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Wave tranquility studies using neural networks
S.N. Londhea, M.C. Deob,*
b
a
Sinhgad College of Engineering, Pune 411041, India
Indian Institute of Technology, Civil Engineering, Mumbai 400076, India
Abstract
Information on heights of waves and their distribution around harbor entrances is
traditionally obtained from the knowledge of incident wave, seabed and harbor characteristics
by using experimental as well as numerical models. This paper presents an alternative to these
techniques based on the computational tool of neural networks. Modular networks were
developed in order to estimate wave heights in and around a dredged approach channel
leading to harbor entrance. The data involved pertained to two harbor locations in India. The
training of networks was done using a numerical model, which solved the mild slope equation.
Test of the network with several alternative error criteria confirmed capability of the neural
network approach to perform the wave tranquility studies. A variety of learning schemes and
search routines were employed so as to select the best possible training to the network. Mutual
comparison between these showed that the scaled conjugate method was the fastest among all
whereas the one step secant scheme was the most memory efficient. The Brent’s search and the
golden section search routines forming part of the conjugate gradient Fletcher–Reeves update
approach of training took the least amount of time to train the network per epoch. Calibration
of the neural network with both mean square as well as the sum squared error as performance
functions yielded satisfactory results.
Keywords: Neural networks; Harbor tranquility; Training algorithms; Numerical wave models; Neural
network performance
1. Introduction
An essential feature of harbor planning is wave tranquility studies in and around a
harbor. When waves advance from deep to shallow water their heights and attacking
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angles change because of the effects of shoaling, refraction, diffraction, breaking and
reflection. Many harbors involve an approach channel dredged to a certain depth.
This is necessitated from the catering to large ships having deeper drafts for
navigation. When waves from open sea invade in and around such channels they are
subjected to abrupt changes in topography and combined action of refraction and
bathymetry-induced diffraction. This can completely change the wave height
distribution around the harbor causing attenuation at some places and concentration
of waves at some other locations.
Information on heights of waves and their distribution around the harbor
entrance is obtained from the knowledge of incident wave and seabed characteristics
by using both experimental modeling as well as numerical modeling.
Experimental or physical models require considerable amount of time and may
not be cost effective. They also suffer from (physical) laboratory effects such as scale
effects. Because of the Froudian scaling involved they often fail to account properly
for wave reflection and dissipation effects which could be dominated by surface
tension and viscosity. However hydraulic modeling may become unavoidable when
complex site conditions are concerned.
Numerical models provide approximate solution of a governing differential
equation describing the wave propagation. The governing equation could be either
of mild slope or of Boussinesq type. Berkhoff’s [1] mild-slope equation was presented
to solve for linear wave attack under steady state or time-dependent conditions.
Finite difference as well as finite element schemes have been used by different
investigators in order to provide for various approximations needed to solve the
mild-slope equations [2]. Available solutions range from Berkhoff’s [1] twodimentional combined refraction–diffraction equation of elliptic form to Copeland’s
[3] hyperbolic and Darlymple and Kirby’s [4] parabolic form.
Application of the mild slope equations is relatively straight but it involves
less satisfactory modeling of the flow physics caused by their underlying
assumptions. Another class of numerical models is therefore alternatively
followed and this is based on the Boussinesq equations. The Boussinesq equations
represent a set of simultaneous, non-linear, hyperbolic equations and contain all
important wave processes, such as shoaling, refraction, diffraction and reflection.
Various modified forms of the original Boussinesq equations have resulted in
accounting for unsteady time-dependent conditions, dispersion characteristics or
wave directionality effects as well as wave non-linearity in the solution procedure.
This has led to increased usefulness of the numerical schemes in a wide range of
water depths unlike the earlier attempts restricted to shallow water regions. Latest
information on the physical and numerical studies can be seen in Young [5] and Edge
and Hemsley [6].
Considerable progress has been made in numerical modeling while simulating as
many real sea conditions as possible like higher order wave theory, complex
boundaries and presence of current. However inherent computational difficulties like
rounding off and truncation errors, speed and memory requirement coupled with
complexity of the phenomenon involved in such a modeling may not always yield
highly satisfactory results. Efforts to tackle the problem of wave propagation into a
S.N. Londhe, M.C. Deo / Marine Structures 16 (2003) 419–436
421
Fig. 1. Location plan.
harbor using new schemes should therefore be welcomed [7]. Artificial neural
networks (ANN) or simply neural networks (NN) could be one such technique.
ANN basically provides mapping between random inputs and outputs irrespectively having knowledge of the underlying phenomenon. Unlike traditional analysis
tools it can learn from the examples or training patterns fed to it, store the
knowledge in its weights and bias values and use them to estimate future values.
Although their use to solve computational problems in civil engineering started since
last two decades, applications of ANNs to tackle ocean engineering contexts are too
scarce and there exists a great room for attempt in this regard.
The aim of this work was to employ the technique of ANN to study wave
attenuation along a dredged approach channel leading to a harbor entrance and
compare the results so obtained with numerical and physical model outputs. The
study also provides analysis of network performances using a variety of training
algorithms, search routines, error goals and error criteria.
The present work is based on data available at two harbor locations, namely New
Mangalore Port and Bina Bay Harbor near Goa located along the western Indian
coastline (see Fig. 1 for location of these sites).
2. Study area
The studies have been carried out with the help of two data sets pertaining to
locations of New Mangalore Port and Bina Bay (Marmugao) port along the west
coast of India (Fig. 1). The available information at the former location was more
exhaustive and hence major studies were concentrated on this data set. As mentioned
above the New-Mangalore port is situated on the west coast of India and its harbor
is connected to the sea by an approach channel which is 5200 m long, 245 m wide and
dredged to –13.5 m below the chart datum with side slopes of 1:10 (Fig. 2). The
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Fig. 2. New Mangalore Harbor (Ref. [10]). (Dots represent the grid 250 m 50 m from seaward boundary
to harbor entrance.)
channel is oriented at 259 with respect to North, so as to make it in-line with
predominant wave direction in the region to achieve maximum wave tranquility in
the harbor.
The developed neural network had inputs in the form of wave heights, wave
periods and angles of wave attack at 9 grid points along a cross section across the
approach channel. The output from the network was the changed wave heights at
each of the 9 grid points located at the subsequent cross section located 50 m on
downstream side. The input–output representation made in this way was adopted for
convenience and according to availability of data for network training in that
manner.
The output of a numerical model REFDIF1 was used in the present studies to
train the neural network. The model involves numerical solution of the parabolic
approximation of mild slope equation, and it explicitly simulates combined effects of
refraction and bathymetric diffraction over an approach channel. This was in the
form of attenuated wave heights at 9 grid points along different cross sections of the
channel, spaced 50 m center to center corresponding to an incident wave height of
3.66 m.
3. Modular neural networks
A network architecture called modular network consisting of several modules with
limited interconnections between them was employed. Its use was found necessary
because initial trials had shown that the network training was extremely slow or
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unattainable due to a large number of training patterns involved. Modularity
permits separate tackling of smaller tasks using small modules and then combination
of these modules in series or parallel form [8]. This helps formation of a network for
the specific task from a set of previously trained modules. Also the system can be
expanded without redeveloping the whole package afresh in order to include new
modules when such a need arises [9]. This approach was found to be most suitable
for designing NN for the present problem because there were as many as 4 input
vectors, each containing 9 values and 1 output vector having 9 values. Ninety-five
such vectors made a total of 4275 values as per the data set.
4. NN models
The modular NN used in the present study had either 3 or 4 modules in the input
layer. Each module pertained to one input vector. Fig. 3 indicates the 4-module
network that was employed. The 3-module network had wave height, wave period
and seabed levels as input vectors while the other one had the angle of wave attack as
an additional input vector. The output of both networks was heights of the wave at
the subsequent cross section. Development of such separate networks was guided by
applications of the numerical model made in advance and by availability of data in
that format.
(Input 1)
[9x1]
Weights[9x9]
weights[9x9]
∑
(Input 2)
[9x1]
Transfer
Function
weights[9x9]
bias
[9x1]
(Input 3)
[9x1]
weights[9x9]
(Input 4)
[9x1]
Fig. 3. 4-Module network.
output
[9x1]
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1.8
network output
target
1.6
wave height (m)
1.4
1.2
1
0.8
0.6
0.4
0.2
500
1000
1500 2000 2500 3000 3500 4000
distance from seaward boundary (m)
4500
5000
Fig. 4. Testing performance (ANN with lab data).
A better way of imparting training to the network would have involved the use of
physical model output for this purpose. The physical model covering the entire
length of 5200 m of the approach channel was constructed to a geometrically similar
scale of 1:150. The model extended upto 13.0 m contour with respect to Chart
Datum on the seaside. The bathymetry extending upto 1200 m on either side of
centerline of the channel was reproduced in order to simulate the transfer of wave
energy across the wave crest properly.
However it is extremely difficult to collect simultaneous observations of wave
heights at several locations as per the training requirement. None the less hydraulic
model experiments reported in Kanetkar et al. [10] were used as a trial to train the
network. The model measurements were available at the centerline of the approach
channel corresponding to incident wave height of 3.66 m and wave period of 9 s.
Fig. 4 shows the results of network testing. It indicates how the wave height
attenuated along the channel axis when physical as well as numerical model based
training was employed. The close proximity between the results obtained through the
two schemes (numerical and ANN) showed that one can train the network using
numerical model output as well. It is expected that owing to the high degrees of
freedom involved, the network could learn information not only as contained in the
numerical model but also something over and above that.
5. Training algorithms
A variety of training schemes was employed to impart training to the network.
This was done to make sure that adequate training is imparted and also to study
relative performance of each and converge to the best scheme. The ASCE Task
committee [11] had earlier appreciated the need to actually compare the network
performance by various algorithms. Thirumalaiah and Deo [12] had found that
network performance as well as prediction accuracy can be improved by appropriate
S.N. Londhe, M.C. Deo / Marine Structures 16 (2003) 419–436
425
choice of training. The methods involved were: (i) gradient descent with momentum
(GDM), (ii) resilient backpropagation (RP), (iii) conjugate gradient Fletcher–Reeves
update (CGF), (iv) conjugate gradient Polak–Ribiere update (CGP), (v) Powell
Beale restarts (CGB), (vi) scaled conjugate gradient (SCG), (vii) Broyden, Fletcher,
Goldfarb and Shanno update (BFG), (viii) one step secant algorithm (OSS) and (ix)
Levenberg–Marquardt algorithm (LM). The details of these algorithms can be found
in Demuth et al. [13]. GDM has provision for faster convergence while RP is aimed
at eliminating problems arising out of smaller magnitudes of the error gradients. A
general conjugate gradient scheme involves performing a search along conjugate
direction in order to determine step size to minimize performance function. Five
search techniques are alternatively used in the current study. They are (i) golden
section search, (ii) Brent’s search, (iii) hybrid bisection cubic search, (iv)
Charalambus search and (v) backtracking. There are four different versions of the
conjugate algorithm (as per the technique of finding a new search direction) namely
CGF, CGP, CGB and SCG. The BFG scheme is based on the Newton’s method but
does not require calculation of the second derivatives of the error gradient. The BFG
scheme updates an approximate Hessian matrix at each iteration of the algorithm.
OSS is a secant approximation falling in between the quasi-Newton’s method and
the conjugate gradient algorithm which assumes at each iteration that the previous
Hessian matrix was the identity matrix. The LM algorithm is designed to approach
second-order-training speed without having to compute the Hessian matrix.
6. Testing results
The first network was trained on the basis of data belonging to a constant wave
period of 11 s. It was then tested for unseen inputs for the case of constant wave
periods of 7 and 9 s, respectively. Fig. 5 shows how the network performed during
testing with respect to the case of input wave period of 9 s. It indicates how the wave
2
wave height (m)
neural network
numerical model
physical model
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
distance from seaword boundary (m)
Fig. 5. Comparison of numerical, physical and ANN model.
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height attenuated along the central axis of the channel. The reduction in wave height
is due to the reduction in propagating wave energy caused by refraction away from
the channel. Fig. 5 also shows network output of wave attenuation along the harbor
channel vis-a" -vis the same yielded by numerical as well as physical model. It may be
noted that the network output agrees very well with the numerical as well as the
physical model outputs. During the testing, the wave height, period and sea bed
elevations at the seaward boundary along the 9 grid points of the first cross section
were provided as input and output wave heights at the next cross section were
obtained using the weights and biases fixed at the end of training. These wave heights
along with the seabed elevations at that section were then used as input for obtaining
wave heights at the next cross section. This process was continued till the harbor
entrance was reached. The mean square error goal for every algorithm was decided
by making sensitivity analysis involving selection of different goals for training and
then comparing the testing results with those of the numerical model. It may be
noted that input and output wave heights are scaled down to half in accordance with
the available data (1.83 m instead of 3.66 m). The training and testing was carried out
on a Pentium 2 processor with 64 MB RAM memory which took 2.36 s for the
testing exercise. Where as the numerical model was run on a silicon graphics work
station with which took about 60 s time.
All the 9 algorithms gave satisfactory results, but the results of CGF with
Charalambous’ search as the default line search routine were the best as indicated by
the scatter plots and accompanying high values of correlation coefficient of 0.99 (see
Fig. 6 corresponding to testing with respect to 7 s) along with achievement of a low
mean square error goal.
When the model was tested with other algorithms the values of correlation
coefficients were not at par with those obtained using the CGF except the algorithm
of BFG. But the CGF algorithm had an added advantage that it was memory
efficient than BFG in that it required lesser time per epoch than the BFG.
2
target wave height (m)
1.8
1.6
1.4
1.2
best fit
Network output
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
actual wave height (m)
Fig. 6. Actual against target wave heights (t ¼ 7 s)
2
427
2200
2
2
1.5
2
2.5
2000
1
2
Lateral Distance (m)
2
1800
1
1600
1400
1200
1000
800
0.5
2
1.5
600
400
2
500
1.51.5
1
0.5
0.5
1000 1500 2000 2500 3000 3500 4000 4500
distance from seaward boundary (m)
Fig. 7. Wave contour plot (t ¼ 7 s).
The wave contour plot for a typical input wave period of 7 s is shown in Fig. 7,
which clearly indicate the wave attenuation along centerline of the approach channel
and also predict the wave heights at any desired location in the model area. Note that
all contours presented are in meter. When the incident wave direction is parallel to
the channel axis (as underlying this figures), waves at the two sides will be higher
than those inside the channel since wave celerity is higher in the channel (because of
deeper water). This supported the observation of Li et al. [14] reported in an earlier
study.
6.1. Effect of change in angle of attack
The study reported in the preceding paragraphs was carried out on the basis of the
3-module network. Thereafter the angle of wave attack, y; made by the wave with the
channel axis was taken as an additional fourth input and the corresponding 4module model was run. The numerical model data were available for a few values of
y ranging from 5 to 35 . However attenuation effect was significant only up to
y ¼ 15 : In the present study the network was trained for y ¼ 10 : Fig. 8 shows wave
attenuation along the harbor channel when y changes from 5 to 15 . Note that all
contour presented are in meter. The final wave heights at the harbor entrance do not
appear to change much indicating lesser refraction effects for higher angles. This
trend is almost similar to the one obtained by the numerical model.
6.2. Effect of initial wave height on wave attenuation
Harbor planning exercise often calls for studying changes in the wave distribution
with respect to small variations in design incident wave heights. The 3-input model
was originally trained for an incident wave height of 3.66 m (at the seaward
boundary) along with a wave period of 11 s. It was now run for different initial wave
heights, retaining the weights and biases of the trained network. This indicated an
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2
wave height (m)
1.5
testing 7 sec.
testing 9 sec.
1
training 11 sec.
0.5
0
0
1000
2000
3000
4000
5000
distance from seaward boundary (m)
Fig. 8. Effect of oblique wave incidence on wave attenuation.
Table 1
Effect of change in initial wave heights
Initial wave height (m) (1)
Final wave height (m) (2)
% attenuation (3)
1.50
1.70
1.75
1.90
0.24
0.14
0.14
0.05
84.28
91.61
93.18
97.40
Note: Wave heights are scaled down to half.
increase in the percentage wave attenuation near the harbor entrance with an
increase in initial wave height as shown in Table 1. As the initial wave height
increases more attenuation at the harbor entrance is noticed because of increased
dissipation for higher waves.
6.3. Effect of change in wave period
The effect of change in the wave period on wave attenuation along the harbor
channel as given by the network is shown in Fig. 9, which is exactly the same as
produced by the numerical model in earlier studies. Larger wave periods result in
lower levels of wave heights along the channel (or higher attenuation) because of the
increase of wavelength and the reduction in the steepness ratio, leading to higher
attenuation of flatter waves.
6.4. Network performance analysis
Relative performances of different training algorithms were drawn in Table 2 (for
3-input network) which give the number of epochs, mean square error (mse) reached,
r7 ; r9 (which are correlation coefficients corresponding to testing with 7 and 9 s
429
2
wave height (m)
1.5
testing 7 sec.
testing 9 sec.
1
training 11 sec.
0.5
0
0
1000
2000
3000
4000
5000
distance from seaward boundary (m)
Fig. 9. Effect of change in wave period on Wave attenuation (ANN approach).
Table 2
Test results for wave periods of 7 and 9 s (training wave period=11 s)
Algorithm (1)
Epochs (2)
mse (3)
r7 (4)
r9 (5)
GDM
CGF
CGP
CGB
SCG
LM
BFG
OSS
RP
18232
520
175
13
50
1
28
2072
2500
0.0099
0.0044
0.0099
0.0099
0.0043
0.0040
0.0097
0.0042
0.0056
0.98
0.99
0.98
0.95
0.95
0.97
0.99
0.98
0.95
0.99
0.99
0.99
0.95
0.98
0.98
0.99
0.99
0.97
Note: r7 ; r9 =correlation coefficient for wave period 7 and 9 s, respectively; Epochs=iterations or pa
period, respectively). A highly satisfactory performance by all algorithms may be
noted. The LM algorithm converged in just one epoch while the GDM converged to
amounts of 18,232 epochs.
6.4.1. Cross-validation
Optimum learning as well as performance of the network can be achieved by crossvalidation where an independent data set is used to assess its performance during
various stages of learning. Cross-validation also results in avoiding overfitting errors,
usually when very large number of training pairs are supplied, as in the present case.
Both the network models mentioned earlier were therefore subjected to crossvalidation. In case of the 3-module model, the network was trained with the help of
data corresponding to a wave period of 11 s and validated and tested using data of 9
and 7 s, respectively.
The 4-module model was trained, validated and tested for data sets belonging to
angles of 10 , 5 and 15 , respectively. The results indicated that in general the mse
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reduced in respect of the training as well as the validation data set during different
stages of training (increasing epochs) shown for example in Table 3 which
additionally shows absence of any overfitting.
6.4.2. Efficiency of algorithms
In the light of various options of network training it was thought to be of help if
some guidance on the algorithm selection is given to the user. The efficiency of an
algorithm will depend on many factors including the complexity of the problem, the
number of data points in the training set, the number of weights and biases in the
network, the error goal and whether the network is used for pattern recognition
(discriminant analysis) or function approximation (regression).
Tables 4 and 5 summarize the results out of the seven training methods of the
network such as algorithms of CGF, CGB, CGP, SCG, LM, BFG and OSS. Each
entry in the tables is a result of 25 different trials, each involving different random
initial weights. The mean time required for each training exercise and their ratio with
respect to the lowest time (columns 2 and 3, Table 4) indicates the speed of the
algorithm. Standard deviation of the time required for training (with respect to mean
time) as well as minimum time and maximum time required for training are also
indicated in columns 4–6 of Table 4. In each case, the network training was
Table 3
Cross-validation for the 3-input network (algorithm: CGF)
Epochs (1)
mse on training set (wave
period 11 s) (2)
mse on validation set (wave
period 9 s) (3)
000
100
200
300
400
520 convergence
0.3914
0.0078
0.0058
0.0050
0.0047
0.0044
0.2453
5.59 105
0.1600
0.1191
0.0728
0.0539
Table 4
Speed comparison
Algorithm
(1)
Mean time
(s) (2)
Ratio of
mean times
(3)
Min. time (s) Max. Time
(4)
(s) (5)
Std.
deviation (6)
Corr. coeff.
(7)
CGF
CGP
CGB
SCG
LM
BFG
OSS
25.56
68.90
10.98
2.49
5.54
30.03
103.18
10.25
27.63
4.40
1.00
2.22
12.04
41.37
14.72
31.37
4.78
2.31
5.39
28.01
76.24
5.380
21.000
3.510
0.428
0.460
1.760
13.980
0.99
0.99
0.99
0.99
0.99
0.99
0.99
34.71
111.55
17.69
4.56
7.80
33.39
123.09
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Table 5
Efficiency of algorithms
Algorithm (1)
Mean epochs (2)
Mean time (s) (3)
Time per epoch (s) (4)
CGF
CGP
CGB
SCG
LM
BFG
OSS
372
1129
140
39
1
71
1848
25.56
68.90
10.98
2.49
5.54
30.03
103.18
0.068
0.061
0.078
0.063
5.540
0.420
0.056
continued till mse was reduced to a low figure of 0.0045. It can also be seen
that the SCG algorithm required the minimum time to train, though the
LM algorithm reached the error goal in 1 epoch. The next fastest algorithm
was the CGB, which was however 4.5 times slower than the SCG. The CGF
and BFG algorithms performed equally in terms of time required for convergence
whereas the CGP and OSS algorithms were much slower; the slowest scheme was
the OSS.
The fastest algorithm for this particular problem was therefore the scaled
conjugate gradient algorithm (SCG). Its speed is possibly due to the fact that it
avoids the time consuming line search, which was more than two times faster
compared with the next fastest algorithm LM as indicated in Table 4, although the
time required per epoch was the least in case of the OSS indicating it as the most
memory efficient as shown in Table 5, which gives mean number of epochs, mean
time to reach the error goal and the time taken per epoch for various training
schemes.
All algorithms performed equally well as far as accuracy in the simulation
of training data set was considered as indicated by the high value of the correlation coefficient, ‘r’ (=0.99). However it is noteworthy that this trend may
not continue in actual application because one additional epoch may adjust
the weights in such a manner that networks may overfit yielding erroneous
results. Hence a sensitivity analysis is recommended to fix the goal, for which the
accuracy of different training algorithms may not be equivalent so far as the testing
is considered.
6.4.3. Model assessment
The work discussed so far involved use of the performance function of mse to
achieve the error goal with correlation coefficient as an indication of the model
performance during testing. Although the criterion of correlation coefficient is very
common in assessing model effectiveness it does not identify specific regions where
the model is deficient. Other error measures were therefore employed to study the
model performance. These are average error (AE), mean absolute error (MAE),
mean relative error (MRE), root mean square error (RMSE), mean squared relative
error (MSRE), the coefficient of efficiency (CE), and coefficient of determination
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Table 6
Model assessment
Location
(1)
Testing
condition (2)
AE
(3)
MAE
(4)
MRE
(5)
RMS
(6)
MSRE
(7)
CE
(8)
r2
(9)
New-Mangalore Port
T ¼ 7s
T ¼ 9s
y ¼ 5
y ¼ 15
T ¼ 10 s
y ¼ 14
2.62
2.05
9.42
33.13
0.063
0.058
0.11
0.28
7.85
7.91
13.25
33.46
0.076
0.075
0.142
0.325
0.0099
0.0103
0.0273
0.1553
0.98
0.98
0.91
0.358
0.98
0.98
0.96
0.91
1.80
0.047
2.12
0.057
0.0006
0.01
0.93
Bina Bay (Marmugao)
Note: T=wave period; y=angle of approach.
(r2 ). The expressions for these error measures are available in Karunanithi et al. [15]
and Dawson [16]. The measures of AE, MAE and MRE give different numerical
values with respect to the original data. However AE gives algebraic deviations while
MAE and MRE present absolute differences disregarding over or under estimation.
The difference between MAE and MRE lies in the non-dimensional evaluation of the
latter, just like the one between RMSE and MSRE, where, in addition, the
distinction between over or under estimation is neglected by taking square of the
differences. Relative errors give a more balanced criterion. The lower the values of
these measures, the greater is the accuracy.
CE and r2 do not depend on data scale and hence are more suited when different
scales are involved. CE indicates prediction capabilities of values different from the
mean and varies from N to +1. CE of 0.9 and above is very satisfactory and that
of below 0.8 is unsatisfactory; r2 varies in the range of (1, 1) and measures the
model-explained variance.
Table 6 shows the values of these error measures during testing of the models. It
involves analysis of data pertaining to New Mangalore as well as Bina Bay harbor.
(The latter is discussed in the subsequent section.) Both neural network models—
with 3- and 4-input modules were considered. The first two rows in Table 6 pertain to
test results with 7 and 9 s wave period (for the 3-module model) while the third and
fourth rows belong to testing with 5 and 15 approach angle (for the 4-module
model). The testing results for the former model can be seen as the best because it
involved lesser errors compared to the latter model. It may be seen that all error
measures collectively indicate lower levels of the discrepancy between the ANN
model and the numerical model.
The neural network models developed in the present work used the performance
function of mean square error or ‘mse’. The network performance with respect to
another function of sum squared error or ‘sse’ was also investigated. Table 7 shows
the result. It gives the number of epochs required to achieve the error goal, the time
taken to do so, the SSE involved, as well as the correlation for the two testing cases
under each model. The high values of the correlation coefficients show that both the
models worked satisfactorily for two performance functions namely the ‘mse’
discussed earlier and the ‘sse’.
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Table 7
Results of using performance function SSE
Model (1)
Epochs (2)
Time (3)
SSE (4)
r7 (5)
r9 (6)
r5 (7)
r15 (8)
3-input
4-input
428
106
32.13
9.33
4.3998
4.9069
0.99
—
0.99
—
—
0.99
—
0.97
Algorithm: CGF, r7 ; r9 =correlation coefficient for testing with respect to wave period of 7 and 9 s;
r5 ; r15 =Correlation coefficient for testing with respect to angle of 5 and 15 .
Table 8
Effect of Line search routine (Algorithm: CGF; 3-input network)
Line search routine (1)
mse (2)
Epochs (3)
Time (s) (4)
r7 (5)
r9 (6)
Time/epoch (s) (7)
Golden section
Brent’s search
Hybrid bisection-cubic search
Charalambous’ search
Back tracking
0.0066
0.0081
0.0078
0.0044
0.0049
427
127
157
520
915
23.73
7.14
16.2
34.49
7.74
0.98
0.99
0.99
0.99
0.99
0.99
0.99
0.99
0.99
0.99
0.056
0.056
0.100
0.066
0.059
Note: r7 ; r9 =correlation coefficient for wave period 7 and 9 s, respectively.
6.5. Effect of line search routines on network performance
Many variations of the conjugate gradient and quasi-Newton algorithms require
that a line search be performed to determine the step size, which will minimize the
performance function along that line. Five different search functions were used in the
present work namely, Golden Section Search, Brent’s Search, Hybrid BisectionCubic Search, Charalambous’ Search and Back tracking. Any of these search
functions can be used interchangeably with all varieties of the conjugate gradient
algorithm. Some search functions are best suited to certain algorithms, although the
optimum choice can vary according to the specific application. Table 8 corresponding to the 3-module network indicates that all search routines performed equally well
as reflected in high values of the correlation coefficients. The Hybrid Bisection-Cubic
Search did more computations for every epoch compared to the Golden Section
Search or Brent’s Search as indicated by its time consumption per epoch. For the
present problem all the search routines performed more or less in equal efficiency as
seen by their relative speed and accuracy.
6.6. Wave attenuation at Bina Bay Port (Marmugao harbor)
The discussion presented under previous sections so far is based on results of data
analysis carried out for the site of New Mangalore port. In order to examine
portability of the developed (trained) network to any another location, the data
available at Bina Bay near Marmugao harbor was analyzed. This is described below.
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Fig. 10. Study area (Bina Bay)—the computational grids have size 10 m 10 m.
6.6.1. Study area
The Bina Bay is located about 5 km south of the existing Marmugao Port along
the west coast of India (Refer Fig. 1). The harbor proposed at this location involves
an approach channel oriented at 256 N in the predominant wave direction. The
proposed channel is 275 m wide, to be dredged to 14.4 m below the chart datum. The
study area is shown in Fig. 10.
6.6.2. Data
Agarwal et al. [17] studied the phenomenon of wave attenuation in the Bina bay
using a numerical model MIKE21-BW developed by Danish Hydraulic Institute,
Denmark. This model involves a different governing equation of Boussinesq type
unlike the earlier numerical model, REFDIF1, that was based on the equation of
mild slope. The numerical model solves the Boussinesq equations after simplification
for water surface elevation at each grid point in the time domain. Hence waveforms
can be simulated with these models. The MIKE 21-BW model was applied within an
area of 3500 m 5000 m. The wave height attenuation at the center line of the
approach channel was plotted through out a distance of 1000 m from the seaward
boundary to harbor entrance located at 2000 m from the sea ward boundary with an
initial wave of height of 2.6 m and period of 10 s.
6.6.3. Training and testing
Considering availability of data the 4-module model was adopted. The network
trained as in the earlier studies was utilized for this application. The initial input
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2.6
ANN model
Target
2.5
wave height (m)
2.4
2.3
2.2
2.1
2
1.9
1.8
1000
1200
1400
1600
1800
2000
2200
distance from seaward boundary (m)
Fig. 11. Wave attenuation using neural network model (Bina Bay Port, Mormugao).
consisted of wave height of 2.6 m, wave period of 10 s, angle of incidence of 14 as
well as depths of sea bed at 9 grid points along the cross-section at starting point of
the seaward boundary. Values of wave heights at the 9 grid points along the
subsequent cross-section located 50 m downstream were the output. The process was
repeated for 1000 m distance, using the preceding cross-section’s output.
Wave attenuation along the centerline of approach channel is shown in Fig. 11,
which follows the same trend as that of the numerical model. The model
performance was also assessed using the various error measures as explained in
earlier section. In this case also all results indicated very less discrepancy between the
ANN model and the numerical model outputs as seen in the last row of Table 6
referred to earlier. All the above results indicate that that the developed neural
network model worked satisfactorily for this harbor also.
7. Conclusions
The foregoing sections presented the development of Artificial neural networks
(ANN) models in order to obtain distribution of attenuated wave pattern at the
entrance of a harbor involving dredged approach channel. The developed networks
were found to follow the expected trend of wave height attenuation along the
approach channel of the selected harbors. The trained network when tested for
unseen inputs yielded satisfactory output of wave heights, which compared well with
the numerical as well as the physical model results. This was confirmed by the scatter
plots and accompanying values of a variety of error measures. The technique of
ANN thus can be used as an option to the conventional numerical scheme with the
advantage that it does not require as much elaborate data as the numerical schemes,
it is relatively less complex in both modeling and computations and also less rigid.
Application of the ANN model shows that larger incident wave heights undergo
higher attenuation as they propagate towards the harbor entrance. Similarly longer
incident wave periods indicate increased attenuation. This trend matches with the
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one observed by earlier investigators while using numerical and physical models. The
effect of variation in the incident wave height, period and angles of wave incidence
was properly simulated by the neural network.
The decrease in error for validation set of data with corresponding decrease in
error for training set of data seen during cross-validation showed that the network
was devoid of overfitting errors. It was found that the fastest algorithm for the
present problem was scaled conjugate gradient algorithm. It required the least time
to train whereas one step secant algorithm was most memory efficient in that it
required least time per epoch. The network training based on the Conjugate
Gradient Fletcher–Reeves update scheme was effected using five different line search
routines in order to compare their performance. All the search routines performed
satisfactorily. The Golden section and Brent’s search took the least amount of time
per epoch indicating their memory efficient nature for the two data sets involved.
Calibration of the model with the help of both ‘mse’ and ‘sse’ as performance
functions yielded satisfactory results.
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