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iahr
Systematic investigation of data analysis methods in wave-ice interaction
problem
Hongtao Li1,2, Ersegun Deniz Gedikli1,3, Raed Lubbad1
1 Department of Civil and Environmental Engineering, Norwegian University of Science and
Technology, Trondheim, Norway
Høgskoleringen 7A, 7491 Trondheim, Norway
hongtao.li@ntnu.no, honglia@space.dtu.dk
2 DTU Space, Technical University of Denmark, Lyngby, Denmark
3 Department of Ocean and Resources Engineering, University of Hawaii at Manoa,
Honolulu, USA
Wave climate and extent of ice cover in the Arctic ocean have changed significantly in recent
decades. Concomitantly, interest in studying wave-ice interaction has rapidly increased in the
last two decades. Laboratory-scale experiments present a fundamental approach to
understand the detailed processes involved in wave-ice interaction and to validate theoretical
models. One key parameter that describes how waves evolve under ice cover is the spatial
attenuation coefficient. To estimate this parameter from laboratory measurements, wave
amplitudes at different positions along the wave propagation direction should be estimated. In
addition to wave amplitudes, it is also critical to confirm that the period of incoming waves
agree with a preset wave period since this parameter is involved in evaluating wave
dispersion relation. In this study, we systematically explore various measurement (signal)
processing techniques to extract wave parameters such as wave amplitudes and periods. The
contamination of wave measurements with noise and wave reflection in addition to the
sometimes use of low-sampling frequency and the possible lack of stationarity in the
measurement require dedicated care when selecting the suitable analysis method. In this
study, we investigate several methods including: Bandpass filtering in frequency domain
(bp), Tikhonov regularization (Tikhonov), two MATLAB built-in functions, fir1 and smooth
(Smooth), Fast Fourier Transform (FFT), Peak Analysis (Peak), Dynamic Mode
Decomposition (DMD), Genetic Algorithm (Genetic) and Prony’s method (Prony). By using
experimental measurements (from Qualisys systems, ultrasound and pressure sensors) and
synthetic signals, we show that bp gives equivalent filtering results as fir1 (difference
expressed by normalized root mean squared error less than 3.5%). Moreover, Tikhonov and
Smooth produces similar smoothing of signal. Additionally, we find that Prony, DMD and
FFT in combination with interpolation quantify incoming wave period accurately
(normalized error less than 2.5%). As for wave amplitudes, FFT combined with interpolation,
Genetic, a variant of DMD, Hilbert transform and its combination with interpolation and
Peak combined with interpolation results in similar estimates with respect to FFT
(normalized error less than 3%).
KEY WORDS: Surface gravity waves, ice, Prony, Dynamic mode decomposition, Hilbert
transform, Genetic algorithm, Tikhonov regularization.
25
th
IAHR International Symposium on Ice
Trondheim, 23 - 25 November 2020
1. Introduction
The interest in wave-ice interaction studies has resurged in recent decades. The driving
factors are multiple. One of them is the increasing concern of potential implications brought
by accelerating melting of ice in waves in both Arctic and Antarctic. To fully comprehend
how waves are changed by ice cover, more high-quality field datasets are in urgent need
(Rabault et al., 2017, Kohout et al., 2015). This is attributed to: (1) diverse proposed
theoretical models for waves propagating in ice need to be calibrated and validated by using
field measurements (Rabault et al., 2017, Mosig et al., 2015, Cheng et al., 2017); (2) various
damping patterns of waves in ice should be investigated (Rabault et al., 2017); (3) dominant
mechanisms in attenuating waves travelling in ice remain to be clarified (Voermans et al.,
2019, Ardhuin et al., 2020). Especially, these information should be systematically and
meticulously measured, i.e., wave elevation, wave travelling direction, ice concentration, ice
types, ice extent, ice thickness, wave speed and wind (Cheng et al., 2017, Kohout et al.,
2015). However, well-controlled laboratory studies are also indispensable, because of its
advantages over the field measurements. Firstly, laboratory studies can be designed to fulfill
the assumptions introduced in those theoretical models. Secondly, ambient conditions could
be comprehensively measured and controlled. Despite these superiorities, proper data
analysis methods are essential in interpreting the laboratory measurements to reveal the
detailed physical processes involved in wave-ice interaction. Hitherto, there have not been
any dedicated systematic investigation of data analysis methods used in wave-ice interaction
community for laboratory study.
Noise is inevitable in real measurements. Smoothing and filtering the recorded data are
necessary pre-processing steps for reliable analysis of data. Physical meaningful signal of
wave and ice motion is typically narrowband. This implies that a narrowband pass filter
should be designed to filter raw measurements. However, narrowband filter design using the
preferred finite impulse response (FIR) approach requires long kernel and hence reduces
temporal precision (Cohen, 2014). In some cases, measurements from small wave flumes
have short time series. Consequently, high-quality narrowband filter designed via FIR is
virtually impossible to be used for short signal. To avoid this kind of caveat of FIR filter, we
propose to use fast Fourier transform (FFT, Cooley and Tukey (1965)) in conjunction with
inverse fast Fourier transform (IFFT) to perform filtering in frequency domain. Despite the
sharp edge of this filtering method in frequency domain gives rise to ripple/ringing effect in
temporal domain (Cohen, 2014), we show that the ripple/ringing effect seems to be minimal
in real application.
Move-mean filter and MATLAB function smooth with option lowess were employed in
previous studies (Zhao and Shen, 2015, Meylan et al., 2015, Yiew et al., 2016). However,
these two methods are severely prone to subjective bias, because span of data (or number of
data points) used by move-mean filter and smooth must be chosen by trial-and-error
procedure. In this study, we introduce the Tikhonov regularization method (Mueller and
Siltanen, 2012) and propose for the first time to use it to smooth data on wave-ice
interactions. Regularization parameter involved in this method can be determined by
generalized cross-validation and L-curve criteria (Aster et al., 2018).
Though Wang and Shen (2010) compared four different estimation methods for wave
frequency, in their study, only pressure sensor measurements from one experimental facility
were analyzed. In contrast, we apply the methods presented in this study on measurements
collected from various devices (i.e., Qualisys systems, ultrasound and pressure sensors) from
four different experimental facilities. Moreover, we introduce for the first time Prony’s
method and dynamic mode decomposition (DMD) to estimate frequency in wave-ice related
studies.
Peak analysis (Wang and Shen, 2010, Sree et al., 2018, Yiew et al., 2019, Li and Lubbad,
2018, Meylan et al., 2015) and the methods based on Fourier transform (Bennetts and
Williams, 2015, Zhao and Shen, 2015, Rabault et al., 2019, Sutherland et al., 2017) have
been used to estimate wave amplitude. To the best knowledge of the authors’, a comparative
study of these two methods have not been performed. In the present study, we evaluate the
difference of amplitude estimates derived from various methods including peak analysis and
those methods based on Fourier transform. Herein, we introduce for the first time a variant of
DMD, and Hilbert transform to wave-ice research community to estimate wave amplitude.
Furthermore, we propose to apply truncated singular value decomposition (TSVD) to
judiciously determine the number of modes to be retained in DMD.
In the following sections, we will start with the description of data used in this study. This is
followed by methods to pre-process data. Methods to estimate incoming wave
period/frequency and wave amplitude are then presented. Thereafter, results of applying
various methods to analyze data are given. The final section summarizes our findings.
2. Description of Data
Table 1 lists the data analyzed in this study.
w
f
is target wave frequency,
w
ka
is wave
steepness and
s
F
is sampling frequency. Cases #1 - # 19 are experimental measurements
collected during four different experimental campaigns that were conducted in four wave/ice
tanks.
Case #20 is a synthetic signal that imitates recorded laboratory measurements. Equation [1]
shows the synthetic signal:
()
10
exp( )sin 2 /
j j jj
j
aT
α πϕ
= − ++
∑
zt ε
[1]
where
1
a
= 18, and
2 3 10
, , , aa a
are sampled from uniform distribution [0, 0.5].
t
represents a time vector.
j
α
is the grow/decay rate and sampled from uniform distribution
[ ]
0, 0.001
. The sign of
j
α
is determined by sample from uniform distribution
[0, 1]
. When
sample exceeds 0.5, a positive sign is assigned for
j
α
, otherwise a negative sign is imposed
on
j
α
.
[4, 1, 1.1, 1.2, 1.5, 1.6, 1.8, 2, 2.2, 8]
j
T=
in seconds, where
14T=
[s] represents the
preset period of the generated waves.
j
ϕ
is sampled from uniform distribution
[ ]
0, 0.02
π
.
ε
denotes 1% white noise.
Table 1. Data description
Case #
w
f
[ ]
Hz
w
ka
3
10
−
s
F
[ ]
Hz
Device Label Remark Source
1-3 0.6 32.58 64 Qualisys Marker 5 disk Montiel
et al.
4-6 0.8 32.99 64 Qualisys Marker 5
thickness
D = 3 mm
(cases #1, 4,
7, 10), 5
mm (cases
#2, 5, 8, 11)
and 10 mm
(cases #3, 6,
9, 12).
(2013)
7-9 1.1 32.87 64 Qualisys Marker 5
10-12 1.3 31.42 64 Qualisys Marker 5
13-15 1.5 62.83 50 Ultrasound
sensors
US4.25,
US4.75,
US5.25
Yiew et
al.
(2019)
16 1.25 91.06 200 Qualisys Disk 2
Yiew et
al.
(2016)
17-18 0.5 12.51 50 Pressure
sensors P7 and P8 test series
3310, see
also Figure
1 in Li et al.
(2019)
Haase
and
Tsarau
(2019)
19 0.5 12.51 100 Qualisys Marker 8
20 0.25 - 200 - - -
3. Data Filtering and Smoothing Methods
Bandpass filtering (bp) implemented by using FFT and IFFT
Bandpass filter can be realized by using FIR filter and infinite impulse response (IIR) filter
(Proakis, 2001). For motion measurements of ice and waves collected in laboratory study, we
recommend another way to implement bandpass filtering, which is easier to use. This
filtering approach is based on Shmuel (2020) and is denoted as bp onwards.
FFT outputs Fourier coefficients that correspond to following frequency consecutively
(Trefethen, 2000, Kutz, 2013).
[0,1, 2, , , - , - ( -1), , -1]
s
F
ff j k k
l
=
[2]
where
21
22
ll
jl
= +− −
,
2
l
k
=
and
l
is the length of selected data (see below).
The steps to achieve bandpass filtering (bp) of measured time series
1n×
∈
z
using FFT and
IFFT are as follows:
(1) Select the cyclic part of
z
(denoted as
l
z
with a size of
1l×
), that is far before the
beginning and well after the ending of steady part of
z
(denoted as
s
z
with a size of
1s×
), with periodic boundary having zero values.
(2) Conduct FFT on data
.
l
z
(3) Select out Fourier coefficients, which are output from FFT, in correspondence with
desired frequency to be retained.
(4) Populate a zero-matrix that has a size of
1l×
with chosen Fourier coefficients having
the same index as output from FFT.
(5) Perform IFFT on the populated matrix and take out the real values.
The requirements for the chosen section of data when using FFT and IFFT results from that
FFT and IFFT assume periodic boundary condition (Kutz, 2013) and use cyclic Fourier series
to do transformation between time and frequency domain. Imposing the condition
ls>
subsides the edge artifact on filtering results of
s
z
(Kohout et al., 2015). The edge
artifact is induced by the possibility that zero periodic boundary condition is not satisfied
(Cohen, 2014) due to noise contamination and low resolution of data. To alleviate the edge
artifact further, tapered cosine (i.e., Tukey) window could be applied at the first step just
mentioned above (MathWorks, 2020e, Kohout et al., 2015, Tucker and Pitt, 2001).
Tikhonov regularization
Tikhonov regularization is one of the possible methods to denoise signal. For reducing noise,
the model used for Tikhonov regularization is:
= +zfε
[3]
where
1
n×
∈z
is the measured time series,
1
n×
∈
f
is the signal that is free from noise and is
to be inversely determined,
1n×
∈ε
is noise. According to Mueller and Siltanen (2012), this
can be formulated as an optimization problem:
() ( )
1
22
2 22
|| || || ||
arg min
n
α
α
×
∈
= −+
b
f Ab m L b
[4]
where
( 2)nn−×
=AI
is an identify matrix;
α
is a regularization parameter that makes a trade-
off between the loss term
( )
2
2
|| ||−Ab m
and the penalty term
( )
2
22
|| ||Lb
;
m
is a vector,
populated with the first
( 2)n−
elements of
z
;
2
L
is a regularization matrix, herein a second
derivative operator based on finite difference methods, and expressed as:
2
2
( 2)
1 21
1 21
1 21
s
nn
F
−×
−
−
=
−
L
[5]
where
s
F
is sampling frequency in Hz. Here, this form of
2
L
implies that no any assumption
about the boundary of signal is introduced. To handle various kinds of boundaries of signal,
e.g., reflexive boundary, other forms of second derivative operator
2
L
are proposed in
previous studies, see more details in chapter 5 in Mueller and Siltanen (2012) and chapter 8
in Hansen (2010). Mathematically, this optimization problem can be solved by using normal
equation (Mueller and Siltanen, 2012):
( )
22
TT T
α
α
+=AA LL f Am
[6]
For computational convenience, the stacked form formulation is used (Mueller and Siltanen,
2012):
2
α
α
=
Am
f0
L
[7]
For large scale problem, Equation [7] is solved by matrix-free method (Mueller and Siltanen,
2012). We propose to use LSQR algorithm (Paige and Saunders, 1982), which is efficiently
implemented to avoid memory size limitations of computer for storing large-dimension
matrices.
In this paper, the regularization parameter
α
is estimated by the robust L-curve method
(Hansen, 2010, Mueller and Siltanen, 2012). As discussed in James et al. (2013), larger
α
results in smoother
α
f
and if
α
→∞
,
α
f
is a straight line. Conversely,
0
α
=
results in that
α
f
converges to
z
.
Note that here we assume
f
is smooth, which is typical the case for the motion signals of ice
and waves recorded in laboratory, therefore second derivative operator is used as the
regularization matrix. Other commonly used regularization matrices include identity matrix
and first derivative operator (Chen and Chan, 2017, Gedikli et al., 2018, Gedikli et al., 2020).
The choice of the regularization matrix is data dependent. It should also be noted that the
implementation of using Tikhonov regularization to denoise signal presented here is novel
and has not been seen in other literatures.
Another approach to denoise the signal is the use of newly developed time-delay-based
differentiation (TDD, Li et al. (2020a)) method which reduces the noise embedded in the
signal in the derivation process.
Apply MATLAB built-in functions to directly filter and smooth data
It is a well-established knowledge that if data are sufficiently long and off-line analysis is
performed, FIR filter is preferable to IIR filter. This is attributed to (1) performance of IIR
filter at best matches with that of FIR filter (Cohen, 2020); (2) FIR filter is stable while IIR
filter maybe not (Cohen, 2014, MathWorks, 2020c); (3) FIR filter results in constant phase
delay for all frequency components whereas IIR filter yields nonlinear phase delay
(MathWorks, 2020b, MathWorks, 2020a); (4) The approaches to construct FIR filter are
primarily linear (MathWorks, 2020c).
MATLAB provides plentiful functions to design filter and smooth data. We only select
commonly used fir1 and smooth functions to compare with the other methods presented in
this study. fir1 is comparable to bp method since transition zones of both methods are tight
(Cohen, 2014). Note that smoothing window is applied to smooth filter kernel in fir1 function
to mitigate ripple generation in time domain (Cohen, 2014).
The MATLAB smooth function with option lowess performs weighted least-squares local
regression of data using first degree polynomial. This approach has been used by earlier
studies to reduce noise (Meylan et al., 2015, Yiew et al., 2016). Here, we recommend using
option rloess which smooths data via local regression of data using second degree
polynomial. This option incorporates an outlier-rejecting mechanism and is used by Li et al.
(2020b) to smooth wave-induced ice-ice interaction signals.
4. Methods to Estimate Wave Period and Wave Amplitude
Prony’s method
Prony’s method is a method to decompose data, which is analogous to Fourier series. The
difference lies in the fact that Prony’s method decomposes data into a complex exponential
series that includes grow/decay rate (see Hu et al., 2013 and references therein). Regarding
formula for discrete data, Prony’s method can be expressed as:
1
exp( ) exp
j
k jj
js
k
z ai F
β
λ
θ
=
=
∑
[8]
where
k
represents
th
k
element of the chosen steady signal
s
z
, the
j
a
and
j
θ
are amplitude
and phase for the
th
j
component, respectively;
2
jj j
if
λα π
=−+
, where
j
α
(1/sec) and
j
f
(Hz) are the grow/decay rate and oscillation frequency for
th
j
component, respectively.
In this study, we employ the implementation of Prony’ method as in Hu et al. (2013), which is
shown by Hu et al. (2013) to be insensitive to sampling frequency and robust against noise.
For more extensive discussions about various implementations, readers are referred to
Rodríguez et al. (2018).
Number of modes
β
is determined by TSVD (Mueller and Siltanen, 2012) method.
Specifically,
β
is chosen as the maximum subspace dimension just before flat noise floor in
singular value spectrum expressed in logarithmic scale (see e.g. Figure 4b).
Dynamic mode decomposition (DMD)
DMD (Schmid, 2010) is another method of decomposing data into complex modes. Here, we
utilize the algorithm presented in Tu et al. (2014) and Kutz et al. (2016). The approach of
estimating average amplitude by means of DMD is developed by Kutz et al. (2016) and this
variant of DMD is referred to as DMDa henceforth. To the best knowledge of the authors’,
this is the first time that DMDa is employed in wave-ice interaction studies.
It should be noted that, both Prony and DMD methods can be related to system identification
techniques, i.e., eigen realization algorithm (Juang and Pappa, 1985), in terms of the way to
identify grow/decay rate and oscillation frequency (Hu et al., 2013, Kutz et al., 2016).
For DMD, we employ TSVD and the optimal hard thresholding method proposed by Gavish
and Donoho (2014) to determine
β
, respectively. To distinguish these two methods in
combination with DMD, the former is denoted as DMD and latter is represented as DMDr
hereafter.
Genetic algorithm to fit a sinusoidal wave
The fitting to a sinusoidal wave can be formulated as a search for a set of parameters to
minimize the objective function (Li et al., 2019):
2
() () ()
1
cos(2 )
s
j jj
j kk k
k
z a ft
ε πϕ
=
=−+
∑
[9]
where
j
represent the
th
j
group of parameters to fit to data
s
z
and
k
denotes the
th
k
element in
s
z
. Genetic algorithm, implemented in MATLAB as function ga, is applied to solve this
optimization problem.
Interpolation in combination with peak analysis and FFT
Inspired by the work of Sree et al. (2017) and Sree et al. (2018), we propose to interpolate
low-frequency sampled data by using makima method in MATLAB. In comparison with
pchip method (applied by Sree et al. (2017) and Sree et al. (2018)) and spline method,
makima produces results that follow measured data trend much better (MathWorks, 2020d).
After high resolution data are obtained by using the makima method, peak analysis and FFT
are employed to estimate amplitude and preset wave period. These two methods are referred
to as IntPeak and IntFFT onwards. Likewise, Int precedes all methods that involve
interpolation henceforth.
Hilbert transform
Hilbert transform can be applied to compute the analytic signal of measurements, whereby
instantaneous amplitude can be further determined as the magnitude of the analytic signal.
The pertinent formulae take the form (Bruns, 2004):
{ }
()Sf z
=
[10]
() ()h f iS f=
[11]
() () sgn()()
H
S f Sf i fhf= −
[12]
1
{ ( )}
HH
z Sf
−
=
[13]
Where
and
1−
represent Fourier transform and inverse Fourier transform, respectively;
f
is frequency,
z
is vertical oscillation signal and
H
z
is the analytical signal of
z
. In
implementation, FFT and IFFT are employed to perform Fourier transform and inverse
Fourier transform. In addition,
s
z
substitutes for
z
. We propose to compute amplitude of
s
z
by taking the mean of the instantaneous magnitude of
H
z
(see e.g. Figure 5), which has the
same length as
s
z
. As mentioned earlier, because FFT and IFFT are involved in taking the
Hilbert transform of
s
z
, periodic boundary of
s
z
should be ensured to ameliorate edge
artifact.
5. Results
Filtering and smoothing results
To demonstrate the efficiency of each method to remove noise, we show the data/signal and
filtered and smoothed signals in frequency domain (Figure 1). It can be seen that bp is most
effective followed by fir1. More importantly, fir1 amplifies low-frequency noise for some
cases (see Figure 1d and Figure 1f). Tikhonov regularization and Smooth are more applicable
to remove high frequency noise. Both Tikhonov regularization and Smooth yield similar
smoothing results.
Note that here we preserve only the frequency components close to the incoming wave
frequency (the most dominant peaks), similar to Sutherland et al. (2017), Rabault et al.
(2019) and Nelli et al. (2017). The other frequency components are regarded as noise and
should be eliminated. Actually, the amplitudes of frequency components in the regions where
the smoothing and filtering methods modify the tails (Figure 1), are at least 5 times smaller
than the predominant peaks (incoming wave frequency). This suggests that those frequency
components with amplitudes changed by smoothing and filtering can be treated as noise.
Figure 1 also demonstrates that both the smoothing and filtering methods effectively
attenuate the high frequency components (noise) when comparing with original signals
(represented by blue lines).
Figure 1. Data and corresponding filtered and smoothed results for different cases. (a) – (d)
for cases #1, 4, 7 and 10. (e) – (l) for cases #13 – 20. Blue line represents original data. Black
and green dash-dot lines denote filtered results by means of bp and fir1 respectively. Red and
orange dashed line represent smoothed results by using Tikhonov regularization and Smooth,
respectively. NOTE: black dash-dot floor around -400 dB in (a) represents numerical
representation of zeros, and those floors in (b)-(l) are not shown.
A point-wise comparison is made for filtering results from bp and fir1 (Figure 2).
Normalized-root-mean-squared errors (NRMSEs) for most cases are below 1.5% except for
case #14, in which fir1 enlarges low-frequency noise (see Figure 1f). In addition, Figure 1
suggests that much of discrepancy between filtered results by bp and fir1 originates from the
remainder after filtering low-frequency noise. To be more specific, the remainder after
filtering high-frequency noise is much lower than that for low-frequency noise.
Considering the facts that bp and fir1 yield similar results and fir1 is devised to alleviate
undesired ripple/ringing caused by sharp edge in narrowband filter design, ripple/ringing
effect is minimal when using bp as well.
Figure 2. Comparison between filtered results by applying bp and fir1. Blue line with red
dots is used to show clearly the NRMSE values.
Figure 3 compares the original signal, filtered and smoothed signals for case #18 using
different filtering (bp, fir1) and smoothing methods (Tikhonov regularization, Smooth). It is
evident that these different filtering and smoothing methods eliminate high frequency
components. This is also apparent in frequency domain (Figure 1j), when comparing the
original signals (blue lines) with the smoothed and filtered signals (other colorful lines).
Figure 3. Case #18 and corresponding filtered and smoothed signals. Red filled dots in (a)
highlight the zero periodic boundary selected for
l
z
. Segment between two vertical dashed
lines in (a) represent steady signal with zero periodic boundary chosen, i.e.
s
z
.
Take case #18 as an example again, Figure 4a - Figure 4b illustrate using L-curve method to
choose regularization parameter
α
for Tikhonov regularization and applying TSVD to
choose number of components
β
to be retained for DMD, DMDa and Prony’s method. The
determined regularization parameter
5
6.40 10
α
−
= ×
. Number of components
2
β
=
since there
is a significant gap between the second and third subspace dimension as exhibited in Figure
4b.
Figure 4. Methods to determine parameters for Tikhonov regularization, DMD, DMDa and
Prony’s method used for case #18. (a) regularization parameter determined by L-curve
method for Tikhonov regularization. (b) number of modes
β
to be used (i.e. TSVD) for
DMD, DMDa and Prony’s method. Red dot in (a) denotes results corresponding to selected
regularization parameter. Inset in (b) highlights that noise floor starts from 3rd subspace
dimension.
Identifying the wave period and oscillation amplitude
Considering that bp and fir1 produce similar results, we perform analysis only on the filtered
s
z
, which is obtained by applying bp on
l
z
. Figure 5 serves as an example to illustrate the
analytic signal obtained by Hilbert transform. As can be seen in Figure 5a, real part of
analytic signal matches with the filtered signal. Figure 5b demonstrates the variation of the
magnitude which is not noticeable in Figure 5a.
Figure 5. Hilbert transform of case #18. (a) compares the filtered signal obtained by bp and
real part of the analytic signal. (b) shows the magnitude of the analytic signal.
Normalized error (NE) of estimated incident wave period by various methods compared with
target wave period is displayed in Figure 6. As shown, all methods except for IntPeak give
accurate estimates of incoming wave period (NE less than 2.5%). Prony’s method, IntFFT
and DMD are preferred methods because results produced by these methods are less
scattered.
Figure 6. Normalized error (NE) of identified incoming wave period in comparison with
target wave period by using various methods. Red plus sign denotes outliers. Inset shows
more clearly the spread of NEs for all methods except for IntPeak.
Regarding amplitudes estimated by various methods, all methods except of Prony, DMD and
DMDr yield similar results as FFT (NEs less than 3%). The significant deviation between
results from Prony, DMDr, DMD and the other results can be explained as follows: (1) the
amplitudes estimated by Prony, DMD and DMDr depend on initial data point due to
grow/decay rate involved in these three methods; (2) other methods yield average amplitudes.
Figure 7. Normalized error (NE) of amplitudes identified in comparison with those obtained
by FFT. Amplitudes here are for components having incoming wave frequency as oscillation
frequency. Red plus sign denotes outliers. Inset displays more clearly the scatter of NEs for
all methods except for Prony’s method, DMDr and DMD.
The same analysis is also conducted on the smoothed
s
z
obtained by using Tikhonov
regularization on
l
z
. Similar spread of NEs for wave period and amplitude is observed. NEs
for wave period are less than 4.5% for the various methods except for IntPeak, which has the
largest NE around 11%. With regards to amplitude, the NEs are within 3% compared with
results from FFT for those methods except for Prony’s method, DMD and DMDr.
To validate the estimation procedures for wave period and oscillation amplitude, we
reconstruct the signals based on the extracted largest oscillation amplitude and corresponding
wave period. Figure 8 (case #18) illustrates that FFT, Genetic, Prony and DMD reliably
reconstruct the filtered signal obtained by bp. The discrepancy quantified as NRMSE is
within 1.7% with respect to the filtered signal.
The same validation steps are repeated for smoothed signal of case #18 obtained by Tikhonov
regularization. The deviation of reconstructed signal by using the same aforementioned
methods relative to the smoothed signal is negligible (NRMSEs less than 2.2%).
Figure 8. Filtered signals obtained by bp and reconstructed signals. The reconstructed signals
are constructed by only utilizing component corresponding with target wave frequency.
6. Conclusion
In this paper, we present a systematic comparative study of methods involved in pre-
processing data and analysis of data in wave-ice interaction studies.
In terms of pre-processing, we find that filtering by using FFT along with IFFT give
equivalent filtering results as MATLAB function fir1. Tikhonov regularization produces
similar results as MATLAB smooth function with option rloess. Tikhonov regularization
denoising is superior to smooth owing to the existence of systematic approach to determine
the regularization parameter involved in the former method.
Comparison of various wave period estimators shows that FFT combined with interpolation,
Prony’s method, DMD and DMDr reliably estimate incoming wave period with small
variance. Lastly, we demonstrate that six methods (IntPeak, DMDa, Genetic, IntHilbert,
Hilbert, IntFFT) give quantitively similar results with those obtained by using FFT
(normalized error less than 3%).
Acknowledgments
The authors wish to acknowledge the support from the Research Council of Norway through
the Centre for Research-Based Innovation SAMCoT and the support from all SAMCoT
partners. The constructive discussions with Per Christian Hansen, and Yiqiu Dong at DTU
Compute, and Zhengshun Cheng from Shanghai Jiao Tong University are greatly
appreciated. The authors would like to extend great thanks to Lucas Yiew and Fabien Montiel
for sharing the data used in this study. Lastly, the leading author feels very grateful towards
the various supports provided by Jens Olaf Pepke Pederson during the author’s research
staying at DTU Space, where part of this work was conducted.
The MATLAB scripts employed in this study are planned to be provided online in near
future.
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