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Detection of changes in rainfall and runoff patterns
Journal of Hydrology, 147 (1993) 153-167
153
0022-1694/93/$6.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
[4]
Detection of changes in rainfall and runoff patterns E.M. Lungu Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana (Received 11 August 1992; revision accepted 22 November 1992)
Abstract This study had two main objectives. First, a simple model has been assumed for time series consisting of trend, periodic, autoregressive and random residual components, which have been used to search for significant changes in the components of a number of runoff time series. Second, the potential of topological conjugacy to identify changes in the dynamics of a system has been investigated. It is shown that the second technique is more sensitive in highlighting short-term changes in the dynamics of a system.
Introduction
Botswana can be divided into two main sections: a narrow strip running along the eastern border with South Africa and the vast western portion. The eastern strip consists of undulating uplands which are crossed by water courses, and the western region, more than 80% of Botswana, is covered by the sands of the Kalahari desert, which are up to 120m deep. Eastern Botswana is developing rapidly. This development has led to transformation of parts of rural areas into peri-urban areas, parts of peri-urban areas into urban areas and further expansion of existing urban areas. Modification of the land surface on a large scale inevitably changes the nature and magnitude of rainfall and runoff processes. This paper is an attempt to look for evidence of this change in existing time series. Historical rainfall series from three stations, Gaborone (No. 037), Mochudi (No. 136) and Kasane (No. 064), and runoff series from three gauging stations, Metsemotlhaba at Thamaga (No. 2423), Metsemotlhaba at Morwa (No. 2124), and Notwane at Gaborone dam (No. 2112), are used in this study. The rainfall stations Gaborone and Mochudi are situated in southeast Botswana
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
154
and Kasane is in the northeast. The average annual rainfall ranges from 650 mm at Kasane to 550 mm at Gaborone. The runoff stations are all in the southeast. In the first part of the study, total runoff sequences were decomposed into trend, periodic and stochastic components and then subjected to spectral analysis, to determine the significant components, according to the method of Kite (1989). Each component was removed step by step, and spectral analysis was applied after each step to indicate the presence of any remaining components. In the second part of the study, the potential for applying topological conjugacy is demonstrated. This involved splitting the time series into subperiods of various lengths (in years), calculating a transition matrix, for each subperiod, and analysing the various transition matrices for topological conjugacy.
Spectral analysis Rather than splitting historical time series into subperiods, we propose, in this section, to search for significant changes in parameters within the whole period of long time series. Spectral analysis of the original log-transformed runoff time series showed an initial spectrum with a significant low-frequency component below 0.1 cycles month -~ , with the peak occurring at 0.08 cycles month-~ for all stations. This result suggested that a low-frequency filter was more appropriate. Hence, the time series X, were decomposed as N/2
X, =
1", + M o + ~ [A k cos(2rckt/N) + B k sin(2rckt/N)] + e,
(1)
k I
where N
N
M o = 1/N Z X,,
(A~, B~) = 2IN ~ X, [cos(Zltkt/N), sin(2rtkt/N)]
1=1
(2)
1=1
N is the number of observations, e, is the stochastic component with mean 2 and T, is the trend component. If a~ is the total variance zero and variance O'e, of the time series X,, then the part of the variance accounted for by the kth harmonic is 2 2 var(k) = (AM + B~)/2ax
(3)
except for the last harmonic (when k = N/2), which has an explained variance (A M+ B~). The significance of the various harmonics is tested using Fisher's g statistic (Yevjevich, 1972), as 2 / N[2
gk = (A~ + B~)/ ~ (A~ + B~)
(4)
As the spectral function is significant at low frequencies, a trend component
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
155
Table 1 Values of variables and model statistics Station
Degree of polynomial
2423 2124 2112
5 3 5
% Variance explained by Trend
Periodic
Autoregressive
Residual
6.8 1.4 8.9
40.0 31.1 19.6
14.8 14.8 32.5
38.4 52.7 39.0
was analysed and removed by using a polynomial regression, i.e. 7", =
bo + b t t
+ b2 t2 + . . .
+ bpt p
(5)
where t is the decimal indication of the corresponding year and month, and bo, b ~ , . . . , bp are constants. The optimal order of (3) is determined as described by Kite (1989). The stochastic component was represented by a model of the type k
e, =
~ aje,_j + a,
(6)
j=0
where aj, j = 0, 1. . . . . k, are constants, and a, is an independent random 2 The task now is to remove the variable having zero mean and variance aa. components step by step and to apply spectral analysis after each step to indicate the presence of any remaining components. Linear regression was used to determine the constants bi, i = 1, 2 , . . . , p. It is found that the trend component accounted for 1-9% of the variance of the log-transformed monthly runoff sequences (Table 1). When the trend component was removed, the spectrum function showed minor reduction in magnitude below 0.1 cycles month-~ (Fig. 1). The periodic component accounted for 19-40% of the variance. The proportion accounted for by the periodic component was found to be smallest for the Notwane river at Gaborone dam, which has the largest drainage area (4200 km2), and largest for the Metsemotlhaba at Thamaga, which has the smallest drainage area (1300 km2). When both the trend and periodic components were removed, the spectrum function showed a drastic reduction in magnitude at low frequencies (Fig. 1), although compared with 95% confidence limits the spectrum function was still significant for Morwa and the Notwane. Autocorrelation and partial autocorrelation functions revealed that e, sequences followed an autoregressive process of order two for the Metsemotlhaba at Morwa and Thamaga and a process of order one for the Notwane at
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
156
STATION: MORWA B,o"
~1
O~OSPECTRAL ESTIMATE OF ORIfINAL TIME SERIES SPECTRAL ESTIMATEOF ORIGINAL TIME SERIES LESS LINEAR TRENDS zY----'~SPE£TRAL ESTIMATE oF ORIGINALTIME SERIES LESS LINEAR TRENDS,LESS PERIODICITIES
6,0.
~
SPECTRAL ESTIMATEOF 0~IGIN/~. TIME SERIES LESS LINEAR TRENDS, LESS PERtODICITIES, LESSAUTOREDRESSIVE COMPONENTS
,<
4,0.
N
\
UPPER 95%
o;1
d,2
05
CONF[DENZE LIMIT
o',~
o;s
FREQUENCY (CYCLES PER MONTH)
Fig. 1. Spectral density functions for Morwa.
G a b o r o n e dam, i.e. 0.4033e,_~ - 0.1202e,_2 + a,,
e, --
f ~0.5246e,_~ -
0.1710e,_2 + a,,
l 0.5420e, ~ + a~
for T h a m a g a for Morwa
(7)
for G a b o r o n e dam
The autoregressive component accounted for 14-33% o f the variance of the total series. When the trend, periodic and autoregressive components were removed, the spectral function was found to be insignificant at all frequencies for all stations. The residual series was found to behave as white noise, and accounted for 38-53% o f the variance of the total series. This analysis showed that there were no significant trends and that the residual component was the most important component for all the stations. However, this analysis does not provide evidence for a climatic change, as the behaviour o f the spectral function is within the range o f normal fluctuations. The occurrence o f one peak only at 0.08 cycles m o n t h - ~ (or once every year) is because there is only one rainy season (November-April), which is the
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
157
STATION: MORWA
LO~
x ~ AUTOCORRELATION FUNCTIQN O-~-OPARTIALAUTOCORRELATION FUN[TION
0,~.1 0,6
95°/*
UPPERCONFIDENCE LIMIT EAGt
~
/
-
95~I*LOWER CONFIDENCELIMIT
-0.21
Fig. 2. Autocorrelation and partial autocorrelation functions for Morwa.
source of this power. This lack of evidence m a y be attributed, among other reasons, to the short record of the time series (62 years). In view o f this, it is proposed to investigate possible changes in the runoff by an approach known as topological conjugacy, which we believe may be more sensitive. Topological conjugacy ~
Its application
We let P = [a~j] be an n x n transition matrix of a M a r k o v process where a o. >1 0 is the probability that an event in class i makes a 'transition' into class j. We denote the matrix obtained from P by raising every non-zero entry to the power n, n e R, by pn. We let S b e an n x n irreducible 0-1 matrix (note that S = p0), and denote a nonempty n-set associated with S, whose elements may be conveniently labelled 1, 2, 3 , . . . , n, by V = {1, 2 , . . . , n). We let E+ = II~ V be the product set. Let us consider the subspace E~ c E + defined by E~ = {X ~. E + :S(xi, X i + l ) = 1 for all i ~> 0} and the shift a~ defined on E~by (~+ X)i
:
Xi+ 1 for x
=
(xi)
(8)
The pair f2 = (V, Y.~-) is called a directed graph, the elements of V are its vertices and the elements of E~ are the arcs, and the pair (Z~-, o-~ ) is called a topological M a r k o v chain (Williams, 1973). Given two M a r k o v chains
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
158 STATION:
THAMAGA
6-
O
O SPECTRAL ESTIMATE
n
I~ SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR T R E N D S
~.
~
O
0 SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR TRENDS, LESS PERIODICITIES.
I~
OF ORIGINAL TIME
SERIES
SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR IRENDS, LESS PERIODICITIES
LESSAUTOREGRESSIVE COMPONENTS.
/
== Q~
0
03
0;2
o:3
0;s
FREQUENCY (CYCLES PER MONTH)
Fig. 3. Spectral density functions for Thamaga.
(E~, a~) and (E~, a~) we shall say that they are topologically conjugate (i.e. they preserve the dynamics of a system) if, for some mapping ~b of Z~ onto E~, we have ~ba~ =
a~b
(9)
For any two 0-1 matrices S and Q, we say that S is a reduction of Q and write S < Q if there exist non-negative integer matrices D and R, where D is a division matrix, such that S
=
DR andQ
=
RD
(10)
Given a matrix S we say that So is a total reduction of S provided (1) So < S, < . . . < Sr = S; (2) S O has no repeated column. It is known (Williams, 1973) that every square matrix S over Z ÷ has a total reduction So, and that ( ~ +s , a2) and (E~ ~r~ ) are topologically conjugate if and only if their total reductions So and Q0 are conjugate by a permutation, that is So =
gQog-'
(11)
where M is a permutation matrix. Equation (11) holds for systems with similar
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167 STATI
1,0
159
ON: THAMAF;A
w O
~ AUTO£ORRELAT0 i NFUNCTION O PARTIALAUTOCORRELATIONFUNCTION
O,(I
0,6 1 01.
0,2
~,/
"
-
~
, ~
~
~
~
--..
LOWER 95°1o CONFIDENCE LIMIT -0, 2
-0t~
-0,6
Fig. 4. Autocorrelation and partial autocorrelation functions for Thamaga.
dynamics, and is used to determine the relationship between subperiods of a time series. The following definitions are required to facilitate the classification o f states: Definition 1. A directed graph f~ is said to be strongly connected if for any ordered pair of distinct states, i and j, there is a path in f~ connecting i to j. Definition 2. If for every pair o f states, we have a,~") > 0 for some n, where a(") 0 denotes the (i, j) entry of P", then we say that all states of P are strongly connected. Definition 3. The integer n >~ 1 for which a~7) > 0 is called the period of the state. If n = 1 the state is said to be aperiodic. Definition 4. If the states of a system are strongly connected, i.e. --q a!") > 0 for some n, then the system is topologically stationary and the integer n >/ 1 is called the degree of stationarity. We let X,, where t = 1, 2, . . . , N, be a time series of observed data. We suppose that this can be subdivided into subsequences of length 2i <~ N, where i = 1, 2, . - . . , k, each of which is subjected to the analysis described above. We suppose further that each subsequence can be divided into three state intervals, such as low events below X~, normal events between X,. and Xj, and high events above Xs, where the truncation levels are fixed and are determined from the observed data for the whole time series. To facilitate the application
.
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
160
of this technique, truncation levels were chosen such that low events (i.e. rainfall or runoff) corresponded to an event with probability of exceedance of at least 0.67, and high events corresponded to one with probability of exceedance of at most 0.33. Transition matrices, computed for each subsequence, were analysed for topological conjugacy as demonstrated below. Rainfall analysis
The long-term behaviour of a system can be determined as described by Bailey (1964), through studying the limiting properties of P" as n --, ~ . The stochastic matrices for the entire rainfall records, obtained by counting the number of observed transitions between states, have the property that a~j > 0, for every i and j, for all stations. The limiting matrix H given by lim pn =
II
(12)
n~oG
is found to be a stochastic matrix with identical rows, satisfying the condition a,j > 0. For example, the transition matrix for the entire rainfall record for the Gaborone rainfall station is
P =
/0.4121
0.3194
0.0960~
|0.3637
0.3403
0.17601
(13)
/
/
\0.2242
0.3403
0.7280/
and the limiting stochastic matrix calculated from (13) is
II =
/0.2257
0.2613
0.5130x~
|0.2257
0.2613
0.51301
/
(14)
/
\0.2257
0.2613
0.5130/
We note that the rows of the limiting matrix II in (14) satisfy the conditions ZiHia~ = IIj, H i > 0 and ZilI~ = 1, a result which is typical for the Mochudi and Kasane rainfall stations. Moreover, the 0-1 matrix S, obtained by raising II to the power zero, is
S =
(ix) 1
1
1
1
1
1
(15)
and its total reduction So, calculated as in the previous section, is given by So =
(3)
(16)
This result suggests the existence, in the long term, of a process whose states
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
161
are strongly connected, aperiodic and stationary, with degree of stationarity one. Analysing subperiods of length 5 years and over for all the rainfall stations yielded various rainfall subsystems which are introduced below. For convenience of notation, we denote S S M [ N ] x as subsystem n u m b e r M, occurring over N subintervals each of length K years. By counting the n u m b e r of observed transitions between rainfall states for various subperiods and using (11), the following distinct subsystems were identified: SS1. This subsystem arises from a family of stochastic matrices for which a o > 0 for every i a n d j . These matrices give rise to a 0-1 matrix S o f t h e form (15) and a total reduction of the type (16). Evidently, the dynamics of this subsystem are similar to the long-term pattern. SS2. This subsystem is associated with a family of stochastic matrices for which %
=0,
for/=
1,j=3
> 0,
otherwise
(17)
(' '!/
These matrices give rise to a 0-1 matrix S of the form
S
=
1
1
1
1
(18)
whose total reduction S o is So =
1
(19)
As a~ ) > 0, where a~ ) are the entries of p2, it follows from the definitions that the states are strongly connected, aperiodic and stationary, with degree of stationarity two. Moreover, by virtue of (16) and (19) the dynamics of SS1 and SS2 are different. SS3. This subsystem arises from a family of stochastic matrices for which = 0,
fori = 1,j = 3andi
> 0,
otherwise
= 3, j = 1 (20)
aij
T h i s gives rise to a 0-1 matrix S of the type
S =
1
1
0
1
= SO
(21)
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
162
It should be noted that a!2 ij ) > 0, so that the states are strongly connected, aperiodic and stationary, with degree of stationarity two. By virtue of (16), (19) and (21), it is evident that the subsystems SS1, SS2 and SS3 are dynamically different. SS4. This subsystem is associated with a family of stochastic matrices for which = 0,
for/ = 1,j = 3andi
> 0,
otherwise
= 2, j = 1
aij
(22)
The 0-1 matrix corresponding to (22) is given by
1 1 0) S
=
0
1
1
= So
(23)
1 1 1 C o m p a r i n g the total reductions of the four subsystems, it is obvious that they are dynamically different. SS5. This subsystem arises from a family of stochastic matrices of the form = 0,
for any diagonal entry
> 0,
otherwise
aij
(24)
These matrices give rise to a 0-1 matrix S of the form = 0,
for any diagonal entry
= 1,
otherwise
S
(25)
and a total reduction So of the type
The states of this subsystem are strongly connected and are stationary, with degree of stationarity two. It should be noted that a}~~ > 0, so that one of the states is now periodic with period two. This is a m a r k e d departure from the long-term behaviour described by (14). We can now analyse the rainfall pattern of the three stations in terms of the subsystems above. In particular, we want to investigate the chronological structures of the rainfall pattern in terms of these subsystems. Gaborone rainfall station
Analysis of the entire rainfall record as described above revealed that
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
163
during the period 1924-1985 four subsystems (SS1, SS2, SS3 and SS5) have occurred. In particular, the analysis for chronological structures for various subperiods yielded the following chains: f o r 2 = 5,
SST =
SS113215 ~ SS2[115 --, SS11315 ~ SS5[115 ~ SS11515 ~ SS21215 SS3[115 ~ SS21215 ~ SS1[1015
(27)
f o r ) , = 6,
SST =
SS114216 ~ SS21416 ~ SS1[1016
(28)
f o r 2 = 7,
SST =
SS114217 ~ SS21317 ~ SS1[1017
(29)
for)` = 8,
SST =
SS114218 ~ SS21218 ~ SSI[IO] 8
(30)
f o r 2 = 9,
SST =
SS1142]9 ~ SS21119 ~ SS1[1019
(31)
for)` = 10,
SST =
SS1152]~0
(32)
These chains revealed two things: (1) for any subperiod of at least 10 years the rainfall states exhibit long-term behaviour (i.e. the rainfall pattern is described by SS1 only); (2) for subperiods of less than 10 years, the rainfall pattern is variable and is described by at least one subsystem. In particular, from 1924 to 1955 the rainfall pattern is described by SS1 only but after 1955 this pattern is replaced by four subsystems (SS1, SS2, SS3 and SS5).
Mochudi rainfall station Analysis of the entire rainfall record (i.e. 1924-1985) revealed that the rainfall pattern is described by three distinct subsystems (SS1, SS2 and SS5). The analysis for chronological structure for various subperiods yielded the following chains: f o r 2 = 5,
SST =
SS113615 ~ SS2[1] 5 ~ SS112] 5 ~ SS21315 ~ SS11415 ~ SS2[115 SS11415 ~ SS51315 ~ SS11415
(33)
f o r 2 = 6, SST
=
SS113916 ~
SS21216 ~
SS1[1016 ~
SS51216 ~
SS11416
(34)
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
164
f o r 2 = 7,
SST
=
SS113917 ~ SS2[117 ---, SS1[1117 -+ SS5[117 ---, SS11417
(35)
f o r 2 = 8,
SST
=
SS115418
(36)
These chains identify the c u t o f f subperiod length (i.e. 2 = 8) above which the rainfall states exhibit long-term behaviour. F o r 2 < 8 we find t h a t the rainfall is described by three distinct subsystems (SS1, SS2 a n d SS5). In particular, we note t h a t f r o m 1924 to 1960 the rainfall p a t t e r n is described by SS1 only but after 1960 the rainfall p a t t e r n is variable a n d is described by three subsystems (SS1, SS2 a n d SS5).
Kasane rainfall station Analysis o f the entire rainfall record (i.e. 1923-1991) revealed t h a t the rainfall p a t t e r n is described by three subsystems (SS1, SS2 a n d SS5) a n d yielded the following chains: f o r 2 = 5,
SST =
SS11221s --+ SS51615 + SS11315 --+ SS2[1]s --+ SSI[12]s --+ SS2[1] s -+ SSI[13]s ---+SS212] s -+ SS114] s
(37)
f o r 2 = 6,
SST =
SS112216 --+ SS51516 --+ SS113116 --+ SS2[116 --+ SS11416
(38)
f o r 2 = 7,
SST =
SS112217 -+ SS51417 --, SS113617
(39)
f o r 2 = 8,
S S T -- SS112218 ~ SS51318 ~ SS1[13618
(40)
f o r 2 = 9,
SST
=
SS112219 ---, SS51219 ---, SS113619
(41)
f o r 2 = 10,
SST
=
SS1122]. 0 ---, SS5[1]m -+ SS1136], 0
(42)
f o r 2 = 11,
SST
=
SS1158],1
(43)
The chains s h o w t h a t for a n y subperiod o f at least 11 years, the rainfall
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
165
pattern exhibits long-term behaviour but for subperiods of less than 11 years, the rainfall pattern is described by three subsystems (SS1, SS2 and SS5). In particular, we observe that from 1925 to 1950 the rainfall pattern is described by SSI only but after 1950 this subsystem is replaced by three subsystems (SS1, SS2 and SS5).
Runoff analysis The long-term behaviour of the system was determined as described in the previous section. For example, for the Metsemotlhaba at Thamaga, the transition matrix for the entire flow record is /0.4333
0.1282
0.0397~
P = /0"3167
0.2180
0.06941
\0.2500
0.6538
(44)
1
0.8909/
and the limiting stochastic matrix calculated from (44) is
1-I --
0.0808
0.1050
0.8142
0.0808
0.1050
0.8142
0.0808
0.1050
0.8142
(45)
The 0-1 matrix & obtained by raising 11 to the power zero, yielded a total reduction S o of the type (16). Evidently, the long-term behaviour of the flow states is that they are strongly connected, aperiodic and stationary, with degree of stationarity one. Analysing the subperiods for topological conjugacy as outlined above, we find that for 2 t> 15, for Thamaga, we have a~') > 0 for n >~ 1, where a~ ) is defined as before. This shows that for any subperiod of the flow record of at least 15 years, the flow states are strongly connected, aperiodic and stationary, with degree of stationarity one. This behaviour is analogous to the long-term behaviour. In the terminology described above, the subperiods possess similar flow dynamics. This result is also true for Morwa for 2 t> 18 and Gaborone dam for 2 ~> 30. It may be noted that the minimum length of flow record required for the subperiods to possess similar dynamics to the long-term pattern is shortest for the Metsemotlhaba at Thamaga, where the drainage area is smallest (i.e. 1300 km 2) and longest for the Notwane at Gaborone dam, where the drainage area is largest (i.e. 4200 km2). A detailed examination of the flow records for Thamaga for 2 < 15 revealed that during the period 1924-1985 the flow pattern was described by three subsystems (SS1, SS2 and SS5). Analysis for chronological structure for
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
166
various subperiods led to identification of the following chains: f o r 2 = 11,
SST
=
SS1130]~ ~ SS214]~, ~ SS115],~ ~ SS513]~ ~ SS119],1
(46)
f o r 2 = 12,
S S T = SS1131112 ~ SS213]~2 --, SS116]~2 --, SS512],2 ---, SS119]~2 for). = 13,
(47)
SST
(48)
=
SS1131]~3 -4 SS212]~3 ~ SS117]t3 ~ SS5[1]~ 3 --~ SS119], 3
for). = 14,
SST
=
SS1131],4 ~ SS2[1],4 --~ SS1[17],4
(49)
for). = 15,
SST
=
SS1148]15
(50)
For ). /> 15 the subperiods possess similar flow dynamics and the flow pattern is described by SS1 only, whereas for 2 < 15 three subsystems (SS1, SS2 and SS5) describe the flow pattern. Moreover, it is found that from 1924 to 1959 the flow pattern is described by SS1 only but after 1959 this subsystem is replaced by three subsystems (SS1, SS2 and SS5). The occurrence of SS5 shows that significant changes in the flow pattern have taken place. These results are found to be true for the Metsemotlhaba at Morwa for 2 < 18 and for the Notwane at Gaborone dam for 2 < 30. Results and discussion Spectral analysis has revealed that the runoff is highly stochastic and that the residual component is the most important component, as it explains about 38-53% of the variance of the total series. However, spectral analysis did not provide evidence for a climatic change. This may be attributed to, among other things, the short record of the time series and probably to the insensitivity of the method. To answer the sensitivity question, a different technique, known as topological conjugacy, was employed. This technique is based on the fact that if two systems are dynamically different, they produce unique graphs. The technique achieved two results. First, the cutoff length in years above which subperiods preserve the dynamics of a system is determined for each station. For runoff, it is found that this length is shortest for the Metsemotlhaba at Thamaga, which has the smallest drainage area, and longest for the Notwane at Gaborone dam, which has the largest drainage area. It is evident from this result that the smaller the drainage area the easier it is to obtain runoff of
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
167
various magnitudes, as the rainfall is usually able to cover the entire drainage area and the threshold rainfall (that is, the initial abstraction of rainfall required to satisfy interception, depression storages and infiltration) is easily exceeded. For a large drainage area, on the other hand, the threshold is not easily exceeded, mainly as a result of the localized nature of rainfall in Botswana. Second, a detailed understanding of the dynamics of a system has been attained through the classification procedure introduced in this study. If we subdivide the rainfall and runoff records into two periods, i.e. 19241950 and 1950-1985, then it is easy to see that both rainfall and runoff patterns have behaved differently in each period. In particular, we find that in the first period of the record, both rainfall and runoff are described by subsystem SS1 only, whereas after 1950 this is replaced by at least three subsystems. We note that these changes take place across the country. A study of climatic change and variability in Southern Africa (Tyson, 1986) has described dry spells during the period 1962-1971 and wet spells during the period 1971-1981. Furthermore, a report on the climate of Botswana (Bhalotra, 1987) has described major droughts in the calendar years 19811982 to 1986-1987. This period of variable rainfall (i.e. 1962-1987) corresponds more or less to the findings of this paper. It is evident from the results of this study that rainfall and runoff patterns have changed. After 1950, the rainfall pattern has become variable, as a result of the occurrence of extreme events such as droughts and wet spells (Tyson, 1986). Changes in runoff pattern suggest, in turn, changes in rainfall pattern, as well as modification of the land surface largely as a result of overgrazing and the covering of parts of the catchment with impervious roofs, sidewalks, roads and parking lots. References Bailey, N.T.J., 1964. The Elements of Stochastic Processes. Wiley, New York, pp. 38-57. Bhalotra, Y.P.R., 1987. Climate of Botswana, Part II. Department of Meteorological Services, Garborone, Republic of Botswana, pp. 1-40. Kite, G.W., 1989. Use of time series analysis to detect climatic change. J. Hydrol., 111: 259-279. Tyson, P.D., 1986. Climatic Change and Variability in Southern Africa. Oxford University Press, Cape Town, pp. 69-92. Williams, R.F., 1973. Classification of subshifts of finite type. Ann. Math., 98:120-153; Errata, Ann. Math., 99 (1974): 380-381. Yevyevich, V., 1972. Probability and Statistics in Hydrology, 1st edn. Water Resources Publications, Fort Collins, CO, p. 50.
153
0022-1694/93/$6.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
[4]
Detection of changes in rainfall and runoff patterns E.M. Lungu Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana (Received 11 August 1992; revision accepted 22 November 1992)
Abstract This study had two main objectives. First, a simple model has been assumed for time series consisting of trend, periodic, autoregressive and random residual components, which have been used to search for significant changes in the components of a number of runoff time series. Second, the potential of topological conjugacy to identify changes in the dynamics of a system has been investigated. It is shown that the second technique is more sensitive in highlighting short-term changes in the dynamics of a system.
Introduction
Botswana can be divided into two main sections: a narrow strip running along the eastern border with South Africa and the vast western portion. The eastern strip consists of undulating uplands which are crossed by water courses, and the western region, more than 80% of Botswana, is covered by the sands of the Kalahari desert, which are up to 120m deep. Eastern Botswana is developing rapidly. This development has led to transformation of parts of rural areas into peri-urban areas, parts of peri-urban areas into urban areas and further expansion of existing urban areas. Modification of the land surface on a large scale inevitably changes the nature and magnitude of rainfall and runoff processes. This paper is an attempt to look for evidence of this change in existing time series. Historical rainfall series from three stations, Gaborone (No. 037), Mochudi (No. 136) and Kasane (No. 064), and runoff series from three gauging stations, Metsemotlhaba at Thamaga (No. 2423), Metsemotlhaba at Morwa (No. 2124), and Notwane at Gaborone dam (No. 2112), are used in this study. The rainfall stations Gaborone and Mochudi are situated in southeast Botswana
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
154
and Kasane is in the northeast. The average annual rainfall ranges from 650 mm at Kasane to 550 mm at Gaborone. The runoff stations are all in the southeast. In the first part of the study, total runoff sequences were decomposed into trend, periodic and stochastic components and then subjected to spectral analysis, to determine the significant components, according to the method of Kite (1989). Each component was removed step by step, and spectral analysis was applied after each step to indicate the presence of any remaining components. In the second part of the study, the potential for applying topological conjugacy is demonstrated. This involved splitting the time series into subperiods of various lengths (in years), calculating a transition matrix, for each subperiod, and analysing the various transition matrices for topological conjugacy.
Spectral analysis Rather than splitting historical time series into subperiods, we propose, in this section, to search for significant changes in parameters within the whole period of long time series. Spectral analysis of the original log-transformed runoff time series showed an initial spectrum with a significant low-frequency component below 0.1 cycles month -~ , with the peak occurring at 0.08 cycles month-~ for all stations. This result suggested that a low-frequency filter was more appropriate. Hence, the time series X, were decomposed as N/2
X, =
1", + M o + ~ [A k cos(2rckt/N) + B k sin(2rckt/N)] + e,
(1)
k I
where N
N
M o = 1/N Z X,,
(A~, B~) = 2IN ~ X, [cos(Zltkt/N), sin(2rtkt/N)]
1=1
(2)
1=1
N is the number of observations, e, is the stochastic component with mean 2 and T, is the trend component. If a~ is the total variance zero and variance O'e, of the time series X,, then the part of the variance accounted for by the kth harmonic is 2 2 var(k) = (AM + B~)/2ax
(3)
except for the last harmonic (when k = N/2), which has an explained variance (A M+ B~). The significance of the various harmonics is tested using Fisher's g statistic (Yevjevich, 1972), as 2 / N[2
gk = (A~ + B~)/ ~ (A~ + B~)
(4)
As the spectral function is significant at low frequencies, a trend component
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
155
Table 1 Values of variables and model statistics Station
Degree of polynomial
2423 2124 2112
5 3 5
% Variance explained by Trend
Periodic
Autoregressive
Residual
6.8 1.4 8.9
40.0 31.1 19.6
14.8 14.8 32.5
38.4 52.7 39.0
was analysed and removed by using a polynomial regression, i.e. 7", =
bo + b t t
+ b2 t2 + . . .
+ bpt p
(5)
where t is the decimal indication of the corresponding year and month, and bo, b ~ , . . . , bp are constants. The optimal order of (3) is determined as described by Kite (1989). The stochastic component was represented by a model of the type k
e, =
~ aje,_j + a,
(6)
j=0
where aj, j = 0, 1. . . . . k, are constants, and a, is an independent random 2 The task now is to remove the variable having zero mean and variance aa. components step by step and to apply spectral analysis after each step to indicate the presence of any remaining components. Linear regression was used to determine the constants bi, i = 1, 2 , . . . , p. It is found that the trend component accounted for 1-9% of the variance of the log-transformed monthly runoff sequences (Table 1). When the trend component was removed, the spectrum function showed minor reduction in magnitude below 0.1 cycles month-~ (Fig. 1). The periodic component accounted for 19-40% of the variance. The proportion accounted for by the periodic component was found to be smallest for the Notwane river at Gaborone dam, which has the largest drainage area (4200 km2), and largest for the Metsemotlhaba at Thamaga, which has the smallest drainage area (1300 km2). When both the trend and periodic components were removed, the spectrum function showed a drastic reduction in magnitude at low frequencies (Fig. 1), although compared with 95% confidence limits the spectrum function was still significant for Morwa and the Notwane. Autocorrelation and partial autocorrelation functions revealed that e, sequences followed an autoregressive process of order two for the Metsemotlhaba at Morwa and Thamaga and a process of order one for the Notwane at
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
156
STATION: MORWA B,o"
~1
O~OSPECTRAL ESTIMATE OF ORIfINAL TIME SERIES SPECTRAL ESTIMATEOF ORIGINAL TIME SERIES LESS LINEAR TRENDS zY----'~SPE£TRAL ESTIMATE oF ORIGINALTIME SERIES LESS LINEAR TRENDS,LESS PERIODICITIES
6,0.
~
SPECTRAL ESTIMATEOF 0~IGIN/~. TIME SERIES LESS LINEAR TRENDS, LESS PERtODICITIES, LESSAUTOREDRESSIVE COMPONENTS
,<
4,0.
N
\
UPPER 95%
o;1
d,2
05
CONF[DENZE LIMIT
o',~
o;s
FREQUENCY (CYCLES PER MONTH)
Fig. 1. Spectral density functions for Morwa.
G a b o r o n e dam, i.e. 0.4033e,_~ - 0.1202e,_2 + a,,
e, --
f ~0.5246e,_~ -
0.1710e,_2 + a,,
l 0.5420e, ~ + a~
for T h a m a g a for Morwa
(7)
for G a b o r o n e dam
The autoregressive component accounted for 14-33% o f the variance of the total series. When the trend, periodic and autoregressive components were removed, the spectral function was found to be insignificant at all frequencies for all stations. The residual series was found to behave as white noise, and accounted for 38-53% o f the variance of the total series. This analysis showed that there were no significant trends and that the residual component was the most important component for all the stations. However, this analysis does not provide evidence for a climatic change, as the behaviour o f the spectral function is within the range o f normal fluctuations. The occurrence o f one peak only at 0.08 cycles m o n t h - ~ (or once every year) is because there is only one rainy season (November-April), which is the
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
157
STATION: MORWA
LO~
x ~ AUTOCORRELATION FUNCTIQN O-~-OPARTIALAUTOCORRELATION FUN[TION
0,~.1 0,6
95°/*
UPPERCONFIDENCE LIMIT EAGt
~
/
-
95~I*LOWER CONFIDENCELIMIT
-0.21
Fig. 2. Autocorrelation and partial autocorrelation functions for Morwa.
source of this power. This lack of evidence m a y be attributed, among other reasons, to the short record of the time series (62 years). In view o f this, it is proposed to investigate possible changes in the runoff by an approach known as topological conjugacy, which we believe may be more sensitive. Topological conjugacy ~
Its application
We let P = [a~j] be an n x n transition matrix of a M a r k o v process where a o. >1 0 is the probability that an event in class i makes a 'transition' into class j. We denote the matrix obtained from P by raising every non-zero entry to the power n, n e R, by pn. We let S b e an n x n irreducible 0-1 matrix (note that S = p0), and denote a nonempty n-set associated with S, whose elements may be conveniently labelled 1, 2, 3 , . . . , n, by V = {1, 2 , . . . , n). We let E+ = II~ V be the product set. Let us consider the subspace E~ c E + defined by E~ = {X ~. E + :S(xi, X i + l ) = 1 for all i ~> 0} and the shift a~ defined on E~by (~+ X)i
:
Xi+ 1 for x
=
(xi)
(8)
The pair f2 = (V, Y.~-) is called a directed graph, the elements of V are its vertices and the elements of E~ are the arcs, and the pair (Z~-, o-~ ) is called a topological M a r k o v chain (Williams, 1973). Given two M a r k o v chains
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
158 STATION:
THAMAGA
6-
O
O SPECTRAL ESTIMATE
n
I~ SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR T R E N D S
~.
~
O
0 SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR TRENDS, LESS PERIODICITIES.
I~
OF ORIGINAL TIME
SERIES
SPECTRAL ESTIMATE OF ORIGINAL TIME SERIES LESS LINEAR IRENDS, LESS PERIODICITIES
LESSAUTOREGRESSIVE COMPONENTS.
/
== Q~
0
03
0;2
o:3
0;s
FREQUENCY (CYCLES PER MONTH)
Fig. 3. Spectral density functions for Thamaga.
(E~, a~) and (E~, a~) we shall say that they are topologically conjugate (i.e. they preserve the dynamics of a system) if, for some mapping ~b of Z~ onto E~, we have ~ba~ =
a~b
(9)
For any two 0-1 matrices S and Q, we say that S is a reduction of Q and write S < Q if there exist non-negative integer matrices D and R, where D is a division matrix, such that S
=
DR andQ
=
RD
(10)
Given a matrix S we say that So is a total reduction of S provided (1) So < S, < . . . < Sr = S; (2) S O has no repeated column. It is known (Williams, 1973) that every square matrix S over Z ÷ has a total reduction So, and that ( ~ +s , a2) and (E~ ~r~ ) are topologically conjugate if and only if their total reductions So and Q0 are conjugate by a permutation, that is So =
gQog-'
(11)
where M is a permutation matrix. Equation (11) holds for systems with similar
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167 STATI
1,0
159
ON: THAMAF;A
w O
~ AUTO£ORRELAT0 i NFUNCTION O PARTIALAUTOCORRELATIONFUNCTION
O,(I
0,6 1 01.
0,2
~,/
"
-
~
, ~
~
~
~
--..
LOWER 95°1o CONFIDENCE LIMIT -0, 2
-0t~
-0,6
Fig. 4. Autocorrelation and partial autocorrelation functions for Thamaga.
dynamics, and is used to determine the relationship between subperiods of a time series. The following definitions are required to facilitate the classification o f states: Definition 1. A directed graph f~ is said to be strongly connected if for any ordered pair of distinct states, i and j, there is a path in f~ connecting i to j. Definition 2. If for every pair o f states, we have a,~") > 0 for some n, where a(") 0 denotes the (i, j) entry of P", then we say that all states of P are strongly connected. Definition 3. The integer n >~ 1 for which a~7) > 0 is called the period of the state. If n = 1 the state is said to be aperiodic. Definition 4. If the states of a system are strongly connected, i.e. --q a!") > 0 for some n, then the system is topologically stationary and the integer n >/ 1 is called the degree of stationarity. We let X,, where t = 1, 2, . . . , N, be a time series of observed data. We suppose that this can be subdivided into subsequences of length 2i <~ N, where i = 1, 2, . - . . , k, each of which is subjected to the analysis described above. We suppose further that each subsequence can be divided into three state intervals, such as low events below X~, normal events between X,. and Xj, and high events above Xs, where the truncation levels are fixed and are determined from the observed data for the whole time series. To facilitate the application
.
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
160
of this technique, truncation levels were chosen such that low events (i.e. rainfall or runoff) corresponded to an event with probability of exceedance of at least 0.67, and high events corresponded to one with probability of exceedance of at most 0.33. Transition matrices, computed for each subsequence, were analysed for topological conjugacy as demonstrated below. Rainfall analysis
The long-term behaviour of a system can be determined as described by Bailey (1964), through studying the limiting properties of P" as n --, ~ . The stochastic matrices for the entire rainfall records, obtained by counting the number of observed transitions between states, have the property that a~j > 0, for every i and j, for all stations. The limiting matrix H given by lim pn =
II
(12)
n~oG
is found to be a stochastic matrix with identical rows, satisfying the condition a,j > 0. For example, the transition matrix for the entire rainfall record for the Gaborone rainfall station is
P =
/0.4121
0.3194
0.0960~
|0.3637
0.3403
0.17601
(13)
/
/
\0.2242
0.3403
0.7280/
and the limiting stochastic matrix calculated from (13) is
II =
/0.2257
0.2613
0.5130x~
|0.2257
0.2613
0.51301
/
(14)
/
\0.2257
0.2613
0.5130/
We note that the rows of the limiting matrix II in (14) satisfy the conditions ZiHia~ = IIj, H i > 0 and ZilI~ = 1, a result which is typical for the Mochudi and Kasane rainfall stations. Moreover, the 0-1 matrix S, obtained by raising II to the power zero, is
S =
(ix) 1
1
1
1
1
1
(15)
and its total reduction So, calculated as in the previous section, is given by So =
(3)
(16)
This result suggests the existence, in the long term, of a process whose states
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
161
are strongly connected, aperiodic and stationary, with degree of stationarity one. Analysing subperiods of length 5 years and over for all the rainfall stations yielded various rainfall subsystems which are introduced below. For convenience of notation, we denote S S M [ N ] x as subsystem n u m b e r M, occurring over N subintervals each of length K years. By counting the n u m b e r of observed transitions between rainfall states for various subperiods and using (11), the following distinct subsystems were identified: SS1. This subsystem arises from a family of stochastic matrices for which a o > 0 for every i a n d j . These matrices give rise to a 0-1 matrix S o f t h e form (15) and a total reduction of the type (16). Evidently, the dynamics of this subsystem are similar to the long-term pattern. SS2. This subsystem is associated with a family of stochastic matrices for which %
=0,
for/=
1,j=3
> 0,
otherwise
(17)
(' '!/
These matrices give rise to a 0-1 matrix S of the form
S
=
1
1
1
1
(18)
whose total reduction S o is So =
1
(19)
As a~ ) > 0, where a~ ) are the entries of p2, it follows from the definitions that the states are strongly connected, aperiodic and stationary, with degree of stationarity two. Moreover, by virtue of (16) and (19) the dynamics of SS1 and SS2 are different. SS3. This subsystem arises from a family of stochastic matrices for which = 0,
fori = 1,j = 3andi
> 0,
otherwise
= 3, j = 1 (20)
aij
T h i s gives rise to a 0-1 matrix S of the type
S =
1
1
0
1
= SO
(21)
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
162
It should be noted that a!2 ij ) > 0, so that the states are strongly connected, aperiodic and stationary, with degree of stationarity two. By virtue of (16), (19) and (21), it is evident that the subsystems SS1, SS2 and SS3 are dynamically different. SS4. This subsystem is associated with a family of stochastic matrices for which = 0,
for/ = 1,j = 3andi
> 0,
otherwise
= 2, j = 1
aij
(22)
The 0-1 matrix corresponding to (22) is given by
1 1 0) S
=
0
1
1
= So
(23)
1 1 1 C o m p a r i n g the total reductions of the four subsystems, it is obvious that they are dynamically different. SS5. This subsystem arises from a family of stochastic matrices of the form = 0,
for any diagonal entry
> 0,
otherwise
aij
(24)
These matrices give rise to a 0-1 matrix S of the form = 0,
for any diagonal entry
= 1,
otherwise
S
(25)
and a total reduction So of the type
The states of this subsystem are strongly connected and are stationary, with degree of stationarity two. It should be noted that a}~~ > 0, so that one of the states is now periodic with period two. This is a m a r k e d departure from the long-term behaviour described by (14). We can now analyse the rainfall pattern of the three stations in terms of the subsystems above. In particular, we want to investigate the chronological structures of the rainfall pattern in terms of these subsystems. Gaborone rainfall station
Analysis of the entire rainfall record as described above revealed that
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
163
during the period 1924-1985 four subsystems (SS1, SS2, SS3 and SS5) have occurred. In particular, the analysis for chronological structures for various subperiods yielded the following chains: f o r 2 = 5,
SST =
SS113215 ~ SS2[115 --, SS11315 ~ SS5[115 ~ SS11515 ~ SS21215 SS3[115 ~ SS21215 ~ SS1[1015
(27)
f o r ) , = 6,
SST =
SS114216 ~ SS21416 ~ SS1[1016
(28)
f o r 2 = 7,
SST =
SS114217 ~ SS21317 ~ SS1[1017
(29)
for)` = 8,
SST =
SS114218 ~ SS21218 ~ SSI[IO] 8
(30)
f o r 2 = 9,
SST =
SS1142]9 ~ SS21119 ~ SS1[1019
(31)
for)` = 10,
SST =
SS1152]~0
(32)
These chains revealed two things: (1) for any subperiod of at least 10 years the rainfall states exhibit long-term behaviour (i.e. the rainfall pattern is described by SS1 only); (2) for subperiods of less than 10 years, the rainfall pattern is variable and is described by at least one subsystem. In particular, from 1924 to 1955 the rainfall pattern is described by SS1 only but after 1955 this pattern is replaced by four subsystems (SS1, SS2, SS3 and SS5).
Mochudi rainfall station Analysis of the entire rainfall record (i.e. 1924-1985) revealed that the rainfall pattern is described by three distinct subsystems (SS1, SS2 and SS5). The analysis for chronological structure for various subperiods yielded the following chains: f o r 2 = 5,
SST =
SS113615 ~ SS2[1] 5 ~ SS112] 5 ~ SS21315 ~ SS11415 ~ SS2[115 SS11415 ~ SS51315 ~ SS11415
(33)
f o r 2 = 6, SST
=
SS113916 ~
SS21216 ~
SS1[1016 ~
SS51216 ~
SS11416
(34)
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
164
f o r 2 = 7,
SST
=
SS113917 ~ SS2[117 ---, SS1[1117 -+ SS5[117 ---, SS11417
(35)
f o r 2 = 8,
SST
=
SS115418
(36)
These chains identify the c u t o f f subperiod length (i.e. 2 = 8) above which the rainfall states exhibit long-term behaviour. F o r 2 < 8 we find t h a t the rainfall is described by three distinct subsystems (SS1, SS2 a n d SS5). In particular, we note t h a t f r o m 1924 to 1960 the rainfall p a t t e r n is described by SS1 only but after 1960 the rainfall p a t t e r n is variable a n d is described by three subsystems (SS1, SS2 a n d SS5).
Kasane rainfall station Analysis o f the entire rainfall record (i.e. 1923-1991) revealed t h a t the rainfall p a t t e r n is described by three subsystems (SS1, SS2 a n d SS5) a n d yielded the following chains: f o r 2 = 5,
SST =
SS11221s --+ SS51615 + SS11315 --+ SS2[1]s --+ SSI[12]s --+ SS2[1] s -+ SSI[13]s ---+SS212] s -+ SS114] s
(37)
f o r 2 = 6,
SST =
SS112216 --+ SS51516 --+ SS113116 --+ SS2[116 --+ SS11416
(38)
f o r 2 = 7,
SST =
SS112217 -+ SS51417 --, SS113617
(39)
f o r 2 = 8,
S S T -- SS112218 ~ SS51318 ~ SS1[13618
(40)
f o r 2 = 9,
SST
=
SS112219 ---, SS51219 ---, SS113619
(41)
f o r 2 = 10,
SST
=
SS1122]. 0 ---, SS5[1]m -+ SS1136], 0
(42)
f o r 2 = 11,
SST
=
SS1158],1
(43)
The chains s h o w t h a t for a n y subperiod o f at least 11 years, the rainfall
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
165
pattern exhibits long-term behaviour but for subperiods of less than 11 years, the rainfall pattern is described by three subsystems (SS1, SS2 and SS5). In particular, we observe that from 1925 to 1950 the rainfall pattern is described by SSI only but after 1950 this subsystem is replaced by three subsystems (SS1, SS2 and SS5).
Runoff analysis The long-term behaviour of the system was determined as described in the previous section. For example, for the Metsemotlhaba at Thamaga, the transition matrix for the entire flow record is /0.4333
0.1282
0.0397~
P = /0"3167
0.2180
0.06941
\0.2500
0.6538
(44)
1
0.8909/
and the limiting stochastic matrix calculated from (44) is
1-I --
0.0808
0.1050
0.8142
0.0808
0.1050
0.8142
0.0808
0.1050
0.8142
(45)
The 0-1 matrix & obtained by raising 11 to the power zero, yielded a total reduction S o of the type (16). Evidently, the long-term behaviour of the flow states is that they are strongly connected, aperiodic and stationary, with degree of stationarity one. Analysing the subperiods for topological conjugacy as outlined above, we find that for 2 t> 15, for Thamaga, we have a~') > 0 for n >~ 1, where a~ ) is defined as before. This shows that for any subperiod of the flow record of at least 15 years, the flow states are strongly connected, aperiodic and stationary, with degree of stationarity one. This behaviour is analogous to the long-term behaviour. In the terminology described above, the subperiods possess similar flow dynamics. This result is also true for Morwa for 2 t> 18 and Gaborone dam for 2 ~> 30. It may be noted that the minimum length of flow record required for the subperiods to possess similar dynamics to the long-term pattern is shortest for the Metsemotlhaba at Thamaga, where the drainage area is smallest (i.e. 1300 km 2) and longest for the Notwane at Gaborone dam, where the drainage area is largest (i.e. 4200 km2). A detailed examination of the flow records for Thamaga for 2 < 15 revealed that during the period 1924-1985 the flow pattern was described by three subsystems (SS1, SS2 and SS5). Analysis for chronological structure for
E.M. Lungu / Journal of Hydrology 147 (1993) 153-167
166
various subperiods led to identification of the following chains: f o r 2 = 11,
SST
=
SS1130]~ ~ SS214]~, ~ SS115],~ ~ SS513]~ ~ SS119],1
(46)
f o r 2 = 12,
S S T = SS1131112 ~ SS213]~2 --, SS116]~2 --, SS512],2 ---, SS119]~2 for). = 13,
(47)
SST
(48)
=
SS1131]~3 -4 SS212]~3 ~ SS117]t3 ~ SS5[1]~ 3 --~ SS119], 3
for). = 14,
SST
=
SS1131],4 ~ SS2[1],4 --~ SS1[17],4
(49)
for). = 15,
SST
=
SS1148]15
(50)
For ). /> 15 the subperiods possess similar flow dynamics and the flow pattern is described by SS1 only, whereas for 2 < 15 three subsystems (SS1, SS2 and SS5) describe the flow pattern. Moreover, it is found that from 1924 to 1959 the flow pattern is described by SS1 only but after 1959 this subsystem is replaced by three subsystems (SS1, SS2 and SS5). The occurrence of SS5 shows that significant changes in the flow pattern have taken place. These results are found to be true for the Metsemotlhaba at Morwa for 2 < 18 and for the Notwane at Gaborone dam for 2 < 30. Results and discussion Spectral analysis has revealed that the runoff is highly stochastic and that the residual component is the most important component, as it explains about 38-53% of the variance of the total series. However, spectral analysis did not provide evidence for a climatic change. This may be attributed to, among other things, the short record of the time series and probably to the insensitivity of the method. To answer the sensitivity question, a different technique, known as topological conjugacy, was employed. This technique is based on the fact that if two systems are dynamically different, they produce unique graphs. The technique achieved two results. First, the cutoff length in years above which subperiods preserve the dynamics of a system is determined for each station. For runoff, it is found that this length is shortest for the Metsemotlhaba at Thamaga, which has the smallest drainage area, and longest for the Notwane at Gaborone dam, which has the largest drainage area. It is evident from this result that the smaller the drainage area the easier it is to obtain runoff of
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various magnitudes, as the rainfall is usually able to cover the entire drainage area and the threshold rainfall (that is, the initial abstraction of rainfall required to satisfy interception, depression storages and infiltration) is easily exceeded. For a large drainage area, on the other hand, the threshold is not easily exceeded, mainly as a result of the localized nature of rainfall in Botswana. Second, a detailed understanding of the dynamics of a system has been attained through the classification procedure introduced in this study. If we subdivide the rainfall and runoff records into two periods, i.e. 19241950 and 1950-1985, then it is easy to see that both rainfall and runoff patterns have behaved differently in each period. In particular, we find that in the first period of the record, both rainfall and runoff are described by subsystem SS1 only, whereas after 1950 this is replaced by at least three subsystems. We note that these changes take place across the country. A study of climatic change and variability in Southern Africa (Tyson, 1986) has described dry spells during the period 1962-1971 and wet spells during the period 1971-1981. Furthermore, a report on the climate of Botswana (Bhalotra, 1987) has described major droughts in the calendar years 19811982 to 1986-1987. This period of variable rainfall (i.e. 1962-1987) corresponds more or less to the findings of this paper. It is evident from the results of this study that rainfall and runoff patterns have changed. After 1950, the rainfall pattern has become variable, as a result of the occurrence of extreme events such as droughts and wet spells (Tyson, 1986). Changes in runoff pattern suggest, in turn, changes in rainfall pattern, as well as modification of the land surface largely as a result of overgrazing and the covering of parts of the catchment with impervious roofs, sidewalks, roads and parking lots. References Bailey, N.T.J., 1964. The Elements of Stochastic Processes. Wiley, New York, pp. 38-57. Bhalotra, Y.P.R., 1987. Climate of Botswana, Part II. Department of Meteorological Services, Garborone, Republic of Botswana, pp. 1-40. Kite, G.W., 1989. Use of time series analysis to detect climatic change. J. Hydrol., 111: 259-279. Tyson, P.D., 1986. Climatic Change and Variability in Southern Africa. Oxford University Press, Cape Town, pp. 69-92. Williams, R.F., 1973. Classification of subshifts of finite type. Ann. Math., 98:120-153; Errata, Ann. Math., 99 (1974): 380-381. Yevyevich, V., 1972. Probability and Statistics in Hydrology, 1st edn. Water Resources Publications, Fort Collins, CO, p. 50.