Mulungushi river flow modelling in the headstreams of the Zambezi basin,
Zambia
J.M. Kampata1,2, B.P. Parida1, D.B. Moalafhi1, D. Mazvimavi3 and S. Ringrose4
1
University of Botswana, Faculty of Science, Department of Environmental Science, P/Bag UB
00704, Gaborone, Botswana.
2
Ministry of Energy and Water Development, Department of Water Affairs, P.O Box 50288, Lusaka.
Zambia.
3
4
University of Western Cape, South Africa. University of Botswana, HOORC, Maun, Botswana.
Abstract
The Mulungushi river basin in central Zambia is the major source of water for demand centres
and/or water users including Kabwe mining township, the Mulungushi University, subsistence
and commercial agricultural enterprises, local communities and the Lunsemfwa Hydropower
Company further downstream. The water managers overseeing the water provision services
over the area experience difficulties particularly during the dry season in rationing water
allocations amongst the various users as the temporal variations in available water (river
flows) for the competing uses is influenced by largely ‘unexplained’ complex interplay of
factors such as climatic variables which subsequently affect the river flow. Since Artificial
Neural Network (ANN) models are capable of mimicking complex processes, an attempt has
therefore been made to develop an appropriate network which can be used for predicting the
observed river flows based on rainfall and evaporation. This then formed the basis from which
short term forecasts of river flow were made to help for onward planning. For this, readily
available data in terms of. two major input variables viz: the monthly rainfall and monthly
evaporation and the output variables which are the concurrent monthly discharges have been
considered. A two layer feed-forward network with one hidden layer with 30 neurons through
which information gets transferred with the help of the sigmoidal transfer function resulted as
the best network with a coefficient of determination (r2) of 0.831 and Mean squared Error
(MSE) of 1.55. The training alogorithm used was the Levenberg-Marquardt back propagation
with early stopping in which data was divided into three groups for training, validation and
testing. The same network using rainfall as the only input resulted in the least MSE of 2.07 and
with r2 of 0.804, while the optimum network using evaporation as the only input gave MSE
and r2 of 14 and 0.630 respectively.
Key words: Artificial neural networks, Levenberg-Marquardt back propagation, Mean
Squared Error, Neurons, Sigmoidal transfer function.

Corresponding Author email: jkampata@yahoo.com
1
1. Introduction
The inadequate knowledge and information on water resources required for decision-making
and long-term planning continues to be a major constraint for effective water resources
management (World Water Assessment Programme, 2009). Modelling of rainfall- runoff or
the prediction of river flows provides water managers with information to quantify the water
resources of a river basin. This is particularly critical for equitable water allocation amidst the
increasing demands from various users.
The deterministic/ conceptual models (that consider the physics of the underlying process) and
systems theoretic/black-box models (that do not consider the underlying physics of the
process) are the two major types of models that have been used for modelling the rainfallrunoff or prediction of river flow (Dawson and Wilby, 2001; Srinivasulu and Jain, 2006; Mutlu
et al., 2008; Akhtar et al., 2009).
The deterministic/conceptual models are more suitable for modelling virgin flows, and are not
able to perform adequately in catchments altered due to manmade or other related activity (e.g.
many storage structures), which essentially affect the true dynamic nature of the rainfall–runoff
relationship in a catchment. The systems theoretic models, on the other hand attempt to
develop relationships among input and output variables involved in a physical process without
considering the underlying physical process (Srinivasulu and Jain, 2006).
Practitioners in water resources have primarily used simple linear regression or time series
models to obtain approximations of the relationships between variables (Maier and Dandy,
2000). One reason for this is that the rules governing sophisticated statistical models have
generally been considered to be too restrictive and to make it too difficult to utilise them for
real-life applications (Maier and Dandy, 2000). Rainfall- runoff process exhibit complex nonlinear dynamics (Abrahart and See, 2007; Aqil et al., 2007). Considering the complexity of this
process, there is a strong need to explore solutions, which can extract the complexity of the
rainfall – runoff process even without having the complete physical understanding of the
system (Aqil et al., 2007).
The use of linear regression however raises a fundamental violation of the characteristics of
water resources variables which are highly non-linear. In addition water resources variables are
often not normally distributed, and their nature is such that it is extremely difficult, if not
impossible, to find suitable transformations to normality (Maier and Dandy, 2001). To
overcome this constraint, ANNs offer an easier solution to solve non-linear problems such as
rainfall-runoff modeling (Maier and Dandy, 2000; Aqil et al, 2007; Mutlu et al. 2008). In
modelling rainfall-runoff relationship in arid and semiarid regions in which the rainfall and
runoff are very irregular, Riad and Mania (2004) found that ANN approach gives much better
prediction than the traditional multiple linear regression (MLR) method.
Artificial Neural Networks (ANNs) fall in the category of systems theoretic models (Dawson
and Wilby, 2001; Srinivasulu and Jain, 2006; Mutlu et al., 2008; Akhtar et al., 2009) and they
have been widely used in the field of hydrology and water resources modelling. Maier and
Dandy (2000) have undertaken a comprehensive review of the use of ANNs in hydrology and
water resources management while Dawson and Wilby (2001) specifically reviewed the use of
2
ANNs in rainfall–runoff modelling and flood forecasting. They indicate that ANNs are well
suited to the challenging tasks of rainfall–runoff and flood forecasting. This is further
ascertained by Abrahart and See (2007) who demonstrated that neural networks are capable of
modelling nonlinear hydrological processes and are therefore appropriate tools for hydrological
modelling. In addition, comparison of the performance of ANN against traditional multi linear
regression models for modelling non-linear hydrology process such as rain-fall and runoff has
found that ANNs perform well (Dawson and Wilby (2001); Abrahart and See, 2007)
ANNs have therefore been chosen over the use of conceptual models as in these headwater
catchments, as is the case in most such catchments in Africa, knowledge of the internal
hydrologic processes is not well known. ANNs therefore offer an advantage as they are capable
of modelling complex nonlinear relationships between input and output data sets (Dawson and
Wilby 2001).
This paper is an attempt to contribute to the practical use of ANNs especially for the semi-arid
environment with limited data. Continued research and development in this field will provide a
stronger understanding and appreciation of the hydrological modelling opportunities that are on
offer such that good scholarship and greater awareness might encourage the wider acceptance
of neural solutions (Abrahart and See, 2007).
Artificial Neural Networks
An Artificial Neural networks (ANN) is a highly interconnected network of many simple
processing units called neurons, which are analogous to the biological neurons in the human
brain (Srinivasulu and Jain, 2006). They are invaluable for applications where formal analysis
would be difficult or impossible, such as pattern recognition and nonlinear system
identification and control (Demuth et al., 2008). ANNs have been used to solve Classification,
Function Approximation, Prediction and Clustering problems (NeuroDimension, Inc., 2007).
An ANN normally consists of two layers, a hidden layer and an output layer. The structure of a
two layer feed-forward ANN is shown in figure 1.
Source: (Demuth et al., 2008).
Figure 1: A two-layer feed-forward neural network
3
More detailed description on ANNs may be found in Maier and Dandy, 2000; Dawson and
Wilby, 2001 and Demuth et al., 2008).
2. Methodology
Based on the problem, a network that is able to predict (and forecast) the runoff based on
climate and derived information is needed. Such a problem falls under the category known in
the literature by various terms namely fitting problem, function approximation, regression
learning, nonlinear regression (Demuth et al., 2008). In this study, the main objective was to
develop an ANN and train it to learn the relationships between climatic variables
(precipitation, evaporation and temperature) as model predictor variables on one hand and river
flow as the dependent variable on the other hand. After learning the relationships, the model
would be used to simulate the river flow patterns.
The study used the feedforward neural network using the backpropagation algorithm for
training as these have been found to be the most suitable and are widely used for the prediction
and forecasting of water resources variables (Sivakumar et al., 2002; Maier and Dandy, 2001;
Maier and Dandy, 2000).
The steps of using ANNs for rainfall-runoff modelling as described by Maier and Dandy
(2001; 2000) and adopted in this study were followed. These are discussed below:
i.
Division and pre-processing of the available data
The available data was spilt into three sub-sets; a training set, a validation set and a testing set
which is implemented as a means of improving generalization under early stopping. In order to
ensure that all variables receive equal attention during the training process, they were
standardised. In addition, the variables were scaled to between 0 and 1 in such a way as to be
commensurate with the limits of the activation functions used in the output layer. Outputs of
the logistic transfer function are between 0 and 1. This is usually taken as a precautionary
measure to avoid too slow or too fast convergence of the performance function and thus this
aids realization of optimum solution.
ii.
Determination of appropriate model inputs
The input variables based on a priori knowledge of available causal variables were rainfall,
temperature and evaporation. Analysis of the influence of the input variables (independent
variables) on the output (dependant variable) was undertaken by determination of the
individual correlations between each input variables and the target variable. This provided the
means of confirming which combination of input variables had significant influence. This
avoided the need to undertake trial and error approach of training separate networks for each
input variable which is computationally intensive.
The input variable with the highest correlation are retained for use in the model. Matignon
(2005) indicates that the draw back to this approach is that it ignores the partial correlation
4
between the other inputs in the model that might result in erroneous inputs added or removed
from the model. To over come this, Matignon (2005) suggests dropping the non-significant
inputs are dropped one at a time. This was done by dropping each input which has its p-value
above the 0.05 alpha- level which means only the input variables that have significant
influence on the output variable are retained.
This avoids the need to undertake trail and error approach of training separate networks for
each input variable which is computationally intensive.
iii.
Network architecture
The number of nodes in the input layer is equal to the number of model inputs, whereas the
number of nodes in the output layer are equal the number of model outputs. A two-layer feedforward network with sigmoid transfer function in the hidden neurons and linear transfer
function output neurons has been used. One hidden layer is used as it has been found to be
adequate in applications of forecasting and predicting water resources variables. The number of
neurons in the hidden layer of the neural network were changed from time to time to
investigate the best results. The network was trained with Levenberg-Marquardt
backpropagation algorithm. The LM algorithm has been found to be very efficient for training
small to medium-size networks (Aqil et al. 2007).
The architecture used was a multi-layer feed-forward neural network (FNN as shown in Figure
1. The hidden layer nodes allow the network to detect and capture the relevant pattern(s) in the
data, and to perform complex nonlinear mapping between the input and the output variables.
The sole role of the input layer of nodes is to relay the external inputs to the neurons of the
hidden layer. The outputs of the hidden layer are passed to the last (or output) layer which
provides the final output of the network.
The results from various combinations of models was analyzed for the best performing model
in order to recommend an appropriate ANN model that best represents the rainfall-run off
process.
The 72 input samples are randomly selected, 50 training, 11 validation and 11 testing. This
approach ensures that the independence assumption of the inputs is retained.
iv.
Optimisation of the connection weights (training)
The input from each node in the previous layer (xi) are multiplied by a connection weight (wji).
These connection weights are adjustable and may be likened to the coefficients in statistical
models. The process of optimising the connection weights is known as ‘training’ or ‘learning’.
This is equivalent to the parameter estimation phase in conventional statistical models. The
connection weights (w) will be adjusted using the gradient descent rule of optimization.

E
wt     
 wt  1 ,
(1)
w
s 1
5
where s is the training sample presented to the network, η is the learning rate, and μ is the
momentum value. The number of training samples presented to the network between weight
updates is called the epoch size (ε).
At each node, the weighted input signals are summed and a threshold value (θj) is added. This
combined input (Ij) is then passed through a nonlinear transfer function (f(.)) to produce the
output of the node (yj). The output of one node provides the input to the nodes in the next layer
(Minasny et al., 2004; Moalafhi, 2004; Maier and Dandy, 2001). This process is summarised in
equations (2) and (3) and illustrated in Figure 2.
I j   w ji xi   j ,
summation
(2)
y i  f I j  ,
transfer
(3)
f(I j)
X0
X1
Ij
w j0
w j1
Yi
X2
Sum
w j2
X3
Transfer
Output path
w j3
Source: (Moalafhi, 2004; Maier and Dandy, 2001)
Figure 2: Operation of hidden layer neuron of a feedforward Artificial Neural Network
The transformation of input into a hidden layer node to an output was done using the logistic
function which is of the sigmoidal type transfer function. Abrahart and See (2007), mention
that “models incorporating sigmoid transfer functions can support improved generalizations
and superior learning characteristics”.
The backpropagation algorithm which is based on the method of steepest descent was used for
optimizing the feedforward ANN. It is extensively used in feedforward networks as such
networks with biases, a sigmoid layer, and a linear output layer are capable of approximating
any function with a finite number of discontinuities (Demuth et al., 2008).
The error function, E, most commonly used is the Mean Squared Error (MSE) function. The
preference of using the MSE as the performance function is mostly because it is calculated
easily, it penalises large errors and its partial derivative with respect to the weights can be
calculated easily. As each input is applied to the network, the network output (modelled value)
6
is compared to the target (observed value). The error is calculated as the difference between the
target output and the network output. The goal is to minimize the average of the sum of these
errors. It is given as:
1
E  MSE 
N
2
N
 obs
i 1
i
 cali 
(11)
Where obsi is the observed data and cali is the calculated output (modelled) predicted by
network. As MSE grows, the accuracy of model reduces.
v.
Model validation and testing.
Data sets not used in training were used for model validation and testing. These are the 20% of
the data that are each kept for validation and testing. The validation data set are used to
measure network generalization, and to halt training when generalization stops improving
while the testing data set has no effect on training and so provides an independent measure of
network performance during and after training. These are checked to see how well the model
performs.
MATLAB 7.6.0.324 (R2008a) (Maths Works Inc., 2008) was used to develop, train and
simulate the ANN.
3. The Study area
The Mulungushi river sub-basin is located between longitude 28.17º to 28.83º East and latitude
14º to 14.5º South on the western part of the Lunsemfwa river basin. It covers an area of 1448
km2 measured upstream of the hydrological station on the Mulungushi river at Great North Road
Bridge (Figure 3).
7
28°9'59"
28°19'58"
28°29'57"
28°39'56"
28°49'55"
28°59'54"
N
W
E
Kapiri Moshi #
14°10'05"
14°10'05"
Great North Road
14°00'06"
14°00'06"
S
14°20'04"
Mulungushi River
28°9'59"
0
28°19'58"
30
28°29'57"
28°39'56"
28°49'55"
60 Kilometers
28°59'54"
14°30'03"
14°30'03"
Kabwe
#
14°20'04"
M
ulu
ng
us
hi
Figure 3: Mulungushi river sub-basin
The town of Kabwe (Provincial Capital of Central Province), the Mulungushi University,
various subsistence and commercial farmers, local communities and Lunsemfwa hydropower
Company (down stream) depend on the Mulungushi river to meet their water needs. Water
demand is high during the dry season when flows are low. Conflicts currently occur between
municipal water, agriculture and hydropower water use. Modelling flows would provide a
better understanding on the flow characteristics. This information can be used by water
managers to improve on water apportionment and monitoring water availability thus mitigate
conflicts among various water users.
Hydro-meteorological data for the period October 1971 to September 1977 was used (72
months). This period was used as it has data with no gaps.
Rainfall data was obtained from the Zambia Meteological Department station at Kabwe and
river discharge data of station Mulungushi River at Great North Road from the Zambia
Department of Water Affairs.
8
4. Results and Discussion
Determination of appropriate model inputs
Table 1 shows the correlations between each of the variables and Table 2 the p-values
Table 1: Lower Correlation coefficients matrix between the variables
Discharge
Rainfall
Temperature
Evaporation
Discharge
1.0000
0.4894
0.1245
-0.3726
Rainfall
Temperature
Evaporation
1.0000
0.4135
-0.2965
1.0000
0.2975
1.0000
As expected the influence of rainfall on the discharge is high which is reflected by the higher
correlation coefficient between them. This means that the higher the rainfall, the higher the
discharge.
The significance of these inputs is then tested to select the inputs that significantly influence
the discharge. The hypothesis of no correlation is tested by calculating the p-values. If p is
small, say less than 0.05, then the correlation r is significant.
Table 2: p-values between variables
Discharge
Rainfall
Temperature
Evaporation
Discharge
1.0000
0.0000
0.2973
0.0013
Rainfall
Temperature
Evaporation
1.0000
0.0003
0.0115
1.0000
0.2975
1.0000
From table 2 the significant correlations are between Discharge-Rainfall, Discharge –
Evaporation, Rainfall-Temperature, Rainfall-Evaporation. Since Temperature is not
significantly correlated to the discharge it does not contribute to explaining the dependant
variable and may thus be dropped from the input variables. The selected inputs to the model
are thus rainfall and evaporation.
ANN model results
Table 3 shows the results of the ANN model using various inputs and with increasing the
number of neurons in its hidden layer. Model number 3 with 30 neurons in its hidden layer
with input rainfall & evaporation gave the best model results with MSE of 1.55 and Coefficient
of determination (r2) of 0.831.
Table 3: Model results
Model no
Neurons in
its hidden
Input rainfall only
Input evaporation
only
9
Input rainfall &
evaporation
layer
10
20
30
1
2
3
r2
0.767
0.748
0.804
MSE
9.82
4.44
2.07
MSE
14.7
14
14.4
r2
0.625
0.622
0.621
MSE
3.20
2.88
1.55
r2
0.778
0.791
0.831
The plot of the modelled discharge resulting from application of rainfall and evaporation as only inputs and with
rainfall and evaporation is shown in Figure 4. It is seen that the models with just one input of rainfall or
evaporation do not simulate the discharge as well as the model with the two inputs of rainfall and evaporation.
This reinforces the importance of evaporation in influencing the rainfall-runoff process.
Q-obs
Q-input rain
Q-Input evap
Q-Input rain and evap
Plot of observed and modelled discharge
- Mulungushi River at Great North road
30
Discharge (m3/s)
25
20
15
10
5
0
-5 0
10
20
30
40
50
60
70
-10
Time (months)
Figure 4: Plot of observed and modelled discharge for various inputs- Mulungushi River at Great North road
10
80
Figure 5: Performance and regression plots for model 3 (input monthly rainfall and evaporation)
11
Plot of observed and modelled discharge
- Mulungushi River at Great North road
Q-obs
Q-Input rain and evap
30
Discharge (m3/s)
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
Time (months)
Figure 6: Plot of observed and modelled discharge for model 3- Mulungushi River at Great North road
Evaluating the performance of model
Apart from the Coefficient of determination and the MSE additional tests were done to check on the performance
of the model. An assumption was made that if the modelled (output) values fall within 95% confidence internal
then the model can be considered to have performed satisfactory. To check this, two plots were evaluated. Figure
1 shows the robust regression line and Figure 2: Plot of residuals both with 95% confidence internal.
25
Output (modelled discharge) m3/s
20
15
10
Data
Robust Regression fit
95% Confidence Interval
5
0
0
5
10
15
Target (Observed discharge) m3/s
12
20
Figure 7: Robust regression
Figure 8 shows the plot of the data residuals which are visually examined to gain an insight into the "goodness" of
the model fit. The residuals are calculated as the difference between observed and modelled discharge. For a good
fit the residuals should be close to zero and have a random scatter. If the residual plot has a pattern which does not
appear to have a random scatter, this indicates that the model does not properly fit the data (Demuth et al., 2008).
This then means there is need for a different model structure or input variable.
Residual Case Order Plot
15
Residuals (modelled-Taget)
10
5
0
-5
-10
-15
10
20
30
40
50
60
70
Time period
Figure 8: Plot of residuals with 95% confidence internal.
The residuals are seen to be random and mostly around zeros as well as within the 95% confidence internal
around zero. The model can thus be concluded to be performing satisfactorily.
There are however, three points that have residuals that do not contain zero (points in red). This indicates that the
residual is larger than expected in 95% of new observations. This may suggest the data points may be outliers
(Montgomery et al., 2001).
13
5. Conclusion
In this study the absence of continues long term river discharge, water level data and meteorological data of
precipitation, temperature, evaporation etc was a constraint. In view of this constrain the inputs available for use
were limited to the monthly precipitation and evaporation. This inevitably affected the length of available data as
well as the types of inputs to use which would have improved the accuracy of the modelling. Even with short
term/ insufficient data the ANN approach has been found suitable to be used to model the rainfall- runoff. It is
better to have some information than none at all to make the necessary water resources management decisions.
Improving the ANN model should be a continuous process of updating it as new data in the basin becomes
available. This will ultimately improve its forecasting capabilities. There is need to reemphasize the importance of
hydro-meteorological data management if water resources are to be effectively managed.
Acknowledgements
Thanks to the Zambia Meteorological Department and Water Affairs Department for providing the rainfall and the
river discharge data respectively; Gratitude to UNESCO who are supporting the SIMDAS project through which
this study has been conducted as input into the ongoing research.
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