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Real Time Estimation of Ship Motions in Short Crested Seas

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Abstract and Figures

The presented research is part of the development of an onboard wave and motion estimation system that aims to predict wave elevation and vessel motions some 60–120 s ahead, using wave elevation measurements by means of X-band radar. In order to validate the prediction model, scale experiments have been carried out in short crested waves for 3 different sea states with varying directional spreading, during which wave elevation and vessel motions were measured. To compare predicted and measured wave elevation, three wave probes were used at different distances from a large set of wave probes that was used as input to the model. At one of the prediction locations, also tests were performed to measure vessel motions. This setup allowed validation of a method that was used for initializing the linear wave prediction and ship motion prediction model. Various observations and conclusions are presented concerning optimal combinations of prediction model parameters, probe set-up and sea state.
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REAL TIME ESTIMATION OF SHIP MOTIONS IN SHORT CRESTED SEAS
P. Naaijen
Delft University of
Technology
Delft, The Netherlands
R.R.T. van Dijk
MARIN
Wageningen,
The Netherlands
R.H.M. Huijsmans
Delft University of
Technology
Delft, The Netherlands
A.A. El-Mouhandiz
MARIN
Wageningen
The Netherlands
ABSTRACT
The presented research is part of the development of an
onboard wave and motion estimation system that aims to
predict wave elevation and vessel motions some 60 - 120 s
ahead, using wave elevation measurements by means of X-
band radar.
In order to validate the prediction model, scale experiments
have been carried out in short crested waves for 3 different sea
states with varying directional spreading, during which wave
elevation and vessel motions were measured.
To compare predicted and measured wave elevation, three
wave probes were used at different distances from a large set of
wave probes that was used as input to the model . At one of the
prediction locations, also tests were performed to measure
vessel motions.
This setup allowed validation of a method that was used for
initializing the linear wave prediction and ship motion
prediction model.
Various observations and conclusions are presented concerning
optimal combinations of prediction model parameters, probe
set-up and sea state.
INTRODUCTION
Within an international joint industry project called OWME
(Onboard Wave and Motion Estimation) a system is being
developed which aims to predict ship motions some 60 seconds
ahead. The main purpose of such a system is to increase safety
and operability during offshore operations that are critical with
regard to vessel motions, e.g. top-site installation (float-over or
lifting), helicopter landing on floating vessel and LNG
offloading connection. Use is being made of newest wave
sensing techniques by means of X-band radar: The Ocean
Waves’ WAMOS II radar image processing software is capable
of providing real-time time traces of wave elevation at a large
number of locations.
This paper describes the validation of a model used to compute
a deterministic prediction of wave elevation and ship motion by
using remote wave elevation measurements in short crested
waves. Linear theory is used resulting in a very simple and
straightforward propagation model. The challenging part is
within the initialization of this model which is the main focus
of the present study.
To assess the accuracy of the prediction, extended model tests
have been carried out at the Maritime Research Institute
Netherlands (MARIN). During these experiments the 2
dimensional wave field was measured by using a large array of
wave probes. The measured wave field is used to predict wave
elevation and ship motion at various distant locations. The
predictions are validated by means of measurements of both
wave elevation and vessel motions at the prediction location.
EXPERIMENTS
Wave measurements
As mentioned the present study aims to predict wave elevation
and vessel motion in a deterministic way using measured time
traces of wave elevation at various locations. To validate the
model that will be described in the next paragraph, model
experiments at a scale of 1:70 were carried out at MARIN, the
Netherlands.
For a selection of 3 short crested wave conditions, wave
elevation was recorded by means of a wave probe array
existing of 10 x 10 wire-type wave gauges.
Figure 1, 10 x 10 wave gauge array
1 Copyright © 2009 by ASME
Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering
OMAE2009
May 31 - June 5, 2009, Honolulu, Hawaii, USA
OMAE2009-79366
Tests were repeated with the wave gauge array positioned at
different locations in the basin, thus obtaining a relatively large
number of wave measurements that could be used to optimize
and validate the prediction model. Figure 2 shows all wave
gauge array positions for which tests were performed. Positions
1, 2 and 3 are the locations for which predictions will be made
and compared with the measurements. The probes at the
remaining locations are used as input for the model. The main
wave direction is in positive X direction as indicated in the
figure.
Figure 2, wave gauge array positions
As the different tests were not performed simultaneously,
checks were carried out in order to confirm that different wave
tests are reproducible. Figure 3 shows a sample of two wave
measurements from different tests, measured at the same
location. As can be seen good reproducibility is obtained.
Figure 3, comparison of measured wave
elevation during different tests at identical
locations
Concerning the wave conditions the choice was made only to
vary the amount of directional spreading. (The effect of wave
steepness was examined in earlier work. (Naaijen et al. [9])).
Tests were performed for a Jonswap spectrum with a peak
period Tp of 9.0 s, significant wave height Hs of 2.5 m and
peakedness factor γ of 3.3.
To include directional wave spreading the following spreading
function was used:
(
)
()
()()()
0
0
2
00
090 20
90
() cos
1
cos
,
s
s
DD
with
D
d
and
SSD
μ
μ
μμμ
μ
μμ
ωμ ω μ
+
=⋅ −
=
=⋅
(1)
X Three different values for the spreading parameter s were used,
being 4, 10, and 50 corresponding to very short crested wind
waves, average wind waves and an average swell respectively.
Y(Measurements at position #5, #6, #10 and #11 as indicated in
Figure 2 were only performed for the tests with s=10.)
For practical reasons concerning the wave maker, the
directionality was cut off leaving the sector of -15 deg – 15 deg
for s=50, -30 deg – 30 deg for s=10 and -45 deg – 45 deg for
s=4.
Ship motion measurements
Apart from wave elevation measurements, motions of a model
of an offshore support vessel, without forward speed, located at
position #2 (see Figure 2) were recorded. The model, being
kept in position by a soft mooring system such that the relative
wave direction was 165 degrees, is depicted in Figure 4.
Figure 4, 1:70 model of offshore support
vessel
The main particulars of the vessel are:
LPP 94.2 [m] - Length between perpendiculars
LWL 101.5 [m] - Length on waterline
2 Copyright © 2009 by ASME
B 21.0 [m] - Breadth max.
TF 6.0 [m] - Draught fore
TA 6.0 [m] - Draught aft
Focusing on the prediction of vertical motions, mainly heave
and pitch motions were considered, of which the RAO’s were
determined using a linear 3D diffraction program. Additional
tests were carried out in irregular (white noise) long crested
waves for relative wave directions varying from bow to bow
quartering. Calculated RAO’s appeared to be in good
agreement with the RAO's determined from the mentioned
experiments.
00.5 11.5 22.5
0
0.5
1
1.5
2
1D Spectrum [m
2
/ s]
00.5 11.5 22.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ω
[rad/s]
Pitch RAO [deg/m]
Heave RAO
[
m/m
]
heave 165 deg
pitch 165 deg
wave spectrum
Figure 5, 1D wave spectrum and RAO's for
relative wave direction of 165 deg.
Figure 5 shows the calculated heave and pitch RAO's together
with the one-dimensional wave spectrum.
PROPAGATION MODEL
The theoretical model used to describe the wave field is a linear
superposition of cosine waves with different frequencies
traveling in different directions:
()
() ()
()
/2 cos sin
11
nnj nm nj nm nm
NM itkx ky
j anm
nm
te
ωμ με
ζζ
−⋅ − +
==
⎧⎫
=ℜ ⋅
⎨⎬
⎩⎭
∑∑ (2)
where:
nm
real part
amplitude of frequency component in
propagation direction ,
frequency of component n
wave number of component n
, co-ordinates of location j
propagation direction of dire
anm
nm
n
n
jj
k
xy
ζ
μ
ω
μ
ℜ=
=
=
=
=
=
nm
ctional
component m for frequency n
initial phase angle of component nm
= number of directional components
per frequency
M
ε
=
From Fourier analysis, frequency components of the measured
time traces at J locations can be obtained, corresponding to the
following representation :
()
()
/2
0
nj
Nit
j anj
n
te
ωε
ζζ
+
=
=ℜ
⎩⎭
%
%%
(3)
where:
index of location for which time trace is provided
index of frequency component
N=number of samples of considered time trace
, amplitude and phase angle for frequency component n
of measured
anj nj
j
n
ζε
=
=
=
%%
time trace at location j following from FFT
From a predefined number M of directional components to be
used in equation (2), the discrete wave directions μnm were
chosen as was suggested by Zhang [1]: the average directional
wave spectrum was determined from the measurements by
means of the MLM method. For each frequency the energy
content was examined and the direction of the most energetic
component was identified. Ignoring on both sides of this most
energetic direction a predefined small amount of wave energy,
a range of wave directions is obtained that is divided into M
segments whose center values are used for μnm. (The values of
μnm determined this way will be frequency dependent which is
why a double index nm is used.) Figure 6 shows an example of
discrete ωn - μnm combinations (marked with dots) determined
as described above for M=10, together with the underlying
two dimensional spectrum for the wave condition with s=10.
Figure 6, directional wave spectrum for s=10
and discrete components used in
representation
As an estimate for the wave component amplitudes ζnm, the
spectral values S(ω,μ) were converted to amplitudes:
()
2,
nm n m
Sdd
ζ
ωμ ωμ
= (4)
3 Copyright © 2009 by ASME
Where:
the band width of each of the M segments mentioned aboved
μ
=
This way the only unknowns left in equation (2) are the initial
phase angles εnm.
By assuming that the above 2D representation of a wave time
trace at location j (equation (2)) should equal the measured
trace represented by (equation (3)), the unknown initial phase
angles in (2) can be solved. This is done by considering a
frequency domain representation of both measurement and 2D
representation.
One method to solve the unknown phase angles is proposed by
Zhang [1]:
For each frequency, an error can be defined as follows:
() ()
()
cos sin
1
n j nm n j nm nm
nj Mikx ky
i
nj anj anm
m
ee
μμ
ε
ζζ
−⋅ − +
=
Δ= ⋅ −
%
%
ε
(5)
The unknown initial phase angles can be solved by minimizing
a target function Rn defined by the sum over all J locations of
the absolute squared error:
{}
()
{}
(
22
2
11
JJ
njn jn jn
jj
R
==
=Δ= ℜΔ +Δ
∑∑
)
(6)
Another way to solve the phase angles is by defining a matrix
vector equation, Ax=b as given in equation (7), and solving
this.
() ()
()
() ()
()
() ()
()
() ()
()
1111 1 1
11
11
cos sin cos sin
1
cos sin cos sin
1
1
nnnn nnMnnM
nJ n nJ n nJ nM nJ nM
nn
nM nJ
ikx ky ikx ky
an anM
ikx ky ikx ky
an anM
ii
an
ii
anJ
Ax b
ee
ee
ee
ee
μμ μ μ
μμ μ μ
εε
εε
ζζ
ζζ
ζ
ζ
−⋅ − −⋅
−⋅ − −⋅
⋅==
⎛⎞
⋅⋅
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⋅⋅
⎝⎠
⎛⎞
⎜⎟
⎜⎟
=
⎜⎟
⎜⎟
⎝⎠
%
%
K
MO M
L
%
MM
%
⎜⎟
(7)
Applying a Singular Value Decomposition (SVD) on A as done
by Janssen et. al. [4] appeared to give the best results. All
results and conclusions mentioned in next paragraphs are based
on the latter method.
Having solved the phase angles from equation (2), the wave
elevation can be calculated at any time and any location by
substituting the desired values for t, xj and yj in the equation.
However, in order to ensure physical significance of the
prediction, t, xj and yj have to be chosen within certain limits.
This will be explained in more detail in the next paragraph.
Using the RAO’s for the ship motions (obtained from linear 3D
diffraction calculations) a ship motion prediction can be made
in a straight forward way:
()
() ()
()
/2 cos sin
11
nnj nmnj nm nmknm
NM itkx ky
kj anm knm
nm
x t RAO e
ωμ μεε
ζ
−⋅ − ++
==
⎛⎞
=ℜ ⋅ ⋅
⎜⎟
⎝⎠
∑∑
(8)
where:
ship motion in mode k (k = 1...6)
frequency and direction dependent response
amplitude operator for mode k
frequency and direction dependent phase angle
for mode k
kj
knm
knm
x
RAO
ε
=
=
=
PREDICTABILITY
Having described the propagation model in the previous
chapter, some attention is paid to the question how the
difference between real surface waves and our representation of
them effects the predictability of those surface waves.
Let's reconsider equation (2). If no restrictions are put to the
domain in space (xj, yj) and time (t) for which we consider this
representation to be valid, we practically assume it to be able to
describe the entire ocean for an unlimited period of time.
Obviously, this isn't a realistic assumption. The validity of the
representation as given in equation (2) will be limited in space
and time.
To discuss these limitations, the one-dimensional case is
revisited here briefly. The space-time diagram for a long
crested wave traveling in positive X-direction showing the
predictable zone is depicted in Figure 7.
X
,
g
low
ct
,
g
high
ct
X0+ΔX
D
Figure 7, predictable zone long crested waves
t
t0 t0 +Δt
X0 t0+D
4 Copyright © 2009 by ASME
In this diagram, introduced by Morris et al [7] and used as well
by Edgar et al [8] and Naaijen et al [9], the triangles indicate
the zone in space and time where the wave elevation can be
predicted or reconstructed using a recorded time trace of the
wave elevation which is represented by the horizontal line at
the base of the triangle.
The second horizontal line represents the prediction based on
this recorded time trace, ΔX away from the measurement
location, shifted Δt seconds ahead. Only its part within the
predictable zone (gray triangle) is supposed to be useful.
In the mentioned publications the slopes of the left and right
boundaries of the predictable zone were considered to equal the
phase velocity of the shortest and longest wave components
present in the recorded time trace. However, as explained by
Wu [10], it is not the phase velocity but the group velocity of
the shortest and longest wave components that governs the size
of the predictable zone. This also explains that it was observed
during the experiments by Naaijen et al [9] that predictions of
the wave elevation could be extended further into the future
than expected based on the predictable zone bounded by the
highest and lowest phase velocities: when applying a long
enough duration D of the recorded wave elevation, only the
fastest wave components will limit the prediction and as their
group velocity is lower than their phase velocity, the steepness
of the right-hand boundary is decreased, meaning that the
prediction can be extended further into the future.
The concept of the predictable zone can be extended for the
three-dimensional case. Considering the three-dimensional
wave field to be a superposition of wave components traveling
in different directions, a similar predictable zone diagram can
be constructed for one specific traveling direction. See Figure
8.
For any point in space and time, the wave elevation due to all
components traveling in the considered direction is a
superposition of all frequency components traveling in that
direction. Depending on which point in space and time is
considered, not all these components might originate from the
measurement. The highest and lowest group velocities of those
components just originating from the measurement for a given
prediction point in space and time (xp, yp, tp) can be defined as
respectively:
()
()
()
()
(
()
()
()
()
()
(
1
2
cos sin /
cos sin /
gp p p
gp p p
cxx yy t
cxx yy t
μμ
μμ
=− +
=− +
%%
%%
)
)
D
(9)
Where a tilde denotes the measurement location.
Denoting ω1 and ω2 as the corresponding wave frequencies and
ωlow and ωhigh as the frequencies of the shortest and longest
wave components that occur in the wave field, an error estimate
for the predicted wave elevation can be defined as the relative
amount of wave energy represented by those components at (xp,
yp, tp) that do not originate from the measurement:
μ
Figure 8, predictable zone for one directional
component of a short crested sea
()
()
()
()
()
2
1
12
2
0
2
0
,
,, 1
,
high
low
ppp
Sdd
Err x y t
Sdd
ωμ
π
ωμ
ω
π
ω
ωμ ω μ
ωμ ω μ
⎛⎞
⎜⎟
⎜⎟
=−
⎜⎟
⎜⎟
⎜⎟
⎝⎠
∫∫
∫∫
(10)
As described in the previous chapter multiple probe records are
used to find the representation of the wave field given by
equation (2). However, keeping in mind the limited
representing capabilities in space and time of this
representation, it can only represent a decomposition of one
probe record.
Imagine that for one specific direction the frequency
components in the three-dimensional wave representation
represent a decomposition of a measurement of length D at
location
(
)
,
x
y
%%.
Figure 8 shows that a measurement at location
(
)
,
pp
x
y can
only be represented by this decomposition between tb and te..
(The vertical axis in Figure 8 corresponds with the spatial
coordinate
x
μ
, parallel to the traveling direction of the
considered wave components.)
The short wave components present in the part before tb do not
originate from the location
(
,
)
x
y
%%. For any point in time of
the wave elevation time trace at
()
,
pp
x
y, the error Err as
x
(
)
(
)
(cos sin )xy
μ
,
g
low
ct
t
p
x
μ
x
μ
%
,
g
high
ct
Dte
tb
μ
=+
t0
5 Copyright © 2009 by ASME
defined in equation (10) can be determined. A representative
mean error value for the whole time trace at
()
,
pp
x
y can
then be defined as follows:
() (
0
0
1
,
tD
meanpp pp
t
Err x y Err x y t dt
D
+
=
)
,,
)
(11)
So when attempts are made to find a three-dimensional wave
field representation (equation(2)) that yields for a certain period
of time D at location
(
,
x
y
%% , only those parts of the
simultaneously measured time traces at surrounding probe
locations (denoted by
()
,
pp
x
y) should be used that are
within the predictable zone. The presented methods to find the
three-dimensional wave field representation are ignoring this
fact since they are frequency domain methods for which it is
not possible to take it into account in a straightforward way.
(Wu [10] describes a time domain method for solving the three-
dimensional representation for which it is possible to account
for limitations in the usable part of the time traces to be used.)
In the area where the wave probes were located, Errmean from
equation (11) is shown for the three wave conditions for
which experiments were carried out in Figure 10. Locations of
the frames containing the probes whose simultaneous
measurements were used for decomposing the wave field are
indicated by the squares. (The dots indicate the used probe
positions that appeared to give the optimal results for the
considered wave condition and prediction location #2.) As can
be seen the probe positions are chosen such that Errmean does
not exceed 0.1 for any of the conditions.
Figure 9, predictable zone for s=10 and
forecast-time of 60 s
Y
X
Y
X
Y
X
Figure 10,
Err
mean
for s=50, 10 and 4 and
available probe-frame positions
When calculating the values for Err according to equation (10)
for a prediction of the wave elevation at a given set of locations
(xp,yp) for a given moment in time tp , the optimal relative
positions of measurement and prediction can be determined.
For a forecast-time of 60 s and wave spreading s=10, the value
of Err is presented in a similar way as was done by Blondel
[11] in Figure 9.
α X
Y
6 Copyright © 2009 by ASME
As shown by Figure 9, prediction location #2, the location
where also the ship model was located, was chosen such that it
would be optimal for a 60 seconds prediction for the wave
condition with s=10.
RESULTS
Several aspects have an effect on the accuracy of the wave
prediction:
The setup of the probes whose measurements are used to solve
the system of equation (7) should be such that their in-
between distances are optimal to identify phase differences for
the whole frequency range that contains wave energy. As
shown by Voogt et al. [12], for one regular wave component
this optimal probe distance amounts to ¼ of the wave length.
Therefore it was aimed to use a probe setup that provided for
all relevant frequencies in-between distances (projected in the
direction of each of the wave directions used in the
representation) of ¼ of the wave length.
The probe setup that appeared to give the best results for a
prediction at location #2 in waves with direction spreading of
s=10 is shown in Figure 9. For each of the 10 discrete wave
directions that were used at the peak frequency in the two-
dimensional representation, intermediate distances between the
used probes, projected in each of the 10 wave directions, were
determined. Figure 11 shows the 2D spectrum based on
wavelength / peak wavelength ratio. The dots indicate all the
available intermediate probe distances divided by ¼ peak
wavelength. This way of presenting should result in a high-
density of dots in the most energetic part of the spectrum for a
favorable probe set-up according to the statement above.
Figure 11, relative probe distances for optimal
setup s=10, prediction position #2
Another important variable that effects the accuracy to a great
extend is the number of directional components that is used in
the representation of the wave field. With a large number of
directional components, the directional spreading is better
covered. However, for a given probe set-up, choosing a too
large number of directional components dramatically decreases
the condition of system matrix A in equation (7) resulting in
poor predictions. For cases where the number of directional
components was too large, it was found that adding more probe
records to the equation (7) not necessarily improves the
prediction. The extra probe records have to add ‘information’ to
the system. Adding probes positioned in frame numbers 4 and
12 (Figure 2) for example did not improve the prediction for
any of the cases. When the requirement to the probe set-up as
stated above is fulfilled, the only way to improve the condition
of the matrix is to decrease the number of wave directions. The
set-ups shown in Figure 10, using 9 probes per frame as
indicated by the dots, were found to give the best results for
prediction at location #2. By extending the number of input
measurements by using all available probes i.e. 100 per frame,
no improvements were obtained.
Apart from the combination of probe set-up and number of
directional components, that determines the condition of the
matrix in (7), the combination of probe set-up and the
prediction location was found to have a significant impact. It
was observed that the use of extending the probe set-up in y-
direction is limited related to the amount of wave spreading and
the prediction location. See Figure 9. The angle α between the
dashed lines starting from prediction location #2 equals the
sector angle that bounds the directionality of the wave
condition: as mentioned, for practical reasons the directionality
was cut off at 30 and -30 degrees for this condition (s=10)
resulting in α = 60 deg. It was observed that using
measurement probes outside this angle did not improve the
accuracy. This observation was found to be consistent for all
combinations of prediction location and directional spreading
as far as the available measurements allowed to verify it. (For
the most short crested condition with s=4, measurements at
positions #5, #6, #10 and #11 that would enable confirmation
of this observation were not available.)
To assess the accuracy of the predictions an error value is
calculated that is defined as:
() () ()
()
2
*
1
*
0
1N
nn
ntt
N
Et m
ζζ
=
= (12)
Where:
(
)
*
*
0
normalized prediction error
( ) realization of predicted wave elevation
( ) realization of measured wave elevation
total number of realizations
RMS of measured wave elevation
n
th
n
th
Et
tn
tn
N
m
ζ
ζ
=
=
=
=
=
Figure 12 shows the normalized error of the wave prediction
averaged over N =100 realizations at location 3, 2 and 1 for the
case with s=10. The vertical dashed lines indicate the
7 Copyright © 2009 by ASME
boundaries of the predictable zone in time. As expected the
error increases outside the predictable zone. The solid vertical
line indicates the boundary between hind cast and forecast.
Error values averaged over the predictable part of the predicted
traces are given in Annex A, Table 1. All values are based on an
average over 100 realizations.
0100 200 300 400 500 600 700
0
0.5
1
1.5
E(t) [-]
0100 200 300 400 500 600 700
0
0.5
1
1.5
E(t) [-]
0100 200 300 400 500 600 700
0
0.5
1
1.5
t [s]
E(t) [-]
envelope
wave elevation
Figure 12, prediction error for locations 3,2
and 1 for s=10
Since for the applications of the onboard wave and motion
estimation system we are interested in the prediction of quiet
periods rather than in an exact deterministic prediction, also the
envelope of the deterministic prediction has been determined:
the predicted deterministic signal has been post processed by
taking the absolute value of its Hilbert transform. Its error,
which is defined similarly to the error of the wave elevation /
ship motion itself is in general significantly smaller.
For location #2, Figure 13 shows time traces of predicted and
measured wave elevation and heave and pitch motion for the
case with s=10. The maximum prediction time has been
determined from the measured mean two-dimensional
spectrum, allowing a value of Err as defined in equation (10)
of 0.1 and amounts to 78 s. The vertical solid line again
indicates the boundary between hind cast and forecast. The
rightmost vertical dashed red line indicates the end of the
predictable zone in time.
In Annex B, samples of time traces are shown for all wave
conditions at all prediction locations. For each prediction
location the most favorable probe set-up is plotted at the right
hand side of each of the corresponding time traces: The black
dots indicate the used probes, the grey cross with the circle
indicates the concerning prediction probe. The color indicates
the Err value as defined in equation (10) for the allowed
maximum forecast time that was aimed for (which is 30 s for
probe 3, 60 s for probe 2 and 120 s for probe 1). As can be seen
the chosen prediction locations match the intended forecast
time well in that sense that for all conditions they are
positioned in the predictable zone (where Err is app. 0.)
1850 1900 1950 2000 2050 2100
-2
0
2
ζ
[m]
1850 1900 1950 2000 2050 2100
-1
0
1
z [m]
measured
predicted
1850 1900 1950 2000 2050 2100
-5
0
5
θ
[deg]
t [s]
Figure 13, sample time traces of prediction
and measurement of wave elevation, heave
and pitch motion, location # 2, s=10
As can be seen from both the mean error values in Table 1 and
from the samples of the time traces in Figure 15, predictions for
s=4 are rather poor especially for locations 1 and 2. This is
caused by the fact that probe measurements at locations further
from the X-axes, that would be required for an optimal set-up
were not available for this condition.
ACKNOWLEDGMENTS
This paper is published by courtesy of all participants and
partners of the OWME-JIP for which they are gratefully
acknowledged.
Special thanks to MARIN for facilitating the experiments.
REFERENCES
[1] Zhang et. al., Deterministic wave model for short-
crested ocean waves: Part I. Theory and numerical
scheme, Applied Ocean Research 21, 1999
[2] Zhang et. al., Deterministic wave model for short-
crested ocean waves: Part II. Comparison wih
laboratory and field measurements, Applied Ocean
Research 21, 1999
[3] Prislin et. al., Deterministic decomposition of deep
water short-crested irregular gravity waves, Journal of
Geophysical Research, Vol. 102 No. C6, 1997
[4] Janssen et. al. Phase resolving analysis of
multidirectional wave trains, Proceedings of the Fourth
International Symposium Waves 2001, September 2-6,
2001, San Francisco, CA
[5] de Jong et. al. A phase resolving analysis technique for
short-crested wave fields
[6] Belmont et. al. The effect of statistically dependent
phases in short-term prediction of the sea: A simulation
study, International Shipbuilding Progress Vol. 51 no.4,
2004
8 Copyright © 2009 by ASME
[7] Morris, E.L., Zienkiewicz, H.K. & Belmont M.R.,
1998 Short term forecasting of the sea surface shape.
Int. Shipbuild. Proqr. 45, no.444, 383-400.
[8] Edgar D.R., Horwood J.M.K., Thurley R., Belmont
M.R. The effects of parameters on the maximum
prediction time possible in short term forecasting of the
sea surface shape, International Shipbuilding Progress
Vol. 47, No.451, 2000
[9] Naaijen P., Huijsmans R.H.M. 2008, Real time wave
forecasting for real time ship motion predictions, Proc.
OMAE 2008
[10] Wu, G., Direct simulation and deterministic prediction
of large-scale nonlinear ocean wave-field, PhD Thesis,
M.I.T, 2004
[11] Blondel, E., Ducrozet, G., Bonnefoy, G., Ferrant, P.,
Deterministic reconstruction and prediction of non-
linear wave systems, 23rd IWWWFB, 2008, Jeju, Korea
[12] Voogt A., Bunnik T., Huijsmans R.H.M. Validation of
an analytical method to calculate wave setdown on
current Proc. OMAE 2005
9 Copyright © 2009 by ASME
ANNEX A
s [-] Error [-] Error Envelope [-] max pred [s]
prediction probe # wave heave pitch wave heave pitch
1 10 0,915 0,703 131
2 10 0,778 0,802 0,748 0,639 0,597 0,547 78
3 10 0,744 0,607 35
s [-] Error [-] Error Envelope [-] max pred [s]
prediction probe # wave heave pitch wave heave pitch
1 4 1,172 0,822 124
2 4 1,179 1,172 1,215 0,785 0,890 0,879 77
3 4 0,899 0,686 34
s [-] Error [-] Error Envelope [-] max pred [s]
prediction probe # wave heave pitch wave heave pitch
1 50 0,783 0,668 138
2 50 0,737 0,827 0,734 0,651 0,686 0,593 83
3 50 0,702 0,566 37
Table 1, Averaged Error values
10 Copyright © 2009 by ASME
ANNEX B
1600 1650 1700 1750 1800 1850 1900
-2
0
2
ζ [m]
measured
calculated
Time trace sample position 1, s=10, max prediction time =131 s, M=14
1850 1900 1950 2000 2050 2100
-2
0
2
ζ
[m]
1850 1900 1950 2000 2050 2100
-1
0
1
z [m]
2
measured
predicted
1850 1900 1950 2000 2050 2100
-5
0
5
θ
[deg]
t [s]
Time traces sample position 2, s=10, max prediction time =78 s, M=10
1680 1700 1720 1740 1760 1780 1800 1820 1840 1860 1880 1900
-2
0
2
ζ
[m]
measured
calculated
Time trace sample position 3, s=10, max prediction time =35 s, M=7
Figure 14, sample time traces for s=10
11 Copyright © 2009 by ASME
1600 1650 1700 1750 1800 1850 1900
-2
0
2
ζ
[m]
measured
calculated
Time trace sample position 1, s=4, max prediction time =124 s, M=8
1600 1650 1700 1750 1800 1850 1900
-2
0
2
ζ
[m]
1600 1650 1700 1750 1800 1850 1900
-1
0
1
z [m]
1600 1650 1700 1750 1800 1850 1900
-5
0
5
θ
[deg]
t [s]
Time traces sample position 2, s=4, max prediction time =77 s, M=12
di i f b 1(X ) 861 3 32)Y ( 69 6 8
1600 1650 1700 1750 1800 1850 1900
-2
0
2measured
ζ
[m]
calculated
Time trace sample position 3, s=4 max predict on, i time =34 s, M=9
Figure 15, sample time traces for s=4
12 Copyright © 2009 by ASME
1750 1800 1850 1900 1950 2000
-2
0
2
ζ
[m]
measured
calculated
Time trace sample position 1, s=50, max prediction time =138 s, M=7
1750 1800 1850 1900 1950 2000
-2
0
2
ζ
[m]
1750 1800 1850 1900 1950 2000
-1
0
1
z [m]
1750 1800 1850 1900 1950 2000
-5
0
5
θ
[deg]
Time traces sample position 2, s=50, max prediction time =83 s, M=7
1700 1750 1800 1850 1900 1950
-2
0
2
ζ
[m]
measured
calculated
Time trace sample position 3, s=50, max prediction time =37 s, M=4
Figure 16, sample time traces for s=50
13 Copyright © 2009 by ASME
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