Hydrological Sciences - Journal - des Sciences Hydrologiques, 30,4, 12/1985

Stochastic characteristics of rainfall-runoff processes in Zambia

T. C. SHARMA Water Resources Research Unit, National Council for Scientific Research, PO Box Ch.158, Che Is ton, Lusaka, Zambia

ABSTRACT Monthly and annual rainfall and runoff sequences from drainage basins in Zambia, central Africa, were subjected to stochastic analyses. The rainfall sequences tended to follow the normal probability dis­tribution and the runoff sequences the lognormal probability distribution. The deterministic periodic component explained more than 60% of the variance in the log-transformed monthly runoff sequences. The stochastic component behaved as a random process in the monthly rainfall sequences and as an autoregressive moving average (ARMA (1,1) or ARMA (1,0)) process in the monthly runoff sequences. The annual rainfall sequences resembled a random process and the annual runoff sequences a Markov or first order autoregressive process (AR (1)). First order linear discrete dynamic models represented the interaction between rainfall and runoff processes on both monthly and annual bases. The runoff models performed satisfactorily for one-step-ahead forecasting.

Caractéristiques stochastiques des processus pluies-débits en Zambie RESUME Des séquences de précipitations mensuelles et annuelles et d'écoulements sur les bassins versants de Zambie, Afrique centrale, ont été soumises à des analyses stochastiques. Les séquences de précipitations tendent à suivre une loi de probabilité normale et les séquences de debits une loi de probabilité lognormale. La composante périodique déterministe explique plus de 60% de la variance dans les séries d'écoulements mensuels (logs). La composante stochastique se comporte comme un processus purement aléatoire dans les séquences de précipitations mensuelles et comme un processus de moyenne mobile autorégressive (ARMA(l.l) ou ARMA (1,0)) dans les séquences d'écoulements mensuels. Les séquences de précipitations annuelles ressemblent à un processus aléatoire et les séquences d'écoulement annuel à une chaîne de Markov ou à un processus autorégressif du premier ordre (AR (1)). Les modèles dynamiques linéaires discrets du premier ordre représente l'interaction entre les processus pluies et débits sur les bases mensuelles et annuelles. Les modèles d'écoulements se comportent de façon satisfaisante pour la prévision "one-step-ahead".

497

498 T.C. Sharma

INTRODUCTION

A number of African countries have well-developed hydrometeorological recording systems. The Republic of Zambia maintains a substantial raingauge network, and records for more than 50 daily rainfall stations cover more than 35 years. Likewise, a reasonable network of river runoff gauging exists, and records of runoff data are available in published form for about 20 years for more than 20 gauging stations.

Zambia is a tropical country (area 746 256 km2) bounded between latitudes 8°01'S and 18°01'S (Fig.l). The outstanding feature of

Fig. 1 River basins of Zambia.

Zambia's rainfall is that the bulk is concentrated into six months (November-April). Most rainfall is associated with the movement of Congo air and the intertropical convergence zone. The weather system is therefore characterized by two seasons, viz. a rainy season in the summer and a dry season in the winter. During the wet season (November-April), the mean monthly temperature is about 21 C, the mean relative humidity is 80%, and wind speeds are 2-3 km h . The

Stochasticity of rainfall-runoff in Zambia 499

months of June-July-August are cool and dry with the temperature dropping to 10°C and wind speeds ranging from 4 to 6 km h- . Maximum temperatures in September-October are upto 35°C and the relative humidity ranges from 33% to 55%.

Zambia comprises the following drainage basins: Kafue, Chambeshi, Luangwa, Lake Tanganyika and Luapula basin (Fig.l). Major copper mining installations, farms and industrial plants are situated in the Kafue basin. The major hydroelectric plant in Zambia is located at the Kafue Gorge. Transportation, communications and other amenities are well developed in this basin. The next basin of importance is the Chambeshi, which offers immense potential for future development. The Government of Zambia has laid a great emphasis on developing agricultural activities in the Kafue and Chambeshi basins. This paper reports studies of the rainfall and runoff to yield information for water resources development projects in these two basins.

DESCRIPTION OF DRAINAGE BASINS

Kafue basin

Upstream of the Kafue hook bridge, the Kafue basin encompasses a drainage area of 93 952 km2. The Kafue Biver is the main drainage channel of the basin with the Lunga River an important tributary (Fig.l). Mean annual rainfall varies from 900 to 1300 mm south to north. Most of the basin lies at an elevation of some 1000-1300 m a.m.s.l. The terrain ranges from undulating to flat but includes hills. A feature of the basin is the Lukanga swamps (area 2500 km2) and marshy lands known as "dambos" which have peaty soils and form about 10-12% of the basin. Most of the basin is underlain by the Katanga-Kandelungu formation. In the northern part of the basin, where the rainfall is high, soils are leached sand veldt. These are sandy to medium textured soils. Moderately fine textured reddish soils can be seen in the regions between the Lukanga swamps and the Kafue hook bridge. The vegetation consists of woodlands or bush with a varying density of grass on drier land. Common species in the woodlands are Brachystegia and Combretum. Aquatic grasses in swamps such as wild rice are attractive stock feed. Much land is suitable for grazing and cropping. The northwestern part of the basin is infested by the tsetse fly. There is some fishing in the Lukanga swamps.

Chambeshi basin

Upstream of the Chambeshi road bridge, the area of the basin is 33 820 km2. The elevation in the basin ranges from 1738 to 1143 m a.m.s.l. The main soil types of the basin are leached sand veldt (75% of the area), seasonally water-logged soils (8% of the area), flood plain soils (8% of the area) and leached red clays (6% of the area). The main lithological units are quartzites and sandstones of the kibran system in the eastern ridge, alluvium in the Chambeshi valley, granites in the headwater region, and shales, mudstones and sandstones in the Chambeshi-Bangweulu plains region. The headwater

500 T.C. Sharma

regions of the Chambeshi basin are undulating uplands which are crossed by water courses. Vast flat areas occur in the alluvial valley bottom of the Chambeshi River, where much of the land is flooded during or after the rainy season. Swamps and dambos are common in the Chambeshi plains. The vegetation of the basin can be broadly divided into two main types: Brachystegia woodlands that dominate the plateau region, and a mixture of Combre turn and dry evergreen forest which predominates over the Chambeshi-Bangweulu swamps.

ACQUISITION OF DATA AND METHODS OF DATA ANALYSES

Monthly and annual rainfalls were abstracted from the publications Monthly and Yearly Rainfall Totals published by the Department of Meteorology, Zambia. Monthly and annual river discharges were abstracted from the Hydroïogical Year Books published by the Depart­ment of Water Affairs, Zambia. Data from the 13 rainfall stations in the Kafue basin and 10 stations in the Chambeshi basin were used for characterizations of the rainfall processes and for discrete dynamic modelling of rainfall-runoff processes. The areal estimates of monthly and annual rainfall depths were obtained through arith­metic averaging of rainfall depths from the raingauge stations. Runoff data from 20 gauging stations in the Kafue basin and eight stations in the Chambeshi basin were subjected to analyses. The results reported herein are for a limited sample of gauging stations. The stations used are thought to represent typical conditions within the basins and to possess satisfactory quality in the data base. Full details are given by Sharma (1982, 1984).

The monthly and annual rainfall and runoff sequences were analysed for their probability distributions. Harmonic and stationary stochastic processes were used to describe the temporal variations of the rainfall and runoff sequences. Stationary stochastic compon­ents in the processes were represented by autoregressive moving average models (Box & Jenkins, 1976). Discrete dynamic linkage relationships between rainfall depths (input) and runoff depths (output) were fitted using autocorrelation and cross-correlation functions. Most of the computations were done on an ICL 2904 computer at the University of Zambia.

RESULTS AND DISCUSSION

Stochastic characteristics of monthly and annual rainfall sequences

The monthly rainfall sequences in the Zambian environments were found to follow approximately the normal probability distribution (Fig.2). It can be assumed that monthly rainfall sequences are composed of deterministic (periodic) and stochastic components. The deterministic and stochastic components in a monthly cyclic series can be separated by subtracting the monthly means and dividing by the monthly standard deviations. Thus:

zi ~ ^xp,t xt)//st (1)

Stochasticity of rainfall-runoff in Zambia 501

(WW) l lVdNIVH AiaVBA

f -«O»

\ . a»<3

UJ

o

t ' S i O l d AlHiNOW

(WW) T W d N l V d AlHiNOW

502 T.C. Sharma

in which ZJ = stochastic component with zero mean and unit variance; xp t = value of rainfall (mm) in the year p and month t; x t = mean of the monthly rainfalls for month t; and st = standard deviation of the monthly rainfalls for month t. In order to obtain the z^ component, one requires 12 means (x^) and 12 standard deviations (s^). However, in the climatic environments of Zambia, six means and six standard deviations would suffice (six months are rainy and the rest are dry). It is to be noted that the mean daily rainfall is nearly zero for the months of May to October in the Zambian environments, and is positively valued for the months November to April. These positive values of mean daily rainfall based on historical data for 46 years for a typical gauging station are shown in Fig.3. An examination of the mean daily rainfall depths

STATION - KATAMB MISSION

10 20 30 9 19 29

MARCH APRIL

Fig. 3 Temporal variation of mean daily rainfall in Chambeshi basin.

in the wet season (Fig.3) indicates similarity of rainfall over the months December to March, while rainfalls in November and April appear significantly less. The similarity of the December to March characteristics was further examined by applying tests for homogen­eity of variance (chi square test, Freund, 1971) and equality of means (analysis of variance tests, Freund, 1971) to the monthly rainfall totals of these months. These tests suggested that, for the majority of the raingauge stations in Zambia, the variances and means of monthly rainfalls from December to March are insignificantly different from each other (Table 1). In other words £3 = X4 = £5 =

Stochasticity of rainfall-runoff in Zambia 503

Table 1 Mean and coefficient of variation for monthly (November to April) and annual rainfall in Zambia

Station

Rosa Mission (50 years of data)

Kayambi Mission (46 years of data)

Chinsali Mission (51 years of data)

Chibwa Mission (45 years of data)

Chalabesa Mission (36 years of data)

Parameter

1 2

1 2

1 2

1 2

1 2

November

129 0.56

96 0.71

91 0.56

140 0.52

130 0.63

December

246 0.36

218 0.36

233 0.37

245 0.34

253 0.37

January

216 0.32

226 0.41

258 0.35

231 0.32

261 0.34

February

219 0.36

218 0.48

201 0.37

227 0.32

243 0.30

March

284 0.37

218 0.39

223 0.38

262 0.37

265 0.36

April

104 0.57

87 0.53

48 0.75

93 0.50

58 0.71

Annual total

1239 0.17

1091 0.17

1088 0.20

1250 0.16

1214 0.18

Parameter 1 = mean (mm); and 2 = coefficient of variation.

xg = x and S3 = S4 = S5 = S6 = s, in which subscript 3 denotes December and subscript 6 denotes March. Thus the non-zero parameters involved in deducing the stochastic component are x2, 5, X7 and S2, s, S7. The stochastic sequence, z±, deduced using equation (1) was subjected to autocorrelation analysis. That analysis indicated a very weak persistence pattern in the stochastic component suggesting that the Zj component behaved like a white noise or random process ai with zero mean and variance s|. The randomness of the zj component was further confirmed by low values of the Portmanteau (chi square) statistic and a run statistic (Freund, 1971).

The annual rainfall, spread over the months November to April, displayed the behaviour of a normally distributed probability function (Fig.2). Autocorrelation analysis of the annual rainfall sequences, X4, indicated a very weak persistence pattern suggesting that Xi is a random sequence with mean X and variance Sx .

Periodic-stochastic characteristics of the monthly discharge sequences

The monthly discharge sequences (m3s ) in the Zambian environment tend to follow the lognormal probability distribution (Fig.4). Therefore the data sequences were log-transformed before subjecting them to stochastic analyses. The log-transformation made the variances of monthly runoff sequences uniform (chi square values in Table 2) and periodicities can therefore be considered reflected in monthly means (Yevjevich, 1972). Denoting qp t as a monthly dis­charge sequence, the following decomposition can be written in terms of deterministic and stochastic components:

q = m + z (2) p, t t 1

in which 1p t = discharge (with log-transformation so as to render the

sequence homogeneous in variance and normally distributed) in month t of year p;

m^ = periodic value reflected in the monthly mean values (t = 1 for October and t = 12 for September for a hydrological year);

504 T.C. Sharma

ZJ = stochastic component which evolves independently of the m^ component; and i takes on values from 1 to 12 times the number of years used for analyses.

Harmonic analyses of monthly means (m^) were done and the first two harmonics were found significant. The temporal variations of monthly means (mt) can therefore be written as:

A + Z2. j=l 3

A. cos ,_ + I 12 j=l "J

B-i sin 2TTtj 12 (3)

in which A0, Aj and Bj are harmonic coefficients. The first harmonic seemed highly dominant (more than 55% variance explained) while the

3-0 -

2 0

1-5

3 10

7-0

5-0

4-0

3.0

2-0

<£ 10 < x % 8-0 o

6 0

5 0

4-0

2 0

LEGEND

• • D a a

a a a •

o

A

•

• * •

X a

a •

0

A

•

• k

m

X

a D a

OCTOBER

DECEMBER

FEBRUARY

APRIL

JUNE

AUGUST

YEARLY

X X

* 1

, x x

' X X , l x x ,

±Ai X X

I A A AA

A A ;

A

? i A *

10 20 30 40 50 60 70 81

PROBABILITY OF EXCEEDANCE (%)

Fig. 4 Distribution of discharges for Kafue hook bridge gauging station.

Stochasticity of rainfall-runoff in Zambia 505

Table 2 Estimates of variances and parameters for the log-transformed discharge sequences in Zambian drainage basins

Station ident i f icat ion (number,area k m 2

sample size years)

Chi square Variance % variance explained % variance % variance Values of Value of statistic of the by periodic explained of the A R M A the for month ly component (first by A R M A whi te parameters Markov un i f o rm i t y runoff and second (1 ,1 ) or noise in the parameter of sequences harmonics) A R I D sequence mon th l y <p1 in the variance* (logs) in the month ly component in the sequences: annua! in log sequences in the month ly <j>l 9 , sequences domain month ly sequences

sequences

Chipoma falls in the 8.98 Chambeshi basin (6-242,632,18)

Chambeshi road 14.08 bridge (6-289, 33820,20)

Kafironda on 11.64 Kafue (4-090, 7066,21)

Machiya ferry on 11.25 Kafue (4-280, 22656,19)

Chi fumpa pontoon 9.05 on Lunga (4-560, 21197,21)

Kafue hook bridge 10.97 on Kafue (4-669, 93952,28)

0.824 79.78(69.47,10.31) 11.78 0.071 AR(1)

1.052 76.31(74.50,1.81) 17.42 0.069 A R I D

1.204 72.72(70.05,2.67) 20.33 0.084 A R M A d . D

1.189 76.36(72.20,4.16) 17.32 0.075 ARMA(1 ,1 )

0.764 60.51(55.56,4.93) 27.84 0.089 AR(1)

1.232 68.88(65.15,3.73) 25.73 0.066 A R M A d . D

0.76 0.00 0.28

0.85 0.00 0.25

0.85 - 0 . 0 5 0.50

0.81 - 0 . 1 6 0.27

0.84 0.00 0.24

0.85 - 0 . 3 0 0.39

'Cr i t i ca l value of chi square statistic at 5% level of significance and 11 d.f. is 19.67.

second harmonic was weak (less than 5% variance explained). The contribution of the remaining harmonics was very weak (less than 1% variance explained) and was therefore disregarded.

Autocorrelation analysis for the z± component revealed that the stochastic component followed an ARMA (1,1) or ARMA (1,0) process, as follows:

ARMA (1,1) i-1 i-1

(4)

ARMA (1,0) i-1

(5)

in which Zj_ = sequential values of the stochastic sequence; §1 = autoregressive parameter; Q± = moving average parameter, and &± = a white noise sequence with zero mean and homogeneous variance

The stochastic component appeared to be greatly dominated by first order persistence trends and explained a variance of the order of 20-45%. The contribution of higher order persistence terms was low and could be represented by a moving average parameter 6j_. Values of <j>i and &i were obtained adopting procedures advanced by Box & Jenkins (1976). The value of <j>i ranged from 0.76 to 0.85 and 9i from -0.30 to -0.05 (Table 2), The stochastic component, zi, followed an ARMA (1,1) process for the gauging stations in the Kafue basin. An ARMA (1,0) or Markov process appeared satisfactory for the gauging stations in both the Chambeshi basin and the Lunga sub-basin

506 T.C. Sharma

of the Kafue basin. The deterministic component (harmonic + ARMA (1,1)) yielded an explained variance exceeding 90% and the remaining 10% component was found to resemble that of a white noise sequence.

Stochastic characteristics of the annual discharge sequences

The annual discharge sequences tended to be lognormally distributed for most of the stations in the basins studied (Fig,4). In most cases, the autocorrelation at lag 1 was high, which is an indication of the first order carryover or persistence tendencies in the discharge sequences. The simplest form of persistence through a Markov process was assumed and checked for its adequacy. This led to considerable reduction in values of autocorrelations at lag 1 and Portmanteau statistics of residuals for the stations in question. Therefore annual discharge sequences can be represented by a Markov process as follows:

<Q± - Q> - VQ i - l " Q> = ai (6>

in which Qi = sequential values of log of annual discharges; Q = mean value of Qi sequences; (t>l = autoregressive parameter; and a^ = a white noise sequence with zero mean and homogeneous variance

2

sa. The persistence term in the stochastic component yielded an explained variance from 5 to 35%. The value of the autoregressive parameter, cf>i, varied from 0.20 to 0.59 (Sharma, 1982 and 1984).

It was noted that the persistence was weak (<f>-̂ close to 0.20) in the headwater regions of basins, where swamps and flood plains were almost absent. Runoff sequences for the gauging stations downstream of swamps (Lukanga Swamps in the Kafue basin) and flood plains (Chambeshi plains in the Chambeshi basin) tended to show higher persistence (Çl^ close to 0.40), which is reasonable because of the natural regulation afforded by these water-storing bodies. In headwater regions, where annual carryover effects are small, one could construct a simple water balance model that annual rainfall = annual runoff + annual évapotranspiration. Annual rainfall and runoff data for 16-20 years for gauging stations in headwater regions were analysed to evolve the aforesaid water balance components. It was found that the ratio of annual runoff to annual rainfall ranged from 0.16 to 0.23 with a mean value of 0.20. In other words about 20% of the annual rainfall is transformed into runoff and the remainder, about 80%, is lost through évapotranspiration.

A discrete dynamic model of monthly rainfall-runoff processes

Linear models were fitted between monthly rainfall depths as input and the logarithms of monthly runoff depths as output. The model-fitting procedures indicated that a first order dynamic model with memory in the output process as well as memory in the input process produced the highest values of R2, the minimum values of residual variance and the most satisfactory nature of the residuals when compared with memoryless and second order dynamic models. The model

Stochasticity of rainfall-runoff in Zambia 507

s t r u c t u r e r e s u l t e d i n t h e f o l l o w i n g form:

y . - v„ y . „ = c + w x . + w„ x . _ + a (7) l 1 l - l o l 1 l - l l

i n which y± = sequential values of output (logs of the monthly runoff depths); xj = sequential values of input (monthly rainfall depths) ; vi, c, w0, wi = system parameters; and a^ = a white noise sequence with zero mean and variance sa uncor­

rected with xi sequences, Values of system parameters vi, c, w0, wi were estimated using a least squares algorithm involving correlation functions of x^ and yi sequences. The level of determination shown by equation (7) was very high (more than 90% variance explained) and the random (white noise) component played a minor role (less than 10% variance explained). The model is parsimonious as only five parameters are needed to describe the dynamic interaction between rainfall and runoff. Values of the system parameters are shown in Table 3.

Table 3 Values of parameters in the linear discrete models of rainfall to log-transformed-runoff processes in Zambian drainage basins

Month ly rainfal! to runof f model Annual rainfal l to runof f model Stat ion number c v, w „ w , sa R2 (%) c

2.521 0.000

1.843 0.000

0.267 0.000

-1.138 0.000

-1.618 0.000

-1.873 0.000

V l

0.170 0.163

0.332 0.321

0.419 0.426

0.332 0.354

0.496 0.493

0.344 0.343

w0

0.005 0.486

0.008 0.622

0.013 0.596

0.024 0.919

0.023 0.787

0.028 0.836

%

0.283

0.395

0.295

0.118

0.148

0.210

R2 (%)

31.80

58.58

62.14

92.52

91.24

85.51

6-242 - 0 . 3 2 0 0.734 0.044 0.004 0.301 89.01 0.000 0.731 0.557 0.054

6-289 - 0 . 4 8 0 0.792 0.020 0.026 0.207 95.85 0.000 0.794 0.288 0.294

4-090 - 0 . 4 1 1 0.854 0.035 0.011 0.206 96.85 0.000 0.861 0.382 0.126

4-280 - 0 . 4 9 8 0.789 0.027 0.020 0.204 96.48 0.000 0.789 0.308 0.228

4-560 - 0 . 3 8 3 0.809 0.027 0.006 0.254 91.54 0.000 0.811 0.391 0.083

4-669 - 0 . 5 8 0 0.821 0.024 0.022 0.231 95.66 0.000 0.823 0.259 0.237

The upper line shows the values of parameters in the non-standardized f o rm and the lower line in the standardized f o r m . Values of the parameters pertain to the rainfall and runof f sequences expressed in cent imetres.

Since the dynamic interaction between rainfall and runoff processes on a monthly basis was highly deterministic (R2 greater than 90%) one could disregard the white noise term in equation (7) in order to understand the dynamic behaviour of the runoff process in response to rainfall. Equation (7) can thus be written in the standardized form as :

y. - v y = w x. + w x (8) 1 1 1-1 O 1 1 1-1

in which v-̂ , w0 and w^ = system parameters in the standardized form. In the backshift operator form, i.e. yi_i = Byj (Box & Jenkins, 1976), equation (8) can be written as:

508 T.C, Sharma

y. - v„ By. = w x. + w„ x. „ (9)

Further decomposition of equation (9) yields:

w0 Vi + w r

y± = wo x i +

direc runoff baseflow

(10)

\ 1 - v-, B direct v x

The fractions in equation (10) show that there are two linear subsystems operating in parallel to yield the output yi. Monthly runoff from the basins can be thought of as occurring from direct runoff and baseflow subsystems. The direct runoff component behaves as a memoryless process and the baseflow component is associated with first-order memory. This implies that the rainfall of a given month would produce the direct runoff in that very month whereas the baseflow would begin appearing from the next month. There is a lag of one month between direct runoff and baseflow components of the runoff process. This finding is of use while carrying out monthly hydrograph analyses for the separation of baseflow and other com­ponents of runoff.

A discrete dynamic model of annual rainfall-runoff processes

Experience in modelling the monthly rainfall to runoff processes encouraged application of system concepts to modelling these proces­ses on an annual basis. Therefore the memoryless linear model was applied to logs of annual runoff depths as output with annual rainfall depths as input. The annual runoff depths were log-transformed to suit the lognormal probability distribution of the annual runoff depth sequences. The R values of the memoryless model ranged from 27 to 80%.

Diagnostic statistics such as R , the cross-correlation structure of the residuals, and autocorrelations of the residuals suggested the possibility of introducing a memory component into the model. Therefore a first order memory term was added to the output process and the model was fitted to the data sequences. Introduction of the memory term increased R values by more than 10% in the majority of cases. The characteristics of the residuals resembled a white noise process very closely. The model structure on a yearly basis therefore resulted as:

y. - v, y. , = c + w x + a. (il) 1 1 1-1 O 1 1

in which yi = sequential values of logs of yearly runoff depths; XJ[ = sequential values of rainfall depths; c> vl> *o = parameters; and a^ = a white noise component with zero mean and variance sa

2, Values of the parameters are shown in Table 3.

Stochasticity of rainfall-runoff in Zambia 509

Potential of runoff models for forecasting

The univariate stochastic model, equations (2) - (5), was used to forecast monthly discharges for a lead time of 12 months adopting procedures advanced by Box & Jenkins (1976) and McKerchar & Delleur (1972). The bivariate rainfall-runoff model (discrete linear input-output relationship), equation (7), was used to forecast the discharge using the mean rainfall for the 12 months ahead. It was observed that forecasts based on a bivariate rainfall-runoff model were closer to observed sequences when compared with a univariate stochastic model (Fig.5). However, large deviations in

•oo-UNIVARIATE MODEL -«--BIVARIATE MODEL -tw—MEAN MONTHLY DISCHARGE -•-•-OBSERVED DISCHARGE

Fig. 5 Forecasting of monthly discharges for 12 months ahead in the Kafue basin.

relation to the observed discharges were evident using both models on long lead-time forecasts. It was, however, noted that the forecasts for many months ahead, particularly the seventh month from the origin and onwards, tended to converge to the mean discharges for those months (Fig.5).

The reliability of the aforesaid models for one-month-ahead adaptive forecasting was investigated. In the case of the univariate stochastic model, the observed discharge for the current month was used to forecast the discharge for the next month. In the bivariate rainfall-runoff model, the observed rainfall depth and discharge for the current month were used to forecast the discharge for the next month. Mean monthly rainfall was used as an estimate of rainfall for the month for which rainfall is yet to be observed. Such forecasting, which is based on the observed rainfall and runoff

510 T.C. Sharma

information of one step back is named herein as adaptive forecasting. Results based on adaptive forecasts indicated that closer fits were offered by the bivariate rainfall-runoff model (Fig.6). The ability of the bivariate rainfall-runoff model to fill gaps in monthly discharge sequences was also investigated. The actual observed rainfall depths were used for estimating discharges from October to September. The estimated and observed discharges compared reasonably well (Fig.7).

CONCLUSIONS

The following conclusions may be deduced regarding the stochastic characteristics of rainfall and runoff sequences in tropical basins of Zambia: 1. Monthly (November to April) and annual rainfall sequences in the Zambian environments can be approximated by a normal probability distribution. A monthly rainfall sequence is comprised of a deter­ministic periodic component and an independent stochastic component. The characteristics of the stochastic component approximated to a white noise process. The characteristics of rainfall from December, to March appeared identical. Annual rainfall sequences can be approximated by random or white noise sequences. 2. Monthly and annual discharge sequences tended to follow a lognormal probability distribution. Logarithms of monthly discharge sequences can be modelled through a linear additive model comprised of periodic and stationary stochastic components. The periodic component, consisting of the first two harmonics of the annual cycle, yielded an explained variance ranging from 60% to 75%. The stochas­tic component was represented by an autoregressive moving average process, ARMA (1,1), or an autoregressive process, AR (1), which accounted for an explained variance ranging from 15% to 30%. The total determinable component from the harmonic and ARMA (1,1) or AR (1) representations jointly explained nearly 90% of the variance in the process, leaving 10% as a white noise component. The ARMA (1,1) process fitted the stochastic component in the discharge sequences from one river whereas an AR (1) process adequately represented the stochasticity of discharges from the other rivers. The model performed satisfactorily in providing one-month-ahead adaptive forecasts. The performance of the model to provide 12-month-ahead forecasts was poor. 3. Logarithms of annual discharge sequences tended to resemble a first order autoregressive AR (1) process or a Markov process. The value of the first order autoregressive parameter (<j>i) , indexing the level of persistence, varied from 0.20 to 0.50. 4. A first order bivariate rainfall-runoff model with memory in the output sequences as well as in the input sequences represented the dynamic interaction between monthly rainfall to log-transformed runoff processes. The model structure, consisting of five parameters, explained nearly 90% of the variance leaving 10% close to a white noise process. The model appeared reliable for one-month-ahead adaptive forecasting and filling gaps in monthly runoff sequences. 5. A first order bivariate model with weak memory in the output process and no memory in the input process adequately represented

Stochasticity of rainfall-runoff in Zambia 511

STATION 4-090 KAFIRONDA

UNIVARIATE MODEL BIVARIATE MODEL OBSERVED DISCHARGE

350-0

30O0

2500

2000

150-0

1000

50-0

0-0

. STATION 4-280 MACHIYA FERRY

1981/82

m

//

\ ft

\Vy VVta

1 2 3 L 5 6 7 8 9 10 11 12

Fig. 6 One-month-ahead adaptive forecasting for the Kafue basin.

BIVARIATE MODEL

OBSERVED DISCHARGE

3500

30OO-

250-0

200-0

150-0

IOO-0

50-0

0-0

STATION 4-280

MACHIYA FERRY

1981/82

Fig. 7 2 3 4 5 6 7 8 9 10 II 12

Filling the monthly discharges in the Kafue basin.

512 T.C. Sharma

dynamic interaction between annual rainfall and log-transformed runoff processes. The determinable portion in the process accounted for an explained variance ranging from 30 to 90% and the remaining portion was found to be indistinguishable from a white noise process.

ACKNOWLEDGEMENTS The study was carried out under the auspices of the project "Water Resources Inventory of Zambia". The author gratefully acknowledges the support of the National Council for Scientific Research, Zambia,in carrying out the studies.

REFERENCES

Box, G.E.P. & Jenkins, G.M. (1976) Time Series Analysis: Fore­casting and Control. Holden Day, San Francisco, California, USA.

Freund, J.E. (1971) Mathematical Statistics. Prentice Hall, Inc., Englewood Cliffs, New Jersey, USA.

McKerchar, A.I. & Delleur, J.W. (1972) Stochastic analysis of monthly flow data-application to lower Ohio t r ibutar i e s . Water Resources Research Tech. Report no. 26, Purdue University, Indiana, USA.

Sharma, T.C. (1982) Water resources inventory of Zambia: Chambeshi basin. Water Resources Research Report no. NCSR/TR50, National Council for Scientific Research, Lusaka, Zambia.

Sharma, T.C. (1984) Characteristics of the runoff processes in the upper Kafue basin. Mater Resources Research Report no. NCSR/TR51, National Council for Scientific Research, Lusaka, Zambia.

Yevjevich, V. (1972) Stochastic Processes in Hydrology. Water Resources Publications, Fort Collins, Colorado, USA.

Received 2 August 1984; accepted 6 June 1985.