceding time instants. The critical drought PDFs are expressed in terms of the conditional probabilities, which are easily converted to the marginal probabilities as in eqn. (10). An application of the methodology developed herein has been shown for observed annual flow series. The following important conclusions can be drawn from this study.

(1) The longest drought duration probability is dependent on the sample size, t runcation level and the transit ional probabilities, but it is independent of the PDF of the hydrologic phenomenon. The transitional probabilities reflect the dependence structure of the phenomenon concerned.

(2) Provided that the sample size, t runcation level and corresponding transitional probabilities are known, one can easily evaluate the risk or probability of critical drought occurrence from the analytical formulation given in this paper.

202 z. SEN

(3) The a n a l y t i c a l f o r m u l a t i o n s m a k e i t poss ib le to f ind the p r o b a b i l i s t i c a n d s t o c h a s t i c p r o p e r t i e s of c r i t i c a l d r o u g h t d u r a t i o n s w i t h o u t r e s o r t i n g to M o n t e Car lo s i m u l a t i o n t e c h n i q u e s , w h i c h a re c u m b e r s o m e a n d expens ive .

REFERENCES

Chin, E.H.A., 1976. A second order Markov chain model for daily rainfall occurrences. Presented at Conf. on Hydromet., Am. Meteorol. Soc., Fort Worth, TX, April 20-22.

Feller, W., 1968. An Introduction to Probability Theory and its Applications, Vol. 1. John Wiley, New York.

Russel, H.C., 1896. On Periodicity of Good and Bad Seasons. Royal Society of New South Wales. Salazar, P.G. and Yevjevich, V., 1975. Analysis of drought characteristics by the theory of runs.

Hydrology Paper No. 80, Colorado State University, Fort Collins, CO. Sen, Z., 1977. Autorun analysis of hydrologic time series. J. Hydrol., 36: 75-88. Sen, Z., 1980a. The numerical calculation of extreme wet and dry periods in hydrologic time series.

Hydrol. Sci. Bull., 21(2): 135-142. Sen, Z., 1980b. Critical drought analysis of periodic stochastic processes. J. Hydrol., 46: 251-263. Sen, Z., 1980c. Statistical analysis of hydrologic critical droughts. J. Hydraul. Div., ASCE, 106

(HY1): 99--115. Sen, Z., 1990. The theory of runs with applications to drought prediction A Comment. J. Hydrol. Yevjevich, V., 1963. Fluctuations of wet and dry years. Hydrology Paper No. I, Colorado State

University, Fort Collins, CO, 110: 383-391. Yevjevich, V., 1967. An objective approach to definitions and investigations of continental

hydrologic droughts. Hydrology Paper No. 23, Colorado State University, Fort Collins, CO. Yule, G:U., 1923. On the time correlation problem with special reference to the incorrect variate

difference correlation method (with discussion). J. R. Statist. Soc., 84:497-537.

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