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Modelling outflow from the Jao/Boro River system in the Okavango Delta, Botswana A. Gieske University of Botswana, Geology Department, P/Bag 0022, Gaborone, Botswana Received 15 January 1996; accepted 1 May 1996 Abstract The mean annual input, including both surface flow and rainfall, into the Okavango Delta (Bots- wana) is about 16 billion m 3 . Between 96 and 98% of this returns to the atmosphere by evapotran- spiration, only 2% leaving the delta by surface outflow through the Thamalakane and Boteti Rivers. It remains unclear how much is infiltrating into the groundwater below the delta. A severe problem is the extreme variability of the outflow, which is of great importance for the water supply of Maun and downstream areas along the Boteti where groundwater is scarce and usually saline. Previous modelling of the surface outflow, using over twenty years of rainfall, evapotranspiration and stream flow data, has revealed fluctuations in outflow which could not be explained with one single parameter set. Attempts were made to interpret this behaviour in terms of flow system changes occurring within the delta, or simply as an effect caused by the poor quality of rainfall measurements. This paper shows that substantial model improvement is possible when both long and short term antecedent climatic conditions are taken into account. The time span of the long term conditions, which is in the order of 10 years, seems to be expressed by shallow groundwater level variations in the lower delta, while the short term conditions, with a memory of one year or less, are linked to a combination of peak rain and flood events. The modelling therefore suggests that the outflow variations observed during the period of record are neither caused by physical changes in the delta nor result as an artefact of errors in the rainfall data. They are indicated to be entirely the result of climatic variations. q 1997 Elsevier Science B.V. 1. Introduction The confluence of the Angolan rivers Cubango and Cuito at the boundary between Namibia and Angola just west of the Caprivi Strip (Fig. 1) marks the beginning of the Okavango River which enters Botswana at Mohembo. After flowing through the Panhandle, a narrow swamp confined on both sides by high shoulders of Kalahari sand, 0022-1694/97/$17.00 q 1997– Elsevier Science B.V. All rights reserved PII S0022-1694(96)03147-2 Journal of Hydrology 193 (1997) 214–239

Modelling outflow from the Jao/Boro River system in the Okavango Delta, Botswana

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Page 1: Modelling outflow from the Jao/Boro River system in the Okavango Delta, Botswana

Modelling outflow from the Jao/Boro River system in theOkavango Delta, Botswana

A. Gieske

University of Botswana, Geology Department, P/Bag 0022, Gaborone, Botswana

Received 15 January 1996; accepted 1 May 1996

Abstract

The mean annual input, including both surface flow and rainfall, into the Okavango Delta (Bots-wana) is about 16 billion m3. Between 96 and 98% of this returns to the atmosphere by evapotran-spiration, only 2% leaving the delta by surface outflow through the Thamalakane and Boteti Rivers.It remains unclear how much is infiltrating into the groundwater below the delta.

A severe problem is the extreme variability of the outflow, which is of great importance for thewater supply of Maun and downstream areas along the Boteti where groundwater is scarce andusually saline. Previous modelling of the surface outflow, using over twenty years of rainfall,evapotranspiration and stream flow data, has revealed fluctuations in outflow which could not beexplained with one single parameter set. Attempts were made to interpret this behaviour in terms offlow system changes occurring within the delta, or simply as an effect caused by the poor quality ofrainfall measurements.

This paper shows that substantial model improvement is possible when both long and short termantecedent climatic conditions are taken into account. The time span of the long term conditions,which is in the order of 10 years, seems to be expressed by shallow groundwater level variations inthe lower delta, while the short term conditions, with a memory of one year or less, are linked to acombination of peak rain and flood events. The modelling therefore suggests that the outflowvariations observed during the period of record are neither caused by physical changes in thedelta nor result as an artefact of errors in the rainfall data. They are indicated to be entirely theresult of climatic variations.q 1997 Elsevier Science B.V.

1. Introduction

The confluence of the Angolan rivers Cubango and Cuito at the boundary betweenNamibia and Angola just west of the Caprivi Strip (Fig. 1) marks the beginning of theOkavango River which enters Botswana at Mohembo. After flowing through thePanhandle, a narrow swamp confined on both sides by high shoulders of Kalahari sand,

0022-1694/97/$17.00q 1997– Elsevier Science B.V. All rights reservedPII S0022-1694(96)03147-2

Journal of Hydrology 193 (1997) 214–239

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the river spreads out into a delta-shaped system of swamps and distributary channelscovering an area between 6000 and 13 000 km2 depending on prevailing floods and pre-cipitation. The Okavango Delta is drained at its distal end by the Thamalakane and BotetiRivers through which the remaining water is finally carried towards the MakgadikgadiPan, the lowest point of the Kalahari Basin.

The Okavango River system dates back to pre-cretaceous times and may have oncelinked to the Orange River and perhaps later to the Limpopo or Zambezi Catchments.(Thomas and Shaw, 1991). Its course has been modified several times by upwarping of theEarth’s crust and fracturing in an extension of the East African Rift still continues. Owingto the very low relief (60 m altitude difference over a distance of 250 km) the waterspreads out over an enormous conically shaped area under structural control of the NEtrending rift faults.

In the Panhandle water is carried through a meandering but fairly stable river system.Below the Panhandle, a number of major distributaries can be distinguished (Fig. 1), themost easterly of which the Magwegqana or Selinda Spillway very occasionally carrieswater into the adjacent Linyanti River and hence to the Zambezi system. The most

Fig. 1. Location map of the Okavango Delta.

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westerly branch, the Thaoge River, drains into Lake Ngami. Historical trends indicate thatthe Thaoge was the major distributary in the last century (Shaw, 1984). In the early part ofthe 20th century, however, papyrus blockages developed and attempts to clear these werenot successful.

Most of the water is presently carried by the central Nqoga channel branching off againinto the Jao-Boro-Kunyere River system south of Chief’s Island and into the Maunachira-Khwai-Mboroga-Santantadibe Rivers north of Chief’s Island. The Santantadibe and BoroRivers drain into the Thamalakane, while the Khwai River carries the water towards theMababe Depression. The Kunyere River only occasionally reaches Lake Ngami. It shouldbe noted that below the Panhandle in the perennial swamps, water is not only carriedthrough the channels but also across the flooded plains. The same occurs in the seasonalswamps when flooded.

Records of river flow at Mohembo and Mukwe (Namibia) are available from 1933,whereas river stage levels at Maun have been monitored since 1950. Reliable flow recordsat Maun, however, start in 1968. Fig. 2 illustrates the average seasonal flood pattern in theOkavango at Mohembo (1933–1995) and in the Thamalakane at Maun (1969–1995).Average annual inflow at Mohembo is 11 billion m3 while the average outflow at Maunis in the order of 250 million m3. Since total input by precipitation is about 5 billion m3,the Maun outflow corresponds to about 2% of the total input. It is usually estimated that afurther 2% leaves the delta by groundwater flow, but because this figure is only based on asurface water balance study in one small catchment, Beacon Island in the seasonal

Fig. 2. Comparison of monthly inflow at Mohembo with monthly outflow through the Thamalakane at Maun.

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swamps, it should be regarded as a tentative estimate only. Total evapotranspiration istherefore close to 98%. Flood peaks in Mohembo generally occur in April but may varyaccording to rainfall patterns in the upper Angolan catchment. River stage levels at Maunusually peak in August, five months later than at Mohembo.

Water levels in the Panhandle vary by 1.7 m between seasons whereas in the perennialswamps, the difference between high and low levels is only 0.15 m. The reason for this isthat in the Panhandle flow is usually confined to the major channels whereas in perennialswamps flood waters are always spread out over a much larger area. In the lower seasonalswamps and Maun, the difference again increases to more than 2 m and channel bedsappear to be meandering through floodplains with many permanent islands which may bemore than 4 m above channel beds. Flow in the swamps is slow and takes place boththrough channels and over floodplains. Had only channel flow been responsible for themovement of the flood across the delta, then its travel time between Mohembo and Maunwould be much less than four months. In spite of this, channels serve as vital arteriessupplying water to outlying areas of the delta. If the supply is cut off, such areas dry outrapidly.

Detailed studies of the swamp’s hydrology were made by SMEC (1987), Dinc¸er et al.(1987), and a new analysis of the available data was recently made by the IUCN (Scudderet al., 1993), while the results of the Witwatersrand University research group with regardto channel evolution and sedimentation processes may be found in the many publicationsby McCarthy and co-workers, e.g. McCarthy and Ellery (1994, 1995). Surface waterchemistry has been studied by Cronberg et al. (1996).

Fig. 3. Basic four-cell model of the Jao/Boro flow system.

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2. Previous modelling

The first model was developed using one cell or compartment representing the entiredelta. This simple model was further developed into a four-cell model for the Boro systemand extended by adding more cells to comprise the entire delta (Dinc¸er et al., 1987, SMEC,1990) (see Fig. 3). Scudder et al. (1993) (IUCN) critically reviewed the most recentmodel and designed an alternative, incorporating changes in both model structure andinput data.

Basically the two models (SMEC and IUCN) use inflow at Mohembo, rainfall andevapotranspiration data to model outflow through the Thamalakane, Shashe and Bororivers, dividing the delta in a number of cells, essentially a series of reservoirs throughwhich the water is routed. A set of distribution parameters determined the flow from eachcell into neighbouring cells.

Some additional assumptions had to be made with regard to cell structure, area–volumerelationships, losses to groundwater and the nature of the hydraulic links between the cells.Differences between the SMEC and the IUCN approaches may be summarized as follows.

2.1. Volume–area relationships

Dincer et al. (1976) determined a volume–area relationship for a small study area insidethe delta near Beacon Island (Fig. 1) as

V =bAn (1)

whereb andn are empirical parameters. Eq. (1) means that the swamp area is modelled asa shallow bowl with a curved surface. Scudder et al. (1993) assumed a trapezoidal shapeand determined maximal an minimal areas from detailed flood maps (SMEC, 1990).

2.2. Cell discharge

Dincer et al. (1987) used

O=k(V −Vmin) (2)

whereO is the outflow andk andVmin are model parameters. The IUCN model used a formof Manning’s formula:

O=AcW2=3p

����

Gp

R(3)

whereAc is the cross-sectional contact area between cells,Wp the wetted perimeter,G thegradient andR the roughness.

2.3. Water balance

The SMEC model used

Vt +1 =Vt + I −O−At(ET −P) −gVt (4)

whereET is evapotranspiration,P rainfall, andg is a groundwater loss parameter.

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Scudder et al. (1993) employed the following relation:

Vt +1 =Vt + I −O−12(At +1 +At)(ET −P) − I loss(At +1 −At) (5)

whereI loss is a parameter relating losses to previous dry or wet conditions. This idea hadalready been discussed by SMEC (1990). Note that the IUCN model does not use lossesowing to groundwater recharge (parameterg). Both models used iterative techniques tosolve the water balance equations for each time step.

2.4. Evapotranspiration

The models used different routines to handle evapotranspiration. Conversion from PETto open water evaporation and actual evapotranspiration is not straightforward and con-siderable effort was put into refining evapotranspiration losses during wet periods. Thisaspect is discussed further in following sections.

2.5. Calibration

The SMEC model parameters had to be optimized manually by trial and error, whereasthe IUCN model used a non-linear optimization routine as in Rosenbrock (1960). Fourmodel parameters were usually optimized by this routine.

During the early modelling by Dinc¸er et al. (1987) it was found that the models pre-dicted a series of Thamalakane flows reasonably well for a number of years, after which,however, the model started to overestimate or underestimate the flows for the next series ofyears. That is, the modelling results obtained with a single parameter set for the entire flow

Fig. 4. Double mass curve of the cumulative flows at Mohembo and Maun.

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series, were not as good as those obtained for individual segments of the series, each withtheir own parameter set. An attempt was made to explain this phenomenon by so-called‘‘regime shifts’’ in the delta. Changes in channels were supposed to lead to redistributionof floods to different parts of the delta. It was proposed to divide the time series from 1969until 1987 into three different parts: a medium-flow regime from 1969 to 1974, a high-flowregime from 1975 to 1982 and a low-flow regime from 1982 until 1989, which was the endof the modelling period (SMEC, 1990). Fig. 4 illustrates the need for this subdivision bymeans of a double mass curve of cumulative Mohembo inflow versus Thamalakane out-flow. The changes in the curve clearly indicate when these regime shifts were taking place.The SMEC modelling was forced therefore to (a) incorporate the empirical dates of regimeshifts and (b) use a different set of distribution parameters (g3 andg4) for each of the threesegments in the time series. This of course drew substantial criticism and the alternativeIUCN model has tried to explain the regime shifts differently. Through double massanalysis of the Maun and Shakawe rainfall records (Fig. 5) together with study of otheravailable rainfall records in the region, it was concluded that the Maun and Shakawerecords needed to be corrected. After correction and re-running of the IUCN modelScudder et al. (1993) concluded that ‘‘good simulation accuracy was obtained withoutmodifying parameters relating to the physical nature of the delta. This demonstrates that areasonable explanation of the ‘regimes’ is that these were an effect caused by errors in themeasurement of rainfall’’. However, it is shown in this paper that good modelling resultsmay also be obtained without changing the rainfall data.

Table 1 gives a summary of the simulation results of both the IUCN and SMEC models.A large number of different parameter and data sets was tried and Table 1 only gives themain alternatives as discussed by Scudder et al. (1993). For option A, the uncorrectedrainfall and PET data were used, with only one parameter set, leading to a monthlycorrelation of 0.840 between modelled and observed values. Use of the corrected rainfall

Fig. 5. Double mass curve of the cumulative rainfall at Shakawe and Maun.

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and potential evapotranspiration data (PET) data resulted in a monthly correlation of0.889. The lower part of Table 1 shows the correlations produced when some form ofregime shift is introduced. IUCN alternative C shows a correlation of 0.908 when addi-tional correction factors were applied to PET before 1975 and after 1983, whereas alter-native D illustrates the IUCN results obtained when two flow regimes are introduced withtwo parameter sets. Finally, option E shows the results of the SMEC model, where use ismade of three flow regimes and three parameter sets.

Considering the differences in modelling approach between IUCN and SMEC, it wasdecided to program a third model for the Boro River Flow system to be able to verify someof the assumptions and calculations of the two existing models. During the development ofthis new model it was found that regime shifts may be modelled in a simple way, and this,therefore, forms the main rationale behind this paper.

3. A four-cell model of the Jao/Boro River flow system

The comparison of SMEC and IUCN modelling approaches focuses on the Jao/BoroRiver flow system, because almost all outflow from the delta into the Thamalakane takesplace through the Boro system where the best calibration data are also available. Instead ofattempting to model the entire delta, it was therefore decided to develop a simple four-cellmodel for the Boro River flow system (Fig. 3), as already described by Dinc¸er et al. (1987).

The basic characteristics of the model are as follows:

1. Water balances are computed monthly in each cell by

Vt +1 =Vt + Inflow −Outflow−Losses (6)

Table 1

Comparison of SMEC and IUCN modelling results

Model Standard error of estimate(MCM)

Correlation coefficient

Monthly Annual Monthly Annual

Withoutregimes

A: uncorrected rainfall andPET IUCN model

16.47 132.3 0.840 0.800

B: corrected rainfall and PETfinal IUCN model

13.48 99.4 0.889 0.891

With regimes C: as above with extra PETcorrection (making use oftwo regime changes)

12.92 90.0 0.908 0.931

D: earlier IUCN model(making use of two regimechanges and two parametersets)

12.90 60.1 0.900 0.957

E: SMEC model (makinguse of two regime changesand three parameter sets)

17.90 89.7 0.860 0.934

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where Vt+1 is the volume in each cell at the end of the month, Vt the volume at the startof the month. Inflow, outflow and losses are computed as averages during the month.Total modelling period was from 1 January 1969 to 30 September 1995.

2. Outflow from each cell to the next is governed byk(V − Vmin) wherek is the routingparameter, which has the same value for all cells.Vmin is the minimum volume in eachreservoir before outflow takes place. This minimum volume is different for each cell.

3. The losses are given by

Losses=Convp Areap (ET −P) (7)

whereConvis equal to 0.001. This conversion factor is required to express all volumesin Mm3 and to be able to enter rainfall (P) and evapotranspiration (ET) in mm. Thevolume–area relationship is given byV = bAn as in Dincer et al. (1987). The evapo-transpiration (ET) is calculated from the potential evapotranspiration (PET) data ofMaun by

ET =aPET (8)

wherea is a parameter to be optimized. A more complicated approach was used by theSMEC and IUCN models as mentioned earlier. The Maun PET was calculated accord-ing to the Penman formula, with the parameters given by SMEC (1987) which yield anaverage annual PET of 2180 mm per year. Some gaps in the PET records were recon-structed using the Maun record of daily maximum temperatures. Initially, precipitation(P) was calculated as the average of the Maun and Shakawe monthly rainfall.

4. Groundwater losses were considered zero as in Scudder et al. (1993). The question ofdeep groundwater recharge remains an unresolved issue.

5. No losses were computed to account for initial dry or wet soil conditions beforeflooding.

6. Several methods are available to solve the water balance Eq. (6). The simplest possibleis the first order explicit Euler method, where use is made of volumes at the beginningof each month. The IUCN and SMEC models use second order iterative schemes.Another possibility is the fourth order Runge-Kutta method (Press et al., 1986). Com-parison of the results obtained with several numerical schemes, however, showed thatthe higher order schemes do not produce better results than the first order Euler methodwhich was therefore chosen. Some technical details are given in Appendix A

7. The automatic calibration procedure by Marquardt (1963) as described by Press et al.(1986) was used to minimize the sum of the squared deviations between model andobservations. Twelve model parameters may be included in the optimization process.

8. The data set consisted of the Thamalakane (Maun bridge), Boro, Shashe and Mohembomonthly flow records, the (unaltered) Maun and Shakawe rainfall data and the MaunPET data for the period 1 January 1969–30 September 1995.

The model described so far is a simplified version of the modelling by Dinc¸er et al.(1987). The number of parameters so far is 11:k, b, n, four minimum volumes, PET factora and three distribution parameters (g2, g3, g4) which model the losses from cells in theBoro system to adjacent systems (Fig. 3). The SMEC model used, is in addition to theseeleven parameters, four groundwater loss parameters, two empirical regime shifts (1974and 1983) and a set of different values forg3 andg4 in each of the three regime periods.

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The results obtained with this simple model are given in Table 2 as result 1. The firstcolumns of Tables 1 and 2 contain the monthly standard error and are, therefore, mostclosely related to the optimization results in terms of the sum of the squared monthlydeviations. Comparison of Tables 1 and 2 shows that result 1 of Table 1 is similar toalternative A of Table 1, although the monthly and annual correlations of result 1 areslightly lower than those obtained by the IUCN. This is not surprising because the modeldescribed so far is a much simplified version of the earlier SMEC model. Leaving outgroundwater losses, regime shifts and corrections for initial dry or wet swamp conditions,has certainly not improved the modelling results.

Now, however, instead of following the Dinc¸er et al. (1987) approach by introducingregime shifts in an empirical way, or the approach by Scudder et al. (1993) to modify therainfall data, a new method was developed introducing the possible effect of shallowgroundwater table fluctuations in the delta on outflow.

4. Groundwater table fluctuations in the delta

The idea of studying shallow groundwater tables in the delta is not new. As far back as1975, three piezometers were installed in a small experimental area in the seasonal swamp,called Beacon Island (see Fig. 1). The study of surface water flow in this area (Dinc¸er et al.,1976) concentrated on determination of water balances and volume–area relationships.The formulaV =bAn and values for constantsb andn were a direct result of this work. Itshould be noted that the floodplains in the area were flooded during their period of study,which was in a high flow regime at that time. The piezometers were installed at the edge ofthe flooded area. The groundwater levels were found to be closely correlated with surfacewater levels, confirming the large infiltration losses. Dinc¸er et al. (1976) called these losses‘‘groundwater losses’’, a somewhat unfortunate term since later investigators haveinterpreted these losses as a groundwater recharge estimate, in the sense that thisrecharge would be permanently lost from the surface flow system and even fromevapotranspiration.

Dincer et al. (1976) noted the high autocorrelations of the observed water levels, and drewthe interesting conclusion ‘‘that it is nearly impossible to have a year with very low deltaoutflow, if preceded by a couple of years with above average hydrologic conditions’’. Thus ifgroundwater tables are high, less infiltration will take place and runoff will be enhanced.

Table 2

Modelling results

Result no. Model Standard error (MCM) Correlation coefficient

Monthly Annual Monthly Annual

1 Basic model 16.61 142.2 0.781 0.7742 CRD factor 11.45 83.3 0.902 0.9213 Flow factors 9.59 47.5 0.932 0.9754 Two outliers

removed8.18 42.7 0.949 0.980

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Unfortunately, however, long-term records of shallow groundwater levels are not avail-able. Work, carried out by the author in the Beacon Island area since 1992, shows thatgroundwater levels may have dropped dramatically. Fig. 6 shows a hydrograph of ground-water levels at the catchment inlet point, where Dinc¸er et al. (1976) had constructed aninlet weir to measure inflow. Whereas during 1976 the water level rose to more than 1.5 mabove the channel bed, it is now 4.25 m below the surface. Water level measurements inauger holes in the area revealed that the groundwater table is nearly horizontal. The drop inwater level therefore affects a large area. Lowering of the water table has also been notedin other places recently. Wells are being dug by the local population in the flood plains justsouth of the delta (October, 1995) to reach a water table 4–5 m below the surface. Thepresent low-flow regime which appears clearly in the very low outflow volumes throughthe Thamalakane during the past couple of years, is thus also expressed in low ground-water tables.

Some new results were recently reported by Bredenkamp et al. (1995) in the modellingof South African groundwater level fluctuations by the so-called cumulative rainfalldepartures (CRD) method. In its simplest form the method is described by

CRDi =Ri −Rav +CRDi −1 (9)

whereCRDi is the value of theCRD at time i (e.g. during monthi), Ri is the rainfallin monthi andRav is the long term average rain. Bredenkamp et al. (1995) extended Eq. (9)to

CRDi =1m

∑i

i − (m−1)Rj

� �

−1n

∑i

i − (n−1)Rj

� �

+CRDi −1 (10)

wherem is the number of months indicating short memory antecedent conditions,n is thenumber of months indicating long-term memory. The CRD can be calibrated to ground-water level fluctuations through simple transformations requiring the aquifer storagecoefficient and adjustment of the short and long memory parametersm andn. The greatadvantage of this method is that it only uses rainfall records. Fig. 7 gives an illustration ofthe method. The CRD method is closely related to modelling of groundwater levels by

Fig. 6. Hydrograph at Beacon Island inlet. The surface water levels are from Dinc¸er et al. (1976). Groundwaterlevels were measured by the author on five occasions.

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ARIMA time series analysis (Box and Jenkins, 1970; Gehrels et al., 1994) and by lumpedparameter approaches (Thiery, 1988). However, detailed comparison of these threemethods is beyond the scope of this paper.

Although neither groundwater level fluctuations nor aquifer storage coefficients areavailable for the delta, the method can still be used in this case. Using a short memoryof 12 months and a long term memory of 120 months, the CRD for the Maun and Shakawerecords can be calculated according to Eq. (10) and then normalized (i.e. divided by themaximum absolute value of the series). The resulting values are shown in Fig. 8. Thesubdivision of the time series into high–medium–low flow regimes by SMEC (1987) is

Fig. 7. Example of cumulative rainfall departure (CRD) method from South Africa after Bredenkamp et al.(1995).

Fig. 8. Normalized CDR curves for Maun and Shakawe rainfall. The bar below indicates high and low flowperiods adopted by the SMEC models (Dinc¸er et al., 1987; SMEC, 1990). Also indicated are the modellingperiods of the SMEC and IUCN models, together with the period considered in this paper.

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also shown. Positive values of the normalizedCRD correspond to high flow regimes andnegative values to low flow regimes. The Shakawe record exhibits the same pattern withlower amplitude. Although the values formandn, respectively 12 and 120 months reflectonly one possible choice, the correspondence with high and low flow regimes was toogood to be ignored and the normalized variableCRDwas built into the model as follows.

It was found by SMEC (1987) that regime shifts could easily be taken into account byadjusting distribution parametersg3 andg4. The idea was that channel changes in the deltawere leading to redistribution of flood waters in the delta. These constant distributionparameters, are now changed into time variable parameters by multiplying them with thefactor

[1+l(eCRD −1)] (11)

where thel is the 12th parameter, and where the transformation ofCRD into eCRD − 1 willenhance the high flows slightly more than reduce the low flows.

Incorporation of the MaunCRD into the basic model and optimization of the twelveparameters, immediately resulted in a monthly standard error of 11.45 MCM with amonthly correlation of 0.902 (result 2, Table 2). That is, the simulation has now becomethe best in terms of the monthly standard error, a clear indication of the strength of thisapproach. Moreover, these results have been obtained with a simpler model, fewer para-meters, without any data alteration and without the need for empirical regime boundaries.Fig. 9 illustrates results 1 and 2 (Table 2) in a cumulative flow diagram. The use of theCRD function has greatly improved the simulation from the thin line (result 1) to the solidline (result 2).

Fig. 9. Double mass curve of the cumulative observed and modelled monthly flows. The thin line gives the resultof the first model without theCRDfunction, while the solid line indicates the result obtained with the CRD model(results 1 and 2 of Table 2).

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The choice of a long memory of 10 years in the evaluation of theCRD function deservessome discussion. In the absence of records of natural groundwater level fluctuations manydifferent choices seem possible. Therefore, fifteenCRD functions were generated withlong-term memories ranging from 5 to 20 years. For each of these functions the parameterset was optimized, producing the result illustrated in Fig. 10, where the monthly standarderrors, annual and monthly correlations are plotted against the long-term memory. Theshort-term memory was kept constant at 12 months. The figure shows that values from 9 to15 years are possible. However, because the annual correlation is slightly better for amemory of 10 years, it was decided to maintain the originalCRD function with a long-term memory of 10 years as the final choice.

Table 3

Optimized parameter values (result 2, Table 2)

Number Parameter Symbol Value j (%)

1 Reservoir constant k 0.720 0.12 Area–volume constant b 0.017 2.13 Area–volume exponent n 2.136 0.24 Distribution coefficient cell 1–2 g2 0.111 1.25 Distribution coefficient cell 2–3 g3 0.616 3.86 Distribution coefficient cell 3–4 g4 0.791 3.27 Minimum volume of cell 1 Vmin,1 20008 Minimum volume of cell 2 Vmin,2 5009 Minimum volume of cell 3 Vmin,3 100

10 Minimum volume of cell 4 Vmin,4 5011 PET ratio a 1.011 0.512 CRD ratio l 0.259 0.5

Fig. 10. Monthly standard error and correlations plotted against possible lengths of the long term memory. Thebest value lies at about 10 years.

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5. Short term antecedent conditions

Fig. 11 further illustrates result 2 (Table 2) in a hydrograph of both observed andmodelled monthly flow volumes. Scatter diagrams of monthly and annual flow valuesare also shown, together with a hydrograph of monthly modelled and observed averages.Table 3 contains the optimized parameter values.

Despite the low monthly standard error, inspection of the monthly hydrograph(Fig. 11(a)) and the scatter diagrams (Fig. 11(b) and (c)) reveals that systematic deviationsdo occur in several years. For example, the modelled hydrograph deviates substantiallyfrom the recorded values in 1974, 1978 and 1989. Because these were all years with higherthan average rain, this raises the question of whether further improvements are possible.The existing IUCN model has built in correction factors to further reduce evaporation afterheavy rainfall. In the present model, which makes use of actual PET data, some reductionalready occurs because temperatures and sunshine duration are generally lower during rainperiods. The problem is to find a reasonable model that would explain the deviationsduring these above average rainfall periods. Second, it is well known that catchment runoffis strongly influenced by antecedent climatic conditions. This aspect has not receivedsufficient attention in relation to delta outflow modelling and it seems important tostudy its relevance to the present problem.

The SMEC model used the following relation to simulate the monthly evapotranspira-tion PET from the available Maun PET averages, calculated by the Penman method:

PET=PET+Pav −P

Pav

CVe

CVpPET (12)

wherePav is the average rainfall for the month considered,CVe andCVp are coefficients ofvariation of respectively the average monthlyPETandP. For the months May–September(with erratic, usually very low rainfall)PETwas taken to be equal to the average value. Itwas found that this PET model compares quite well with the actual PET record. A smallimprovement was found by applying the rule that for monthly rain greater than 180.0 mmthe correction according to Eq. (12) is calculated withP still equal to 180 mm. This is inaccordance with the findings of Scudder et al. (1993). Comparing the SMEC model withthe Maun PET record gives a standard error of 12.7 mm and a correlation of 0.94.

Surprisingly, improvement in the modelling (over result 2, Table 2) can be obtained ifthe PET data are lagged by one month in relation to the other data, that is if for thecalculations in the interval (t,t + 1) the PET data of the interval (t − 1,t) are used. Thisimproves the fit of the average modelled hydrograph to the recorded hydrograph (seeFig. 11(d).

Good results were also obtained if instead of the average PET in Eq. (12) the followingtheoretical relation is used

PET=180+65 cosp

6(i −0:5)

h i

(13)

wherei is the month in the year.As mentioned earlier, extreme rainfall events may enhance runoff in the delta, leading

to larger outflow than predicted by the model. The flow in the delta consists of two flow

228 A. Gieske/Journal of Hydrology 193 (1997) 214–239

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Fig

.11.

(a)

Mod

elle

dan

dob

serv

edhy

drog

raph

for

the

perio

d19

69–

1995

(Par

amet

erse

tofT

able

3w

asus

ed);

(b)

scat

ter

diag

ram

ofob

serv

edve

rsus

mod

elle

dm

onth

lyflo

wva

lues

;(c

)sc

atte

rdi

agra

mof

obse

rved

and

mod

elle

dan

nual

valu

es;

(d)

aver

age

annu

alhy

drog

raph

com

pare

dw

ithth

em

odel

led

hydr

ogra

ph.

229A. Gieske/Journal of Hydrology 193 (1997) 214–239

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components, flow produced by rain in the current delta rainy season, and flow generated byrains in Angola one year earlier. Normally Okavango River flow dominates and theseparate peak owing to rainfall is not noticeable. However, in 1974 and 1978, one clearlyobserves a double peak (Fig. 11), the first one owing to delta rainfall and the second owingto floods coming down from Angola.

When the rains in the delta are good, favourable antecedent conditions are createdleading directly to more runoff but also to increase of outflow by subsequent floods.Conversely, if floods have been good, favourable antecedent conditions for the followingrainy season will be produced. This dual nature of the event response mechanism com-plicates analysis and modelling of peak delta outflows. Despite the paucity of data, anattempt was made to model peak flow responses by generating antecedent rain and flowconditions. The approach is not physically based. No data are available from the delta withregard to, for example, the question whether peak event response takes place throughHortonian flow, saturation overland flow or subsurface storm flow. The latter two seemmore likely because groundwater levels are generally very close to the surface. In theabsence of data, a heuristic numerical approach was adopted here.

The algorithm to model peak event response consists of three steps (see Appendix B fora technical description), illustrated in Fig. 12.

1. Calculation of cumulative values for both flow and rain with memories of respectivelyone year and half a year.

2. Use of a threshold value to retain only peak events, leading to step functions for bothflow and rain.

3. A set of rules and relations, through which the step functions are modified and throughwhich the new flows can be calculated.

The block diagram of Fig. 12 illustrates the combination of rain and flow eventsrequired to bring flow factor 3 into play. In the model, flow factor 1 was used to modifythe inflow into cell 1. Flow factors 2 and 3 are taken into account by modifying theexpression for the losses (Eq. (7)) as follows

L =convp areap [ET − (1+ flofac2) p (1+ flofac3) p P] (14)

Fig. 13 illustrates the effect on the modelled outflow for the crucial period from 1973 to

Fig. 12. Block diagram illustrating the three steps in the heuristic algorithm to compute short term antecedentconditions for flow and precipitation.

230 A. Gieske/Journal of Hydrology 193 (1997) 214–239

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1982. Flow factors 1 and 2 increased the flow during the peak rain years of 1974, 1977 and1978, while flow factor 3, which models enhanced flow during good rainy seasons aftervery good inflow, was different from zero only in 1977. It is clear from Fig. 13 that themodelled flow pattern has much improved. The same holds for the 1989 season whenantecedent conditions were high owing to very good rains. It should be noted thatenhanced flow may not only originate from processes within the Jao/Boro four cell system,but also as spillover from adjacent delta areas, i.e. the flow system may be altered duringpeak flows.

The constants used for calculating flow factor 3 were also checked for the outflowrecords in the 1952–1961 period. These outflows were reconstructed by Dinc¸er et al.(1987) from water level measurements and were not considered very accurate. In theabsence of the original data, the flow data were again reconstructed from the hydrographgiven by Dincer et al. (1987). Outflow was modelled with these flow data using theparameter set for the period 1969–1995. Fig. 14 illustrates the modelling both with and

Fig. 13. Modelling of the high flow period 1973 to 1982 to illustrate the use of the three flow factors.

Fig. 14. Modelling of the period 1952 to 1960 using reconstructed flow records from Dinc¸er et al. (1987). Thebottom diagram indicates the importance of the flow factors for modelling the shape of the 1955 hydrograph.

231A. Gieske/Journal of Hydrology 193 (1997) 214–239

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Fig

.15.

Mod

ellin

gre

sults

are

show

nin

(a)

whe

rebo

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RD

and

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nditi

ons

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232 A. Gieske/Journal of Hydrology 193 (1997) 214–239

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without the flow factors. It should be noted first that a reduction of 25% of the recon-structed flows was necessary to obtain a good fit (standard error 10.84 MCM, annual andmonthly correlations 0.956 and 0.907 respectively). However, the analysis of the shape ofthe 1955 hydrograph is the essential part here, which does not depend on the outflowreduction. The 1954 inflow at Mohembo was the highest on record with an annual flow of14520 MCM, while the 1955 inflow was one of the lowest with 8170 MCM. The 1954/1955 rainy season produced peak rains for Shakawe (1091.4 mm) and Maun (863.5 mm).Because of the high antecedent flow and rain conditions, flow factor 3 proved to beessential in modelling the shape of the 1955 hydrograph. Although technically it wouldhave been possible to model the period 1969–1995 without this factor, it was decided toinclude it because of the consequences for the 1955 hydrograph. Analysis of the ante-cedent conditions for the entire flow and rain record since 1933, showed that flow factor 3has only been really important in 1955. The consequences of rare combinations of eventsfor delta outflow, underline the fact that the flow record length of 63 years is too short for acomprehensive analysis.

6. Results

Fig. 15 illustrates the results obtained when both the long-term CRD function and theshort-term antecedent conditions are used. The standard errors and correlations have beenlisted in Table 2 as result 3. The monthly standard error has been reduced to 9.59 MCMwhile the monthly and annual correlations have gone up to 0.932 and 0.975 respectively.Fig. 15(b) shows the existence of two months for which the modelling results inexplicablydiffer by a large amount from the observed flows. When these two outliers are removed byadjusting the flows from 98 to 34 MCM (June 1969) and from 103 to 62 MCM (July 1987),the monthly standard error goes down to 8.18 MCM. This result has been listed in Table 2as number four.

The results are summarized in Fig. 16, which clearly shows the decrease in standarderror from the basic model to the model with the long-term cumulative rainfall departure

Fig. 16. Results obtained by the models discussed in this paper in terms of the monthly standard error.

233A. Gieske/Journal of Hydrology 193 (1997) 214–239

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(CRD) method, and to the model incorporating both the CRD function and the short-termantecedent conditions.

Further extension of the modelling to include the period 1952–1960 has shown that if areduction of 25% is applied to the reconstructed flow records, it then becomes possible tosimulate the flows with a monthly standard error of 10.84 MCM, i.e. better than the resultsof the existing IUCN and SMEC models for their calibration period. The annual values forthese period are plotted in Fig. 15(c) for comparison with the period 1969–1995.

Whereas in the IUCN model the Thamalakane (Maun bridge) monthly outflow datawere used, the SMEC model made use of the combined outflow records of the Shashe andBoro rivers (Fig. 1), on the assumption that this reflected the real outflow from the Jao/Boro system better. The results shown in Table 4 (result 1) show that it does not makemuch difference which set of outflow data is used. It could also be argued that the flowrecords of the Thamalakane and Shashe rivers should be added, as this represents a betterpicture of the total outflow of the delta into the Boteti. Again, the results (Table 4, result 2)show little difference with modelling of the Thamalakane outflows.

Finally, it is also possible to use weighted averages of the Maun and Shakawe rainfalldata as input, instead of the simple average of the Maun and Shakawe data which was usedas input into each cell. For example, in cell 1 the Shakawe rain could be used (Ps), in cell 4the Maun rain (Pm), in cell 2, P = (2Ps + Pm)/3 and in cell 3,P= (Ps +2Pm)/3. Result 3(Table 4) shows that this does not make much difference either.

7. Discussion

It has been shown that good simulation results of the delta outflow can be obtained iflong term climatic variations are taken into account through a transformation of the Maunrainfall data. Because this type of transformation is similar to those used in the modellingof groundwater level fluctuations, the modelling suggests that long term fluctuations indelta outflows, the so-called high and low flow regimes, are caused by shallow ground-water fluctuations in the delta. The fact that the Maun rainfall data gives the best trans-formation and modelling results, as opposed to using, for example, the Shakawe rainfalldata or the Mohembo inflow data, suggests that delta outflow is modified more by events inthe lower than in the upper delta. However, in view of the absence of long-term andregionally representative records of groundwater fluctuations, the identification of theprocess with groundwater level changes remains hypothetical and needs confirmation.

Table 4

Modelling results obtained with modified input files

Result no. Input file Standard error Correlation coeff.

Monthly Annual Monthly Annual

Table 2, result 3 9.59 47.5 0.932 0.9751. Boro+ Shashe 9.69 47.7 0.933 0.9712. Tham+ Shashe 10.54 53.2 0.937 0.9763. Weighted rain 9.67 50.1 0.931 0.972

234 A. Gieske/Journal of Hydrology 193 (1997) 214–239

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It has also been shown that response of the delta to peak inflow and rain events can betaken into account by a set of antecedent conditions through a heuristic algorithm. Prac-tical hydrological study of the delta has not paid much attention to peak event responsethus far. The modelling has shown that peak events have a profound influence on deltaoutflows.

The fact that good results may be obtained on the basis of climatic data only, withoutaltering rainfall data or without having to introduce empirical regime shifts, implies that nomajor physical changes have taken place in the delta during the period of record. This doesnot mean of course that no physical changes are taking place in the delta at all. Channel,floodplain and island evolutionary processes have all been documented and researched andthese will undoubtedly lead to changes in outflow patterns. However, they seem to act on alonger time scale than usually assumed. From a regional, large-scale point of view, deltaoutflows seem to have been stable for the last thirty to forty years, modified only byclimatic influences.

The modelling has also shown that it is not necessary to make the assumption of flowlosses occurring as deep groundwater recharge. Shallow recharge does seem to occurbut this recharge component may disappear in the long term by evapotranspiration andre-entry into the surface flow system.

Prediction of future outflow seems to be limited by the difficulty in predicting peak flowand rainfall events. One event like the 1973/74 rainy season may significantly changeoutflow patterns for a number of years. However, prediction of outflow for a period of 6–8months seems now possible at the end of the rainy season in April or May.

Acknowledgements

The kind permission of the Director of the Department of Water Affairs through theBotswana Ministry of Mineral Resources and Water Affairs, to make use of Botswanadata, is gratefully acknowledged. The views expressed in this paper only reflect theauthor’s opinion. The author would also like to acknowledge the discussions with DrM. McFarlane and Dr F. Sefe (both University of Botswana) and the assistance by MrsJane Prince Nengu (Maun, Dept. of Water Affairs).

Further details of the model with data files are available from the author on request.

Appendix A. First order explicit numerical scheme

For a single reservoir the differential equation of the type considered is given by

dydt

= f (y) (A1)

The formula for the explicit Euler method is

yt +h =yt +hf (yt) (A2)

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which advances the solution fromt to t + h. The formula uses derivative information atthe beginning of the interval only. The error of this first order method is in the orderof O(h2). Euler’s method is usually not recommended for practical use, because it isnot very accurate when compared to other methods of equivalent stepsize, nor is it verystable.

A second order scheme can be found by combining the implicit and explicit approaches:

yt +h =yt +hf (yt,yt +1) (A3)

where the right hand side also containsyt+h. The solution to Eq. (A3) can be founditeratively as was done by Dinc¸er et al. (1987) and Scudder et al. (1993), or it can firstbe linearized and then solved (Press et al., 1986).

Higher order schemes are also possible. For example, the fourth order Runge-Kuttascheme (Press et al., 1986) is formulated as

yt +h =yt + 16(k1 +2k2 +3k3 +4k4) +O(h5) (A4)

where

k1 =hf (yt)

k2 =hf (yt +k1=2)

k3 =hf (yt +k2=2)

k4 =hf (yt +k3)

(A5)

In view of the fairly large stepsize of one month, it was expected that higher order schemessuch as Eqs. (A3), (A4) and (A5), would yield better results than the Euler explicitapproach. Surprisingly, however, it turned out that the higher order schemes do notyield better results than the first order explicit scheme. There seem to be several reasonsfor this. First, inadequacies in the numerical scheme can to some extent be compensatedfor by changing parameter values. Second, the problem studied here is non-linear in thesense that negative flow values are not allowed, and therefore derivatives are not contin-uous. It was found that phase shifts and changes in outflow amplitude occur owing to thiseffect. Lastly, following the previous modelling attempts, the delta outflow problem wasstudied within the constraints of a four reservoir series. Different results may be obtainedwith a longer series of reservoirs.

Therefore the Euler scheme was finally adopted, leading to the following numericalscheme for the delta flow problem with the four reservoir model:

yit +1 =yi

t + I i −1t −Oi

t −Lit (A6)

where variabley denotes a volume of water, where the timestep sizeh has been taken as 1(one month), and wherei ranges from 2 to 4. The inflowI into reservoiri is evaluated in thefollowing way.

I it =gik(yi −1

t −yi −1min) (A7)

whereg i is the distribution parameters, wherek is the routing parameter and whereymin

is the threshold volumes. Inflow into cell 1 (i = 1) is given by the monthly Okavango

236 A. Gieske/Journal of Hydrology 193 (1997) 214–239

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flow volume at Mohembo (betweent and t + 1). The outflowO from reservoir i isgiven by

Oit =k(yi

t −yimin) (A8)

and lossesL are determined through the relation

Lit =CAi

t(ETi −Pi) (A9)

where Ait is the area of celli which is determined byy=bAn, where ET and P are

respectively evaporation and precipitation from celli in the time interval (t,t + 1) andwhereC is a conversion factor of 0.001, allowingETandP to be expressed in millimetersof water.ET is calculated asaPET wherePET is the Maun Potential Evapotranspirationanda is a factor to be determined in the optimization process.

Finally, after determiningy4t+1, the outflow from cell 4 is calculated by

O4t +1 =k(y4

t +1 −g4min) (A10)

which is to be compared with the accumulated recorded outflow in the period fromtto t + 1. This last step does not seem correct at first sight, because the integratedoutflow from time t to t + 1 is compared with the actual flow rate at timet + 1. Itwould seem better to compute the outflow as

�t +1

tO dt < k

y4t +y4

t +1

2−y4

min

" #

(A11)

using the trapezium rule to integrate the flow fromt to t + 1. However, in practice thisoffsets the phase of the response leading to worse results. Thus Eq. (A10) is to be preferred.It should be noted that Dinc¸er et al. (1987), SMEC (1990) and Scudder et al. (1993) alsoused Eq. (A10) rather than Eq. (A11).

The numerical scheme given by Eqs. (A6)–(A10) is easy to implement and fast inoperation which is a distinct advantage in the optimization procedures.

Appendix B. Modelling antecedent conditions

Referring to the block diagram (Fig. 13), the heuristic algorithm consists of the follow-ing three steps

Appendix B.1. Step 1

The cumulative short memory functions are calculated as follows

1. Antecedent conditionAPi for rain P in month i is calculated as

ARi = ∑i

j = i −6Pj (B1)

If a dry month is encountered (Pj =0) thenAPi is divided by 10. Finally allAPi aredivided by 1175 for normalization.

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2. Antecedent conditionAFi for flow F in month i is calculated as

AFi = ∑i

j = i −12Fj (B2)

Finally all AFi are divided by 2 4000 for normalization

Appendix B.2. Step 2

The threshold is simply implemented as follows

1. if APi , 0.5 thenAPi = 02. if AFi , 0.5 thenAFi = 0

Appendix B.3. Step 3

The flow factors are determined through

1. CRi = exp(CRDi) − 1 whereCRDi is the long term cumulative rain function describedby Eq. (12), main text.

2. flofac1= 0.30APi

3. flofac2= 0.55APi (1 + exp (1.9CRi)4. flofac3= APi AFi (1 + exp (2.4CRi)

Thus the flow factors are zero except during peak rainy seasons. Flow factor 3 is onlyactive when bothAPi andAFi are greater than zero.

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