Neural network parameterisation of the
mapping of wave–spectra onto nonlinear
four–wave interactions
Kathrin Wahle ∗ , Heinz Günther, Helmut Schiller
GKSS Research Center, Max–Planck–Str. 1, 21502 Geesthacht, Germany
Abstract
A new approach to parameterise the exact nonlinear interaction source (snl) term
for wind wave spectra is presented. Discrete wave spectra are directly mapped onto
the corresponding snl–terms using a neural net (NN). The NN was trained with
modeled wave spectra varying from single mode spectra to highly complex ones. The
specification of training data was based on a classification of the wave spectra by
cluster analysis. In course of the structuring of the NN the intrinsic dimensionalty of
the spectra was estimated with an auto-associative neural net (AANN). The AANN
might be used for a scope check of the method.
Key words: Neural networks, Nonlinear wave–wave interaction, Wave–spectra,
Wave models, Cluster analysis
1
Introduction
Wave–models compute the dynamics of the sea–surface by solving the action balance equation describing the evolution of interacting weakly–nonlinear
waves, e.g. Komen et al. (1994). Besides propagation of waves this involves
growth of waves due to wind, the dissipation of wave energy due to whitecapping and the transfer of wave energy due to nonlinear wave–wave interaction. The computation of the last–mentioned process is by far the most
time consuming step in the modelling. Operational wave–models therefore use
approximations as the well known discrete interaction approximation (DIA)
Hasselmann et al. (1985). The DIA is computational fast but has also known
deficiencies.
∗ Corresponding authors.
Email address: kathrin.wahle@gkss.de (Kathrin Wahle).
Preprint submitted to Elsevier
2 April 2009
The further development of third generation operational wave–models requires
a replacement of the approximative methods for the calculation of the nonlinear wave-wave interaction by fast and accurate methods. Attempts have
been made with extensions of the DIA but they are so far not generally applicable since tuning for different classes of wave spectra is needed. The same
problem occurs for methods based on diffusion operators. For a comprehensive
discussion see Cavaleri et al. (2007).
In this paper a neural net (NN) parametrisation of the mapping between
wave–spectra and the corresponding nonlinear wave–wave interaction will be
presented. The idea of applying neural networks in this context was first introduced by Krasnopolsky et al. (2001), Krasnopolsky et al. (2002). There the
wave spectra as well as the nonlinear interaction source terms are assumed
to be separable functions of frequency and direction which are approximated
by expansion series, respectively. The neural network is used to map the two
sets of expansion coefficients. This assumption was dropped in a successive
work Tolman et al. (2005) where the authors used twodimensional Empirical
Orthogonal Functions for the expansion of single peaked spectra. In contrast
to this approach the feasibility of the direct mapping of the discretised wave
spectra onto the nonlinear interaction source term will be demonstrated here.
The study will not be restricted to single peaked spectra but will also include
multi–modal ones.
Two design decisions are crucial for the successfull construction of a NN: the
choice of the training data and of the NN architecture, respectively.
The wave spectra used for the training of the neural network (NN) are simulated spectra from a hindcast with the wave model WAM cycle 4 WAMDI
group (1988). Although WAM uses the DIA the simulated spectra are suited
to investigate the approach of a direct mapping. (Once the feasability has
been shown one should make the effort of calculating all spectra with a wave
model that uses the exact nonlinear interaction.) We expect that the training
for wave spectra representing multi–modal wave systems will be more difficult
then for single peaked spectra. Therefore the more complex cases should be
well represented. To be able to enrich the number of multi–modal spectra in
the training data set we perform a classification of the wave spectra. For the
classification we use a cluster algorithm.
To fix the NN structure it is important to have an estimate of the intrinsic
dimensionality of the spectra, i.e. the minimum number of variables needed
to represent the spectra. For this we constructed auto-associative neural nets
(AANN) which map the spectra onto themselves while compressing them in
between. The difference of the original spectrum with the reproduced one gives
a quality measure of the reduced representation. This difference can be used
to decide wether the compression was successfull for a given spectrum (wether
2
it is in the scope of the algorithm). The finding of the intrinsic dimensionality
led to the decision about the actual NN structure for the mapping of the wave
spectra onto the nonlinear interaction source terms.
The results of the cluster analysis are given in section 2 whereas the technical
details of the method are described in appendix A. Section 3 starts with a
short introduction to neural nets and the special case of auto-associative neural nets. The concrete applications follow: in 3.1 some AANN architectures
are tested in order to get an estimate of the intrinsic dimensionality of wave
spectra. The design and training of the actual neural network for the nonlinear
interaction source term is described in section 3.2. It includes the application
of the AANN for outlier identification. We then discuss the relevant steps towards an operational usage of the procedure in an outlook given in section 4.
Finally, the results are summarised in some concluding remarks in section 5.
2
The cluster analysis of wave–spectra
The wave spectra used for the training of the neural network (NN) are simulated spectra from a hindcast with the wave model WAM cycle 4 WAMDI
group (1988). The data span over 22,000 points in the north-atlantic for the
one month period of january 1995. Spectra were saved every three hours at an
equidistant resolution of 15◦ in direction and an equidistant relative resolution in frequency ranging from approximately 0.04 Hz to 0.4 Hz. The resultant
dimensionality of the spectra is 600 (24x25).
From this dataset of over five million spectra a representative subset containing
all classes of spectra had to be selected. By random sampling the simple single–
and bi–modal wave system cases would outweight any other class of spectra.
As a result the neural networks would well map these cases but would badly
perform for other classes. Thus a classification method should first of all be
applied to the whole dataset.
Important characteristics for the classification of the wave spectra are the
number and shapes of wave systems in them. The classification method should
identify these wave systems. This clearly is a pattern recognition problem
which we solved by applying a cluster algorithm.
As the spectra are represented using an azimuthal angle it happens that a
wave system crossing 0◦ /360◦ is splitted artifically. To avoid this the wave
spectra are preprocessed: the point of the origin of the direction is shifted if
necessary to the direction with lowest energy. This is the only preprocessing
needed here. Tolman et al. (2005) showed possiblities how to normalize the
spectra further for other approaches.
3
Input to the cluster algorithm are the 600 points of a wave spectrum. Each
of these points is defined by its coordinates (frequency and direction indices)
and by the energy at this coordinate bin of the spectrum. The energies serve
as weights of the points. The algorithm assigns the points into groups (called
clusters) so that points from the same cluster are more similar (according to
a distance measure) to each other than points from different clusters.
The cluster algorithm CLUCOV Schiller (1980) is used which was adapted
for the twodimensional problem of parametrising the wave spectrum. Each
cluster is characterised by a twodimensional Gaussians. For each cluster its
respective center and covariance matrix are estimated and the resulting Mahalanobis distance is used to assign points to the closest cluster. This allows
for varying orientations of the different clusters. In the iteration procedure it
is checked if the clusters are unimodal. If not, the cluster is split into two. The
number of clusters also can decrease: if two clusters strongly overlap these
clusters are combined. In the appendix A the algorithm is described in more
detail. One can summarize the method as performing a point density function
approximation with Gaussians as radial basis functions (RBF).
Figure A.1 shows three examples. The left panel shows the input to the algorithm (original wave spectra). The right panel shows its parametrisation by
the cluster algorithmus. It can be seen that the main characteristics are well
reproduced. But also it should be noted that the number of clusters is often
greater then the number of wave systems. In the first example the number of
wave systems is two, but three clusters were found, the second example shows
three wave systems and four clusters and the last spectrum has at least three
wave systems in it and six clusters were found. This is due to the structure of
the wave systems which are best approximated by two Gaussians, one around
the peak and one for the tail (see again figure A.1).
To summarize the cluster algorithm is well suited to classify wave spectra.
The main characteristics are well reproduced. The number of clusters is often
larger than the number of wave systems in a spectrum but the two numbers
are highly correlated. High number of clusters indicate complex structures in
the corresponding spectrum.
Table 1 lists the result of the cluster analysis, i.e. the number of clusters
found for each spectrum in the dataset. On this basis a representative subset of
791,570 spectra was choosen. The number of selected spectra increases with the
number of clusters (more complex wave spectra). Above a number of clusters
of five all spectra were taken and included twice into the dataset. Table 1 also
quantifies the selection. This representative dataset was randomly subdevided
in a training set (755,173 spectra) and a testing set (36,397 spectra) for the
neural networks.
4
number of clusters
number of spectra
percentage selected
1
406,332
9.8
2
1,536,057
5.2
3
1,907,143
6.3
4
1,033,269
15.5
5
354,712
56.4
6–12
95,804
200
Table 1
Spectra used for the representative dataset.
3
Neural Networks
Wave spectra and the nonlinear interaction source term are related via a six–
dimensional Boltzmann integral. A computational efficient method to parametrise
this complex functional relation is the usage of artificial neural networks (NN).
This is possible since a NN with at least one hidden layer – a layer between the
input and the output layer – is able to approximate any continuous function
(Universal Approximation Theorem) as was shown by Haykin (1999).
So a NN — in this context — is a computational tool for function approximation whose generalities will be described briefly (see Bishop (1995)).
A NN is organized in layers: one input layer, one output layer and one or
more hidden layers inbetween. Each layer consists of neurons. The number of
neurons in the in– and output layer are given by the dimensions of the in–
and output vectors. (In our application the two–dimensional wave spectra and
nonlinear interaction source terms will therefore be arranged as vectors with
600 components, respectively.) The number of neurons in the hidden layer(s)
are not preset and depend upon the problem. Each neuron in a layer is linked
to each neuron in a neighboring layer with a weight.
Neural nets work sequentially: each element of the input vector serves as entry
for one of the neurons of the input layer. The output of the first hidden layer
is computed by summation of the weighted inputs, shifting it by a bias and
applying a nonlinear function (a sigmoid here). The procedure is repeated
itself until the output layer is reached where the outcome of each neuron gives
one element of the output vector.
Weights and biases are the free parameters of the approximation. They are
fixed during the training phase of the NN. To do so a dataset — the training
set — is needed consisting of N pairs of input vectors (the 600 components of
the wave spectra) and corresponding desired output vectors (the correspond5
ing 600 components of the exact nonlinear interaction source term). At the
beginning of the training the outcome of the NN will differ largely from the
desired output. Let ~y and y~′ be the m–dimensional output vectors as emulated
by the NN and as given in the dataset respectively. Then the difference of the
two will result in an error e defined by
e=
N
1 X
ep
N p=1
m
y ′ i − yj
1 X
ep =
m i=1 max(y ′pi , p ∈ [1, N]) − min(y ′pi , p ∈ [1, N])
!2
.
(1)
This mean squared relative error per neuron e is iteratively minimized during
the training by backpropagating it through the NN and adjusting the biases
and weights according to a gradient descent scheme. It is measure for the
quality of the function approximation by the NN.
It is good practise to have an additional independent dataset — the testing
set — which is needed to check the generalization–power of the NN after the
training, i.e. to test if reasonable output is produced for input not included in
the training.
The training and testing phase is time consuming. But it needs to be done
only once. Whereas the usage of a NN is very fast.
The abovementioned holds true for an auto–associative NN which is a particular NN. Its especialness is that it maps the input vector onto itself. But at one
stage of the mapping — in the so called bottleneck layer — the dimensionality
is reduced. So when applying AANN’s with different degrees of dimensionality
reduction to the 600 components of the wave spectra the number of variables
needed to capture the information contained in the spectra can be determined.
Technically speaking the number of neurons in at least one hidden layer in
an AANN is less than the dimension of the in- and output vector ~x and x~′ .
The AANN part m(~x) which maps the input vector onto the neurons in the
bottleneck layer p~ is also called mapping part and the backprojection x~′ = d(~p)
is called demapping part, accordingly (see figure A.2).
The mapping part can be considered as performing a nonlinear generalization
of the Principal Component Analysis (Nonlinear Principal Component Analysis, NLPCA) Kramer (1991). It retains the maximum possible amount of information from the original data, for a given degree of compression. Like PCA
also NLPCA can serve important purposes, e.g. filtering noisy data, feature
extraction, outlier identification, the compression of data, and the restauration
of missing values Hsieh (2001), Schiller (2003).
6
3.1 Auto–associative neural network of wave–spectra
In order to achieve the goal of directly mapping the wave spectra onto the
corresponding nonlinear interaction source term it is essential to find their
intrinsic dimensionality (the number of independent variables) to be able to
fix an appropriate NN structure.
The cluster analysis in the last section was not only useful for the creation of a
representtive dataset but also it gives a hint about the intrinsic dimensionality
of the wave spectra. This can be reasoned since the parametrisation of the wave
spectra by gaussians is a first compressed representation. Since each spectrum
can be parametrised by a few clusters and each cluster is characterised by
six numbers (energy of the cluster, its center, its orientation and shape) the
intrinsic dimensionality is expected to be of the order of a few tens. This
will now be further investigated by means of auto–associative neural networks
(AANN’s).
To find an optimal AANN–parametrisation of the wave spectra AANN’s with
different number of neurons in the bottleneck layer had to be trained. The
optimization problem consists of balancing the loss of information with decreasing bottleneck neuron number and the usability of the NN. At optimum
the compression is as loss–free as possible with additional bottleneck neurons
giving only slight improvement.
For the training the neural net program developed by Schiller (2000) was
used. The AANN’s have 600 neurons in the in– and output layer since the
wave spectrum is interpreted as a 600-dimensional vector. Each of the trained
AANN’s had three hidden layers with 80 neurons in the first and third hidden
layer and with differing number of neurons in the middle bottleneck layer.
Figure A.3 shows the mean squared relative error per neuron e (see equation
1) of the AANN’s for the testing data set as a function of the number of
bottleneck neurons.
As expected the error e decreases with increasing number of bottleneck neurons. Clearly two different regimes are visible (the straight lines were fitted
by linear regression): one for the number of bottleneck neurons below approximately 33 and one above. From this it was decided to choose the smallest
AANN belonging to the second regime with 39 bottleneck neurons in it.
Figure A.4 shows three examples of the performance of this AANN. The left
panel shows the original wave spectra. The middle panel shows the output
of the AANN — the mapping of the original wave spectra onto themselves.
The right panel shows the directionally integrated wave spectra. The examples exhibit an increasing complexity of the wave spectra. The AANN output
strongly resembles the original wave spectra throughout the spectral space.
7
The residual average error
√
e is 1.4, 1.5 and 1.8% from top to bottom.
To summarize the construction of different AANN’s has shown that the intrinsic dimensionality of the analysed wave spectra is about 40. An AANN with
39 bottleneck neurons gives a good parametrisation for all different classes of
√
wave spectra with a residual average error of e =1.3%.
3.2 Nonlinear wave–wave interaction
So far we have selected a dataset which incorporates all classes of wave spectra
(see section 2) and we have found the typical number of independent variables
describing each of these wave spectra (see section 3.1).
After these preparatory works one can now start with the construction of a
neural net (NN) for the direct derivation of the nonlinear interaction source
terms from wave spectra. As a first approximation we decided to use the same
architecture already used for the AANN: In– and output layers consist of 600
neurons as the discretized wave spectra and the snl–terms are interpreted as
600–dimensional vectors. Inbetween there are again three hidden layers with
80, 39 and 80 neurons, respectively.
To train and test the NN the (exact) nonlinear interaction source (snl) terms
corresponding to each of the spectra of the selected dataset of 791,570 wave
spectra (see section 2) had to be calculated first. The method first suggested
by Webb (1978) and known as the WRT method with further improvements
by Vledder (2006) was applied for this purpose.
The pairs of wave spectra and snl–terms were again subdivided
√ into the training and testing dataset. The resulting residual average error e (see equation
1) of the resulting NN was 1.1% for both the testing– and training.
Figure A.5 shows results of the performance of the NN. The top row shows
the original wave spectra which served as input for the NN. The next row
shows the corresponding exact snl–terms calculated with the WRT–method
and below it the NN results (the WRT emulation by the NN) are shown. The
bottom row shows the directionally integrated snl–terms. In all cases the NN
emulation is very similar to the exact solution. The directionally integrated
plots highlight the quality of the fit. The three wave spectra vary from single
to three wave systems to demonstrate the broad applicability of the method
to different classes of wave spectra.
The choice of a represantative training dataset is crucial since a NN has good
interpolation properties but produces unpredictable output when forced to
extrapolate. The approach described here offers a ’in range’ check Schiller et
8
al. (2001), i.e. the AANN can be used to check if a given spectrum belongs
to the class of spectra chosen for the training of the AANN. If the error e
of the AANN is big for a given wave spectrum then it did not succeed to
represent the spectrum through the bottleneck layer. Then it is questionable
that the outcome of the snl–term NN will be in good agreement with the exaxt
snl–term.
Figure A.6 exemplifies the resulting
√ outlier identification: A wave spectrum
which gave a large of error (here e = 3.6%) when parametrising it with
the AANN was choosen randomly (upper panel). The emulation of the corresponding snl–term with the NN is poorly (lower panel). The
√ resultend relative
error when comparing the exact with the NN solution is e = 10.2%.
The NN method described here differs in mainly two points from the NN
based method first presented in Krasnopolsky et al. (2001) and with latest
results from Tolman et al. (2005): There the wave spectra as well as the snlterms are first decomposed and represented by a linear superposition of two
sets of orthogonal functions. Two different approaches are used for the basis
functions. The former approach is based on a mathematical basis where the
frequency and direction dependence was separated (an assumption beeing not
satisfied by spectra having more than one wave system at one frequency). The
later one is based on principal components but it is restricted to single peaked
spectra. In either case the neural net is used to map the expansion coefficients
onto each other. The training and testing sets used were generated based on
theoretical spectral descriptions.
So firstly, the approach presented here is not restricted to special cases of wave
spectra with e.g. only one wave system in it since a set of modeled spectra of
a high variety of complexity has been used for its development. And secondly,
no assumptions (as separabilty or convergence of an expansion series) about
the wave spectra and nonlinear interaction terms are made.
Finally an important advantage of any NN parametrisation over other implementations of complex physical processes is its computational efficiency. It is
very time consuming to train a NN and to find an appropriate net architecture
but its usage is fast. A runtime comparison of the exact WRT–method with
the NN method presented here gives a speedup factor of roughly 500.
4
Outlook
We have presented results of a neural net (NN) mapping directly wave spectra
onto the corresponding exact nonlinear interaction source term. We now want
to describe how we imagine to distinctly improve the method in order to
9
operationally incorporating it.
To start with, a new training/testing dataset of wave spectra should be calculated with a model that uses the exact nonlinear interaction. The datasets
should again contain all classes of complexity of wave spectra.
With these datasets the training of the neural nets should be repeated and
different net architectures (e.g. different mapping/demapping part) should
be tried. In particular we assume it promising to train a AANN for both,
the wave spectra and the nonlinear interaction source terms and to map the
both bottleneck layers onto each other. This ansatz has some advantages:
Firstly, more complex net architectures can be tried out for the NN which maps
the two bottleneck layers onto eachother since the number of in– and output
variables is one order of magnitude smaller than in the present NN. Secondly,
with two AANN’s the ’in range’ check can be performed twice: one time for
the wave spectra before applying the actual NN and one time afterwards for
the emulated nonlinear interaction. Still, this is a necessary condition but not
a sufficient one. Thirdly, when changing the resolution of the wave model only
the two AANN’s have to be retrained, the NN for the actual mapping stays
the same. This can be easily done, because the number of neurons in the
bottleneck layers of the AANN’s is already known since the complexity of the
spectra and the source terms does not change with resolution.
Finally, it has to be shown that the NN emulation gives robust and accurate results when implemented in a numerical wave model. As suggested by
Krasnopolsky et al. (2005), Krasnopolsky et al. (2007) parallel runs of the wave
model with the original parametrization of the exact nonlinear interaction and
with its NN emulation should be performed to do so. When operationally running the wave model a quality control block as suggested by Krasnopolsky et
al. (2008) could determine whether the NN emulation will be used or not.
5
Conclusions
The feasability of a neural network (NN) based method for direct mapping
of discrete wave spectra onto the corresponding (exact) nonlinear wave–wave
interaction source (snl) terms has been demonstrated.
In a preparatory step the intrinsic dimensionalty of the wave spectra was
estimated by means of an AANN to be about 40, i.e. 40 variables allow to
capture the variability of the wave spectra with a residual average error of
1.3%. The composition of the training data set for the NN was based on
automated classification of the wave spectra by means of a cluster analysis.
10
The NN for the mapping of the wave spectra onto the corresponding snl–terms
shows good performance. It is able to emulate the WRT method calculations
for single and multi mode wave spectra with a much higher accuracy then
the approximations implemented in nowadays operational wave models. The
quality of the emulation might be controlled using the corresponding AANN.
6
Acknowledgements
The authors like to thank Gerbrant Ph. van Vledder for providing his improved
version of the WRT–method for calculating the nonlinear interaction source
terms and for offering his help on running it. Further, we thank the reviewers
for their valuable comments on an earlier version of this paper.
A
The cluster algorithm CLUCOV: twodimensional case
The CLUCOV algorithm is used assign the 600 points of a wave spectrum
to clusters. The k–th cluster Gk is characterized by the moments of order
zero, one and two of the distribution of the N points contained in this cluster.
The points are defined by their frequency and direction coordinates X m =
(Xfm , Xθm ) and their energies which serve as weights w m . Then the three moments are
– the total weight I k of points X m contained in the cluster Gk
k
k
I =
N
X
wm
m=1
– the centroid Qk of the cluster Gk
Qki
Nk
1 X
w m Xim
= k
I m=1
with X m ∈ Gk , i ∈ {f, θ}
– the covariance matrix C k of the cluster Gk
Nk
1 X
k
Cij = k
w m (Xim − Qki )(Xjm − Qkj )
with X m ∈ Gk , i, j ∈ {f, θ}
I m=1
The eigenvectors of the covariance matrix point into the direction of the main
axes of the ellipsoids by which the shape of the clusters is approximated. The
square root of the eigenvalues of the covariance matrix denote the lengths of
the main axes, and the square root of the determinant of the covariance matrix
thus measures the volume of the clusters.
11
k
All these three moments enter the definition of the distance fm
of a point m
m
k
at X from the k–th cluster G . Thus, each cluster builds its ‘own’ metric:
The (Mahalanobis) distance ρkm of a given point X m from the centroid Qk of
the k–th cluster enters the exponent of a Gaussian g k (X m )
g k (X m ) =
ρkm
1
exp[− (ρkm )2 ] with
2
2π |C k |
1
q
q
= (X m − Qk )T (C k )−1 (X m − Qk ).
The total weight I k of points in cluster k is included into the distance measure
k
fm
as a linear weight factor, so that big clusters ‘attract’ further points
k
fm
= I k · g k (X m ).
In the direction of the main axes distances are measured in units of the square
root of the corresponding eigenvalue of the covariance matrix (which is suggested by the quadratic form in the exponent of the gaussian). This distance
measure is also invariant under linear transformations such as translation and
rotation.
k
The determinant in the denominator of fm
favours compact clusters against
voluminous clusters of the same content.
In the algorithm CLUCOV it is also possible to split and merge clusters, that
is to change the number of clusters.
To achieve this, a measure t for the overlap of two clusters Gk and Gl is
defined:
t= √
h0
hk hl
The quantities hk and hl are superpositions of the gaussians f k (X) and f l (X)
of the clusters Gk and Gl in their centroids Qk and Ql :
hk = f k (Qk ) + f l (Qk )
and
hl = f k (Ql ) + f l (Ql )
The quantity h0 is the minimum of this superposition along the distance vector
(Qk − Ql ) between the clusters:
h0 = min[f k (X) + f l (X)]
12
If for a pair of groups this measure t exceeds a limit tmerge (parameter of the
merging procedure) this pair of groups is united into one group.
The compactness of the clusters is tested by arbitrarily subdividing the clusters
with straight lines through the cluster. If the overlap of the two parts of the
clusters is smaller than tsplit (parameter of the splitting procedure) this cluster
is split by the corresponding line.
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V.M. Krasnopolsky, M.S. Fox-Rabinovitz, D.V. Chalikov (2005) New Approach to Calculation of Atmospheric Model Physics: Accurate and Fast
Neural Network Emulation of Long Wave Radiation in a Climate Model.
Monthly Weather Review 133(5), p.1370–1383
V.M. Krasnopolsky (2007) Neural Network Emulations for Complex Multidimensional Geophysical Mappings: Applications of Neural Network Techniques to Atmospheric and Oceanic Satellite Retrievals and Numerical Modeling. Reviews of Geophysics 45, RG3009
V.M. Krasnopolsky, M.S. Fox-Rabinovitz, H.L. Tolman, A.A. Belochitski
(2008) Neural network approach for robust and fast calculation of physical
processes in numerical environmental models: Compound parameterization
with a quality control of larger errors. Neural Networks 21, p.535–543
14
RBF parametrisation
0.
1
0.01
0.01
0.5
1
3
°
0
0.1 .5
0.05
0.1
0.
05
0.
0.5
5
10
05
0.
0.1
0.5
3
1
0.05
0.06
01
3
5
6
1
0.41
5
0.1
0.0
0.10
0.16
0.26
frequency [Hz]
45
0
0.1
°
5
1
1
0.
90
0.01
0.5
5
10
3
0.1
1
0.0
05
0.5 .1
0.01
0.05
0.1
0.5
1
1
00.05
.1
3
5 5
1
0.5
135
1
0.5
direction [ ]
180
0.01
.015 1
00.
3 5
10
6
6
3
0.5
225
10
5
0.01
1
1 .5 0..015
0
0
1
0.0
01
0.05 0.01
0.1
1
0.01
0.05
0.1
1
0.41
0.
0.5
0.5
3 6
0.10
0.16
0.26
frequency [Hz]
RBF parametrisation
270
1
0.06
0.06
360
315
10
6
3
1 .5
0
0.1
0.05
0.01
1
3
5
01
0.5
1
3
5
0.41
0.5
1
0.
5
1
3
0.1
05
0.
3
direction [ ]
0.01
5
0.0 1
0.0
0.1
5
0.0 0
0.05 .01
0.5 0.1
1
3
6
0.
05
05
0.
1
45
0
0.01
3
0.5
0. 0
05 .0
1
1 0
.5
direction [°]
0.1
3
5
0.0
0.0
5
5
°
1
0.5
0.05
direction [°]
0.0
5
0.5
0.0
0.1
5 0.
01
0.05
0.01
5
0.
0. 0.
05 1
0.01
°
0.05
0.10
0.16
0.26
frequency [Hz]
original wave spectrum
270
direction [ ]
5
0.1
135
0.01
6
0.1 0.05 0.01
0.0
0.06
360
315
45
0
1
0.5
1
3
0.5
0.5
0.1
1
0.0
05.1
0. 0
45
0
.1
90
5
90
5
05 0
180
01
0.
0.05
0.1
3
0.
0.41
0.01
0.0
0.1
5
0.5
1
225
1
0.1
0.5
05
0.15
0.0
0.0
0.1
5
0.5
0.5
1
0.01
135
0.01
0.05
0.1
0.0
direction [ ]
180
0.1
0.01
0.5
0.10
0.16
0.26
frequency [Hz]
RBF parametrisation
0.
270
0.01
0.510.1
3
1
0.00
.501
1
0.0
5
0.01
0.
1
0.06
360
315
1
0.
1
1
0.1
225
0.41
1
0.5 00. .05
1
0.0 .05
0 0.1
0.5
0.05
270
45
0
5
0.
3
0.01
0.10
0.16
0.26
frequency [Hz]
original wave spectrum
1
135
0.5
0.06
0.5
180
0.1
0.05
0.05 0.1
0.01
225
90
1
0.0
0.5
0.15
360
315
90
5
0.0
0.01
1
0.
45
0
135
0.1
0.01
0.05
0.1
0.1
0.1
90
180
5
0.
1 3
135
270
0.1
1
180
0.01
0.5
0.01
225
225
0
0.1 .05
0.05
0.1
0.01
0.01
270
360
315
1
00.
.05
0.01
original wave spectrum
360
315
0.10
0.16
0.26
frequency [Hz]
0.41
Fig. A.1. Examples of cluster analysis: left panel shows original wave spectra, right
panel shows corresponding RBF parametrisation.
15
Fig. A.2. Autoassociative NN with bottleneck: the input is mapped by p~ = m(~x)
onto a lower dimensional (dim(~
p) < dim(~x)) space and is approximately reconstructed by the demapping NN part x~′ = d(~
p) ≈ ~x.
16
−4
squared relative error
4
x 10
3
2
1
0
10
20
30
40
50
number neurons in bottleneck
60
Fig. A.3. Squared relative error of the AANN’s as function of the number of bottleneck neurons.
17
original wave spectrum
0.5
AANN parametrisation
315
0.00.1 0.1
5 05
0.
0
3
0.1
0.
5
1
6
5
3
5
0.5 1
0.5
0.06
315
0.1
0.1
00.1
.1
.1
0.10
0.1
0.05
0.05
1
°
direction [ ]
0.1
1 1
0.5
00.0
.055
.1
0.41
05
0.05
0.
original
AANN
1.2
0.05
0.05
5
.05
00.0
1
0.0
1
0.10
0.26
0.01
0.01
1
0.8
0.6
0.4
0.2
0.01
0.01
0.26
.055
00.0
0.1
1
0.5
0.10
0.16
frequency [Hz]
1
05
0.
0.5
1
0.
45
0.10
0.16
frequency [Hz]
0.06
0.0
0.5
90
45
0.1
0.01
0.1
135
0.10.1
1
0.0
0.01
180
90
0.06
spectral energy density
0.01
°
direction [ ]
0.01
0
0. .05
1
5
0.0
0.01
225
55
0.
0.
055
0.0.0
0.5
0.1
0.
1
0.0
0.5
1
1
1 0.555
0.0
0.0
0.01
135
270
5
0.0
0.1
0.1
0.1
0.1
0.0
5
00.1
.1
.05
0.050
00.
.005
5
055
0.0
0.
0.5
1
0.1.1
0
.1
00.1
0
0.01
0.5
1.4
00.
5.5
1
.05
1
directionally integrated
1
0.01.1
1.5
0
0.41
0.0
0.01
0.1
0.1
0.1
0.050.05
5.05
0.00
0.5
225
0.1
0.1
50.05
0
0.0
1 .5
1
00.
.005
0.5
50.01
315
1
0.01
0.5
0.01
0.1
0.26
0.055
0.0
00.0
.055
0.01
original
AANN
2
AANN parametrisation
0.0
270
0.0.55
0.10
0.16
frequency [Hz]
360
0.1
11
0.050.05
0.1
315
0.05
0.05
.1
00.1
0.06
.11
00.
0
0.41
0.5
0.5
5
5.0
0.00
0.26
original wave spectrum
360
1
11
45
0.1
0.0.55
00.5.5
1
0.1
0.10
0.16
frequency [Hz]
0.0
0.1
11
0.0
0.06
90
05
0.
1
0
135
0.0
1
90
0.1
1
0.41
directionally integrated
1
0.1
0.1
0.0
0.26
0.01
180
.1
00.1
0.0
5
135
05 0.1
5
05.0
0.0
055
0.0
0.
.11
00.
01
0.
225
0.10
0.16
frequency [Hz]
5
0.0
0.0
0.
0.1
5
00.5.
1
0.1
0.06
2.5
0.1
1
05
0.
1
0. 5
5
0.0
0.0
0.0
180
0
0.41
0.1
0.0
0.1
05
0.
270
55
0.0
0.0
0.01
0.26
0.01
0.5
0.5
45
0.0
5
0.01
00.1.1
225
20
10
0.0
5
5
0.0
1
0.01
1
00.
.005
5
0.05
0.05
270
°
55
0.0
0.0
0.1
0.0
0.1
0.0
315
30
AANN parametrisation
360
0.05
0.05
1
0.1
0.10
0.16
frequency [Hz]
original wave spectrum
360
direction [ ]
3
1
0.0
0.1
0.5
0
0.41
0.0
0.1 0.0
55
5
0.26
0.1
10
5
0.10
0.16
frequency [Hz]
5
.50
0.0
1
0.0
0.0
1
0.0
0.5
0.06
5
6
10 6
0.1
0.1
0.5
10
6 5
0.1
0
45
5
0.01
0.05
0.1 0.05
0.5 0.5
1
1
3
90
0.0
3
0.05
0.1
0.51
40
0.
05
1
1
11
11
0.5
05
0.01
135
10
3
0.1
5
6
180
0.5
0.01
00.0.0
55
0.1
3
1
0.0
6
0.1
0.5
11
0.
0.1
05
1
1
0.5
0.1
1
0.5
1
0.0 0.1
5
0.
3
0.1
0.1
0.5
1
0.5
°
°
1
01
0.
0.01
0.05
0.05
5
0.0
0.01
0.05
225
0.1
0.0
0.0
0.
55
5
0.0155
00.0.0
135
direction [ ]
180
0.
°
50
0.5
0.5
0.1
0.01
direction [ ]
0.100.0
.15
1
0.00.1
5
90
direction [ ]
original
AANN
1
0.0
270
0.01
0.05
0.5
225
0
0.1
0.1.05
.005
0.01
270
180
0.5
0.5
spectral energy density
1
0.1
0.05
0.1
0.05
1
1
0.01
45
directionally integrated
60
1
0.0
0.1 .05
0
1
0.0
0.5
1
0.
0.0
315
05
360
1
1
0.
spectral energy density
360
0.41
0
0.06
0.10
0.16
frequency [Hz]
0.26
0.41
0
0.06
0.10
0.16
frequency [Hz]
0.26
Fig. A.4. Examples of the performance of the AANN: left panel shows original wave
spectra, panel in the middle shows corresponding AANN parametrisation, and right
panel shows the directionally integrated spectra.
18
0.41
original wave spectrum
original wave spectrum
360
°
5
0.0
0
−1e
−1e
−1−08
1e
0 −08
1e−06
−0.0001
−1e−05
1e−10 1e−06
1e−05
0.0001
001
−0.0 −5e
−1 −0
1e−10
1e−05 1ee−−055
1e−05
0−1e−
6 06
0.001
−1e
10−08
0−1e−
1e−
101e−06
5
1e−0
1e
−1−0
e−80
6
1e−
8
08
0
e− 1e
−1e−10
−06
−10
e−1e−06
−0 0
1e8−1
1e−06
0
6
−1
−0
1e
1e
6
1e−0
−0
1e
1e−10
6
135
1e−08
°
direction [ ]
1e
−10
0
08
e−
−1
06
1e−
1e−0
−180
1e−05
08
−1e−101e−
−0.0005
6
0
1e
1e−−0
10
8
1e
−0
8
1e−10
1e−10
°
direction [ ]
5
1e−0
0
1e
−1
0
0
°
−1e−
−1e−10
−1e−
0806
direction [ ]
8
e−−
10e8
−1
0 −1
1e
−0
−5e−05
5e−06
01
0
−5e−06 .00
05
5e−
0.1
direction [ ]
0.1
5
8
−0
1e
8
1e−0
°
1e−10
−0.0003
direction [ ]
0.00
03
5e−0.0002
−005
6
5e−05
06
10
0.1
0.01
0.0
°
direction [ ]
0.0
5
0.5
°
direction [ ]
°
direction [ ]
5e−
5e−05
5e−06
1e−05
00
0
08
°
068
−−0
11ee
direction [ ]
0.05
−0.001
e8−
1−010
e1−e
.00e05
−−016−
−006
−1
−1e−05
−5e
1e−−05
0.0001
0
0.41
directionally integrated
1e−06
1e−
1e−05
05
1e−05
0.06
0.10
0.16
frequency [Hz]
0.26
x 10
exact
NN
2
2
1
0
−2
0.41
directionally integrated
−3
3
nonlinear transfer rate
nonlinear transfer rate
nonlinear transfer rate
0.05
0.1
0.5
1
0
−1
0
10
0.41
005
0.26
e−
0.10
0.16
frequency [Hz]
1e−05
−1
0.26
−1
0.06
0.0
1
−4
0.41
−0.0001
0
.0
exact
NN
4
−2
1e
−0
1088
10
−1e−08
e0−−0
−11e
1e−
180
45
1
0.10
0.16
frequency [Hz]
3
0
225
8
0
e−
exact
NN
2
0.000
0
−1
x 10
0.26
90
1e
−5
4
1e−05
−1e
06
−1e−
−0e−
10
1e−10
−1
80
1
00
0.0
0
10
directionally integrated
−4
x 10
5
001
0.0
005
0.10
0.16
frequency [Hz]
08
8
0.06
1e−0
−0
1e−
−0
0
1e−06
0
1e−05
1e
0.41
1e
−1−e10
−−1
1e
06e−
−010
08
8
0.0
−1e−10
0.06
1e−
0.26
08
0.10
0.16
frequency [Hz]
1e−06
1e−08
5e−05
1e−05
1e−060
1e−10
1e−08
−1e−10
−1e−08
−1e−06
1e−10
1e−06
05
1e−
1e−06
270
8
0
−0 −1e−
110
1e0−
45
0.06
e−
135
1
−5e− 0.0
05
−−011e
1ee−−−10
−0e0185−e−
080e−
5−1
6 10
5
1e
315
1e−06
0
−1e−06
−1e−08
−1e−10
1e−08
90
0.0001
45
0
135
1e
−1e−05
5e−06−5e−06
1e−05
5e−05
5e−1e
06−05
1e−06
−1e−05
−1e−06
1e−08
−1e−10
1e−10
0−1e−08
1e−06
1e−05
5
1e−10
−1e−06
−1e−10
−1e−08
1e−08 −1e 1e−0
1e−0
0
−05
6 0
−1−e −1
−111−eee0 e−
−−−61001e8180 05
−e0−61
0
0
360
−1e−
1e−
08 0810
−1e−
−1
e−
05
−1e−
−1106
ee−−
100
8
08
005
00
nonlinear interactions from NN
−1
05
−5e−
5
−0
1e
−5e−−06
1e−06
6
5
−0
1e−10
0
0
−1
10e−06
−1e
e−
−1
02
.00
−0
5e
06
−05
8
−0 10
−1e−
1e
−0.0
5
−1e−0
−1e
−1e
−1e
−06
1e−
−08
−10
081e−1
1e−0
60
0
1e−06
1e−
1e−05
5e−06
−5e−06
−1e−05
−5e
180
0
05
0
05
−1e−
1e−05
3
00
0.0 0001
0.
1e−
225
1e−1
1e−05
1e−05
5e−06
05
180
0.41
1e−06
−0
270
e−
6
6
−0
1e
0.26
0
1e
270
−5
−0
nonlinear interactions from NN
315
6
5e5−0
e−0
6
5
−0
1e
e−06
−5−05
−1e
−5e−05
−0.0002
−0.0003
1e−08
08
0.10
0.16
frequency [Hz]
0
05
6
−0
1e
0.06
0.41
.00
05
00 0
0e.60−011
−0.001 00
1
0
8
−−1.0
e−05
−0
−0
1005
05
−08
1e
1e
−.0
−5e−05
0 −0
−1e−
−1e
0.0
−1e−06
−1e−08
1e−10
0
−1e−10
1e−08
1e−06
1e−08
1e−10
−1e−10
01e−10
−1e−08
−1e−10
0 −1e−08
8
225
45
0
315
5e−06
3
06
360
90
1
−1e−
e−
01
1e
1−e1
−00
6
−0
0.41
0.26
1e
90
nonlinear interactions from NN
135
5
05
−1
90
360
5e−0
6 05
1e−
0. 5e−05
00
0.0001
02
0.0
002
5e−06
5e−05
1
1e−08
06
e−
−5
0.26
0.0001
0
0.0
8
0
e−
270
08−1e
1e−0
1e−10
−1e−1
−
0
1e
−1−1
e−
010 0
1e−08
−0
05
−0
135
0.0
0.1 5
1
0.10
0.16
frequency [Hz]
−1e−
1e
0.01
0.5
3
5
1e−0
1e−0
6
−1e
1e−08
1e−10
0
−1e−08
−06 −1e−10
e−
0
8 −1
1e−0−1e−e08
−1
0
0
360
1e−10
−1e−10
−1e−08 0
−1e−08
−1e−10
0
1e−10
1e−08
45
0.10
0.16
frequency [Hz]
0.06
315
−1
8 10 0
e−
01e−1
−1
e−
−1
10
0
180
−06
1e
1e−08
−
1e
1e−06
5
5
1e−10
0
10
6
6 −0
−08 1e
1e1e−0 −
0
−506
−1e−0
−5e
1e−0
5e−06
−5e−06
−1e−05
05
e−
225
1e−06
2
00
5
−5
−5e−05
5
−5e−06
1e−05 −1e−05
−0.0002
−1 e−0
5e−05
e− 5
5e
0.0001
05 1
−0−1−
e−
6 e5−e0
05
−50
5e−05
6
5e−06
1e−0
0
0.41
5 6
10
1e−06
−1
1e−1
−1e−08
1e
6
1e−0
05
1e−
−0
45
180
0.10.05
0.5
1
exact nonlinear interactions
08
0
1e−1
05
0.0
5e−
1e
06
05
065e−
−1−e−
0.0001
1e−
−5e
05
−0.0002−5e−05
0.26
1e−
0
5e−06
1e−05
6
1e−055e−0
−1−0.0
00 5e−0002
1e5
0.0
−0
5e−
5
05
225
5
0.0
0.10
0.16
frequency [Hz]
270
0.06
3
90
0.1
0.06
1e−1
270
0
01
0.
1
exact nonlinear interactions
315
06
0.1
0.0
45
0
0.41
315
5e−
0.5
0.1
0.01
0.01
0.26
360
90
6
90
exact nonlinear interactions
135
0.01
5
5 0.01
45
0.10
0.16
frequency [Hz]
1e−05e−06
5
5e−0
0.0 5
00
1
5
0.0
0.1
5
135
0.01
0.1
360
180
180
0.5
0.1
0.5
5 6
1
0.0
0.1
1
0.1
0.05
0.01
225
0.0
0.5
0.5
0.06
1
0.05
0.01
0.05
1
3
0.0
45
0
135
5
3
0.05
0.0
1
0.1
65
10
1
1
0.
90
1 3
3
00.0.5
5 1
0.0
0.5
1
6
180
1
0.05
0.1
10
0.1
0.5
0.1
56
135
3
0.01
0.1
1
225
0.5
3
0.05
0.01
180
1
5
1
5
0.0
0.0
0.5
225
0.01
0.5
1
0.01
0.1
0.05
10
1
270
0.5
0.01
0.1
0.
1 5
3
6
10
0.05
0.05
225
1
5
0.0
3
270
0.
05
0.
270
315
0.1
5
0.0
315
0.5
315
0.1
0.01
0.05
5
0.01
6
original wave spectrum
360
6
360
1
0
−1
−2
0.06
0.10
0.16
frequency [Hz]
0.26
0.41
−3
0.06
0.10
0.16
frequency [Hz]
0.26
Fig. A.5. Examples of the performance of the NN for emulating the WRT method:
upper row shows original wave spectra, next row the corresponding exact snl–term
and third row shows its emulation by the NN. In the plots of the snl–term the
black areas correspond to most negative values and the white areas to the biggest
positive values. The gray areas inbetween cover ranges of four orders of magnitude
for the snl–terms for either sign. The lower row shows the directionally integrated
snl–terms.
19
0.41
AANN parametrisation
360
0.1
5
0.1
0.0
0.0
5
0.1
0.0
1
1
00.5.5
0.0
1
5
0.0
90
4
3
2
0.06
6
−0
5e
−1
nonlinear transfer rate
1e−08
1e−
07
0.06
5
−0
1e
1e−06
0.10
0.16
frequency [Hz]
0.41
x 10
0.26
0.41
0
−0.5
−1
−1.5
6
−0
0
−1e−05
−5e−06
−1e−06
−1e−07
0
1e−08
e−−071
1
e−10
1e
0.26
0.5
1e−06
6
5e−0
e−
1e−06
−10e6 −1e−06
1e−07
1e−08
−1e−07
−05
5e
0.41
1
0.10
0.16
frequency [Hz]
directionally integrated
−4
−5
1e−06
1e−10
0.26
07
1e−
45
1e−06
6
1e−0
1e−08
0
−1e−07
−1e−08
8
−0
6
1e−07 1e−10
1e−0
6
1e−0
6
5e−
06
−1e−10
1e−−
10e6−
08−1
−5e−05
e−0
7
−0
7
1e−07
1e
0
−1
1e
°
direction [ ]
135
1e−1
1e
06
−1e−−06
0.1
°
direction [ ]
0.5
0.1
0.1
00.0
.05
5
5e
−0
5 −05
1e−08
−1e−10
−5e−0
−1e
6
1e−05
1e
−0
6
6
−0
−
−1e 5e−
−06 06
1e−10
1e−06
5e
−5e−05
−0
6
1e
08
07
1e−
1e−
−1
0
1e
180
6
−0
1e
1e
−
−−00081780
−11ee
−1e−10
1 1
0.
05
0.5
0.
1
0.0
1
°
direction [ ]
°
5
0
−1e−07
07
−1e−08
−1e−1
1e−10
e−
0
−1
−06 1e−0
−1e
8
1e−07
08
e−
−1
06e−07 1e−06
8−−1
−0−e50701−6e0
10
−1−e5e−−
6
1e
−0
5e
225
08
e8−
−0
−10
1e
1e−1
7
−0
1e
direction [ ]
0.41
08
10
1e−10
−1e−
−1e−
8
1e−0
1e−0
0.10
0.16
frequency [Hz]
0.1
270
−10
−11ee−10
07
−1
e
1e−10
−1e−10
1e−08
1e−07
1e−07
1e−08
−1e−10
1e−10
−1e−08
−1e−07
7
6
0.26
5
−0
e
−1
07
e−
−1e
−1−08
315
0
0−71
1−e
1−e
5
−0
−0
1e
1e−0
0.06
360
6
−0
1e
0
1e−08
10
1e−
45
0
1e−1
08 −10
08
e−−11e−
e
90
−1
7
06 e−−008
5e−70−81−0−11e
0−e
1−−e1
1e
−5e−06 −1e−
6
nonlinear interactions from NN
−1
−5e−
e− 06
1e−07
1e−08
1e−10
05
0
135
0.10
0.16
frequency [Hz]
1e
0
7
1
0.01
0.05
0
1e−1
180
1e−1
06
e−
08 1e−07
1e−05
−07
−106
1e−
−1e−
5e−06
−1e
7
0.1
5e−06
6
−0
0.50.5
0.06
1e−1
−0
06
1e
5
05
0. 0.1
0
0.41
0
1e−1
7
0−0608
ee1−−e
81−011−
−e0−
1−e1
5e
5e−
08
0.0
06
0078
1−10
1e−e−
−1−1e
1e−
0.01
0.26
−1
e−
−1
0 −1e−10 −10 1e−08
e−
1e
1e−07
1e708
−1e−10
1e−07 1e−08
−1−07
e−
06
−1e−07
−1e−08
1e−06
5e−06
1e−05
−5
e−
5e−06
5e−0
1e−07
1e−10
06
5
−1e−05
5
−0
1e
1
0. 0.05
0.05
0.01
8
−0
1e
7
−0
1e
e−
270
225
−5
e−
06
1e
1e−−0
087
1
1
1e−07
1e−08
−−11−0
−1e5e
−06
ee−−60
1708
−1
−
1e
.5
00.5
0.0
exact nonlinear interactions
10
315
0.1
135
90
0.10
0.16
frequency [Hz]
360
1
0.5
0.5
0.01 0.05
0.1 0.05
0.1
0.5
0.06
0.05
0.01
45
0.05
0.05
.1
0.5
00.1
11
5
0.0 0.1
0.0
0
180
0.01
1
0.0
0.01
5
0.1
0.1
05
0.05 0.05
0.105
1 .
0. 0
1
00.0
.055
45
1
0.0
225
original
AANN
8
0.
0.1
1
90
0.5
135
0.0
0.1
0.5
1
1
0.5
0.5
0.0
0.5
5
0.5
270
.055
00.0
0.1
0.01
180
3
1
0.05
0.05
0.01
225
1
0.
01.0
50.1
0.0
0.1
270
315
0.05.5
0.1
05
0.
3
0.1
directionally integrated
9
0.1
5
1
0.
0.55
5
0.1
.5
00.5
315
10
6
.5
1 .5
00 0.1
55
00.0.0
1
0.0
1
spectral energy density
original wave spectrum
360
exact
NN
1e−06
1e−0
8
0.26
0.41
−2
0.06
0.10
0.16
frequency [Hz]
Fig. A.6. The upper panel shows the orinal wave spectra and the output from the
AANN for a case where the error of the AANN is large. The lower panel shows the
corresponding exact and NN emulated snl–term (contours as in figure A.5).
20