Mwakalila Feyen Hydro Logical Processes

8/6/2019 Mwakalila Feyen Hydro Logical Processes

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HYDROLOGICAL PROCESSESHydrol. Process. 15, 22812295 (2001)DOI: 10.1002/hyp.257

Application of a data-based mechanistic modelling (DBM)approach for predicting runoff generation in semi-arid

regions

S. Mwakalila,1* P. Campling,1 J. Feyen,1 G. Wyseure2 and K. Beven3

1 Institute for Land and Water Management (ILWM), K.U.Leuven, Belgium2 Faculty of Agricultural and Applied Biological Sciences, K.U.Leuven, Belgium

3 Institute of Environmental and Natural Sciences, Lancaster University, UK

Abstract:

This paper addresses the application of a data-based mechanistic (DBM) modelling approach using transfer functionmodels (TFMs) with non-linear rainfall filtering to predict runoff generation from a semi-arid catchment (795 km 2)in Tanzania. With DBM modelling, time series of rainfall and streamflow were allowed to suggest an appropriatemodel structure compatible with the data available. The model structures were evaluated by looking at how wellthe model fitted the data, and how well the parameters of the model were estimated. The results indicated that aparallel model structure is appropriate with a proportion of the runoff being routed through a fast flow pathwayand the remainder through a slow flow pathway. Finally, the study employed a Generalized Likelihood UncertaintyEstimation (GLUE) methodology to evaluate the parameter sensitivity and predictive uncertainty based on the feasibleparameter ranges chosen from the initial analysis of recession curves and calibration of the TFM. Results showedthat parameters that control the slow flow pathway are relatively more sensitive than those that control the fast flowpathway of the hydrograph. Within the GLUE framework, it was found that multiple acceptable parameter sets give

a range of predictions. This was found to be an advantage, since it allows the possibility of assessing the uncertaintyin predictions as conditioned on the calibration data and then using that uncertainty as part of the decision-makingprocess arising from any rainfall-runoff modelling project. Copyright 2001 John Wiley & Sons, Ltd.

KEY WORDS data-based mechanistic modelling approach; transfer function models; Generalized Likelihood UncertaintyEstimation; parameter sensitivity and predictive uncertainty

INTRODUCTION

Rainfall-runoff models are important tools for water resource planning, development and management.

However, limitations of both model structures and the data available on parameter values, initial conditions

and boundary conditions, generally make it difficult to apply any hydrological model without some form of

calibration or conditioning of model parameter sets.

The data-based mechanistic (DBM) modelling addressed in this study is based on the concepts of lettingthe data suggest an appropriate model structure compatible with the inputoutput data available, but then

evaluating the resulting model to see if there is a mechanistic interpretation that might lead to insights

not otherwise gained from modelling based on theoretical reasoning. These concepts have been applied

successfully in humid climates, e.g. see Young and Beven (1994), Young (1998), and Beven (2001). Similarly,

in the application of an IHACRES model the data are also allowed to suggest the form of the transfer function

used for flow routing rather than specifying a fixed structure beforehand (see, Jakeman et al., 1990, 1993;

Jakeman and Hornberger, 1993; Schreider and Jakeman, 1996; Post and Jakeman, 1996; Sefton and Howarth).

* Correspondence to: Dr. S. Mwakalila, Institute for Land and Water Management Katholieke Universiteit Leuven, Vital Decosterstraat 102,B-3000 Leuven, Belgium. E-mail: [email protected]

Received 28 June 2000

Copyright 2001 John Wiley & Sons, Ltd. Accepted 16 January 2001


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2282 S. MWAKALILA ET AL.

The results generally show that the parallel transfer function is a suitable structure for rainfall-runoff simulation

at the catchment scale, with one fast flow pathway and one slower flow pathway. The fast flow pathway

provides the major part of the predicted storm hydrograph, and the slower pathway the major part of the

recession discharge between storm periods.

This study focuses on extending the application of these concepts to semi-arid catchments and also to assess

the parameter sensitivity and discharge prediction limits of the identified model structure.

Compared with humid climate regions, in semi-arid regions that cover a large part of the world there are

a lot of problems in predicting runoff generation. In general, the two interrelated underpinning problems

in rainfall-runoff modelling in such regions are due to scarcity of data and data uncertainties. The success

of any catchment rainfall-runoff model depends critically on the data available to set it up and drive it.

Nearly all rainfall-runoff models require data on some combination of rainfall, discharge, evapotranspiration

and catchment morphology. The basic sources of error and uncertainty in each of these data types and

methods to account for these uncertainties are discussed in detail in Melching (1995). He noted that the

primary uncertainties in using point-rainfall measurements to describe the true rainfall input for a catchmentare: depth measurement error in the gauge due to equipment malfunction; representativeness of ground-

level precipitation at the gauging point; gauge location; the areal-mean rainfall; gauge-network areal-mean

rainfall versus true areal-mean rainfall; rainfall spatial variability; rainfall temporal variability; lack of

synchronization between time clocks for rain and stream gauges; and lack of synchronization between time

clocks for the various rain gauges in the catchment. The availability of discharge data is important for

the model calibration process. Discharge data are, however, generally available at only a small number of

sites in semi-arid regions. It is also an integrated measure, in that the measured hydrograph will reflect

all the complexity of flow processes occurring in the catchment. However, the discharge data available

are generally point data and may not be error free. If any model is calibrated using data that are in

error, the effective parameter values will be affected and the predictions for other periods that depend on

the calibrated parameter values will be affected. This will be an additional source of uncertainty in the

modelling.

The main objective of this paper therefore, is to address the application of a DBM modelling approach

to predict runoff generation from a semi-arid catchment. Furthermore, the paper employs a Generalised

Likelihood Uncertainty Estimation (GLUE) methodology using Monte Carlo simulation (MCS) as key

component to assess the model parameter sensitivity and predictive uncertainty. This type of technique has

been used on several occasions (e.g. Beven and Binley, 1992; Beven, 1993; Romanowicz et al., 1994; Freer

et al., 1997; Franks et al., 1998; Cameron et al., 1999), but not in the context of semi-arid regions.

METHODOLOGY

The methodology in this study has three distinct stages. At the first stage a DBM approach was used to suggest

an appropriate transfer function model (TFM) structure compatible with the available set of daily rainfall-

runoff data. At this stage it is the idea that the data should be allowed to indicate which model structure is

most appropriate rather than specifying a structure beforehand. Therefore model structures were evaluated by

looking at how well the model fitted the data, and how well the parameters of the model were estimated.

The second stage involves the separation of the slow flow component from the total discharge in order to get

a hydrologically more realistic model structure. At the third stage the GLUE methodology approach using

MCS as a key component was employed for performance evaluation of the identified model structure for the

purpose of parameter sensitivity analysis and uncertain estimation.

Data used

The data used in this study were rainfall and streamflow records available on a daily basis for 197080

period for the Great Ruaha catchment at Salimwani gauging station (IKA 8A) in Tanzania. The study area

Copyright 2001 John Wiley & Sons, Ltd. Hydrol. Process. 15, 22812295 (2001)


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PREDICTING RUNOFF GENERATION 2283

covers an area of about 795 km2. The gauging station is located at 34 07 E longitude and 8 54 S latitude.

It is a sub-basin of the Great Ruaha River Basin, which is draining an area of about 68 000 km2. The annual

mean temperature varies from about 18 C at the higher altitudes to about 28 C at the lower and drier part

of the basin. The rainfall regime in the basin is typical of the unimodal type, with a single rainy season from

November through May and no rainfall during the rest of the year. As the main process causing rainfall is

convection, the rainfall is highly localized and with the incoming wind direction and the topography being

the main determining factors of spatial patterns. The mean annual rainfall ranges from 800 mm in the lower

part of the catchment to 1000 mm in the highlands. Owing to the generally high potential evaporation rates

(14001800 mm year1) compared with the yearly rainfall, most of the catchment has an annual rainfall

deficit. The runoff patterns in the basin correspond closely to the unimodal rainfall patterns prevailing in

the whole catchment. The streams start rising in NovemberDecember, experience a maximum flow in

MarchApril and have their recession period from May to October. As rainfed cultivation is very unreliable

in the area due to the low and unreliable rainfall, farmers in the area depend very much on irrigation. The

major part of irrigation takes place downstream of the gauging station. In all types of irrigation, rice is thedominant crop, which is grown during the rainy season (DANIDA, 1995).

Initial evaluation of general linear TFM

In this stage of the methodology, the general linear TFM was evaluated to identify an appropriate

model structure. Within the DBM approach a non-linear rainfall filter function [Equation (1)] was used to

produce an effective rainfall, which was then related to discharge using a generalized linear transfer function

[Equation (2)].

Ut / RtQtn 1

where Ut is the effective rainfall, Rt is the rainfall input at time t, Qt is the discharge; parameter n controls

the non-linearity of fast runoff generation (if n D 0, the rainfall filter is linear as for subsurface recharge).Here, the current discharge was used as an index of the contribution of antecedent moisture status of the

catchment on runoff generation. Therefore, the interpretation of RtQtn can be referred to as a contributing

area function.

The general form of the discrete time linear transfer function that forms the basis of the TFM package can

be presented as follows:

Qt Db0 C b1z

1 C C bMzM

1 a1z1 a2z2 C C aNzNUt 2

where Qt is a simulated discharge, Ut is the effective rainfall, z is a backward difference operator, a and b

are coefficients treated as parameters, such that parameter a is related to the mean residence time of fast flow

pathway and parameter b is a gain parameter that scale the differences in total volumes of input and output;

M and N represent the total number ofb and a parameters respectively, and is the time delay. The derivationof Equation (2) can be found in Beven (2001). Parameter estimation is carried out using a simplified refined

instrumental variable (SRIV) technique which has been shown to be robust to data errors. For more detail

see Young (1984), Young and Beven (1994) and Beven (2001).

The higher the model order (i.e. the greater the number of a and b coefficients), the larger the standard

errors of the estimated parameter values will tend to be. Very large standard errors on the coefficients are an

indication that a model is over-parameterized. One way of checking for over-parameterization is by looking

at how well the coefficients of the model are estimated. The aim in this study, is to find a model structure

that gives a good fit but is parsimonious in having a small number of coefficients.

Therefore, using a daily rainfall-runoff data set an appropriate TFM structure was identified, by finding the

best values of (N, M, ) and the corresponding coefficients.



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Performance evaluation of TFM structures

Underlying the model identification step is the idea that the data should be allowed to indicate which model

structure is most appropriate, rather than specifying a structure beforehand. There may be many models giving

an acceptable fit to the data. In this study, therefore, model structures were evaluated in terms of the following

criteria: NashSutcliffe efficiency R2, which indicates how well the model fits the data; Young Information

Criterion (YIC), which combines a measure of goodness of fit with a measure of how well the parameters

are estimated.

R2, introduced as the efficiency measure for rainfall-runoff modelling by Nash and Sutcliffe (1970), is

statistically known as the coefficient of determination indicating the proportion of the variance in the data

explained by the model, and can be defined as:

R2 D 1 e

2

o2

3

where o2 is the variance of the observed output data calculated over all time steps used in fitting the model

and e2 is the variance of the differences between observed and simulated outputs at each time step. As the

model fit improves, the value of R2 will approach unity. If the model is no better than fitting the mean of the

observed outputs e2 D o

2, the value of R2 will be zero or less.

The YIC is defined as:

YIC D ln

e

2

o2

C ln

1

k

k

e2 Pii

i

4

where i, i D 1 . . . k , are the model coefficients, and Pii is the ith diagonal element of a scaled parameter

covariance matrix. From its definition, the first term of YIC is simply a relative logarithmic measure of

how well the model explains the data: the smaller the model residuals the more negative the term becomes.

The second logarithmic term, on the other hand, tends to become large (less negative) when the model is

over-parameterized and the parameter estimates are poorly defined. Consequently, as whole, the criterionattempts to identify a model that explains the data well but with the minimum of statistically well-defined

(low-variance) parameters.

Separating a slow flow component from total discharge

Since it was learnt that the general linear TFM is underestimating the long-term recession flows, therefore,

the slow flow component was separated off from total discharge in order to get a hydrologically more realistic

model structure. Therefore, to allow for non-linearity the linear storageoutflow relationship was generalized

by an exponential reservoir model as presented by Beven et al. (1995):

Qb D Qoes/m 5

where Qb is the outflow from the saturated zone considered as baseflow, Qo is the discharge when recessionof streamflow commences, s is the local storage deficit and m is a model parameter considered as the scaling

depth of the effective catchment soil profile.

Solution of Equation (5) for a pure recession in which inputs are assumed to be zero shows that discharge

has an inverse or first-order hyperbolic relationship to time as:

1

QbD

1

QoC

t

m6

A plot of 1/Qt against time t should plot as straight line with a slope of 1/m, where Qt is the recession flow

at time t. Therefore, Equation (5) was used for separating the slow component based on the following major

steps.



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(1) The catchment average storage deficit before each time step was updated by subtracting the unsaturated

zone recharge and adding the baseflow (slow flow component) calculated for the previous time step, thus:

StC1 D St C Qo eSt/m cRt 7

where Rt is the observed rainfall at time t and parameter c represents the proportion of rainfall that

percolates as recharge to the groundwater.

(2) If an initial discharge QtD0 is known and assumed to be only the result of drainage from the saturated

zone, Equation (5) can be inverted to give a value for S at time t D 0 as:

S D m

ln

QtD0

Qo

8

Equations (7) and (8), therefore, form the basis for separating the baseflow from the total discharge during

storm events. Then, given the time series of discharge Qt and rainfall Rt at time t, the time series of directrunoff (fast flow component) Qd was calculated as:

Qd D Qt Qo eSt/m 9

Performance evaluation of identified model structure

The performance evaluation of the identified model structure was based on the framework of the GLUE

methodology for the purpose of parameter sensitivity analysis and predictive uncertainty estimation. The main

objective here was to examine those parameters to which the model simulation results were most sensitive

and to assess the probability of simulated discharge being within a certain prediction interval. The prediction

interval was defined by 5 and 95% limits (i.e. a 90% probability that the predicted discharge value lies within

the interval).

The main procedure was based on the following major steps:

(1) determining a feasible parameter range from initial analysis of recession curves and calibration of TFM;

(2) using a MCS method to choose parameter values from uniform distributions spanning specified ranges of

each parameter;

(3) using a likelihood value to divide acceptable simulations from unacceptable simulations;

(4) normalizing likelihood values for the parameter sets that yield acceptable simulations, such that the sum

of the normalized likelihood values equals unity;

(5) using the same generated parameter sets and associated likelihood value and model outputs to assess

parameter sensitivity and predictive uncertainty.

The modelling efficiency R2 in Equation (3) was used as a likelihood measure for determining acceptable

simulations. With the MCS technique, 5000 runs of the model were made with different randomly chosenparameter sets. The feasible range of parameters to be sampled was based on an initial evaluation of the

TFM and recession flow analysis. Those simulations with R2 > 0 5 were retained, those that did not meet

this performance criterion were deleted. Finally, 4000 simulations were retained as behavioural. In this study,

therefore, R2 D 0 5 was considered as a behavioural threshold indicating the proportion of the variance in

the data explained by the model. This value was considered as a minimum value to accept the simulations

(Beven, 2001).

In the GLUE procedures a prior likelihood estimate can be updated with a new (posterior) likelihood

measure calculated from the prediction of a new set of observations. The idea here is to check if the model,

which performs well during the conditioning or calibration period, will continue to perform well in other

periods. If this is not the case then its combined likelihood measure will be reduced. The Bayes equation



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was used here to combine likelihood measures, the type of combination using the Bayes equation may be

expressed in the form:

Lp ijY DLo iL ijY

C10

where Loi is the prior likelihood of the ith parameter set, LijY is the likelihood calculated for the

current evaluation given the set of observations Y, LpijY is the posterior likelihood and C is a scaling

constant to ensure that the cumulative posterior likelihood is unity. The special characteristic of the Bayes

equation is its multiplicative operation, if any evaluation results in a zero likelihood, the posterior likelihood

will be zero regardless of how well the model has performed previously. This may be considered as an

important way of rejecting non-behavioural models: it may cause a re-evaluation of the data for that period

or variable; it may lead to the rejection of all models.

The GLUE methodology used here is one technique for conditioning of model parameter sets in the face

of rejecting the idea that there is an optimum parameter set in favour of the concept of the equifinalityof different models and parameter sets. Parameter interactions and non-linearity in the model responses are

handled implicitly. Errors in input data and the observation data are also handled implicitly. Thus the likelihood

measure reflects the ability of a particular model to predict a particular series of observations (which may

not be error free) given a particular set of inputs (which may not be error free). There is thus an implicit

assumption that, in prediction, error structures will be similar in some broad sense to those in the evaluation

period.

RESULTS AND DISCUSSIONS

Initially, models were calibrated for five periods of 2 years in length within the decade (197080). The second

sub-period, 197273, was found to give the best calibrations. Therefore, the time series of daily rainfall and

discharge (for the 197273 observation period) were allowed to suggest an appropriate model structure withinthe general transfer function modelling framework. The results in Table I indicate that a first-order (1, 1, 0)

transfer function is an appropriate model structure with parameter n in Equation (1) ranging between 05 and

Table I. DBM model results of total observed rainfall and discharge in 197273 period

Model ordera R2 YIC a parameters b parameters

1, 1, 0 079 873 a1: 09592(00017) b0: 00202(00008)

1, 1, 1 078 843 a1: 09542(00022) b0: 00223(00010)

1, 1, 2 077 841 a1: 09561(00021) b0: 00213(00009)

1, 2, 2 078 602 a1: 09610(00020) b0: 00473(00043)b1: 00281(00044)

1, 2, 1 079 473 a1: 09554(00026) b0: 00350(00042)b1: 00133(00044)

1, 2, 0 078 383 a1: 09335(00024) b0: 00096(00042)b1: 00129(00044)

2, 2, 0 079 261 a1: 06958(02429) b0: 00080(00043)a2: 02471(02316) b1: 00199(00061)

2, 1, 1 079 170 a1: 08573(01549) b0: 00239(00035)a2: 00937(01482)

a Model order D N, M, (N is the number of a parameters, M is the number of b parameters, is adelay).



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10.00

0.00

10.00

0 100 200 300 400 500 600 700 800

Time (days)

Error

0.00

4.00

2.00

6.00

10.00

12.00

0 200 400 600 800

Discharge(mm/da

y)

Qo

Qs

Time (days)

0

20

40

60

80

0 200 400 600 800

Rainfall(mm)

Time (days)

Figure 1. Rainfall, observed discharge Qo and simulated discharge Qs and error

10 for a non-linear rainfall filter function. However, this model structure is underestimating the long-term

recession flows, as depicted in Figure 1.

As can be noted from Table I, it seems a second-order transfer function model (2, 2, 0 and 2, 1, 1) does

better in terms of R2 and YIC, presumably because errors are dominated by peaks during the wet season

and are relatively small in the dry period recession (see Figure 1). This indicates that, in order to get a

hydrologically more realistic model, the slow flow component should be separated off first before evaluating

the TFM structures.

The DBM model results after separating the slow flow component are presented in Figure 2 and Table II,

and the synthesized total and baseflow hydrograph is given in Figure 3. A graphical comparison between the

observed and simulated fast flow time series shows a good fit. This justifies that the slow flow component

should be separated off first before calibrating the TFM.

In connection with Figure 3, this implies that, from the general linear TFM [Equation (2)], the effec-

tive rainfall can be linearly related to the fast flow component of the hydrograph as presented in the

Equation (11):

Qst D aQst1 C bRtQdtn 11



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Table II. DBM results after separating the slow flow Qb component

Model ordera R2 YIC a parameters b parameters

1, 1, 0 092 1096 a1: 09580(00009) b0: 00257(00005)

1, 1, 1 091 1074 a1: 09559(00010) b0: 00268(00006)

1, 1, 2 090 1046 a1: 09550(00011) b0: 00271(00006)

1, 2, 2 090 565 a1: 09565(00012) b0: 00378(00034)b1: 00115(00035)

1, 2, 1 091 523 a1: 09568(00011) b0: 00342(00032)b1: 00080(00033)

1, 2, 0 091 377 a1: 09570(00010) b0: 00227(00031)b1: 00036(00032)

a Model order D N, M, (N is the number of a parameters, M is the number of b parameters, is adelay).

0.00

2.00

4.00

6.00

8.00

10.00

0 200 400 600 800

Time (days)

D

ischarge(mm/day)

Qd (mm)

Qd(mm)

Figure 2. Observed Qd and simulated Q0d fast flow component

0.00

4.002.00

6.00

8.00

10.00

12.00

0 200 400 600 800

Time (days)

Dischar

ge(mm/day)

Qt(mm)

Qb(mm)

Figure 3. Total discharge Qt and slow flow component (baseflow) Qb



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where Qst is a simulated fast flow component, Rt is the observed rainfall, Qdt is the estimated (from

Equation (9)) fast flow component, t is time, and a, b, and n are optimized model parameters. The slow flow

component can be represented by Equation (12) as:

Qbt D Qo eSt/m 12

where Qbt is a simulated slow flow component, St is the catchment storage deficit estimated from

Equations (7) and (8), and Qo, m and c are optimized model parameters.

The identified final model structure can be presented as shown in Figure 4, which indicates that a parallel

model structure is appropriate with a proportion of the runoff being routed through a fast flow pathway and

the remainder through a slow flow pathway.

As an example, the synthesized model performance of behavioural simulations for feasible parameter

ranges using the framework of the GLUE approach is shown in Table III. Table III, shows that there may

be many representations of a catchment that may be equally valid in terms of their ability to produceacceptable simulations of the available data. Multiple acceptable parameter sets give a range of predictions.

This may actually be an advantage, since it allows the possibility of assessing the uncertainty in predictions

Effective rainfall

Fast Flow Pathway(Surface Flow)

Qs(t)= aQs(t1)+ bRt(Qd(t))n

Recharge

DischargeQtRainfallRt Nonlinear rainfallfilter function

Slow Flow Pathwya(Base Flow)

Qb(t) = Qoe-st/m

Figure 4. Final block diagram of the identified model structure



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Table III. A sample of identified model performance after using a GLUE approach

Set of parameters Model performance (likelihood measure)

c m Qo n a b 1971 1972 1973 1974 1975 1976

040 5804 462 096 0742 0043 091 092 096 088 084 084033 4989 440 084 0770 0044 091 092 096 087 085 084033 5014 446 085 0777 0040 090 093 096 087 084 083040 6233 715 084 0768 0041 091 092 095 086 084 083026 5130 461 077 0781 0046 090 091 095 086 084 083028 5705 567 093 0686 0049 085 093 095 088 082 082033 5447 741 073 0739 0049 089 093 094 085 083 083034 4917 626 069 0855 0034 094 089 094 083 084 083042 4527 677 082 0796 0031 092 091 093 083 082 082048 4970 405 099 0575 0047 090 089 093 084 082 081

041 5870 523 063 0807 0041 093 090 093 082 083 083009 6038 451 055 0840 0049 090 088 094 082 084 082036 5491 488 091 0807 0027 086 091 092 083 080 081054 5393 521 088 0767 0037 094 083 094 081 085 083020 6358 416 075 0767 0044 083 092 092 084 080 081051 5044 747 086 0846 0029 095 083 094 080 084 083042 5349 606 063 0877 0031 096 085 093 079 084 082040 6189 477 082 0815 0028 088 090 092 082 080 081014 5375 797 065 0869 0032 088 090 093 081 081 081035 5686 429 075 0809 0032 088 092 091 082 079 081029 4920 629 090 0653 0046 082 092 092 085 079 079026 5712 718 060 0826 0037 088 092 091 081 079 081020 4870 793 064 0780 0044 084 092 090 081 078 080018 4891 627 088 0782 0033 080 091 092 083 079 079009 6032 489 056 0801 0048 084 092 090 081 079 080

047 4616 786 071 0708 0041 092 088 090 079 079 080053 6064 775 085 0848 0026 094 081 092 078 083 082033 5032 663 075 0803 0030 086 091 090 080 078 079042 6306 710 057 0894 0028 096 084 091 077 082 081045 5688 697 073 0733 0037 089 088 089 079 077 079020 6055 510 092 0675 0043 077 091 090 084 077 077049 4746 549 061 0846 0030 095 083 091 076 080 081044 4745 619 086 0526 0049 087 089 089 079 076 078026 4566 460 072 0752 0039 083 092 089 080 076 078039 4716 718 100 0883 0014 086 088 090 079 077 078011 5633 767 055 0880 0030 086 091 089 078 077 080050 4684 675 095 0815 0019 090 085 090 078 078 079044 5338 724 060 0886 0023 094 085 090 076 078 080042 4628 671 077 0670 0039 087 090 088 078 076 078

as conditioned on the calibration data and then using that uncertainty as part of the decision-making process

arising from any rainfall-runoff modelling project.

The dotty plots in Figure 5 represent a projection of a sample of points on the goodness of fit response

surface onto individual parameter dimensions. Each dot represents one run of the model from an MCS out

of 4000 simulations with different randomly chosen parameter values. It will be seen that for each parameter

there are good simulations across a wide range of parameters. The good models are those that plot near the

top. This indicates that for each parameter set there are good simulations across a wide range of parameters.

There are generally also poor simulations across the whole range of each parameter sampled. Whether a

model gives good or poor results is not a function, therefore, of individual parameters, but of the whole set



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1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

0 0.5 1 0.5 0.7 0.9

c-Parameter value n-Parameter value

Likelihoodmeasure

Likelihoodmeasure

* Simulations

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.8 0.9

a-Parameter value

Likelihoodmeasure

* Simulations

0

0.2

0.4

0.6

0.8

1

0.01 0.02 0.03 0.04 0.05

b-Parameter value

Likelihoodmeasure

* Simulations

* Simulations

1

0.8

0.6

0.4

0.2

0

45 65 85

m-Parameter value

Likelihoodmeasure

* Simulations

1

0.8

0.6

0.4

0.2

0

4 5 6 7 8

Qo-Parameter value

Likelihoodmeasure

* Simulations

Figure 5. Parameter distributions of behavioural simulations after being conditioned with the 197071 period data of G. Ruaha catchmentat IKA 8A station

of parameter values and the interactions between parameters. As a projection of the response surface, the

dotty plots cannot show the full structure of the complex parameter interactions that shape that surface. In

one sense, however, that does not matter too much, since it primarily allows us to identify where the good

parameter sets are as a set.

An impression of the parameter sensitivity can be gained from Figure 6, which depicts the cumulative

likelihood weighted distributions for each parameter after evaluating each behavioural parameter set on data

from the 197075 simulation period. Lacking any prior information about the covariation of the individual

parameters, each was sampled independently from uniform distributions across the feasible range identified



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Figure 6. Rescaled likelihood weighted distributions for parameters after being conditioned with the 1970 71 period data at IKA 8A station

from initial evaluation of the TFM and recession flow analysis. A straight diagonal line on each plot would

represent the prior distributions. Parameter c and parameter m show the strongest departures from the prior

distributions. The other four parameters (Qo, n, a and b) are much less well conditioned by the observations.

The possible reasons for parameter sensitivity can be related to the physical interpretation of parameters

controlling the slow flow and fast flow pathways of the hydrograph: parameter n controls the non-linearity of

fast runoff generation, parameter a control the residence time of the fast flow pathway; parameter b controls

the differences in total volumes of input and output; parameter m controls the effective catchment soil profile;

parameter c controls the proportion of rainfall that percolates as recharge to the groundwater; parameter Qo



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represents the discharge when the recession of streamflow commences. Therefore, owing to the shallow wet

soil in the catchment, evapotranspiration would not be expected to be strongly limited (indirectly controlled

by the n parameter), neither would the residence time for surface flow (controlled by the a parameter). Thus

any changes in the c and m parameters would result in variations in recharge, which has a major influence

on streamflow.

Since dotty plots show that for each parameter there are good simulations across a wide range of parameters,

all of these good parameter sets will, however, give different predictions. In this study, therefore, the resulting

uncertainty in the predictions was estimated by weighting the predictions of all the acceptable models by their

associated degree of belief (as quantified by the likelihood measure values).

Figures 7 and 8, therefore, present prediction limits for stream discharge from the selected catchment after

conditioning in the 197071 simulation period and updating with the likelihoods on individual years for the

entire 1970 75 period data. Figure 9 depicts the prediction limits after testing the behavioural parameter sets

with new rainfall and discharge data for the 19761978 observation period. For most of the simulations the

observations are bracketed by the prediction limits, but there are periods when the observations fall outsidethe prediction limits, especially during the wet period. This indicates some deficiencies in the input data or

model structure used. As noted from Table I, it seems that a second-order transfer function model (2, 2, 0

and 2, 1, 1) does slightly better in terms of R2. However the YIC has the most negative value for a first-order

0.00

2.00

4.00

6.00

8.00

10.00

13-Dec-69 01-Jul-70 17-Jan-71 05-Aug-71 21-Feb-72 08-Sep-72

Date

Discharge(mm/day) Observed discharge

95% Prediction limit5% Prediction limit

Figure 7. Prediction limits after being conditioned and updated with the 197072 data

0.00

2.00

4.00

6.008.00

10.00

12.00

27-Nov-72 15-Jun-73 01-Jan-74 20-Jul-74 05-Feb-

75

24-Aug-

75Date

Discharge(mm/day) Observed discharge

95% Prediction limit5% Prediction limit

Figure 8. Prediction limits after conditioning and updating with the 197375 data



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0.00

2.00

4.00

6.00

8.00

10.00

12.00

02-Dec-75 19-Jun-76 05-Jan-77 24-Jul-77 09-Feb-78 28-Aug-78Date

Discharge(mm/day)

Observed discharge95% Prediction limit5% Prediction limit

Figure 9. Prediction limits after testing with the new 197678 discharge data

transfer function (1, 1, 0) with high R2 (good fit). Therefore, the second-order transfer functions were not

tried owing to the fact that they have larger standard errors of the estimated parameter values, an indication

that a model is over-parameterized. It should be noted here that, the aim in this study was to find a model

structure that gives a good fit but is parsimonious in having a small number of coefficients.

The authors suggest that much of the errors may be due to an inadequate representation of the spatial

patterns of rainfall within the catchment. Although the models used in this study are not fully distributed

process-based models, these results are perhaps representative of the type of accuracy that might be achieved

in predicting the discharge while representing the state of the catchment as a single set of process-related

parameters. Further, it proves to be surprisingly difficult in rainfall-runoff modelling to bracket all the discharge

observations for at least 90% confidence interval.

CONCLUSIONS

With a DBM modelling approach, the simplest TFM structure that is consistent with the observations is made

up of storage elements for fast flow (surface flow) and slow flow (baseflow) components. The advantage of

the DBM approach is that the data are allowed to suggest the form of the transfer function used rather than

specifying a fixed structure beforehand.

On the basis of the GLUE approach used in this study and the findings, it can be concluded that for

any rainfall-runoff model there is no single best parameter set that represents the catchment for a range

of rainfall-runoff responses, but rather a range of different sets of model parameter values may representthe rainfall-runoff process equally well. Therefore, there may be many representations of a catchment that

may be equally valid in terms of their ability to produce acceptable simulations of the available data. The

parameter sensitivity analysis with the GLUE approach is essentially a non-parametric method, in that

it makes no prior assumptions about the variation or covariation of different parameter values, but only

evaluates sets of parameter values in terms of their performance. This study has found that recession flow

parameters that control the slow flow component are relatively more sensitive than those of the fast flow

component.

Within the GLUE framework, multiple acceptable parameter sets give a range of predictions. This may

actually be an advantage, since it allows the possibility of assessing the uncertainty in predictions as

conditioned on the calibration data and then using that uncertainty as part of the decision-making process



15/15


arising from the modelling project. However, the present study has found that during the wet period the

rainfall-runoff modelling does not bracket all the discharge observations for at least 90% confidence interval.

This is due to errors in the input data used being dominated by peaks during the wet season and being

relatively small in the dry period recession. Most of these errors may be due to an inadequate representation

of the spatial patterns of rainfall within the catchment. Although the models used in this study are not fully

distributed process-based models, these results are perhaps representative of the type of accuracy that might

be achieved in predicting the discharge while representing the state of the catchment as a single set of

process-related parameters.

ACKNOWLEDGEMENTS

This work was performed during the doctoral study programme of the first author at the Institute for Land

and Water Management (ILWM), Katholieke Universiteit Leuven (K.U.Leuven), Belgium. The study was

funded by the Belgian Administration for Development Co-operation (BADC/ABOS) and the Universityof Dar es Salaam (UDSM), Tanzania. The first author is pleased to acknowledge this financial support.

He is also grateful for the accessibility to the streamflow data and rainfall data provided by the Water

Section of the Department of Civil Engineering at UDSM and the Directorate of Meteorology respec-

tively.

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