Lihua Xiong, Kieran M. O’Connor and Monomoy Goswami
Department of Engineering Hydrology, National University of Ireland, Galway, Ireland
Correspondence address:
Dr. Lihua Xiong
Department of Engineering Hydrology
National University of Ireland
Galway, Ireland
Fax: 353-91-524913
E-mail: Lihua.xiong@nuigalway.ie; kieran.oconnor@nuigalway.ie
Abstract:
In this paper, the artificial neural network (ANN)
model is employed in the application of flood forecasting on a karstic catchment,
namely the Fergus catchment. Normally the rainfall-runoff relationship on the
karstic catchments appears to be highly non-linear and is very difficult to
simulate. Since the ANN is mainly designed for the approximation of non-linear
relationships, then its use is indicated for rainfall-runoff modeling on karstic
catchments. The Fergus catchment is located at Ballycorey in County Clare, Ireland,
with an area of 512 km2. Daily data is used to test the ANN model,
covering the period from June 01, 1977 to December 31, 1999.
The calibration period is from June 01, 1977 to May 31, 1994, and the
verification period is from June 01, 1994 to December 31, 1999. The Nash model
efficiency index
of the ANN is shown to be 98.37%
in the calibration period and 97.44 in the verification period, which is very
impressive and can be considered successful. The conclusion is drawn that the
ANN model can be used for flood forecasting in karstic catchments that have
a very strong non-linear rainfall-runoff relationship.
Keywords: flood forecasting, artificial neural network (ANN), karstic catchment
The application of the artificial neural network (ANN) in hydrology, especially in the context of rainfall-runoff modeling, has become very popular in recent years, largely because the ANN models have the inherent ability to capture the non-linear relationships involved (Hsu et al, 1995; Imrie et al., 2000). Examples of such applications can be found in many papers recently published (e.g. French et al., 1992; Zhu et al., 1994; Hsu et al, 1995; Smith and Eli, 1995; Minns and Hall, 1996; Shamseldin, 1997; Campolo et al., 1999; Gautam et al., 2000; Imrie et al., 2000). From these works, it has been demonstrated that the ANN models are indeed very flexible and efficient enough to successfully simulate the rainfall-runoff processes, at least with respect to the calibration (i.e. training) period of the models. However, as Hsu et al. (1995) pointed out, the ANN models are by nature “black-box” models and therefore cannot provide any physically realistic structure of the component processes or the identification of any physical parameters in their representation of the true hydrologic processes, even though such models are capable of reflecting complex non-linear relationships between the rainfall and the runoff time series. In this sense, the ANN models are not an alternative to the conceptual or the physically based distributed models for the purposes of analyzing the physical mechanisms of the hydrological processes in the watershed. Nevertheless, they do have real value as operational tools in the context of river flow forecasting.
In this paper, the
ANN model is applied for flood forecasting on an Irish karstic catchment, namely
the Fergus catchment. As generally acknowledged, the rainfall-runoff
relationship on karstic catchments appears to be very strongly non-linear and is
difficult for some rainfall-runoff models to simulate. Considering that the ANN
has such potential to approximate the non-linearity, it is worth testing the
ability of the ANN model to simulate the rainfall-runoff relationships on such a
karstic catchment, e.g. the Fergus catchment.
This paper is organized as follows. Firstly,
some basic knowledge about the ANN is presented. Secondly, the basic framework
of employing the ANN in the rainfall-runoff modelling is described. Thirdly, the
results of testing the proposed ANN model on the Fergus catchment are
considered. Finally, the conslusions are drawn.
The architectures of artificial neural networks are motivated by models of biological neural networks that can recognize patterns and learn from their interactions with the environment (Haykin, 1994). Since the 1950s, many ANN structures have been proposed and explored for tasks such as pattern recognition, learning, and control. Some of the structures most widely researched and reported on include the multilayer feed forward networks (Rumelhart et al., 1986), self-organizing feature maps (Kohonen, 1982), and Hopfield networks (Hopfield, 1982). Among these different structures, the multilayer feed forward networks have been found to have the best performance in the context of input-output function approximation (Haykin, 1994; Friedman and Kandel, 1999). As a matter of fact, almost all ANNs explored in rainfall-runoff modeling are multilayer feed forward networks (Campolo et al., 1999).
A
feed forward neural network can have many layers. For example, a typical multilayer
neural network with a single hidden layer is illustrated in Fig.1. It has input
nodes
(excluding a bias), hidden nodes
(excluding a bias), and output
nodes
, where X, Z,
and Y represent the input, hidden, and output layer respectively, n,
l, and m
represent the number of the nodes in each layer, and p
denotes the training pattern. Each of the nodes is connected to the nodes of
neighboring layers. The parameters associated with each of these connections
are called weights. The weights associated with the connections between the
input and the hidden nodes are denoted by
,
,
and those between the hidden and
the output nodes are denoted by
,
,
. All connections are designed
to be “feed forward”, which means that information transfer is allowed to proceed
only from an earlier layer to the next consecutive layer. Nodes within a layer
are not interconnected, and nodes in nonadjacent layers are likewise not connected.
The
function of each node in the ANN is assumed to be like a neuron. For a typical
node in the hidden or the output layers, it receives information from every node
in the previous layer, and its effective income is just the weighted sum of all
the incoming signals. Then, the effective income is passed through a nonlinear
activation function to produce an output signal, which is sent to all nodes in
the next layer at the same time.
Fig.1
A multilayer neural network with a single hidden layer
For example,
for the node
in the hidden layer (see Fig.1),
its effective income signal, denoted by
, is calculated as
,
(1)
where
,
, represents the input to each node in the input layer.
For the node
, its corresponding output signal, denoted by
, is obtained by using an activation (or transfer) function
,
(2)
The most widely used activation function is the sigmoid function (Haykin, 1994; Friedman and Kandel, 1999). The characteristics of a sigmoid function are that it is bounded above and below, it is monotonically increasing, and it is continuous and differentiable everywhere (Haykin, 1994; Hsu et al., 1995; Friedman and Kandel, 1999). With such important features, the sigmoid function is particularly useful in ANNs trained by the backpropagation method (Friedman and Kandel, 1999).
Among the several different sigmoid functions available, the most often used for the ANNs is the logistic function defined by
(3)
where
is an adjustable parameter used
in the activation function
.
Since any watershed has certain storage capacity, the runoff at its outlet is related not only to the current rainfall rate but also to the past rainfall and runoff situations. For a discrete lumped hydrological system, the rainfall-runoff relationship can be generally expressed as (Chow et al., 1988; Hsu et al., 1995)
(4)
where R
represents rainfall, Q represents
runoff at the outlet of the watershed, F
is any kind of model structure (linear or nonlinear),
is the data sampling interval,
and
and
are positive integers numbers reflecting
the memory length of the watershed.
Global search
methods are used to find a set of optimum values for those weights used in the
ANN, which are denoted by
,
,
and by
,
,
. In this paper, the selected optimization algorithm is the Simplex method (Press
et al., 1989). The estimated runoffs, denoted by
, are determined as a function of those optimum weights of the ANN, which is
expressed as
(5)
When the ANN is implemented to approximate
the above relationship between the watershed average rainfall and runoff, there
will be a number of
nodes in the input layer, i.e.
n =
, while there is only one node in the output, i.e. m
= 1. The general ANN structure can be represented by
and the corresponding ANN block
diagram is plotted in Fig.2.
Fig.2 The block diagram of the ANN for rainfall-runoff modeling
(where
B is the unit delay operator or the backward shift operator, defined by Br
=
for r = 0, 1, 2,…)
The proposed ANN model is tested on an Irish karstic catchment, namely the Fergus catchment. This catchment is located in County Clare in the west and has an area of 512 km2. The catchment is predominantly flat, with karstic nature in the highest parts. As far as land use is concerned, most of the catchment is farmland, with some proportions of scrubland, coniferous plantation, natural woodland, and mixed woodland around the catchment outlet. The Burren National Park is located at the center of the catchment. Caves, scattered ponds, lakes, and man-made drains are the major hydro-geological features of the catchment. The stream networks and the lakes are concentrated near to the lower half part of the catchment, while the upper part is karstic land. The data used for this test include daily rainfall and runoff, from June 1, 1977 to December 31, 1999. The mean annual rainfall is found to be around 1460mm, and the mean flow rate at the outlet is around 10.40 m3s-1 (Demissie, 1999). The chosen calibration period is from June 01, 1977 to May 31, 1994, and the verification period is from June 01, 1994 to December 31, 1999. In this catchment, because of it karstic characteristics, it is found that it normally takes seven days or more for the rainfall to cause flow at the outlet of the catchment.
The main model efficiency criterion selected for the
application of the ANN model is the widely used Nash-Sutcliffe model efficiency index (1970), which is
defined as
(6)
where
is the observed runoff,
is the corresponding simulated
runoff, and
is the mean value of the observed runoff series for the calibration period.
The value of objective function
is
expected to approach unity for a perfect simulation of the output data series.
The second index employed to assess the model performance is the simple index of volumetric fit (IVF), which is expressed as the ratio of the simulated runoff volume to the corresponding observed one, i.e.
(7)
When
the ANN is implemented to approximate the rainfall-runoff relationship on the
Fergus catchment, an optimum ANN structure, represented by
, should be determined by varying the numbers of the neurons in each layer to
determine which structure gives the best model efficiency in simulating the
observed discharges over the calibration period. As a result of such comparisons,
the optimum ANN structure for the Fergus catchment is found to be
, i.e. there are 8 nodes in the input layer, 3 nodes in the hidden layers, and
one node in the output layer. These 8 nodes in the input layer simply reflect
the fact that it normally takes around seven days or more for the rainfall to
produce flow at the outlet of the Fergus catchment.
The ANN model efficiency value, denoted
by
, is found to be 98.37% in the calibration period and 97.44% in the verification
period. As regards to the volumetric fit, the value of the IVF is 1.01 in the
calibration period and 0.99 in the verification period. From these results,
there is no doubt the ANN model is very successful in simulating the non-linear
rainfall-runoff relationship on the Fergus catchment. The simulated hydrograph
of the period from June 01, 1993 to May 31, 1994 is plotted, together with the
corresponding observed hydrograph, in Fig.3.
Fig.3 Comparison of the observed and simulated hydrographs
A earlier study carried out at NUI, Galway by Demissie (1999) involved the use of three black box models (i.e. the linear SLM, the non-linear VGFM, and the seasonally based LPM), and one conceptual rainfall-runoff model (i.e. the SMARG model), to simulate the same discharge series on the Fergus catchment. The results are presented in Table 1.
Table
1 Simulation results of
on the Fergus catchment by Demissie
(1999)
|
|
Calibration |
Verification |
|
SLM |
59.72 |
64.22 |
|
VGFM |
58.87 |
64.61 |
|
LPM |
83.82 |
82.44 |
|
SMAR |
92.45 |
91.04 |
(*The
data used by Demissie is from 1973 to 1996 inclusive, sixteen years for
calibration and eight years for verification.)
From
Table 1, it is found that the best model efficiency values reported by Demissie
are those of the SMARG model, the value of
being 92.45% in the calibration
period and 91.04% in the verification period. Clearly, even the SMARG model
did not perform nearly as well as the ANN model, in terms of the model efficiency
index
.
Normally the rainfall-runoff relationships of the karstic catchments appear to be substantially non-linear and are perceived to be very difficult to simulate. Since the ANN is mainly designed for the approximation of non-linear relationships, then it is indicated for application in rainfall-runoff modeling on karstic catchments.
In this paper, the artificial neural
network (ANN) model is employed for flood forecasting on an Irish karstic catchment,
namely the Fergus catchment, located at Ballycorey in County Clare, Ireland
and having an area of 512 km2. The daily data used to tested the
ANN model cover the period from June 01,1977 to December 31, 1999, the calibration
period being from June 01, 1977 to May 31, 1994, and the verification period
from June 01, 1994 to December 31, 1999. The Nash model efficiency index
of the ANN is found to be 98.37%
in the calibration period and 97.44% in the verification period, which is very
impressive and the modelling exercise can thereby be considered to be successful.
The conclusion is drawn, on the basis of this study, that the ANN model can
be used for flood forecasting for a karstic catchment that has a very strong
non-linear rainfall-runoff relationship.
