Partitionnement de spectres et statistiques sur
l’acuité () des systèmes de vagues observés sur
le site d’expérimentation EMR SEM-REV
J-Baptiste SAULNIER
Ecole Centrale de Nantes, LHEEA (France)
Ile de Berder – 05/07/2013
(Comm. OMAE2013-11470)
Introduction
• Marine Renewable Energy needs fine characterisation of environmental
parameters, and sea state ones in particular (design, survivability,
commissioning/decommissioning…)
 Wave spectra from in situ measurements (wave buoys, ADCPs…)
• Statistics of Hs, Tp… and spectral peakedness (bandwidth/narrowness)
required in particular for simulating extreme sea states (fatigue and
survivability) using e.g. JONSWAP spectra  effect of wave groups
• A sea state is the combination of several independent wave systems
(swell(s) and wind-sea)
 Sea state partitioning for considering wave systems individually
and the peakedness characterising each system
-I-
WAVE SYSTEM IDENTIFICATION
AND MODELLING
Goal
Complex
sea state
Individual
components ‘i’
Simple
(swells, wind-sea)
methodology
Hm0
i,
Hm0,i
Tp, T02…
Tp,i, T02,i…
θp, θm…
θp,i, θm,i…
γ (shape)...???
γi
…
…
 More relevant physically
 Not relevant if more
than 1 peak in the spectrum
(simulations, design…)
STEP 1
Partitioning of the discrete spectral matrix Ŝ(fi,θj)
(source: dir. wave buoy, ADCP, array of sensors… or numerical models)
 Simplified watershed technique [Hanson et Phillips, 2001]
SWELL
Ŝ(f,θ)
WIND-SEA
Bimodal directional spectrum
estimated from buoy
measurements (with smoothing)
Watershed partitioning algorithm
= path of steepest ascent technique
(e.g. Hanson & Phillips, 2001)
STEP 2
fs = separation frequency
Partitions grouping:
 Partitions with fp > fs
 Partitions with fp <= fs & Hm0 >= Hmin
 Partitions with fp <= fs & Hm0 < Hmin
 PARTITION 1 = WIND-SEA
 PARTITION = SWELL j
 GROUPED WITH SWELL
WITH CLOSEST fp
STEP 3
 P partitions identified (1 wind-sea + (P-1) swells)
Fitting of analytical shapes (least-squares minimisation)
 JONSWAP for Sj (f) (∫partition_j(f,θ) dθ) [Hasselmann et al., 1973]
 Cos^2s for Dj (θ) (∫partition_j(f,θ) df) [e.g. Mitsuyasu et al., 1975]
JONSWAP
Set of parameters
for each identified
wave system
Cos^2s
=1
 = 3.3
=7
10
Frequency fitting
shapes…
8
JONSWAP spectra
(gamma = 1, 3.3, 7)
6
4
2
0
0.05
0.1
0.15
0.2
0.25
Frequency[Hz]
Cos^2s function
(s = 2, 10, 50)
… Directional fitting
shapes
(not crucial here)
Dir. spreading function[1/rad]
Wave spectral density[m2/Hz]
12
0.3
2
s=2
s = 10
s = 50
1.5
1
0.5
0
0
45
90
135 180 225 270 315 360
Direction[°]
STEP 4
Correction of mutual influences [Kerbiriou et al., 2007]
 Correction of Hm0,j so as to minimise the area difference
of the total reconstructed density S(f) with target Ŝ(f)
SEM-REV - 05-Jan-2011 07:00:00
1.6
Original
Not corrected fit
1.4
Corrected fit
 S1  f1  S2  f1   Sp  f1  








S
f
S
f
S
f
 1 2
2
2
p
2 
Σ



 


 S  f  S  f   S  f 
2
n
p
n 
 1 n
2
/Hz]
1.2
1
Spectral density[m
Σ  C  Sˆ
0.8
0.6
e ~ 25%
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Frequency[Hz]
n
Goodness-of-fit estimator:
e
 Sˆ  f   S  f f
i
i 1
i
n
 Sˆ  f f
i 1
i
i
i
 100%
- II -
SEM-REV WAVE DATA
SEM-REV location
Nantes
(50km)
Loire
estuary
SEM-REV location
BMTO2
E WAVE BUOY
W WAVE BUOY
ADCP
Datawell directional buoy
and spectral processing:
•
•
•
•
•
•
Measurements of {x,y,z} motions (continuous)
1.28Hz sampling rate
HF radio transmission + onboard storage
1h-based signals for cross-spectral analysis
36 non-overlapping 100s periodograms (72 dof)
Cos^2s directional reconstruction (based on 1stand 2nd-order dir. Fourier coefficients)
• Δf = 0.01Hz, Δθ = 10°
• Spectral smoothing (3x3 cell moving average)
 8748 hourly directional spectra in 2011
(easternmost buoy) over 8760 expected
(99.9% success rate)
- III -
RESULTS AND DISCUSSION
Processing of SEM-REV 2011 hourly dir. spectra
 Separation frequency swell/wind-sea:
Interpolated (1h) ECMWF ERA-Interim 10m-height wind
speed for location (4.75°N, 3.0°W) close to SEM-REV
In practice here: fs = min(g/2πβU10 , 0.20Hz)
 Min. threshold for swell partition grouping : Hmin = 0.20m
Algorithm
performance
e mean = 17,7%
(95% | e ≤ 30%)
f , f [Hz]
s
0.3
p
0.2
0.1
H m 0 [m],
U
/2[m/s]
ECMWF wind data
 p ,  w,10 [º]
w,10
4
3
2
1
0
360
270
180
 [-]
Correlation to
ECMWF wind data
(ERA-Interim)
 [º]
No time tracking
0.4
0
90
0
5
4
3
2
1
0
60
45
30
15
0
60
e[%]
Time evolution of
wave system
parameters
(~18600 systems
extracted,
~2.1 syst./s.s.)
0.5
45
30
15
0
07/02
08/02
09/02
10/02
11/02
12/02
13/02
14/02
15/02
Sea states type in SEM-REV (2011) for different Hm0 thresholds
(i.e., wave systems with Hm0 lower than this value are disregarded in the counting)
/8748
Sea states may be considered as
unimodal
only 25% to 64% of time!
(according to threshold)
Peakedness statistics (γ < 10, -3%)
[0.08;0.12Hz[
Hm0 > 0.5m, 0.04Hz<f<=0.08Hz
Hm0 > 0.5m, 0.08Hz<f<=0.12Hz
Hm0 > 0.5m, 0.12Hz<f<=0.15Hz
40
40
5
[-]
0
0
10
Hm0 > 1m, 0.04Hz<f<=0.08Hz
5
[-]
Hm0 > 1m, 0.08Hz<f<=0.12Hz
40
0
0
5
[-]
20
0
0
10
Hm0 > 3m, 0.04Hz<f<=0.08Hz
5
[-]
 = 1.36
0
0
5
[-]
10
40
20
0
0
SWELLS
Occurrence rate[%]
0
0
10
5
[-]
Hm0 > 1m, 0.2Hz<f<=0.5Hz
Hm0 > 1m, 0.15Hz<f<=0.2Hz
30
20
10
0
0
10
10
 = 2.28
5
[-]
10
20
10
0
0
5
[-]
10
Hm0 > 3m, 0.12Hz<f<=0.15Hz
 = 1.90
Occurrence rate[%]
10
10
5
[-]
10
 = 1.65
20
 = 1.21
Occurrence rate[%]
20
Hm0,i > 3m
5
[-]
20
60
60
30
10
0
0
10
30
0
0
10
Hm0 > 3m, 0.08Hz<f<=0.12Hz
40
20
 = 1.38
Occurrence rate[%]
10
40
 = 2.14
30
 = 1.27
Occurrence rate[%]
20
5
[-]
40
 = 1.38
Hm0,i > 1m
10
Hm0 > 1m, 0.12Hz<f<=0.15Hz
60
30
20
0
0
10
Occurrence rate[%]
10
30
 = 1.81
30
Occurrence rate[%]
10
20
Hm0 > 0.5m, 0.2Hz<f<=0.5Hz
Hm0 > 0.5m, 0.15Hz<f<=0.2Hz
 = 1.46
Occurrence rate[%]
Hm0,i > 0,5m
30
[0.20;0.50Hz[
30
 = 1.47
Occurrence rate[%]
Occurrence rate[%]
20
0
0
Occurrence rate[%]
[0.15;0.20Hz[
30
 = 1.60
Occurrence rate[%]
[0.12;0.15Hz[
Occurrence rate[%]
[0.04;0.08Hz[
f
5
[-]
10
40
no
data
20
0
0
5
[-]
no
data
10
?
WIND-SEAS
• Again, statistics vary according to Hm0 threshold
• Mean peakedness values found within [1;2] (except HF)
Values range from 1 to 5 mostly, even for swells
• In ]0.04; 0.12Hz] (swells) γ decreases with fp on average
 consistent with theory of swell evolution
[e.g. Gjevik et al., 1988]
• Above 0.15Hz γ (wind-seas) increases with fp on average
 consistent with JONSWAP observations as peakedness
decreases during sea growth [Hasselmann et al., 1973]
• [5% bias to be deducted from γ here approx. due to sampling
variability in the spectral estimation with 72 dof
(see paper OMAE2013-10004, same author)]
Peakedness in severe sea states
 fatigue, survivability, certifications…
Severe sea state: Hm0 > 3m (> 8m Joachim storm in December 2011)
6
Regression line:
γ (biased) against Hm0
for Hm0 > 3m
5
[-]
(100% sea states are unimodal)
 More data required
]0.04;0.08Hz]
]0.08;0.12Hz]
]0.12;0.15Hz]
]0.15;0.20Hz]
]0.20;0.50Hz]
4
3
2
Storms with low fp
within ]0.04Hz;0.12Hz]
1
0
 = 0.219*H
2
4
6
Hm0[m]
+0.43
m0
8
10
- IV -
CONCLUSIONS & FURTHER WORKS
• On average, JONSWAP peakedness γ decreases and increases
with peak frequency within [1;2] – from swell to wind-sea
frequency range (most values within [~1;5] for both)
• Partitioning algorithm successful: In SEM-REV in 2011, sea
states could be considered as unimodal 64% of time at best
 partitioning required for metocean and engineering
studies
• Further work 1: JONSWAPs adapted to the spectral
modelling of swells?... (preliminary results available now)
• Further work 2: dynamic tracking of wave systems for better
system type identification
Merci de votre attention
Contact: jbsaulni@ec-nantes.fr (< août 2013)
toupaixil@yahoo.fr (ensuite)
Interval
0.04-0.08Hz
0.08-0.12Hz
0.12-0.15Hz
0.15-0.20Hz
0.20-0.30Hz
0.30-0.50Hz
0.20-0.50Hz
Hm0 min
0.2m
9.2%
29.6%
10.5%
8.5%
16.5%
15.4%
31.9%
0.5m
7.0%
21.6%
7.1%
6.0%
10.1%
2.3%
12.4%
1m
3.9%
10.0%
4.8%
9.9%
1.1%
0
1.1%
3m
0.2%
0.6%
<0.1%
0
0
0
0
SEM-REV location
Cable route
Le Croisic
town
SEM-REV
Salt evaporation
ponds of Guérande