Coastal Dynamics 2013
PREDICTION OF WAVE TRANSMISSION TROUGH A NEW ARTIFICIAL NEURAL
NETWORK DEVELOPED FOR WAVE REFLECTION
Sara Mizar Formentin1, Barbara Zanuttigh2
Abstract
This paper presents the results of the application of an Artificial Neural Network for the prediction of the wave transmission coefficient from low crested structures.
The model essentially works with 13 input parameters, which describe the wave attack conditions and the main feature
of the structures. It has been firstly created to estimate the wave reflection coefficient, by training and validating the
model against nearly 6’000 data, including a wider range of coastal and harbor structures under perpendicular and
oblique wave attacks.
Afterwards, the Artificial Neural Network has been employed for the prediction of the wave transmission coefficient,
by training it against 3’379 additional tests on low crested structures. The results of this application are pretty satisfactory, also in comparison with another existing Artificial Neural Network.
Key words: wave reflection, wave transmission, neural network, database, low crested breakwaters, training
1. Introduction
For design purpose, the assessment of coastal structure performance requires the accurate analysis of
the wave-structure interaction processes, which can be essentially described through three quantities: the
overtopping discharge (qt), the wave reflection coefficient (Kr) and the wave transmission coefficient (Kt).
Besides the existing empirical formulae, a tool which has already proved to be useful and efficient in
the prediction of such parameters is represented by the Artificial Neural Networks (ANNs). The neural
network modeling is a kind of mathematical modeling which essentially consists in reproducing the relationships among a set of input data and one (or more) output(s) based on a “learning” process of experimental or prototype tests.
Successful examples of such kind of models are the wave overtopping ANN realized within the European project CLASH (Van Gent et al., 2007; Verhaeghe et al., 2008), the ANN for the prediction of Kt behind
Low Crested Structures LCSs (Panizzo and Briganti, 2007) and the wave reflection ANN for the prediction
of Kr from coastal and harbour structures (Zanuttigh et al., 2013, under review).
Since the wave reflection and wave transmission processes are physically correlated, it is made the assumption that both can be described by means of the same parameters (wave attack conditions and structure features). Therefore aim of this paper is to apply the existing wave reflection ANN to the estimation of
Kt. In order to check if this ANN can be an appropriate tool to predict both Kr and Kt, the data used for
training the reflection ANN are the same used by Panizzo and Briganti (2007) for setting-up the first ANN
for LCS.
The database employed to train the reflection ANN for the prediction of Kt is synthetically described in
Section 2, while the mean features characterizing the ANN are presented in Section 3. The discussion and
analysis of the results are provided in Section 4. The performance of this ANN is compared with Panizzo
and Briganti ANN in Section 5. Finally, some conclusions are drawn in Section 6.
1
2
University of Bologna, DICAM, Viale del Risorgimento 2, Bologna 40136, Italy; [email protected]
University of Bologna, DICAM, Viale del Risorgimento 2, Bologna 40136, Italy; [email protected]
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2. The wave transmission database
The database used in this research, which consists of 3’379 data, comprehends the whole dataset of
2’285 tests employed by Panizzo and Briganti (2007). The further tests has been gathered, as well as
Panizzo and Briganti tests, from the wide database of the European project DELOS (Seelig, 1980; Allsop,
1983; Daemrich and Kahle, 1985; Powell and Allsop, 1985; Ahrens, 1987; Van der Meer, 1988; Daemen,
1991; Daemrich et al., 2001; Kramer et al., 2005; Van der Meer et al., 2005). A synthesis of all data type is
given in Table 1.
Table 1. Tests included in the ANN training database; where reference is not given,
data were kindly provided by private communications.
Database section
A – LCS breakwaters
Structure type
#
Smooth
215
Rocks
1292
References
Daemrich and Kahle (1985),
Seelig (1980)
Seelig (1980), Allsop (1983),
Ahrens (1987), Van der Meer
(1988), Daemen (1991), Seabrook
and Hall (1998)
Rubble mound
215
B - Aquareefs
Acquareefs
1062
Powell and Allsop (1985), Kramer et al., (2005), Daemrich et al.
(2001), Calabrese et al. (2002)
Hirose et al. (2002)
C – Armour units
Core-Locs
Tetrapods
122
267
Melito and Melby (2000)
Daemrich and Kahle (1985)
Accropods
10
Rocks
112
Van der Meer et al. (2003), Kramer et al., (2005)
Smooth
84
Van der Meer et al. (2003)
E – Oblique attacks
3. The Artificial Neural Network
ANNs are “black-box” mathematical models which elaborate experimental input and output data and
“learn” the relationships between them, working apart from the knowledge of the physical process. Once
an ANN has “learned” how the input are linked to the outputs, it is said to be “trained” and it is ready to
elaborate new input and predict the corresponding outputs.
The architecture of an ANN model essentially consists of layers: an input set composed by all the input
parameters involved, an hidden layer –the proper “black-box” and the core of the model, which encloses
the learning and training algorithms – and an output layer containing the quantities to be predicted. The
numerical information contained in the input elements is elaborated by the ANN and passed to the hidden
layer through the “hidden layer transfer function”; similarly, the output neuron receives the information
from the hidden layer through the “output neuron transfer function”.
The ANN presented in this work has been firstly developed to reproduce the wave reflection process
and estimate Kr. The optimization of the ANN has been carried out through an in-depth sensitivity analysis
both to the different structures of the ANN architecture and to the input parameters (Zanuttigh et al., 2013).
These have been selected in order to describe the physics of the wave-structure interaction, taking into account the most significant effects of the structure type (geometry, permeability, submergence) and of the
wave attack (wave steepness, breaking index, shoaling factor, wave obliquity). The best ANN layout has
been definitely chosen by comparing the model performance by changing a single parameter in each run.
Being the wave transmission and the wave reflection strongly interdependent and essentially representing two features of the same physical process, the input parameters are supposed to be exactly the same for
the estimation of both the coefficients, Kr and Kt.
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Synthetically, the ANN architecture is characterized by the following features:
- multilayer network, based on a “feed-forward back-propagation” learning algorithm;
- the input vector consists of 13 input elements, all non-dimensional;
- the hidden layer comprehends 40 hidden neurons; this number has been defined after a specific
sensitivity analysis;
- the output neuron corresponds either to Kr or to Kt;
- training algorithm: Levenberg – Marquardt (Levenberg, 1994; Marquardt, 1963);
- hidden neurons transfer function: hyperbolic tangent sigmoid function;
- output neuron transfer function: linear transfer function.
The 13 non-dimensional structural and hydraulic parameters composing the input vector are listed hereafter, their values are summarized in Table 2, while the meaning of the symbols (based on CLASH project,
Van der Meer et al., 2008) is illustrated in Figure 1:
1. Hm,0,t/Lm-1,0,t: it is proportional to the wave steepness and it is part of the breaking parameter, which
represents the breaker level of energy.
2. ht/Lm-1,0,t : it accounts for shoaling effects associated to incident waves;
3. Rc/Hm,0,t: the lower the relative crest freeboard, the greater the wave transmission and the lower the
wave reflection;
4. cotαd: off-shore structure slope in the run-up area; together with (1), it completes the description of
the Iribarren parameter . The cotangent form has been privileged to the tangent to prevent infinite
values of the tangent in case of seawalls, i.e. when αd = 90°;
5. γf: armour layer roughness factor, index of wave energy dissipation induced by the roughness of the
structure during the run-up process;
6. Dn,50/Hm,0,t: this term essentially represents the wave pressure inside the structure pores; it is involved in the definition both of design conditions and of the ranges of validity for existing
formulae (Davidson et al., 1996; Zanuttigh and Van der Meer, 2008; Calabrese et al., 2008);
7. β: this parameter describes oblique wave attacks. In order to skip the use of coefficients that may
be inaccurate, it has been selected to use directly β rather than the factor γβ.
8. Gc/Lm-1,0,t: this parameter is introduced to represent dissipation over the crest and percolation
through the crest that both reduce wave transmission;
9. B/Lm-1,0,t: this parameter, as well as (10) and (11), is used to describe the effects induced by a berm.
It is conceptually similar to (8), accounting for the same effects induced by the berm instead of the
crest;
10. hb/Hm,0,t: this parameter accounts for the process of waves breaking on the berm, dissipating energy
and reducing the run-up process over the structure upper slope.
11. cotαincl: the angle αincl differs from αd when the structure has a berm, therefore this parameter is included to account for modification of the slope;
12. m: a foreshore in front of a breakwater might enhance shoaling effect and increase wave steepness
while waves travel from offshore to the structure toe;
13. Spreading: the directional wave spreading tends to increase the effects induced by wave obliquity,
and therefore the greater the directional spreading, the lower the wave transmission.
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Figure 2. Hydraulic and structural parameters involved in the wave transmission database.
Symbols and notations have been derived from the project CLASH.
Table 2. Ranges of values of the ANN 13 input parameters.
Input Element
Reflection database
min
max
Transmission database
min
max
Hm,0,t/L m-1,0,t
0.001
0.121
0.004
0.099
h t/Lm-1,0,t
0.008
1.892
0.049
0.544
Rc/Hm,0,t
-8.087
25.391
-10.000
8.824
cotαd
0.000
7.000
0.200
5.0000
γf
0.340
1.000
0.380
1.0000
Dn,50/Hm,0,t
0.000
6.537
0.000
3.5454
β
0.000
83.490
0.000
83.000
Gc/Lm-1,0,t
0.000
2.238
0.712
2.825
B/L m-1,0,t
0.000
1.024
0.000
0.171
h b/Hm,0,t
-1.972
6.091
-0.732
35.000
cotαincl
0.000
10.012
0.000
5.000
m
0.000
1000.000
0.000
1000
Spreading
0.000
50.000
0.000
50.000
4. Results
This Section presents and discusses the main results obtained by the ANN within the prediction of Kt.
The methodology of analysis of the results (briefly synthetized in Subsection 4.1) follows precisely the
work already done for the wave reflection ANN (Formentin et al., 2012; Zanuttigh et al., 2013). Just a difference occurs between the two application of the ANN: an additional routine has been included here (developed in Matlab language) to prevent the prediction of negative values of Kt.
The discussion about the larger errors in estimating Kt is presented in Subsection 4.2.
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4.1 Analysis of the ANN results
The performance of the ANN is qualitatively shown by the comparison of Kt,ANN values with
(Fig. 2,
to the left) and by the dispersion of the absolute error e = Kt,s - Kt,ANN as a function of Kt,s (Fig. 2, to the
right). A quantitative estimate of the ANN accuracy is provided by the average values (resumed in Tab. 3)
of 3 error indexes: the root mean square error (rmse), the Willmot index (WI, see Wilmott, 1981) and the
coefficient of determination (R2).
To allow an easier comparison, Table 3 reports also the error indices associated to:
- the ANN trained on the narrower database of 2’285 data (the same employed by Panizzo and Briganti, 2007, hereafter, PB and ANNPB);
- the existing transmission ANNPB, where the rmsePB value has been derived from the Dataset 07 reported in Tab. 1 of PB;
- the reflection ANN.
In order to provide an assessment of the uncertainty associated to the performance of the model, the results are shown as average values obtained from several different train and run of the ANN. The stochastic
independence of the results is guaranteed by the re-initialization of the ANN which has been performed before any training process and by the employment of a bootstrap technique to resample the training database
each time.
The random selection of data through the bootstrap resampling is driven by the application of a weight
factor (WF) which is associated to each test of the database. The WF takes into account the reliability of
the datum itself and the complexity of the structure: the more reliable the datum is and the less complex
geometry the structure has, the higher value of WF is associated to the test (Van der Meer et al., 2008).
Therefore the most reliable and simplest tests are the most likely to be included in the training dataset.
As a result of the sensitivity analysis, a number of 20 training-testing-simulation processes proved to be
sufficiently large to fully describe the actual uncertainty of the ANN error distribution. This value of the
optimal number of simulations is sensibly lower than for reflection (40), probably due to the greater extension of the wave reflection database (5’781 data against 3’379), and therefore to the wider ranges of values
of the 13 input elements.
Within the application for the estimation of , the ANN shows the tendency to produce some negative
values of the predicted coefficient
. On average 25 values
occur at each simulation, especially in correspondence of very low experimental values
(see Subsection 4.2). To eliminate any negative value of
, a routine reads the predictions performed by the ANN after each simulation and substitutes each
with a “NaN” (“Not a Number”, in Matlab language). The addition of this routine in
the code represents the only modification applied to the original ANN.
2.1. Discussion about the results
From both the graphs of Figure 2, it can be appreciated the very good agreement of computations and
measurements and above all the great degree of symmetry in the error values distribution. Both the highest
and the lowest values of are pretty well represented by the ANN, which therefore does not appear to be
affected by systematic errors. Just a few scattered values, out of 95% confidence level bands (dashed lines),
are detectable.
From a quantitative viewpoint, the rmse value (0.037, referring to the complete database, see Tab. 3) is
slightly lower than the one which characterizes the prediction of Kr (0.038). It represents a pretty satisfactory result, either in comparison to ANNPB (see more details in following Section 5), either considering that
the ANN architecture was optimized for wave reflection and not for wave transmission.
Similarly, the very high value of WI (larger than 0.990) confirms the great symmetry in the errors distribution qualitatively appreciated in the diagrams of Figure 2. Also WI value is greater than the value associated to the prediction of Kr (0.985).
The values of the standard deviation associated to the average indexes have been computed in order to
assess the uncertainty of the errors and quantify the ANN stability: consistently with the wave reflection
case, each index is characterized by a standard deviation value of about 10-3.
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1
1
0.9
0.8
0.8
0.6
0.7
0.4
0.6
0.2
e=K t,s -Kt,A NN
K t,ANN
Coastal Dynamics 2013
0.5
0
0.4
-0.2
0.3
-0.4
0.2
-0.6
0.1
-0.8
0
0
0.1
0.2
0.3
0.4
0.5
Kt,s
0.6
0.7
0.8
0.9
1
-1
0
0.1
0.2
0.3
0.4
0.5
Kt,s
0.6
0.7
0.8
0.9
1
predicted values (
, ordinate) and corresponding experimental values
Figure 2. Left: comparison among
(
, abscissa); the continuous bisector represents the ideal condition (
=
), while the dashed lines refer to
(ordinate) as a function of
(abscissa).
the 95% confidence levels. Right: difference
Table 3. rmse, WI and R2 average values and corresponding standard deviations derived from the 20 simulations;
transmission ANN (trained on the complete database of 3379 and on the 2285 data employed by PB) is compared to
reflection ANN and ANNPB (data refer to Set07 in Tab. 1 of PB work).
ANN for Kt (#3379)
mean
stand. dev.
ANN for Kt (#2285)
mean
stand. dev.
ANNPB for Kt (#2285)
mean
stand. dev.
ANN for Kr (#5781)
mean
stand. dev.
rmse
WI
0.037
0.993
0.003
0.001
0.033
0.9941
0.002
0.0006
0.065
-
-
0.038
0.985
0.003
0.003
R2
0.973
0.004
0.978
0.002
0.983
-
0.943
0.006
A specific analysis of the errors has been carried out, in order to detect and define the “largest” errors
computed by the ANN, which are not supposed to fall within the random uncertainty. The “threshold” value e ≥ |0.15| has been identified, i.e. elarge = {e ≥ |0.15|}.
All the “large” errors computed during the 20 simulations are shown in Figure 3 as a function respectively of the progressive test indices (left diagram) and of Kt,s (right diagram). The concentration of “large”
errors associated to a particular test (plot to the left) or to a particular value of Kt,s (plot to the left) are
therefore immediately visible. If the same test is systematically affected by “large” error, it may imply either that the test itself is less “reliable” (for example, due to error measurements, especially for the lowest
and the greatest values of Kt), or that the ANN is not able to correctly represent that test.
By observing the plot to the left of Figure 3, it is clear that some datasets are never affected by “large”
errors: in particular, the dataset identified by indices values <1000, referring to Acquareefs, and the dataset
associated to 1300÷2000, referring to rock LCSs collected by Seabrook and Hall (1998). By observing the
plot to the right of Figure 3 it is clear as well that the ANN captures with similar accuracy both high or low
values of Kt, since the “large” errors distribution is randomly spread around the whole range of Kt values.
The results of this qualitative analysis may lead to the conclusion that “large” errors occur especially for
specific datasets, which correspond to oblique and 3D wave attacks (indices 1000÷1300), Melito and Melby (2002), tests (indices 2000÷2100) and Daemrich, May and Ohle (2001) tests (indices >3200).
To provide a quantitative estimate of the frequency of occurrence of “large” errors and an assessment of
the “worst” condition of prediction, the following indexes have been defined:
(1)
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,
(2)
In Eq. (2), the figure ‘308’ is the total number of “large” errors computed over the 20 simulations of the
ANN, i.e. over more than 67’000 data.
The indexes and
(Eq. 1) respectively represent the absolute and the relative errors computed on avand
(Eq. 2) represent the same average quantities comerage by the ANN, while the indexes
puted considering the 308 “large” errors only. The values of all the indexes are resumed in Table 4: the
“worst” condition is represented by
= 48%, i.e. the ANN may produce errors equal or larger than
48% with a frequency of 0.4% (just 4 times over 1’000 data).
Table 4. Values of the average absolute and relative errors provided by the ANN over the 20 simulations.
4.3%
0.223
48%
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
e > |0.15|
e > |0.15|
0.020
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
0
500
1000
1500
Figure 3. Left:
right:
2000
2500
data index
3000
3500
0.462
4000
-1
0
0.1
0.2
0.3
0.4
0.5
Kts
0.6
0.7
0.8
0.9
1
(ordinate) as a function of database test indexes (abscissa);
(ordinate) as a function of
(abscissa).
5. Comparison with the existing transmission ANN
The aim of this section is to provide a comparison between the performance of the present ANN and the
one associated to ANNPB, both referring to the evaluation of Kt. ANNPB was trained against 2’285 test
which have been all included in the training dataset of the present ANN.
The comparison is qualitatively given by the analysis of the respective diagrams Kt,ANN vs Kt,s and quantitatively by the respective values of rmse and R2 indexes (Subsection 5.1).
Following the work of PB, in Subsection 5.2 is presented a further analysis to discuss the distributions of
the errors (the quantity e) and of the predicted values Kt,ANN as functions of some specific input elements,
such as the relative crest freeboard Rc/Hm,0,t and the non-dimensional structure crest width Gc/Hm,0,t.
Finally, a discussion about strengths and weaknesses of the new ANN is drawn in Subsection 5.3
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5.1 Overall performance
The performance of the present ANN and ANNPB is compared by means of the respective diagrams
Kt,ANN vs Kt,s (plot of Fig. 7a in PB paper, Fig. 1 to the left in the present work). Both ANNs show a narrow
distribution around the ideal condition (represented by the bisector of the diagrams), however the scatter of
the predicted values Kt,ANN by ANNPB is greater than the one obtained by ANN, especially for Kt,s >0.85.
Regarding the error indexes, ANNPB is characterized by rmsePB = 0.063 (see Tab. 3) and R2PB = 0.983,
denoting generally a similar performance (in effect, rmsePB suggests a worse performance, while at the
same time R2PB a better one).
The results of the present ANN are therefore satisfactory, also considering that ANNPB architecture was
specifically calibrated against wave transmission tests and trained over a narrower database. For example,
the present database contains 299 tests related to smooth structures, which were excluded in ANNPB.
To investigate more in depth the role played by the additional data, the new ANN has been also trained
and simulated against the same narrowest dataset employed by PB (see Tab. 3) All the indexes show in this
case an improved performance, also remarked by the decrease of the values of the standard deviations,
which drop even below 10-3.
5.2 Analysis of the error distribution
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
e=K t,s -Kt,ANN
e=K t,s -Kt,ANN
As a further comparison with PB work, the analysis of the distribution of both Kt,ANN and e = Kt,s - Kt,ANN
as functions of the same selected parameters, i.e. Rc/Hm,0,t and Gc/Hm,0,t, is carried out.
The error distributions are shown in Figure 4. The plot to the left (Rc/Hm,0,t on the abscissa) is compared
with Figure 8a in PB, while the plot to the right (Gc/Hm,0,t on the abscissa) with Figure 9a in PB. Both the
present diagrams show a noteworthy reduction of the scatter.
In the left plot, e increases up to and over |0.1| just within Rc/Hm,0,t ≈ [-1 ; 1], while, in the corresponding PB
diagram, errors lower than -0.1 are detectable within the whole range of Rc/Hm,0,t.
In the right panel, the distribution of e is symmetric and narrower around the ideal condition e = 0 for
each value of Gc/Hm,0,t. More in details, Fig. 9a in PB shows some amount of points gathered around a specific value of Gc/Hm,0,t, while in the present work (Fig. 4 to the right) the points are more smoothly distributed on the abscissa, i.e. the error distribution is more uniform over the range of Gc/Hm,0,t.
0
-0.1
0
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-4
-3
-2
Figure 4. Left:
right:
-1
0
Rc/Hm,0,t
1
2
3
4
-0.5
0
10
20
30
40
50
Gc/Hm,0,t
60
70
80
90
(ordinate) as a function of the relative crest freeboard
(abscissa);
(ordinate) as a function of the non-dimensional structure crest width
.
Figure 5 presents Kt, as function of Rc/Hm,0,t (Kt,s values on the ordinate to the left panel, Kt,ANN to the
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Coastal Dynamics 2013
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
K t,ANN
K t,s
right), and corresponds to the panels (a) and (b) of Figure 10 in PB paper. The experimental distribution is
pretty well reproduced, even taking into account the lowest and the highest values of Rc/Hm,0,t.
The dependence of Kt,s and of Kt,ANN on Gc/Hm,0,t is reported respectively to the left and to the right of
Figures 6 and 7. Similarly to PB, the data have been divided in classes of Rc/Hm,0,t and only the structures
with Rc/Hm,0,t = 0, i.e. the ones showing the greater scatter, have been used in this analysis. The data have
been distinguished according to the values of the armour stone diameter (Hm,0,t/Dn,50 in Figure 6 and according to the values of the breaking parameter
in Figure 7.
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-4
Figure 5.
-3
-2
-1
0
Rc/Hm,0,t
1
2
(ordinate, left panel) compared to
3
4
0
-4
-3
-2
-1
0
Rc/Hm,0,t
(ordinate, right panel) as functions of
1
2
3
4
(abscissa).
By comparing the plots to the right of Figures 6 and 7 respectively with the panels (a) and (b) of Figure
12 of PB, two main issues are immediately detectable:
- the new ANN shows an improved predicting capacity, especially for high values of . While
ANNPB never predicts Kt, values approximately greater than 0.55, the new ANN can correctly reproduce also values greater than 0.7;
- the new ANN is instead not able to reproduce Kt values lower than 0.02÷0.03, probably due to the
routine developed to prevent negative predictions (see Subsection 3.2). More precisely, the minimum vale Kt,ANN = 0.027 ≈ 0.03 can be defined as the lower minimum of validity of the ANN. This
problem – which does not affect ANNPB – is particularly evident for Gc/Hm,0,t > 40 (compare left
and right panels of both Figs. 6 and 7).
Therefore, if by one hand ANNPB seems to be upper-limited (≈ 0.55), by the other hand ANN is lowerlimited by ≈ 0.03.
Likewise ANNPB, the new ANN overcomes the discontinuity of Van der Meer et al. formulae (2005) for
Gc/Hm,0,t = 0 and is able to represent the dependence on Dn,50 and ξ0. Furthermore, the experimental values
which do not follow Van der Meer distribution (see left panels of Figs. 6 and 7) are pretty well reproduced
by the new ANN (corresponding right panels). On the contrary, the values Kt,ANN,PB are generally aligned
with Van der Meer predictions (panels (a) and (b) in Fig. 12, PB paper), showing not significant improvement with respect to the formulae.
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1
1
0<Hm,0,t/Dn,50<1
0<Hm,0,t /Dn,50<1
1<Hm,0,t/Dn,50<2
0.9
1<H
0.9
m,0,t
2<Hm,0,t/Dn,50<3
3<Hm,0,t/Dn,50<5
0.8
5<Hm,0,t /Dn,50<6
VDM 2005
0.7
0.6
K t,ANN
K t,s
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
10
20
Figure 6.
30
40
Gc/Hm,0,t
50
60
70
(ordinate, left panel) compared to
different classes of
0
80
0
10
20
30
40
Gc /Hm,0,t
50
60
(ordinate, right panel) as functions of
. Only data characterized by
80
(abscissa) at
are included.
Csi op<3
Csi op<3
3<Csi op<5
0.9
3<Csiop<5
0.9
5<Csi op<7
5<Csiop<7
7<Csi op<9
0.8
7<Csiop<9
0.8
VDM 2005
Csi op>9
0.7
0.6
0.6
K t,ANN
0.7
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
70
1
1
K t,s
<2
n,50
3<Hm,0,t /Dn,50<5
0.8
5<Hm,0,t/Dn,50<6
0.7
/D
2<Hm,0,t /Dn,50<3
0
Figure 7.
10
20
30
40
Gc/Hm,0,t
50
60
70
80
0
0
10
20
30
40
Gc/Hm,0,t
50
60
70
80
(ordinate, left panel) compared to
(ordinate, right panel) as functions of
(abscissa, left
and right) at different classes of
. Only data characterized by
are included.
5.3 Discussion about the new ANN
The new ANN generally demonstrates an improved performance with the respect to ANNPB, considering
both the indexes and the distribution of the errors. However it is worthy to discuss the following key issues
that may affect the ANN applicability:
- the number of input parameters required (13 instead of the 6 needed by ANNPB), which means that
more detailed experimental/prototype information is needed to run the ANN;
- the number of the hidden neurons (40 instead of 6), which represents an increased complexity of
the ANN architecture, and requires extended databases to correctly train and calibrate the model;
- the limitation in representing low values: Kt,ANN < 0.03.
It has to be noted that the high dimension of the hidden layer may affect the ANN applicability only during its training, i.e. only in case the model has to be re-initialized. As the ANN has been already trained, it
should be suitable to be directly applied at least to similar LCSs, and therefore this issue does not affect an-
636
Coastal Dynamics 2013
ymore the ANN.
The lower limit of 0.03 is actually a very low value, which in most cases falls within the measurement
uncertainty. Instead, the upper-limit of about 0.55 in ANNPB is a more restrictive condition, since the number of tests presenting Kt,s < 0.03 is sensibly lower than the number of Kt,s > 0.55 (see Tab. 5).
In conclusion, the main drawback related to the new ANN is represented by the challenging high number of information required to completely compose the input set.
Table 5. Number of tests presenting Kt,s < 0.03 and Kt,s > 0.55 within the complete database (3’379 data) and
PB database (2’285 data) and relative percentages with the respect of the total number of data.
Database
ANN # 3379
ANNPB # 2285
Kt,s < 0.03
61; 2%
44; 2%
Kt,s > 0.55
1266; 37%
890; 39%
6. Conclusions
An ANN for the prediction of the wave transmission coefficient Kt behind LCSs has been presented. The
model has been directly and completely derived from the wave reflection ANN (Zanuttigh et al., 2013),
which was originally trained, calibrated and validated against a wide database (5’781 data).
The ANN architecture, which has been maintained, consists of 13 non-dimensional input parameters,
one hidden layer of 40 neurons and an output layer represented by the output neuron Kt. The application to
wave transmission has been carried out through the re-training of the ANN against 3’379 additional tests,
mostly referring to LCSs.
This ANN provides accurate predictions of Kt, being characterized by an average value of rmse ≈ 0.037
(with an average percentage error of 4.3%), and providing, as worst condition, errors larger than 48% with
a frequency of 0.4% (just 4 times over 1’000 data).
The ANN has been compared to the other existing ANN for wave transmission behind LCSs (Panizzo
and Briganti, 2007), either discussing the values of the error indexes (WI and R2, besides rmse), either
comparing the distribution of the errors. The new ANN demonstrates an improved capability of reproducing the functional relationships among Kt and some of the most significant parameters (such as the relative
crest freeboard Rc/Hm,0,t and the relative crest width, Gc/Hm,0,t). Nevertheless it is characterized by an increased complexity of the architecture and of the number of input elements.
These first promising results suggest that this ANN may be employed for the prediction of both Kr and
Kt. However, to say something more conclusive, further research should be performed to test the ANN
against different structures than LCSs.
Acknowledgements
The support of the European Commission through Contract 244104 THESEUS (“Innovative technologies
for safer European coasts in a changing climate”), FP7.2009-1 Large Integrated Project, is gratefully
acknowledged.
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