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Critical drought analysis of periodic-stochastic processes
Journal of Hydrology, 46 (1980) 251--263
251
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[3] CRITICAL D R O U G H T ANALYSIS OF PERIODIC--STOCHASTIC PROCESSES
ZEKAI SEN*
Civil Engineering Faculty, Department of Hydraulics and Water Power, Istanbul Technical University, Istanbul (Turkey) (Received August 1, 1979; accepted for publication October 1, 1979)
ABSTRACT Sen, Z., 1980. Critical drought analysis of periodic--stochastic processes. J. Hydrol., 46: 251--263. A general method is presented for finding the exact probability distribution function of the longest drought duration in a finite sample of any process on the basis of runs theory and enumeration technique. The methodology developed is applied to some hypothetical and observed monthly river flow sequences at various parts of Turkey.
INTRODUCTION
In general drought is related to deficit or shortage of some kind. For instance, in water resources system design and operation the relative positions of demand pattern and available water play an important role in drought definition. The definition adopted in this paper has been suggested by Yevjevich (1967) and is based on runs. A good aspect of such an approach is that the definitions of run-lengths, run-sums and run-intensities are clear and they can be examined analytically. For convenience a continuous time series, X ( t ) as available water sequence and a functional demand level, Xo(t) are shown in Fig. 1. The truncation of the time series at the Xo(t) level forms positive and negative deviations. Hence, the sequence of consecutive negative deviations preceded and succeeded by positive deviations is called a negative run which is an objective representative of the drought duration. Negative run-length and drought duration are interchangeably used throughout the paper. Depending of the circumstances it may be important that the truncation level is either a constant or a deterministic or stochastic variable. Generally, the available water sequence is of stochastic nature with stationary or nonstationary properties. However, *Present address: Institute of Applied Geology, King Abdulaziz University, P.O. Box 1744, Jeddah, Kingdom of Saudi Arabia.
252 x(t) l x(t)
Xo(~) . . . . .
Xo(t)
o o
time
~t
Fig. 1. Available water sequence and functional demand level. only the nonstationary periodic--stochastic processes will be considered in this paper. The drought analysis of periodic stochastic processes is much more complex than the same analysis for stationary stochastic processes. This is because of the fact that, in the former case the time position becomes one of the important factors in evaluating drought characteristics. However, drought investigations of periodic--stochastic processes are essential in finding fluctuations of water deficit within a year, especially for a growing season. Once the analytical expressions for drought characteristics of periodic-stochastic processes are derived then they can be easily reduced to the case of stationary and/or independent processes. Among the several droughts the critical one, i.e. the maximum, is of special interest since it will create the most severe conditions under which the system will operate. Throughout this paper the critical drought will mean the longest drought duration to be observed during the economic life of the project concerned. PREVIOUS ANALYTICAL STUDIES In the past, the analytical investigation of runs has been achieved only for independent and some simple dependent cases for which more often the number of runs, and the run-lengths were considered. A summary of the present state of art is given by Salazar and Yevjevich (1975) who also studied distributions of the longest run-length in both univariate and bivariate independent as well as dependent processes. The expression which they have obtained is extremely complicated and only valid for stationary processes. Although Tase (1976) has studied periodic--stochastic processes for their drought behaviours no explicit formulation has been given for the probability distribution of the longest drought duration. However, the first study on the longest run-length in a sample of size n for independent series is due to Feller (1968) w h o gave an approximate formula on the basis of recurrence events as:
(y-1)(1fn(L) =
qy)
(L + l ) - L y p
1 yn~,"
(1)
253 with y = 1 + p q L + ( L + l ) ( p q L ) 2 + ( L + l ) 2 ( p q L ) 3 + ...
(2)
where L represents the longest run-length. Millan and Yevjevich (1971) found experimentally the longest run-length distributions by using Monte Carlo techniques. On the other hand, David and Barton (1962) presented an analytical expression for the probability of the longest run of successes in n independent Bernoulli trials conditioned on the occurrence of say rn successes. Later Millan (1972) succeeded to find the exact probability of the longest negative run-length in small samples for simple dependence on the basis of dependent Bernoulli trials and combinatorial approach. His result is also very complicated and only valid for stationary Markov chains. Although he obtained some clues about the numerical probability values for the truncation levels in the form of trends he gave no explicit formulation of this situation. PERIODIC AND DEPENDENT BERNOULLI TRIALS The exact probability distribution function (PDF) of the longest drought duration in periodic--stochastic processes can best be obtained through dependent Bernoulli trials and enumeration technique. To the best of the author's knowledge, the PDF of the longest negative run-length in nonstationary processes is n o t available in the past literature. In the following text this PDF is f o u n d exactly for the case in which the result of each trial is dependent only on the outcome of the previous trial. If the process is periodic then the dependent Bernoulli trials will also be periodic, constituting a nonhomogeneous Markov chain with two states whose transition probabilities, Pi, between any two consecutive time instants i - 1 and i can be written in a matrix form as: +
+ [Pi(+[+) Pi = _ LP/(+[-)
Pi(-[+) ] Pi(-I-)
(3)
In eq. 3 the two states are denoted by + and - corresponding to surplus (success) and deficit (failure), respectively. The elements in the transition matrix are the conditional probabilities of passing from one state to the following state. The sum of probabilities in each row is equal to one, therefore: P/(+l+) + Pi(-[+) = 1
and
Pi(+[-) + P i ( - [ - ) = 1
(4)
If the periodicity of the process considered extends over k time units then i = 1, 2 . . . . . k and there exist k transition matrices, defined similar to eq. 3, which are P2, P3, ..., Pk a n d Pk +1. Here, Pi (i = 2, 3, ..., k ) represent transition probabilities within a cycle whereas Pk+l is the transition probability matrix between two successive cycles. The conditional probabilities in eq. 3 can be obtained either theoretically from bivariate probability distribution of two consecutive variables or empirically from a data file giving the truncation
254 levels. They depend on the autocorrelation coefficient, p, the truncation level and the t y p e of underlying distribution. In particular, the bivariate normal probability distribution is published b y the N.B.S. (1959). For periodic-stochastic processes, even if the truncation level is constant, the four conditional probabilities in eq. 3 show a periodic pattern. In the case of stationary stochastic processes with periodic truncation levels drought durations are also periodic. Hence, it is possible to conclude that the conditional probabilities do reflect n o t only the properties of the stochastic process concerned b u t also the behaviour of the truncation level. If there are jumps and/or trends either in the truncation level or in the stochastic process they can be easily accounted by appropriate definitions of the conditional probabilities. LONGEST DROUGHT DURATION Millan (1972) has employed stationary transition probabilities of an irreducible Markov chain with t w o ergodic states to derive the probability of the longest drought duration. However, the m e t h o d o l o g y presented herein bases on a direct application of the enumeration technique with the considerations of: (1) that in each sequence there exists one and only one longest negative run-length; (2) the longest negative run-length probability in sample size say, i can be related to the similar probability in the sample of size ( i - 1 ) leading to a recurrence type of relationship; and (3) in transition from one sample of ( i - 1 ) dependent Bernoulli trials to the following sample only the two state probabilities of the final stage in the previous sample are necessary in addition to the conditional probabilities in eq. 3. In the following derivations even the runs with elements at the first and/or final locations of the sample are taken into consideration. In this way Feller's (1968) definition of runs has been adopted. The derived probability of the longest positive run-length can be easily converted to the probability of the longest negative run-length, provided that the probability sets Pi (+i+), Pi (+l-), Pi (-I+), Pi ( - I - ) are replaced by Pi (-I-), Pi i-I+), Pi (+i-), Pi (+1 +) in the given order. This is equivalent to interchanges of + and - signs in expressions. The probability of the longest positive run-length, L, to be equal to an integer j-value in a sample size of i with + state at the final stage will be denoted by P~ (L =] ~. At the initial state there exists no transition. The state probabilities of the initial stage will be denoted by P, (+) and P ( - ) for the + and - stages, respectively. It is obvious that P~ (+) + P~ (-) = 1.0. After all of these definitions the enumeration technique can be applied step by step as follows. For sample size i=l it is straightforward to write:
P[{L=O} = P,(-)
and
P~(L=I} = Pl(+)
(5)
In the case of i = 2 there are 22 combinations of surpluses and deficits which can be written by transition from i=l to i = 2. For the purpose of finding maximum run-length this transition can be decomposed into the following
255 probabilities:
P;(L=O} = P~(L=O}.P2(-I-) P;{L=I } = P-~{L=O} .P2(+l-) P~{L=I} = P~{L=I}.P2(-I+) p•{L=2}
(6)
= P:{L=l}.p2(+l+)
In the same way the transition from i=2 to i=3 leads to:
P~{L=O} = P~{n=o}'P3(-I-) p;{n=l}
= p~ { n = o } . p 3 ( + l - ) + P ; ( L = I } "P~(+l-)
p ; { L = I } = p ; { L = l } . p 3 ( - I + ) + P~{L=I} "P3(-I-) P;{L=2} = P~{L=2}.P3(+I+)
(7)
P;{L=2} = P~{L=2}.P3(-I+) P~(L=3} = P'~{L=2}.P3(+I+) and finally for i = 4 similar type of relationships can be obtained as:
PT, {L=O} = P~ {L=O}'P4(-I-) p~{L=I} = p~{L=O}.p4(+I - ) + P~{L=I}.P4(+I+) p~{L=I} = p~{L=I}.p4(-I+) + P ~ ( L = I } . P 4 ( - I - ) p~{L=2} = p~{n=l}.p4(+l+) + P~{L=2} .P4(+I-) P[, {L=2} = P~ {L=2}.P4(-[+) + P~{L=2}'P4(-I-)
(8)
P~,{L=3} = P;{L=2}.P4(+I+) P~{L=3} = P~{L=3}.P4(-I+) P~{n=4} = P~{L=3}.P4(+l+) If desired it is possible to continue the enumeration, but for the sake of brevity it has been avoided herein. However, a close inspection through eqs. 5--8 reveals some features of the general pattern of equations. For instance, the probability of no run-length in a sample of size n can be easily detected from the first expressions in eqs. 5--8. Hence, in general:
Pn{L=O} = Pn_1{L=O}Pn(-[-)
(9)
By inspection the probability of the full run-length can be obtained from the final expressions in eqs. 5--8 as: P},{L=n} = Pn-, {L=n-1}Pn(+l+)
(10)
All other probabilities in the interval 1 ~ j~< n - 1 have two components one for + final state and one for the - state. In fact:
Pn{L=j} = Pn{L=j} + Pn{L=j}
256 with the explicit expressions of components as,
k, j-1 Pn{L=j} = ~ Pn_j(L=m}Pn_j+l(+l -) 11 Pn_j+k+l(+l+) m=O k=l ks
m-1
+ ~ Pn_m{L=j}Pn-m÷,(+l -) II Pn_m+k÷~(+l+) m=l k=l
(11)
and
Pn{L=j} = Pn-1 {L=j}Pn(-]+) + Pn-1 {L=j}Pn(-J-), if n - 1 = j then P ( - i - ) -- 0
(12)
where k, = m i n ( n - j - l , j ) and k2 = m m ( n - l - l , l - 1 ) . Hence, eqs. 9--12 give the full description of the longest positive run-length in n dependent Bernoulli trials. For the longest negative run-length, N, probability with the aforementioned sign interchanges one obtains: +
+
Pn{N =0} = Pn-~{N=O} Pn(+J+) k,
j-I
Pn(N=j} = ~
Pn_j{N=m} Pn_j+,(-I+) H Pn-j+k+,(-J-)
m=0
k=l
k2
rn - 1
+ ~_j Pn_m{N=j}Pn_m+i(-J+) H Pn-m+k+l(-J-) m=l k=l
(13)
Pn{N=J} = Pn-, {N=j} Pn(+l-) + Pn-1 {N=J} Pn(+l+), if n - 1 = j then Pn(+l+) = 0
Pn{N=n} = Pn-1 {N=n-1} Pn(-i-) Eqs. 13 are exact probabilities for nonhomogeneous two-state Markov chains. Of course, the sum of probabilities Pn{N = 0}, Pn(Y=l }, Pn{N=I } , . . . , Pn{N =n } is equal to one for any n-value. The results obtained so far are exact for the case in which the Markov property applies, namely, that the probability of a surplus or deficit is dependent only on the outcome of the previous event. However, it is important to point out at this stage that the results of the Markov chain can be applied fairly well to some other time dependences, especially when the observations come out of Markovian processes. In such cases, periodic-stochastic processes can be modelled by employing the now conventional Thomas and Fiering model. The expected value of the longest drought duration, E(N) can be calculated
257
numerically by: n
E(N) = ~ iPn{N=i }
(14)
i=1 APPLICATION TO MONTHLY FLOW SEQUENCES
The applications of exact formulas derived in the previous section can be achieved provided that 2k +1 probability values are given or estimated from an available observed sequence of any periodic---stochastic process. Among these probabilities there are one initial state probability P~ (+) or P~ (-) and any of the n o n c o m p l e m e n t a r y transition probabilities out of the total 4k transition probabilities. Given 2k transition probabilities the rest can be evaluated simply from eq. 4. In the case of m o n t h l y flows 24 transition probabilities are necessary in addition to the initial state probability. On the other hand, the transition probabilities are directly related to the persistence and/or periodic properties of the underlying process concerned. It is well known that in persistent processes high flows follow high flows and low flows are followed by low flows as a result of which the conditional probabilities Pi(+I+) and P i ( - I - ) are expected to be relatively higher than the crossing probabilities Pi(-i+) and Pi(+i-). A similar statement is valid also for periodic but independent processes. Two sets of applications are deviced herein. In the first set only the hypothetical initial and conditional probabilities of periodic but independent processes are considered as given in Table I. In all these cases the truncation levels are assumed to be at the median value which means t h a t the initial probabilities are equal to 0.5. Hence, by considering the process, X t , to have either +1 or - 1 deviations from the truncation level one can easily note t h a t the mean, variance and first-order autocorrelation coefficient are E ( X t ) = 0, V ( X t ) = 1, and: 12
p = ~
[Pi(+i+)Pi(+) -Pi(-i+)Pi(+) - P i ( + l - ) P i ( - ) + P i ( - I - ) P i ( - ) ]
(15)
i=l
respectively. Applications of eq. 15 to the four hypothetical cases is sufficient to prove t h a t the processes considered are independent. Due to the differences in transition probabilities within a basic cycle the processes are only periodic. In the first hypothetical case (case 1 ) the transition probabilities Pi(+i+) increase steadily from a value of 0.1 up to 0.9 and then decrease at the same rate to 0.1 whereas P/{-I-)'s have just the opposite pattern. The interpretations of these patterns in terms of periodicity is t h a t within the cycle of 12 months the surplus-to-surplus transitions between two successive m o n t h s increase and then decrease to the same value. Practically speaking, within a year there is a steady transition from water-poor to water-rich m o n t h s again
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.
1 2 3 4 5 6 7 8 9 10 11 12
Averages
month
(i)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
0.430
0.100 0.200 0.400 0.600 0.800 0.900 0.800 0.600 0.400 0.200 0.100 0.100 0.570
0.900 0.800 0.600 0.400 0.200 0.100 0.200 0.400 0.600 0.800 0.900 0.900 0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Pi(-I-)
Pi(+)
Pi(+I+)
Case 2
Case 1
H y p o ~ e t i e M probabilities
TABLEI
0.570
0.900 0.800 0.600 0.400 0.200 0.100 0.200 0.400 0.600 0.800 0.900 0.900
Pi(+I +)
0.430
0.100 0.200 0.400 0.600 0.800 0.900 0.800 0.600 0.400 0.200 0.100 0.100
Pi(-I-)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Case 3
0.500
0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900
Pi(+I +)
0.500
0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100
Pi(-I-)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Case 4
0.500
0.900 0.900 0.900 0.900 0.900 0.900 0.100 0.100 0.100 0.100 0.100 0.100
Pi(+I+)
0.500
0.100 0.100 0.100 0.100 0.100 0.100 0.900 0.900 0.900 0.900 0.900 0.900
Pi(-I-)
O0
b~
259
to water-poor months. The mean longest drought duration of this situation has been evaluated by using eq. 14 and plotted vs. the sample length in Fig. 2. It is evident that the periodic structure in the conditional probabilities is reflected in the expected value of the longest drought duration. In small sample sizes the periodicity of droughts is much more accentuated than the longer samples. As the sample size increases the periodicity damps out and approaches to an asymptotic value. A different periodic pattern of transitions has been considered in case 2 where this time the transition probabilities form a deficit-to-deficit increase followed by decrease and an opposite pattern for surplus-to-surplus transitions. On the other hand, the average of 12 deficit-to-deficit transition probabilities in case 2 is less than those in case 1 as a result of which one expects more severe longest drought durations in this case as is obvious from Fig. 2. Case 3 includes alternate surplus-to-surplus and deficit-to-deficit transition probabilities w i t h o u t any distinction between surplus and deficit patterns. Both deviations are given the same weight. An alternate truncation level yields such a pattern similar to the stationary processes. In fact case 3 in Fig. 2 reflects a stationary process drought property appearance with no periodicity at all. Finally, in case 4 a jump is considered in the middle of the water year. The effect of such a jump in the expectation of the longest drought duration in Fig. 2 is also periodic almost with the same cycle as in the original transition probabilities. The transition probabilities in this case can be viewed as coming from an alternate truncation level with epoch length equal to six time units. Therefore, the comparison of cases 3 and 4 leads to the conclusion that in Years i
iO
2
l
i
i
I
3
!i
i
4
5
6
i
i
i i
i
i
I
N
/,.
4
..................... ........
~
Io
20
i
'
i I
,
i
/"/............ X t r i 0
7
~
8
i
30
40
50
60
v
r
!L
J
l
l
i ;
I
Ca.se. ~ ..........
70
80
90
L Months
Fig. 2. Hypothetical expected longest drought duration vs. sample length.
260
an alternate truncation level epoch length is important and periodic longest droughts are to be expected when this length increases. The following general conclusions can be drawn from hypothetical considerations that: (1) if the transition probabilities include a periodic pattern so does the expectation of the longest drought (or wet) period; (2) the effect of periodicity in the longest drought duration decreases with increasing sample size; and (3) the order of transition probabilities is much more important than their values. For instance, both in cases 3 and 4 the number of probabilities with values 0.1 and 0.9 are six, b u t in case 4 their order gives rise to a longer drought duration than in case 3. This last statement is equivalent to the epoch effect mentioned above. The second set of applications concentrates on some actual monthly flow values observed in various parts of Turkey as presented in Fig. 3. The properties of four monthly flow measurement stations are furnished in Table II. The actual transition probabilities can be calculated from Pi(+l+)=Pi(+,+)/Pi(+), Pi(-[+ )=Pi(-, +)/Pi( +), Pi( +l-)=Pi( +,-)/Pi(-) and Pi(-I-)=Pi(-,-)/Pi(-) where Pi(+, +), Pi(-, +), Pi(+,-) and P i ( - , - ) are the joint occurrence probabilities of various deficit and/or surplus combinations. For instance, Pi(+, +) shows the BLACK
A4~D,~
.SEA
~r"fi--~(~
&,S"
•
Measurement
stations
Fig. 3. Monthly flow measurement station locations considered in the study.
TABLE II Flow measurement station characteristics River name
Station name
Drainage area (kin 2)
Record length (yr.)
B. Menderes Manavgat Kizilirmak Firat
Cine Manavgat Kizilirmak Keban
948 928 15,582 63,874
33 31 32 34
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.
1 2 3 4 5 6 7 8 9 10 11 12
Averages
month
(i)
Pi(-]-)
0.535
0.364 0.455 0.845 0.485 0.545 0.576 0.576 0.515 0.545 0.515 0.545 0.455
0.795
0.833 0.466 0.687 0.500 0.555 0.842 0.894 1.000 0.889 1.000 0.944 0.933 0.801
0.762 0.722 0.765 0.529 0.600 0.857 0.857 0.875 0.933 0.937 1.000 0.777 0.515
0.415 0.484 0.613 0.484 0.484 0.516 0.451 0.516 0.548 0.548 0.548 0.581 0.823
0.855 0.667 0.737 0.733 0.733 0.750 0.929 0.875 0.882 0.941 1.000 0.777 0.819
0.606 0.813 0.916 0.500 0.750 0.800 0.823 1.000 0.928 0.928 1.000 0.769 0.429
0.375 0.406 0.406 0.312 0.468 0.500 0.500 0.469 0.406 0.406 0.437 0.469
0.725
0.833 0.769 0.846 0.800 0.533 0.500 0.750 0.600 0.847 0.769 0.857 0.600
Pi(+t +)
Pi(+)
Pi(+l +)
Pi(+)
Pi(-I-)
Pi(+)
Pi(+l +)
Kizilirmak
Manavgat
B. Menderes
River n a m e
Observed probabilities
T A B L E III
0.772
0.750 0.840 0.842 0.773 0.882 0.562 0.750 0.588 0.789 0.842 0.944 0.705
Pi(-]-)
0.421
0.441 0.323 0.353 0.323 0.323 0.529 0.470 0.441 0.471 0.471 0.500 0.441
Pi(+)
Firat
0.756
0.933 0.818 0.750 0.636 0.636 0.333 0.875 0.867 0.812 1.000 0.882 0.533
Pi(+[+)
0.822
0.947 0.739 0.909 0.782 0.826 0.687 0.777 0.842 0.889 1.000 0.941 0.526
Pi(-l-)
b.a
b0
262
joint probability of surpluses between months i and i+1 given respective truncation levels. Other joint probabilities have similar interpretations. The estimations Pi(+) and Pi(+]+) of Pi(+ +) cml be achieved through the following steps: (1) The monthly flows of months i and i+1 in each year are truncated at given respective demand levels. (2) The number (nss)i of joint surplus occurrences are counted throughout the observation period. In addition the number (ns) i of surpluses in the ith month is determined. (3) Finally the estimates can be calculated as: /3i(+1+1 = (nss)i/n
/3i(+) = (ns)i/n
and
(16)
where n is the record length in years. The transition probability estimates can then be found as Pi(+[+) = Pi(+,+)/Pi(+). Other estimates can be determined in a similar manner. The actual records are truncated at a periodic level determined by the average m o n t h l y flows. The necessary parameter estimates for the application of eq. 13 are given in Table III. The e m p l o y m e n t of these parameters in eqs. 13 and 14 lead to the mean longest drought durations shown in Fig. 4. Since these actual records are periodic as well as dependent the curves in Fig. 4 do Y~ars
1
2
t
i
,
3 l
4t
,
,
~ ,
i
1&
.
t
6
,
t7
,
.
,
.
t v
i
,
.
i
. , . ,....,-"
....... ' ..............
[
i
-J"
!
:u
-
i : Compuei d
4
2
0 ~'~
/
~
/!// /
.
-
.
.
,:~y //
.
.
1
...............
Kiz=hrrnak
. . . . . .
F,rat !
.
10
.
~.
20
.
.
30
.
.
.
o
i
i '
40
•
~ i
i
50
J
'
60
i I
""
.
.
70
.
.
.
~0
L Months
Fig. 4. Actual expected longest drought duration vs. sample length.
90
263
reflect their joint effects. The longest drought durations are all periodic and the periodicity diminishes with increasing sample length. On the other hand, observed and computed longest drought durations are very close to each other. Under the light of aforementioned procedure the longest drought duration properties of any monthly, weekly, daily, etc., flow sequences observed at any part of the world can be evaluated with no major difficulty. CONCLUSIONS
The PDF of the longest drought duration in periodic--stochastic process have been rigorously derived and applied to some hypothetical periodicindependent as well as actual monthly flows. The following conclusions can be drawn from the study in this paper. (1) The longest drought duration of periodic--stochastic processes can be expressed in terms of marginal and conditional probabilities. (2) The parameters such as the expectation, variance, etc., of the longest drought duration in periodic--stochastic processes are also periodic. (3) Not only the numerical values of transition probabilities but more effectively their order along the time axis play an important role in the longest drought probabilities and parameters. (4) The effect of periodicity on the longest drought duration decreases markedly with increasing sample size.
REFERENCES David, F.N. and Barton, D.E., 1962. Combinatorial Chance. Griffin, London. Feller, W., 1968. An Introduction to Probability Theory and its Application, Vol. 1, Wiley, New York, N.Y. Millan, J., 1972. Use of regional economic models in the evolution of drought impact. Proc. 2nd Int. Syrup. on Hydrology, F o r t Collins, Colo. Millan, J. and Yevjevich, V., 1971. Probabilities of observed droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. Pap. 50. N.B.S. (National Bureau of Standards), 1959. Tables of the bivariate normal distributions and related functions. Appl. Math. Ser. 50, U.S. Government Printing Office, Washington, D.C. Salazar, P.G. and Yevjevich, V., 1975. Analysis of drought characteristics by the theory of runs. Colo. State Univ. F o r t Collins, Colo., Hydrol. Pap. 80. Tase, N., 1976. Area--deficit--intensity characteristics of droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. Pap. 87. Yevjevich, V., 1967. An objective approach to definitions and investigations of continental hydrologic droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. PaP. 23.
251
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[3] CRITICAL D R O U G H T ANALYSIS OF PERIODIC--STOCHASTIC PROCESSES
ZEKAI SEN*
Civil Engineering Faculty, Department of Hydraulics and Water Power, Istanbul Technical University, Istanbul (Turkey) (Received August 1, 1979; accepted for publication October 1, 1979)
ABSTRACT Sen, Z., 1980. Critical drought analysis of periodic--stochastic processes. J. Hydrol., 46: 251--263. A general method is presented for finding the exact probability distribution function of the longest drought duration in a finite sample of any process on the basis of runs theory and enumeration technique. The methodology developed is applied to some hypothetical and observed monthly river flow sequences at various parts of Turkey.
INTRODUCTION
In general drought is related to deficit or shortage of some kind. For instance, in water resources system design and operation the relative positions of demand pattern and available water play an important role in drought definition. The definition adopted in this paper has been suggested by Yevjevich (1967) and is based on runs. A good aspect of such an approach is that the definitions of run-lengths, run-sums and run-intensities are clear and they can be examined analytically. For convenience a continuous time series, X ( t ) as available water sequence and a functional demand level, Xo(t) are shown in Fig. 1. The truncation of the time series at the Xo(t) level forms positive and negative deviations. Hence, the sequence of consecutive negative deviations preceded and succeeded by positive deviations is called a negative run which is an objective representative of the drought duration. Negative run-length and drought duration are interchangeably used throughout the paper. Depending of the circumstances it may be important that the truncation level is either a constant or a deterministic or stochastic variable. Generally, the available water sequence is of stochastic nature with stationary or nonstationary properties. However, *Present address: Institute of Applied Geology, King Abdulaziz University, P.O. Box 1744, Jeddah, Kingdom of Saudi Arabia.
252 x(t) l x(t)
Xo(~) . . . . .
Xo(t)
o o
time
~t
Fig. 1. Available water sequence and functional demand level. only the nonstationary periodic--stochastic processes will be considered in this paper. The drought analysis of periodic stochastic processes is much more complex than the same analysis for stationary stochastic processes. This is because of the fact that, in the former case the time position becomes one of the important factors in evaluating drought characteristics. However, drought investigations of periodic--stochastic processes are essential in finding fluctuations of water deficit within a year, especially for a growing season. Once the analytical expressions for drought characteristics of periodic-stochastic processes are derived then they can be easily reduced to the case of stationary and/or independent processes. Among the several droughts the critical one, i.e. the maximum, is of special interest since it will create the most severe conditions under which the system will operate. Throughout this paper the critical drought will mean the longest drought duration to be observed during the economic life of the project concerned. PREVIOUS ANALYTICAL STUDIES In the past, the analytical investigation of runs has been achieved only for independent and some simple dependent cases for which more often the number of runs, and the run-lengths were considered. A summary of the present state of art is given by Salazar and Yevjevich (1975) who also studied distributions of the longest run-length in both univariate and bivariate independent as well as dependent processes. The expression which they have obtained is extremely complicated and only valid for stationary processes. Although Tase (1976) has studied periodic--stochastic processes for their drought behaviours no explicit formulation has been given for the probability distribution of the longest drought duration. However, the first study on the longest run-length in a sample of size n for independent series is due to Feller (1968) w h o gave an approximate formula on the basis of recurrence events as:
(y-1)(1fn(L) =
qy)
(L + l ) - L y p
1 yn~,"
(1)
253 with y = 1 + p q L + ( L + l ) ( p q L ) 2 + ( L + l ) 2 ( p q L ) 3 + ...
(2)
where L represents the longest run-length. Millan and Yevjevich (1971) found experimentally the longest run-length distributions by using Monte Carlo techniques. On the other hand, David and Barton (1962) presented an analytical expression for the probability of the longest run of successes in n independent Bernoulli trials conditioned on the occurrence of say rn successes. Later Millan (1972) succeeded to find the exact probability of the longest negative run-length in small samples for simple dependence on the basis of dependent Bernoulli trials and combinatorial approach. His result is also very complicated and only valid for stationary Markov chains. Although he obtained some clues about the numerical probability values for the truncation levels in the form of trends he gave no explicit formulation of this situation. PERIODIC AND DEPENDENT BERNOULLI TRIALS The exact probability distribution function (PDF) of the longest drought duration in periodic--stochastic processes can best be obtained through dependent Bernoulli trials and enumeration technique. To the best of the author's knowledge, the PDF of the longest negative run-length in nonstationary processes is n o t available in the past literature. In the following text this PDF is f o u n d exactly for the case in which the result of each trial is dependent only on the outcome of the previous trial. If the process is periodic then the dependent Bernoulli trials will also be periodic, constituting a nonhomogeneous Markov chain with two states whose transition probabilities, Pi, between any two consecutive time instants i - 1 and i can be written in a matrix form as: +
+ [Pi(+[+) Pi = _ LP/(+[-)
Pi(-[+) ] Pi(-I-)
(3)
In eq. 3 the two states are denoted by + and - corresponding to surplus (success) and deficit (failure), respectively. The elements in the transition matrix are the conditional probabilities of passing from one state to the following state. The sum of probabilities in each row is equal to one, therefore: P/(+l+) + Pi(-[+) = 1
and
Pi(+[-) + P i ( - [ - ) = 1
(4)
If the periodicity of the process considered extends over k time units then i = 1, 2 . . . . . k and there exist k transition matrices, defined similar to eq. 3, which are P2, P3, ..., Pk a n d Pk +1. Here, Pi (i = 2, 3, ..., k ) represent transition probabilities within a cycle whereas Pk+l is the transition probability matrix between two successive cycles. The conditional probabilities in eq. 3 can be obtained either theoretically from bivariate probability distribution of two consecutive variables or empirically from a data file giving the truncation
254 levels. They depend on the autocorrelation coefficient, p, the truncation level and the t y p e of underlying distribution. In particular, the bivariate normal probability distribution is published b y the N.B.S. (1959). For periodic-stochastic processes, even if the truncation level is constant, the four conditional probabilities in eq. 3 show a periodic pattern. In the case of stationary stochastic processes with periodic truncation levels drought durations are also periodic. Hence, it is possible to conclude that the conditional probabilities do reflect n o t only the properties of the stochastic process concerned b u t also the behaviour of the truncation level. If there are jumps and/or trends either in the truncation level or in the stochastic process they can be easily accounted by appropriate definitions of the conditional probabilities. LONGEST DROUGHT DURATION Millan (1972) has employed stationary transition probabilities of an irreducible Markov chain with t w o ergodic states to derive the probability of the longest drought duration. However, the m e t h o d o l o g y presented herein bases on a direct application of the enumeration technique with the considerations of: (1) that in each sequence there exists one and only one longest negative run-length; (2) the longest negative run-length probability in sample size say, i can be related to the similar probability in the sample of size ( i - 1 ) leading to a recurrence type of relationship; and (3) in transition from one sample of ( i - 1 ) dependent Bernoulli trials to the following sample only the two state probabilities of the final stage in the previous sample are necessary in addition to the conditional probabilities in eq. 3. In the following derivations even the runs with elements at the first and/or final locations of the sample are taken into consideration. In this way Feller's (1968) definition of runs has been adopted. The derived probability of the longest positive run-length can be easily converted to the probability of the longest negative run-length, provided that the probability sets Pi (+i+), Pi (+l-), Pi (-I+), Pi ( - I - ) are replaced by Pi (-I-), Pi i-I+), Pi (+i-), Pi (+1 +) in the given order. This is equivalent to interchanges of + and - signs in expressions. The probability of the longest positive run-length, L, to be equal to an integer j-value in a sample size of i with + state at the final stage will be denoted by P~ (L =] ~. At the initial state there exists no transition. The state probabilities of the initial stage will be denoted by P, (+) and P ( - ) for the + and - stages, respectively. It is obvious that P~ (+) + P~ (-) = 1.0. After all of these definitions the enumeration technique can be applied step by step as follows. For sample size i=l it is straightforward to write:
P[{L=O} = P,(-)
and
P~(L=I} = Pl(+)
(5)
In the case of i = 2 there are 22 combinations of surpluses and deficits which can be written by transition from i=l to i = 2. For the purpose of finding maximum run-length this transition can be decomposed into the following
255 probabilities:
P;(L=O} = P~(L=O}.P2(-I-) P;{L=I } = P-~{L=O} .P2(+l-) P~{L=I} = P~{L=I}.P2(-I+) p•{L=2}
(6)
= P:{L=l}.p2(+l+)
In the same way the transition from i=2 to i=3 leads to:
P~{L=O} = P~{n=o}'P3(-I-) p;{n=l}
= p~ { n = o } . p 3 ( + l - ) + P ; ( L = I } "P~(+l-)
p ; { L = I } = p ; { L = l } . p 3 ( - I + ) + P~{L=I} "P3(-I-) P;{L=2} = P~{L=2}.P3(+I+)
(7)
P;{L=2} = P~{L=2}.P3(-I+) P~(L=3} = P'~{L=2}.P3(+I+) and finally for i = 4 similar type of relationships can be obtained as:
PT, {L=O} = P~ {L=O}'P4(-I-) p~{L=I} = p~{L=O}.p4(+I - ) + P~{L=I}.P4(+I+) p~{L=I} = p~{L=I}.p4(-I+) + P ~ ( L = I } . P 4 ( - I - ) p~{L=2} = p~{n=l}.p4(+l+) + P~{L=2} .P4(+I-) P[, {L=2} = P~ {L=2}.P4(-[+) + P~{L=2}'P4(-I-)
(8)
P~,{L=3} = P;{L=2}.P4(+I+) P~{L=3} = P~{L=3}.P4(-I+) P~{n=4} = P~{L=3}.P4(+l+) If desired it is possible to continue the enumeration, but for the sake of brevity it has been avoided herein. However, a close inspection through eqs. 5--8 reveals some features of the general pattern of equations. For instance, the probability of no run-length in a sample of size n can be easily detected from the first expressions in eqs. 5--8. Hence, in general:
Pn{L=O} = Pn_1{L=O}Pn(-[-)
(9)
By inspection the probability of the full run-length can be obtained from the final expressions in eqs. 5--8 as: P},{L=n} = Pn-, {L=n-1}Pn(+l+)
(10)
All other probabilities in the interval 1 ~ j~< n - 1 have two components one for + final state and one for the - state. In fact:
Pn{L=j} = Pn{L=j} + Pn{L=j}
256 with the explicit expressions of components as,
k, j-1 Pn{L=j} = ~ Pn_j(L=m}Pn_j+l(+l -) 11 Pn_j+k+l(+l+) m=O k=l ks
m-1
+ ~ Pn_m{L=j}Pn-m÷,(+l -) II Pn_m+k÷~(+l+) m=l k=l
(11)
and
Pn{L=j} = Pn-1 {L=j}Pn(-]+) + Pn-1 {L=j}Pn(-J-), if n - 1 = j then P ( - i - ) -- 0
(12)
where k, = m i n ( n - j - l , j ) and k2 = m m ( n - l - l , l - 1 ) . Hence, eqs. 9--12 give the full description of the longest positive run-length in n dependent Bernoulli trials. For the longest negative run-length, N, probability with the aforementioned sign interchanges one obtains: +
+
Pn{N =0} = Pn-~{N=O} Pn(+J+) k,
j-I
Pn(N=j} = ~
Pn_j{N=m} Pn_j+,(-I+) H Pn-j+k+,(-J-)
m=0
k=l
k2
rn - 1
+ ~_j Pn_m{N=j}Pn_m+i(-J+) H Pn-m+k+l(-J-) m=l k=l
(13)
Pn{N=J} = Pn-, {N=j} Pn(+l-) + Pn-1 {N=J} Pn(+l+), if n - 1 = j then Pn(+l+) = 0
Pn{N=n} = Pn-1 {N=n-1} Pn(-i-) Eqs. 13 are exact probabilities for nonhomogeneous two-state Markov chains. Of course, the sum of probabilities Pn{N = 0}, Pn(Y=l }, Pn{N=I } , . . . , Pn{N =n } is equal to one for any n-value. The results obtained so far are exact for the case in which the Markov property applies, namely, that the probability of a surplus or deficit is dependent only on the outcome of the previous event. However, it is important to point out at this stage that the results of the Markov chain can be applied fairly well to some other time dependences, especially when the observations come out of Markovian processes. In such cases, periodic-stochastic processes can be modelled by employing the now conventional Thomas and Fiering model. The expected value of the longest drought duration, E(N) can be calculated
257
numerically by: n
E(N) = ~ iPn{N=i }
(14)
i=1 APPLICATION TO MONTHLY FLOW SEQUENCES
The applications of exact formulas derived in the previous section can be achieved provided that 2k +1 probability values are given or estimated from an available observed sequence of any periodic---stochastic process. Among these probabilities there are one initial state probability P~ (+) or P~ (-) and any of the n o n c o m p l e m e n t a r y transition probabilities out of the total 4k transition probabilities. Given 2k transition probabilities the rest can be evaluated simply from eq. 4. In the case of m o n t h l y flows 24 transition probabilities are necessary in addition to the initial state probability. On the other hand, the transition probabilities are directly related to the persistence and/or periodic properties of the underlying process concerned. It is well known that in persistent processes high flows follow high flows and low flows are followed by low flows as a result of which the conditional probabilities Pi(+I+) and P i ( - I - ) are expected to be relatively higher than the crossing probabilities Pi(-i+) and Pi(+i-). A similar statement is valid also for periodic but independent processes. Two sets of applications are deviced herein. In the first set only the hypothetical initial and conditional probabilities of periodic but independent processes are considered as given in Table I. In all these cases the truncation levels are assumed to be at the median value which means t h a t the initial probabilities are equal to 0.5. Hence, by considering the process, X t , to have either +1 or - 1 deviations from the truncation level one can easily note t h a t the mean, variance and first-order autocorrelation coefficient are E ( X t ) = 0, V ( X t ) = 1, and: 12
p = ~
[Pi(+i+)Pi(+) -Pi(-i+)Pi(+) - P i ( + l - ) P i ( - ) + P i ( - I - ) P i ( - ) ]
(15)
i=l
respectively. Applications of eq. 15 to the four hypothetical cases is sufficient to prove t h a t the processes considered are independent. Due to the differences in transition probabilities within a basic cycle the processes are only periodic. In the first hypothetical case (case 1 ) the transition probabilities Pi(+i+) increase steadily from a value of 0.1 up to 0.9 and then decrease at the same rate to 0.1 whereas P/{-I-)'s have just the opposite pattern. The interpretations of these patterns in terms of periodicity is t h a t within the cycle of 12 months the surplus-to-surplus transitions between two successive m o n t h s increase and then decrease to the same value. Practically speaking, within a year there is a steady transition from water-poor to water-rich m o n t h s again
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.
1 2 3 4 5 6 7 8 9 10 11 12
Averages
month
(i)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
0.430
0.100 0.200 0.400 0.600 0.800 0.900 0.800 0.600 0.400 0.200 0.100 0.100 0.570
0.900 0.800 0.600 0.400 0.200 0.100 0.200 0.400 0.600 0.800 0.900 0.900 0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Pi(-I-)
Pi(+)
Pi(+I+)
Case 2
Case 1
H y p o ~ e t i e M probabilities
TABLEI
0.570
0.900 0.800 0.600 0.400 0.200 0.100 0.200 0.400 0.600 0.800 0.900 0.900
Pi(+I +)
0.430
0.100 0.200 0.400 0.600 0.800 0.900 0.800 0.600 0.400 0.200 0.100 0.100
Pi(-I-)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Case 3
0.500
0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900
Pi(+I +)
0.500
0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100 0.900 0.100
Pi(-I-)
0.500
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500
Pi(+)
Case 4
0.500
0.900 0.900 0.900 0.900 0.900 0.900 0.100 0.100 0.100 0.100 0.100 0.100
Pi(+I+)
0.500
0.100 0.100 0.100 0.100 0.100 0.100 0.900 0.900 0.900 0.900 0.900 0.900
Pi(-I-)
O0
b~
259
to water-poor months. The mean longest drought duration of this situation has been evaluated by using eq. 14 and plotted vs. the sample length in Fig. 2. It is evident that the periodic structure in the conditional probabilities is reflected in the expected value of the longest drought duration. In small sample sizes the periodicity of droughts is much more accentuated than the longer samples. As the sample size increases the periodicity damps out and approaches to an asymptotic value. A different periodic pattern of transitions has been considered in case 2 where this time the transition probabilities form a deficit-to-deficit increase followed by decrease and an opposite pattern for surplus-to-surplus transitions. On the other hand, the average of 12 deficit-to-deficit transition probabilities in case 2 is less than those in case 1 as a result of which one expects more severe longest drought durations in this case as is obvious from Fig. 2. Case 3 includes alternate surplus-to-surplus and deficit-to-deficit transition probabilities w i t h o u t any distinction between surplus and deficit patterns. Both deviations are given the same weight. An alternate truncation level yields such a pattern similar to the stationary processes. In fact case 3 in Fig. 2 reflects a stationary process drought property appearance with no periodicity at all. Finally, in case 4 a jump is considered in the middle of the water year. The effect of such a jump in the expectation of the longest drought duration in Fig. 2 is also periodic almost with the same cycle as in the original transition probabilities. The transition probabilities in this case can be viewed as coming from an alternate truncation level with epoch length equal to six time units. Therefore, the comparison of cases 3 and 4 leads to the conclusion that in Years i
iO
2
l
i
i
I
3
!i
i
4
5
6
i
i
i i
i
i
I
N
/,.
4
..................... ........
~
Io
20
i
'
i I
,
i
/"/............ X t r i 0
7
~
8
i
30
40
50
60
v
r
!L
J
l
l
i ;
I
Ca.se. ~ ..........
70
80
90
L Months
Fig. 2. Hypothetical expected longest drought duration vs. sample length.
260
an alternate truncation level epoch length is important and periodic longest droughts are to be expected when this length increases. The following general conclusions can be drawn from hypothetical considerations that: (1) if the transition probabilities include a periodic pattern so does the expectation of the longest drought (or wet) period; (2) the effect of periodicity in the longest drought duration decreases with increasing sample size; and (3) the order of transition probabilities is much more important than their values. For instance, both in cases 3 and 4 the number of probabilities with values 0.1 and 0.9 are six, b u t in case 4 their order gives rise to a longer drought duration than in case 3. This last statement is equivalent to the epoch effect mentioned above. The second set of applications concentrates on some actual monthly flow values observed in various parts of Turkey as presented in Fig. 3. The properties of four monthly flow measurement stations are furnished in Table II. The actual transition probabilities can be calculated from Pi(+l+)=Pi(+,+)/Pi(+), Pi(-[+ )=Pi(-, +)/Pi( +), Pi( +l-)=Pi( +,-)/Pi(-) and Pi(-I-)=Pi(-,-)/Pi(-) where Pi(+, +), Pi(-, +), Pi(+,-) and P i ( - , - ) are the joint occurrence probabilities of various deficit and/or surplus combinations. For instance, Pi(+, +) shows the BLACK
A4~D,~
.SEA
~r"fi--~(~
&,S"
•
Measurement
stations
Fig. 3. Monthly flow measurement station locations considered in the study.
TABLE II Flow measurement station characteristics River name
Station name
Drainage area (kin 2)
Record length (yr.)
B. Menderes Manavgat Kizilirmak Firat
Cine Manavgat Kizilirmak Keban
948 928 15,582 63,874
33 31 32 34
Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.
1 2 3 4 5 6 7 8 9 10 11 12
Averages
month
(i)
Pi(-]-)
0.535
0.364 0.455 0.845 0.485 0.545 0.576 0.576 0.515 0.545 0.515 0.545 0.455
0.795
0.833 0.466 0.687 0.500 0.555 0.842 0.894 1.000 0.889 1.000 0.944 0.933 0.801
0.762 0.722 0.765 0.529 0.600 0.857 0.857 0.875 0.933 0.937 1.000 0.777 0.515
0.415 0.484 0.613 0.484 0.484 0.516 0.451 0.516 0.548 0.548 0.548 0.581 0.823
0.855 0.667 0.737 0.733 0.733 0.750 0.929 0.875 0.882 0.941 1.000 0.777 0.819
0.606 0.813 0.916 0.500 0.750 0.800 0.823 1.000 0.928 0.928 1.000 0.769 0.429
0.375 0.406 0.406 0.312 0.468 0.500 0.500 0.469 0.406 0.406 0.437 0.469
0.725
0.833 0.769 0.846 0.800 0.533 0.500 0.750 0.600 0.847 0.769 0.857 0.600
Pi(+t +)
Pi(+)
Pi(+l +)
Pi(+)
Pi(-I-)
Pi(+)
Pi(+l +)
Kizilirmak
Manavgat
B. Menderes
River n a m e
Observed probabilities
T A B L E III
0.772
0.750 0.840 0.842 0.773 0.882 0.562 0.750 0.588 0.789 0.842 0.944 0.705
Pi(-]-)
0.421
0.441 0.323 0.353 0.323 0.323 0.529 0.470 0.441 0.471 0.471 0.500 0.441
Pi(+)
Firat
0.756
0.933 0.818 0.750 0.636 0.636 0.333 0.875 0.867 0.812 1.000 0.882 0.533
Pi(+[+)
0.822
0.947 0.739 0.909 0.782 0.826 0.687 0.777 0.842 0.889 1.000 0.941 0.526
Pi(-l-)
b.a
b0
262
joint probability of surpluses between months i and i+1 given respective truncation levels. Other joint probabilities have similar interpretations. The estimations Pi(+) and Pi(+]+) of Pi(+ +) cml be achieved through the following steps: (1) The monthly flows of months i and i+1 in each year are truncated at given respective demand levels. (2) The number (nss)i of joint surplus occurrences are counted throughout the observation period. In addition the number (ns) i of surpluses in the ith month is determined. (3) Finally the estimates can be calculated as: /3i(+1+1 = (nss)i/n
/3i(+) = (ns)i/n
and
(16)
where n is the record length in years. The transition probability estimates can then be found as Pi(+[+) = Pi(+,+)/Pi(+). Other estimates can be determined in a similar manner. The actual records are truncated at a periodic level determined by the average m o n t h l y flows. The necessary parameter estimates for the application of eq. 13 are given in Table III. The e m p l o y m e n t of these parameters in eqs. 13 and 14 lead to the mean longest drought durations shown in Fig. 4. Since these actual records are periodic as well as dependent the curves in Fig. 4 do Y~ars
1
2
t
i
,
3 l
4t
,
,
~ ,
i
1&
.
t
6
,
t7
,
.
,
.
t v
i
,
.
i
. , . ,....,-"
....... ' ..............
[
i
-J"
!
:u
-
i : Compuei d
4
2
0 ~'~
/
~
/!// /
.
-
.
.
,:~y //
.
.
1
...............
Kiz=hrrnak
. . . . . .
F,rat !
.
10
.
~.
20
.
.
30
.
.
.
o
i
i '
40
•
~ i
i
50
J
'
60
i I
""
.
.
70
.
.
.
~0
L Months
Fig. 4. Actual expected longest drought duration vs. sample length.
90
263
reflect their joint effects. The longest drought durations are all periodic and the periodicity diminishes with increasing sample length. On the other hand, observed and computed longest drought durations are very close to each other. Under the light of aforementioned procedure the longest drought duration properties of any monthly, weekly, daily, etc., flow sequences observed at any part of the world can be evaluated with no major difficulty. CONCLUSIONS
The PDF of the longest drought duration in periodic--stochastic process have been rigorously derived and applied to some hypothetical periodicindependent as well as actual monthly flows. The following conclusions can be drawn from the study in this paper. (1) The longest drought duration of periodic--stochastic processes can be expressed in terms of marginal and conditional probabilities. (2) The parameters such as the expectation, variance, etc., of the longest drought duration in periodic--stochastic processes are also periodic. (3) Not only the numerical values of transition probabilities but more effectively their order along the time axis play an important role in the longest drought probabilities and parameters. (4) The effect of periodicity on the longest drought duration decreases markedly with increasing sample size.
REFERENCES David, F.N. and Barton, D.E., 1962. Combinatorial Chance. Griffin, London. Feller, W., 1968. An Introduction to Probability Theory and its Application, Vol. 1, Wiley, New York, N.Y. Millan, J., 1972. Use of regional economic models in the evolution of drought impact. Proc. 2nd Int. Syrup. on Hydrology, F o r t Collins, Colo. Millan, J. and Yevjevich, V., 1971. Probabilities of observed droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. Pap. 50. N.B.S. (National Bureau of Standards), 1959. Tables of the bivariate normal distributions and related functions. Appl. Math. Ser. 50, U.S. Government Printing Office, Washington, D.C. Salazar, P.G. and Yevjevich, V., 1975. Analysis of drought characteristics by the theory of runs. Colo. State Univ. F o r t Collins, Colo., Hydrol. Pap. 80. Tase, N., 1976. Area--deficit--intensity characteristics of droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. Pap. 87. Yevjevich, V., 1967. An objective approach to definitions and investigations of continental hydrologic droughts. Colo. State Univ., F o r t Collins, Colo., Hydrol. PaP. 23.