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Identifying Wave Loads During Random Seas Using Structural Response

Michael Vigsø*, Rune Brincker** and Christos T. Georgakis*

*Department of Engineering, Aarhus University, Denmark.

**Department of Civil Engineering, Technical University of Denmark, Kgs. Lyngby, Denmark

ABSTRACT

Structural monitoring is increasingly becoming everyday business in

the offshore industry. The monitoring may target the strain estimation

or focus on tracking the changes in the dynamic properties of the

structure in order to predict damages at remote / or possibly subsea

locations. This paper will show that by monitoring the structural

response, it is also possible to indirectly estimate the wave loading

acting on the system. This information can be used to increase

conﬁdence in the load probability models for the structural design or

aid the health monitoring procedure. During ambient vibration, the

principles of operational modal analysis (OMA) are applied to harvest

the dynamic properties of the structure. Successively, a dynamic

model is formulated and used to calculate the loading from a random

sea state using the response of the structure. A laboratory experiment

is conducted in a wave ﬂume at LASIF, Marseille, France, where a

scaled offshore model is equipped with accelerometers to monitor the

structural response during a random sea. The study shows that it is

possible to use the structure as a dynamic load cell and monitor the

loads occurring in actual conditions. Both the short time variations

and the load spectra can be computed successfully using the structural

response.

KEYWORDS: Wave Loading; Indirect Measurements; Opera-

tional Modal Analysis; Offshore Structures

INTRODUCTION

In the ﬁeld of offshore structures, an increase is seen in the subject of

monitoring. Recently, TOTAL announced that as for the redevelop-

ment of the Tyra ﬁeld, the platform Tyra East will be equipped with no

less than 100000 sensors (Beck, 2018). Most of these will, of course,

target the production processes, but the monitoring scope will also in-

clude the structural performance. The aim of structural monitoring

may be plentiful, for instance with regards to operational limitations

such as heading, static deformation or vibration level. The vibration

pattern can be used for health diagnostics, and since offshore structures

are prone to fatigue damages, monitoring their well-being is essential

for ensuring safety and reliability.

Due to the nature of offshore structures, it is common to monitor the

surrounding sea state, but more interesting is the wave loading itself

rather than the surface elevation. The wave loading depends on the ge-

ometry and surface properties of the structure, but also on its dynamics.

Therefore, it is challenging to obtain the load measurements directly

and hence we need to rely on indirect methods. By monitoring the

response of a structure, it is, however, possible to identify the loading

during operational conditions with the actual boundary conditions.

Indirect (wave) load identiﬁcation has been used in ship design, but

it is becoming more common in the ﬁeld of ﬁxed offshore structures

as well. Within industry practice, wind turbines or oil and gas instal-

lations are exploited for this. For example the "digital twin" concept

introduced by Ramboll Oil and Gas as part of their monitoring pro-

gramme (Tygesen et al., 2018a), (Tygesen et al., 2018b), (Perisic et al.,

2014a) and (Perisic and Tygesen, 2014b). Within academia, early stud-

ies (Jensen et al., 1992) suggest the feasibility of wave load estimates

using the structural response on a simulated case study. Later studies

(Noppe et al., 2016; Fallais et al., 2016; Petersen et al., 2019; Maes

et al., 2018; Vigsø et al., 2018a) demonstrate how wave loads can be

estimated using existing load identiﬁcation techniques. The studies in-

clude both numerical and experimental work and aim to quantify loads

from a variety of wave types, including breaking waves.

In this paper, we will add to the study of indirect methods for wave

load identiﬁcation by monitoring a scaled model in a wave ﬂume. The

aim of this study will be to estimate the load variations during random

seas.

The mathematical notation used is denoting matrices by a double un-

derline and vectors by a single underline.

THEORY

For the load identiﬁcation process, we shall rely on the time domain

technique using the modally reduced model. The method is adopted

from (Lourens and Fallais, 2019) with reference to (Lourens et al.,

2012). The principle is that a modal analysis is performed and modal

parameters are obtained for the structure at in-place conditions. The

modal parameters are then fed into a recursive modal description to-

gether with response measurements from the structure, and by using

the means of a linear Kalman ﬁlter, an estimate of the wave loading is

obtained in real-time.

Published in:

Procedings of the International Ocean and Polar Engineering Conference, ISOPE

Publishing date: 2019

The performance of the method is sensitive to assumptions regarding

the spatial distribution of the load and hence a wave gauge is used to

ﬁx the loading to the wetted area of the structure during the random

sea.

The algorithm is recapped here in a condensed form suited for the need

of this study. We start out by deﬁning the state modal coordinates:

x(t) = "Φ0

0Φ#ζ(t)(1)

Here ζ(t)is the state vector in modal coordinates while x(t)is the state

vector in physical coordinates. Φcontains the mode shapes, expanded,

mass normalized and arranged in columns. We arrange the natural

frequencies ωiand the modal damping ξiin diagonal matrices such

that: 1

Ω=

ω10 0

0...0

0 0 ωn

,and Γ=

2ξ1ω10 0

0...0

0 0 2ξnωn

(2)

For the classic continuous state space in modal coordinates we have:

˙

ζ(t) = "0I

−Ω2−Γ#

| {z }

Ac

ζ(t) + "0

ΦTSp#

| {z }

Bc

p(t)(3)

Here, we assume that the hydrodynamic added mass and damping are

embedded in the modal parameters and thus do not appear directly in

the state equation. Another important aspect of this formulation is the

separation of the load into the spatial distribution Spand the scaling

p(t). Note that the loading may be composed of multiple sources and

thus the spatial distribution will be a matrix with load sources arranged

in columns, and the scaling will be a corresponding vector. For this

study, however, we assume that only one load source is present, i.e.

the load caused by the waves. The subscript cindicates that the state

matrices are on continuous form.

Since the spatial distribution is not constant during the wave loading,

but instead will relate to the water runup on the pile, we will adjust the

spatial distribution accordingly using the wave gauge measurements.

The position of the wave gauge may be seen in Fig. 2. Consequently,

the modal load is adjusted as:

q(t) = ΦTSpp(t)→q(t) = ΦTSp(η(t)) p(t)(4)

Here, η(t)is the water surface elevation near the structure. The prin-

ciple is sketched in Fig. 1. The distribution of the load is derived as

an inertia dominated load from a linear wave, given a peak frequency

and signiﬁcant wave height matching the wave spectrum. The distri-

bution is then (with similarity to Wheeler stretching (Wheeler, 1969))

stretched such that it follows the surface elevation as shown in Fig. 1.

Augmented state

Now, since we wish to obtain an estimate on the input rather than the

response, we expand the state vector to include the load (referred to as

augmented state vector). That is:

ζa(t) = "ζ(t)

p(t)#(5)

1This formulation is based on proportionally damped systems and thus might be vio-

lated by the hydrodynamic contribution.

Sp (t)

a)

h (t)Sp (t + Dt)

b)

h (t + Dt)

Fig. 1. Principle of time varying load distribution. The spatial

distribution Sp(t)is stretched by the information from the

wave gauge η(t).

In Eq. (5) p(t)shall be interpreted as the load scalar function. The

subscript aindicates augmented state.

We assume that the augmented state equation in discrete form can be

written for time step k+1 as:

ζa(k+1) = Aaζa(k) + w(k)

γ(k)(6)

Here, w(k)and γ(k)are the noise processes which account for dis-

crepancies in the relation (more about these later). The discrete state

transition matrix Aais written as:

Aa="A B

0I#(7)

where

A=eAcdt ,and B=A−IAc

−1Bc(8)

Now we have deﬁned the augmented state equation in discrete form

and hence we can implement the Kalman ﬁlter for a real-time input

estimate.

Kalman ﬁlter

We assume that the load scalar p(k)remains constant during the time

steps and that variations are caused only by a stochastic process also

known as a random walk. Hence:

p(k+1) = p(k) + γ(k)(9)

Here, γis assumed to be Gaussian.

Since the resolution of the mode shapes is greater than the number

of sensors, we must deﬁne the selection matrix. In general, we wish

to link the measurement z(k)to the system state ζ(k). Since only

accelerations are measured, the state observation equation reduces to

(in discrete form):

z(k) = h−SaΦ Ω2−SaΦ Γ SaΦ ΦTSp(η(k))iζa(k) + v(k)

(10)

The selection matrix Sais a matrix that picks the degrees of freedom

(DOFs) which coincide with the sensors. For example, sensor 1 coin-

cides with DOF 8 and sensor 2 with DOF 7. This yields:

Sa=

00000001. . .

00000010. . .

.

.

....

nsensors×nDOF

(11)

An important ingredient for the Kalman ﬁlter algorithm is the error

covariance matrices (noise models). In lack of information, we often

assume that the noise processes are Gaussian and mutually uncorre-

lated. Hence for the augmented error covariance matrix (with single

loading) we get:

Ew(k)

γ(k)wT(k)γ(k)="Q0

0S#δkl (12)

Here, δkl is the Kronecker delta to ensure that the covariance matrix

remains diagonal. We will assume a constant value in the system error

covariance matrix Q.

In Eq. (9) we have deﬁned the load as a Gaussian random walk. In

order to capture any sharp variation in the load, the associated error

covariance Smust be increased. Several power of magnitude is needed

compared to the values in Qand R.

The measurement noise is also assumed to provide an error covariance

matrix with constant values on the diagonal. That is:

Ev(k)vT(k)=Rδkl (13)

The remaining part of the algorithm follows the general linear Kalman

ﬁlter, but with an update in state matrices for each time step given the

wave gauge measurements. (Kalman, 1960)

EXPERIMENT

An experiment campaign was conducted at the wave ﬂume facility at

LASIF, Marseille University in France, 2018. The laboratory frame-

work covers a bottom-ﬁxed mono-pile structure subjected to wave

loading. Although the model does not descend from any physical

(full scale) counterpart, it is roughly scalable as 1/100 given the con-

ditions in the North Sea. The model with key dimensions is shown in

Fig. 2. The pile is made from hollow sections leaving the inside dry

at all times. The sections are bolted together through internal ﬂanges.

The model is made from plexiglass and equipped with 16 uniaxial ac-

celerometers (Brüel & Kjær type 4508-B 100mV/g). Eight of these are

facing the incoming waves (red), and eight are facing the transverse

direction (blue) as indicated in Fig. 2 b). A wave gauge is installed

to monitor the surface elevation near the structure. It is positioned at

the front of the pile with a transverse offset of 200 mm from the pile

centre. The model is attached to an ATI load cell which captures the

global loading on the system including dynamic ampliﬁcation. This is

used for veriﬁcation purposes only. The dynamic characteristics may

be seen in Table 1.

A unidirectional random sea is released towards the structure from the

front, and the corresponding vibrations are captured by the sensors

along with the surface elevation of the passing waves.

Wave Spectrum

The waves are created as a series of random waves generated from a

JONSWAP (Joint North Sea Wave Project) wave spectrum. A sample

of the wave train is shown in Fig. 3 along with the spectrum. The

structure is tested for 20 minutes at a sampling rate of 1024 Hz. The

spectrum is made using the Welch averaging technique with segments

of 32 s and a 50 % overlap (Welch, 1967).

Load cell

Max water level

Accelerometer

200

400

150

3

5

200

typ

900

385

a) b) c)

50

Wave gauge

AFT

FWD

STB

PORT

1

2

6

8

10

14

16

12

3

5

7

9

4

15

11

15

13

13

4

Fig. 2. Experimental setup. a) shows the side view cross section including the overall dimensions of the model, b) indicates the sensor position and

direction, c) shows a picture from the experiment. Dimensions are in mm. The ﬁgure is taken from (Vigsø et al., 2018b).

0 0.5 1 1.5 2 2.5 3

Frequency [Hz]

0

1

2

3

4

5

6

Wave Spectrum S(f) [m2/Hz]

Peak Period Tp =1.5 s

Significant wave height Hs = 6.4 cm

520 522 524 526 528 530

Time [s]

-4

-2

0

2

4

Surface elevation [cm]

a) b)

Fig. 3. Random sea state: a) shows the wave spectrum of the random waves measured from a wave gauge in close proximity to the pile. b) shows a

sample of the unﬁltered wave gauge time signal. The signiﬁcant wave height is calculated as the mean of the 1/3 highest waves using zero

down-crossing separation.

246810

Frequency [Hz]

-110

-100

-90

-80

-70

-60

-50

Singular values of spectral matrix

[dB rel. to 1 g2/Hz]

520 522 524 526 528 530

Time [s]

-2

-1

0

1

2

3

Acceleration of sensor 9 [g]

10-3

a) b)

Fig. 4. Structural accelerations: a) shows the acceleration spectrum in terms of the singular values (Brincker and Ventura, 2015). Only the three

most signiﬁcant singular values are shown in descending order. b) shows a sample of the corresponding (unﬁltered) accelerations for

sensor 9 during the random sea. Sensor 9 is positioned above the water level and is facing the incoming waves.

Acceleration Spectrum

As a consequence of the random waves, the structure will initiate some

vibrations. The vibrations are shown in Fig. 4 in both the frequency

domain and the time domain. The frequency domain representation is

made using the Welch averaging technique with segments of 32 s and a

50 % overlap. The spectra from the 16 accelerometers are decomposed

into singular values and the three most signiﬁcant are shown.

From Table 1 and Fig. 3 a) we note that the natural frequency of the

structure is well above the main frequency content of the waves and

thus expect the response to be mainly quasi-static. When we examine

the frequency content of the accelerations shown in Fig. 4 a), we see

how the wave peak frequency fpis scattered between 1.0fpand 2.5fp.

The second thing that appears is how the natural frequency is repre-

sented in the response at approximately 8.4 Hz. The remaining modes

between 35 and 50 Hz are barely visible in the signal.

From the acceleration spectrum in Fig. 4 a) it is noted that cross vibra-

tions are present during the experiment as the second singular value

near 8.6 Hz corresponds to the ﬁrst bending mode in the transverse

direction. See also Table 1.

Modal Analysis

The acceleration spectrum generated from the one-directional random

waves yields a poor basis for operational modal analysis (OMA). In-

stead, the OMA is based on additional loading, either by including

wind or brush strokes by the testing team. The modal analysis is omit-

ted from this paper and the reader may refer to (Vigsø et al., 2018b) for

further details. The ﬁrst ﬁve modes are summarized in Table 1. The

Table 1. Estimated modal parameters. See (Vigsø et al., 2018b) for

more details.

Natural frequency Damping ratio Comment on mode shape

fn[Hz] ξn[%]

8.37 3.7 First bending mode

8.66 3.5 First bending mode, trans.

37.6 4.9 First torsional mode

40.3 3.7 Second bending mode

47.0 4.4 Second bending transverse

246810

Frequency [Hz]

-100

-80

-60

-40

-20

0

20

Load Spectrum

[dB rel. to 1 (Nm)

2/Hz]

520 522 524 526 528 530

Time [s]

-10

-5

0

5

10

Overturning moment [Nm]

Estimated

Measured

a) b)

Fig. 5. Global loading: a) shows the load spectrum (moment) during the random sea. Red is the estimated load while black is measured by the load

cell. b) shows a sample of the corresponding time signal. The load cell data is low-pass ﬁltered at 50 Hz.

experimental mode shapes are scaled and expanded using a surrogate

FE model such that the full ﬁeld deformation of the structure can be

estimated. The effect of the hydrodynamic added mass is included as a

non-structural mass that only affects the horizontal inertia and not the

torsional. The added mass is included in the updated FE model and is

mainly used to scale the mode shapes.

Wave Loading

The global loading on the structure is measured by a load cell at the

base. The load cell measures all six degrees of freedom, but we shall

conﬁne ourselves to the overturning moment in the direction of the

waves. Once again, the spectrum is computed by means of Welch

averaging with segments of 32 s and a 50 % overlap, and shown in

Fig. 5 a). The data from the load cell is contaminated by harmonic

noise at 50 Hz. Hence the data shown in Fig. 5 has been low-pass

ﬁltered to compensate for this. When comparing the time signal in

Fig. 5 b) and Fig. 3 b), we see that the structure experiences the max-

imum loading before the apparent wave reaches its maximum crest

height.

When examining the spectrum in Fig. 5 a), it is seen that the resonant

frequency of the structure is repeated in the reaction forces. This is

due to the dynamic interaction between the loading and the system.

RESULTS

During random sea, the structural accelerations and wave gauge mea-

surements are recorded and fed into the algorithm. The algorithm then

provides a real-time estimate on the wave load (red). The estimated

wave load is compared to the measurements from the load cell at the

base (black). The comparison is shown in Fig. 5 in terms of the over-

turning moment. We see that the overall trend is captured through the

indirect method, although some discrepancies are seen:

When examining the load spectra in Fig. 5 a), we see that the estimated

spectrum includes some low frequency content that is not seen by the

load cell. We expect that this is a result from the performance of the

accelerometers at low frequency (<0.5 Hz).

Next, we observe how the energy is less near the resonant frequency

of 8.4 Hz. We recall the objective of the algorithm which is to identify

the input, i.e. the wave loading, while it should omit any dynamic

ampliﬁcation from the structure. Since the peak at 8.4 Hz represents a

dynamic ampliﬁcation from the ﬁrst mode, it also represents a ﬂaw in

the identiﬁcation algorithm as it should not have been represented.

In Fig. 5 b), a comparison is made between the estimated input (wave

load) and the measured reaction force in the time domain. Again, since

the measured force includes the dynamics of the structure and the es-

timates do not, it is not completely comparable. However, since the

primary part of the response is of a quasi-static nature, the difference

should be limited and we will not pursue to adjust for this.

When using a real-time algorithm as presented in this paper, it is com-

mon to encounter a phase shift in the estimate compared to the mea-

sured. Note that the result shown in Fig. 5 b) is adjusted for this by an

offset of 0.09 s.

DISCUSSION

When applying this type of Kalman ﬁlter approach for load identiﬁ-

cation, it is a challenge when only accelerations are available as the

state estimates are likely to drift. Filtering and detrending may im-

prove this issue. As an alternative, one can convert an accelerometer

into a pseudo displacement sensor through integration in the frequency

domain, however, this makes the real-time implementation more chal-

lenging.

The performance of the Kalman ﬁlter is dependent on the tuning of

the noise models, i.e. the error covariance matrices in Eq. (12) and

Eq. (13). The parameters chosen for this study are as follows:

Qii

∼100,S∼109and Rii ∼10 (14)

We admit that access to the measured load is very convenient when

tuning these parameters for the Kalman ﬁlter.

In Eq. (4), we have assumed that the spatial distribution is constant

in time apart from the stretching. This is a crude simpliﬁcation for

a random sea as waves with shorter wavelengths will concentrate the

load near the surface and vise versa. The spatial distribution could also

have been reﬁned in such way that it adds a contribution from a drag

dominated scenario and an inertia dominated scenario. This has not

been pursued in this paper.

If wave breaking or slamming occur near the structure, the frequency

content of the accelerations is expected to broaden and hence yield

modal activity from higher modes. In order to estimate these loads,

more modes are likely needed and possibly another local load model

for the impact area. This will be the focus for future study.

CONCLUSION

Through this study, we have experienced the challenges associated

with both indirect and direct measurements of wave loading when

dealing with both quasi-static and dynamic response. Although chal-

lenges remain, we have demonstrated the feasibility of indirect esti-

mation of wave loads on a structure during random seas. The results

are based on experimental system identiﬁcation (OMA) and driven by

acceleration measurements. Through simultaneous monitoring of the

surrounding sea, the wave loading was constrained to the wetted area

of the structure.

We have examined a scenario where the natural frequency of the struc-

ture is well above the wave peak frequency. This yields a high portion

of quasi-static response. Note that for actual offshore installations, the

frequency gap between the peak wave and the structure may be smaller

and thus the dynamic response may be larger, i.e. the Valdemar plat-

form in the North Sea, (Skafte et al., 2014).

ACKNOWLEDGEMENTS

The authors acknowledge the funding received from the Centre for Oil

and Gas – DTU/Danish Hydrocarbon Research and Technology Centre

(DHRTC).

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