82
Water Resources Systems--Hydrological Risk, Management and Development (Proceedings of symposium
HS02b held during IUGG2003 al Sapporo. July 2003). IAHS Publ. no. 2 8 1 . 2003.
A "consensus" real-time river flow forecasting
model for the Blue Nile River
ASAAD Y. SHAMSELDIN
Department
Birmingham
of Civil Engineering,
BIS 2TT, UK
The University
of Birmingham,
Edgbaston,
[email protected]
KIERAN M. O'CONNOR
Department
of Engineering
Hydrology,
National
University
of Ireland,
Galway,
Ireland
Abstract The efficacy of using a consensus real-time river flow-forecasting
model for the Blue Nile River is investigated. The selected consensus model
combines the river flow forecasts of two individual multiple-input singleoutput river flow routing models, both operating in simulation non-updating
mode, the first being a non-parametric linear storage model and the second
having the parametric structure of a multi-layer feed-forward neural network.
The upstream inflow to the Blue Nile and the outflows of its two major
tributaries are used as inputs to both models in order to provide the simulationmode river flow forecasts just upstream of Khartoum, the capital city of Sudan.
The weighted average method (WAM) is used to combine the simulation-mode
forecasts of these two models. The consensus real-time river flow forecasts are
obtained by updating the combined simulation-mode forecasts using an auto­
regressive (AR) model error updating procedure. Disappointingly, the results
show that the performance of the consensus model, operating in the simulation
mode, is not different from that of the best individual model, i.e. that the linear
model is given practically zero weight in the consensus model. However,
significant improvements in the forecasting performance are obtained after
updating the simulation-mode consensus forecasts.
Key words Blue Nile; consensus real-time forecasting; linear model; neural network
INTRODUCTION
The essence of the "consensus" model concept is that the synchronous discharge
forecasts of a number of structurally different river flow forecasting models are
optimally combined to provide an overall "consensus" discharge forecast. In this
approach, each of the individual models contributing to the combination is regarded as
providing a source of information which is different from that provided by the other
models so that the consensus forecast, obtained by a judicious combination of these
different sources, would be expected to be more accurate and reliable than that of the
best of the individual models used in producing that forecast (cf. Shamseldin et al.,
1997). Results of previous studies (Shamseldin et al, 1997; Shamseldin & O'Connor,
1999; See & Openshaw, 2000; See & Abrahart, 2001) indicate that, generally, this
hypothesis holds true.
There are a number of different methods, of varying degrees of complexity, which
can be used for producing consensus forecasts. These methods include linear
weighting, neural network-based and fuzzy-based combination methods. In the
A "consensus " real-time river flow forecasting model for the Blue Nile River
83
previous applications of these combination methods for river flow forecasting, using
either the flow forecasts of a set of rainfall-runoff models (e.g. Shamseldin et al,
1997) or of river flow routing models (e.g. See & Abrahart, 2001), only small or
medium-sized rivers have been considered. In the present study, the Weighted Average
Method (WAM, a linear regression-type combination method) is applied for river flow
forecasting on a large river, namely, the Blue Nile, the results presented being those of
a preliminary investigation of the efficacy of consensus flow forecasting for that river.
For the WAM, the combined discharge forecast is obtained as the weighted sum of
the corresponding discharge forecasts of the individual constituent models, the weights
being estimated by the method of ordinary least squares (OLS), whereby the sum of
squares of the differences between the consensus and corresponding observed
discharges are minimized.
The form of WAM used here combines the forecasts of just two constituent
models: those of a linear river flow routing model and those of a nonlinear neural
network river flow routing model, both of which operate in simulation (non-updating)
mode, i.e. without their forecasts being updated using recently observed discharges as
feedback. These two models are described briefly in later sections of this paper. The
final updated consensus river flow forecasts are obtained by updating the WAMcombined simulation mode forecasts using an auto-regressive (AR) model error
updating procedure. This updating procedure is based on forecasting the errors in the
simulation-mode consensus discharge forecasts, the final updated real-time discharge
forecast at each time step being the sum of the non-updated (simulation-mode)
discharge value and the corresponding error forecast.
The paper is organized as follows: first, a brief description of the catchments and
the data used in the study is given. Secondly, the constituent models used in producing
the consensus forecasts are briefly described. Third, the application of the consensus
WAM and that of its constituent models are discussed. Finally, the conclusions of the
study are provided.
CATCHMENT AND DATA
The Blue Nile originates in Lake Tana, on the Ethiopian plateau, in East Africa. It has
a basin area of 324 530 km , which covers most of Ethiopia west of longitude 40°E
and between latitudes 9°N and 12°N (Shahin, 1985, p. 42). This plateau is characterized
by diversity in climate, geology, topography and vegetation. The average annual
rainfall in the basin, upstream of Eldiem, is 1600 mm (Sutcliffe & Parks, 1999, p. 130).
When the Blue Nile leaves Ethiopia, it flows through Sudan where it joins the
White Nile (one of the main tributaries of the River Nile) at Khartoum (the capital of
Sudan) to form the River Nile (Fig. 1). The Blue Nile contributes about 59% of the
annual flow of the Nile—the average annual flow being 84 km . Thus, it is regarded as
the main source of flooding on that river. Such flooding causes loss of life and massive
scale damage to the agricultural sector and to riparian property.
The Blue Nile has two main tributaries: the Dinder and Rahad rivers, which join it
in the reach between Eldeim, near the Sudanese-Ethiopian boarder, and Khartoum.
Both tributaries, which originate on the Ethiopian plateau, about 30 km west of Lake
Tana, only flow for four to five months in any year, reducing to a series of scattered
2
3
Asaad Y. Shamseldin & Kieran M. O 'Connor
84
Fig. 1 The Blue Nile River and its tributaries.
6000 r
4500
-
3000
-
o 1500
tf)
-
\
O)
c5
JZ
a
0 '
J
1
F
1
M
'
A
—
1
1
M
1
J
J
i
i
t
i
i
A
S
O
N
D
Month
Fig. 2 Average annual discharge hydrograph of the Blue Nile River at Eldeim.
water pools for the rest of the year. The upstream flow hydrograph of the Blue Nile,
measured at the Eldeim, the flow hydrograph of the River Dinder, measured at Gwasi,
and also the flow hydrograph of the River Rahad, measured at Hawata, are used as
inputs to the flow routing models considered here.
Nineteen years of daily flow of the Blue Nile flow values, covering the period
1976-1994, are used. The average annual flow at Roseries/Eldeim is 48.75 km , with
the annual flow varying between 28.68 km and 69.80 km . The Blue Nile is a very
seasonal river with the peak flow occurring in late August (Fig. 2). The total flow
3
3
3
A "consensus " real-time river flow forecasting model for the Blue Nile River
85
during the flood season (June-October) constitutes, on average, 80% of the total
annual flow in the river. During the flood season, the maximum daily flow can reach a
value of 10 000 m s" .
3
1
LINEAR RIVER FLOW ROUTING MODEL
The Linear River Flow Routing Model (LRFRM) is based on establishing a linear
time-invariant relationship between the Blue Nile flow, just upstream of Khartoum,
with the three input time series referred to above, i.e. the upstream flow hydrograph at
Eldeim and the outflow hydrographs from its two major tributaries. Thus, the LRFRM
can be regarded as a multiple-input single-output model. Previous studies have shown
that the LRFRM is very successful in flood forecasting for large rivers (Liang & Nash,
1992), including the River Nile (Abdo et al, 1992; Elmahi & O'Connor, 1995).
The overall operation of the LRFRM, for the z'-th time period, incorporating a
residual error, e,., can be expressed mathematically as:
3
q(j)
7=1
k=\
where Q denotes the observed discharge, q(j) is the order/memory length of the y'-th
input time series Xj, co^- is the model coefficient/parameter corresponding to Xj, and b(J)
is the input lag time for Xj. Equation (1), describing the overall transformation operation
may be regarded a multiple linear regression type of model. Thus, the parameters of
the LRFRM can be estimated directly using the method of ordinary least squares.
i
NEURAL NETWORK RIVER FLOW ROUTING MODEL
The Neural Network Riverflow Routing Model (NNRFRM) is based on the structure
of the feed-forward Multi-Layer Perceptron (MLP) which constitutes a flexible
mathematical modelling technique inspired by research on biological networks. In the
present study, the NNRFRM is visualized as a nonlinear multiple-input single-output
river flow routing model.
The MLP, which has dominated the applications of neural networks in
hydrological modelling (cf. Dawson & Wilby, 2001; Maier & Dandy, 2000), is
characterized by its powerful capabilities in the modelling of complex nonlinear inputoutput relations. It is simply a network of interconnected computational elements, i.e.
the neurons, linked together by connection pathways, which are arranged in a series of
layers (Fig. 3), each layer performing a distinctive function in the operation of the
network (Fig. 1). The neuron layers of the MLP are the input layer, the output layer,
and at least one hidden layer between the input and output layers. The input layer
receives the external input array to the network, each input array element being
assigned to only one neuron. The elements of the external input array are those of the
same three input time series used in the LRFRM. The output of each neuron in the
input layer, which is equal to its external input element (corresponding to a unit-
86
Asaad Y. Shamseldin & Kieran M. O 'Connor
D i s c h a r g e forecast
Input Neuron
Fig. 3 Schematic diagram of the Neural Network River Flow Routing Model (NNRFRM).
identity transformation), then becomes the input to each of the neurons in the first
hidden layer. Thus, each neuron in this hidden layer has an input array consisting of
the outputs of the input layer neurons.
In this study a single hidden layer is used, as the use of more than one is hardly
ever beneficial (Masters, 1993). Each hidden layer neuron produces only a single
output which becomes an element of the input array to each neuron in the subsequent
(output) layer. In the present flow forecasting context, the output layer has only one
neuron, which produces the final network output. The required number of neurons for
the input and output layers is usually found by trial and error.
For all hidden and output layer neurons, the process of the transformation of the
input array to a single output is quite similar. In contrast to the simple unit-identity
transformation used for the input layer neurons, this process is basically a nonlinear
transformation of the total sum of the products of each of its input array elements with
its corresponding weight (or re-scaling factor), plus a constant "baseflow" term. This
constant term is known as the neuron threshold value, the function used in such a
transformation being known as the neuron transfer function. The same transfer
function is used for all of the hidden and the output layer neurons. Effectively, the
weights and the threshold values constitute the parameters of the network, which are
estimated by calibrating (or training) the network. This is done by minimizing the sum
of the squares of the differences between the network output series, and the
corresponding re-scaled observed discharges, using nonlinear optimization algorithms.
The transfer function used in this paper is the logistic function, which has been
widely used in neural network studies (Blum, 1992). The logistic function has an "S"
shape and its range varies between 0 and 1, which implies that the estimated network
output values are likewise bounded within this range (0,1). As the actual observed
discharge values are usually outside this range, re-scaling of these discharge values is
required in order to compare the actual observed discharges and the final output series
of the network. In the present study, in the case of the N N P v F R J V I , a simple linear
rescaling function is used for this purpose.
A "consensus " real-time river flow forecasting model for the Blue Nile River
87
APPLICATION
The available period of 19-years of flow data from the Blue Nile catchment is split into
two non-overlapping periods, the first 13 years being used for model calibration and
the following (i.e. remaining) six years being used for model verification/validation
purposes.
For chosen values of memory length and input lag times, the optimum parameter
values of the LRFRM are estimated by the method of ordinary least squares. The
optimum input orders and input lag times, estimated by trial and error, are shown in
Table 1. The same input orders and input lag times are used in conjunction with the
NNRFRM. The neural network parameters of the NNRFRM are estimated using the
sequential optimization procedure of the successive use of the genetic algorithm values
for the conjugate gradient method, as adopted by Shamseldin et al. (2002). The
number of neurons in the hidden layer of the NNRFRM is fixed at two, as it was found
that there was no real improvement in the overall performance of the network by
further increasing that number. As the three input variables have different orders of
magnitude, each of the three input time series is rescaled by dividing each element of
the series by its maximum value in the calibration period so as to facilitate model
calibration and to improve model performance.
The combination weights of the WAM are estimated by the method of ordinary
least squares and their optimum values are shown in Table 2. Inspection of Table 2
shows that the weight assigned to the NNRFRM is 0.9944, while the corresponding
weight assigned to the LRFRM is only 0.0061. The implication is that the simulation
mode consensus forecasts are virtually the same as those of the NNRFRM.
The model performance is evaluated quantitatively using the R criterion of Nash
& Sutcliffe (1970), which is defined by:
2
R = 3dL \oo
2
x
%
(2)
F
Table 1 Input orders and lag times.
Input
Input order
Lag time
Eldeim (Blue Nile)
Dinder River
Rahad River
3
1
1
6
1
1
Table 2 The optimum weight of the WAM consensus model.
Model weight LRFRM
0.0061
Model weight NNRFRM
0.9944
2
Table 3 The R efficiency values of the different models.
2
LFRM
NNRFRM
WAM Consensus model (Simulation mode)
WAM Consensus model(updating mode)
2
Calibration R (%)
Verification R {%)
91.97
95.29
95.29
98.74
85.35
87.15
87.19
98.74
88
Asaad Y. Shamseldin & Kieran M. O 'Connor
'Observed •••••••••'•'•••'NNRFRM
10000
l-Jun-1988
LRFRM
Consensus M o d e l (updating mode)
r
l-Jul-1988
31-Jul-1988
30-Aug-l988
29-Sep-l9S8
29-Oct-1988
Fig. 4 Observed and estimated discharge hydrographs of the Blue Nile River at Khartoum.
where F is the sum of the squares of differences between the observed discharges and
the mean discharge over the calibration period.
The R~ values of the LRFRM, the NNRFRM and the consensus model are shown
in Table 3. The R performance of the NNRFRM is significantly better than that of the
LRFRM model, in both the calibration and the verification periods and, consistent with
the weights of Table 2, the R~ value of the simulation mode forecasts of the consensus
model is virtually the same as that for the NNRFRM, the best individual model. Table 3
also shows that the updated real-time forecasts of the consensus model are
significantly better, in terms of R , than the simulation mode consensus forecasts. This
shows that the updating of the simulation mode discharge forecasts of the consensus
model (or of the NNRFRM) by the AR model error updating procedure is quite
successful. Figure 4 shows comparisons of the observed and the estimated discharge
hydrographs of the different models used in the study in two high flood years, which
reflect the numerical results discussed above.
0
2
2
SUMMARY AND CONCLUSIONS
The present study explores the uses of a consensus river flow model for real-time
forecasting on the Blue Nile upstream Khartoum. The real-time consensus forecasts
are obtained by updating the simulation-mode consensus forecasts using the standard
AR model error updating procedure. The simulation mode consensus forecasts are
A "consensus " real-time river flow forecasting model for the Blue Nile River
89
obtained by using the Weighted Average Method (WAM), which combines the
simulation mode forecasts of two models, namely, the Linear Flow Routing Model
(LRFRM) and the Neural Network River Flow Routing Model (NNRFRM).
The results show that, for the Blue Nile data set, the nonlinear NNRFRM has
significantly better performance than the more primitive LRFRM. Thus, the NNRFRM
can be viewed as an effective tool for flood forecasting and hence as a component of a
decision support system for the mitigation of the hazardous impacts of floods in the
Blue Nile. The consensus forecasting results obtained using the WAM are certainly
disappointing, in that the performance of the consensus model operating in simulation
mode is basically the same as that of the best individual model (the NNRFRM). While
considerable improvement in the forecasting performance is obtained after the
simulation-mode consensus forecasts are updated using the AR model error updating
procedure, a similar improvement can be expected by updating the forecasts of the
NNRFRM. Clearly, in retrospect, the use of just two constituent models in the WAM
form of consensus model was quite inadequate in this case and, in future applications
of the consensus model to river flow forecasting on the Blue Nile, consideration might
also be given to the use of more sophisticated combination procedures, such as the
neural network-based and fuzzy-based combination methods, for producing the
consensus forecasts. The use of such methods, incorporating a greater number of
constituent models, may lead to improvements in the performance of the simulation
and the real-time mode consensus river flow forecasts.
Acknowledgements The authors are pleased to express their sincere appreciation and
thanks to Eng. Abderhman S. Elzein, Ministry of Irrigation and Water Resources
Sudan, for his efforts in the preliminary processing of the Blue Nile flow data.
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