**Crest height calculations**

By Arne Kvitrud, Sondre Nordheimsgate 9, 4021 Stavanger.

The
document was made 2.9.1997, but put on Internet 15.9.2002.

My interest of the shape of
the waves was born when discussing ringing; where the shape of the waves is of
major interest. The interest is connected both to horizontal and vertical
asymmetry in the waves. The subsidence of the Ekofisk, has called upon many
studies to be performed by several companies and persons and having several
clients.

Here I will restrict myself
to the discussed connected to the vertical asymmetry or the height of the wave
crest. I will briefly describe:

a) observed damages on platform decks

b) theoretical thoughts on crest heights

c) the ringing crest height discussions

d) the Ekofisk crest height discussions

e) I will try to make a conclusion and give a text draft for
the ISO-standard

__a) Incidents on platform decks__

I have not done any survey
for this purpose, but I have earlier written down the following incidents
related to waves on deck structures.

a) Eldfisk 2/7 Bravo-
jacket- 24.11.1981 was the northern wall of the "Maintance Shop" hurt
by waves; 20m above LAT. The significant wave height were about 11.8m.

b) Ekofisk 2/4 Tank - Doris
tank - 2.1.1984 was several containers on the deck (17m above LAT) moved and
one was trown into the sea. Signifikant wave height was about 9m.

c) Ekofisk 2/7 Alfa -
jacket - 3.2.1984. Many
damages on the deck 15m above LAT. The significant wave height was about 10m.

d) Ekofisk 2/4 Tank -
28-29.2.1988 a deck plate was buckled upward at 15.8m above LAT. Largest
measured wave height was 21.8m and the largest measured crest was 10.6 above
LAT.

e) Ekofisk - most of the
platforms - 12.2.1990 wind and waves caused damage to several mainly secondary
installations on the decks, at a cost of about 10 million dollars.

f) Veslefrikk
semisubmercible - December 1992 a wave hit the deck without making any damage.

g) Snorre - TLP - 4.1.1993
a wave hit the deck and damaged a burner boom.

h) Veslefrikk
semisubmercible - 31(?).1.1995 a wave hit the deck and buckled some beams
upwards. Destroyed some gratings.

i) Snorre - TLP - 31.1.1995
a wave hit the deck and damage was done on secondary structures.

j) 16/11-E - jacket -
31.1.1995 a wave of 20.6m hit the deck (?) putting out lights and sweeping away
scaffolding.

Many of the incidents have
then occurred in the Ekofisk area, which is the area with the lowest water
depth on our shelf.

__b) Theory__

The skewness and the waves
have be an interest for research through several decades. Here I have listed a
few of the previous results which have been refered to in the Norwegian
discussions.

1. Haring and Heideman
(1978) examined the skewness of ocean waves in the Gulf of Mexico. They
presented a formula for the short term distribution and calculated a set of
parameters: b1 = 4.37 and b2 = 0.57.

2. In the mid 1980-ies a
lot of work was done by Kjeldsen and Myrhaug. Kjeldsen (1984) showed that a
restriction in steepness of waves in irregular sea states of 0.1411 is not
valid. The number of 0.1411 is achieved in wave trains. He demonstrate that
transient waves can occur with amplitudes of 73% of the wave height in
irregular sea states. The transient waves can have a steepness of 0.21.

3. Longuet-Higgins (1986)
described theoretically the difference between a measuring system which is
looking at the waves from above (as laser/radar) and measurements done by a
buoy floating free in deep water. The down looking instruments give a Eulerian
observation and the buoy give a Lagransian observation. The Eulerian
observation are in general the correct case. He than examines the differences
with a buoy. The buoy will not follow the wave-elevation path and always give a
to low skewness.

Arguments against the use
of his results are that the buoys are moored and not free floating.

4. Longuet-Higgins (1988)
follows up with what is happening at finite water depths. The shape of Lagrangian
waves depend of the water depth and are most symmetric when k·d = 1.93.

5. Winterstein and
Martinsen (1992) examined the differences between the buoy and radar
measurements and revealed a major difference between the instruments with
regard to skewness. They calculated the relationship between skewness and
kurtosis as :

skewness = 3 * s * k * ( D1 + D 2 )

where D i are dimensionless
parameters dependent of the water depth.

k is the wave number and s is the variance of the surface elevation

kurtosis = 3 + ( 1,33 * skewness ) 2

If the skewness is known
the kurtosis is also known. The results fits well with the results from Ekofisk
radar. They claim that the buoys are giving wrong results because of an error
in the software.

The skewness coefficient is
expressing the symmetry properties of the distribution. An increase in the
coeffisient is telling us that the extremes are increasing. The distribution is
symetrical if the skewness is zero. Positive values are giving a skewness
toward higher values.

The kurtosis coefficient is
expressing the ratio between small and large values. An increase is indicating
that the number of large values are increasing. For a Gaussian prosess the
kurtosis koefficient is 3.0. An increase in the value from 3.0 to 3.1 will increase
the wave height with 0.3 to 0.4 m at Gullfaks (Haver).

6. Even if it is related to
the discussion at Ekofisk I will briefly summarize some of the discussion from
Smed et al (1994). They discuss the ability of different instruments to measure
sea spray.

Waveriders will not measure
wave spray but might not measurer all the crests. Wavestaff measurements will
depend on the water content in the sea spray. EMI lasers are heavily influenced
by sea spray. The pressure cells will not measure sea spray and give time
series closely to those measured by the wavestaff and laser, except for the
most peaked crests.

__c) Asymmetry for ringing__

Model testing of Heidrun
(TLP), Draugen (concrete monotower) and Troll A (concrete - 4 legs) showed that
the loading was very dependent on the shape of the waves. A discussion was
rised if the laboratory waves reflected the offshore waves. An investigation
was performed by Barstow et al (1992); followed by Haver (1993) and Krogstad
and Barstow (1994).

1. Barstow, Bern og Krogstad
(1992) got a factor of 0.7 for Draugen between wave crest and wave height as a
conservative value. The show that the buoys give a completly different picture
compered to the laser measurements. The buoy is consistently on the low side.
They regard the buoy data to be unuseful for the purpose. The average factor
increase by increasing wave height, but the standard deviation is reduced.

2. Barstow and Krogstad
(1993) examined waves measured by laser at Gullfaks C, Tern, WADIC experiment
(at Ekofisk) and Hutton. They also examined the skewness and kurtosis of wave
buoy data showing that there were a significant difference between the two
measuring instruments. They concluded on numbers for skewness which were even
more asymetric than the laboratory waves. They had a skewness of 0.18 and a
kurtosis value of about 3 or slightly higher.

3. Sverre Haver (1993) went
through a similar exercise based on the laser measurements at Gullfaks C.
Haver's results was used as design basis for Troll A. He recommended to use a
corelation between skewness and kurtosis as developed by Winterstein and
Martinsen (1992). Han got a skewness of 0.15 with a standard deviation of 0.07.
Further a kurtosis of 3.09 with a standard deviation of 0.24. He also
demonstrated that the kurtosis varies little with the steepness.

According to Haver (1993)
has Rodenbush found a similar relation. Vinje has in 1987 also found a relation
between steepness and waveheights. Haver says that Vinjes relation give a
significant higher steepness than observed on Gullfaks C.

__d) Asymmetry
at Ekofisk__

At Ekofisk, Phillips
Petroleum (PPCoN) ordered Danish Hydraulic Institute (DHI) to make a metocean
study. This study was verified by DnV on behalf of PPCoN. Later Statoil (Sverre
Haver) made a study and got it verified by Norwegian Hydrodynamic Laboratorie
(NHL) in Trondheim. In addition I have made some calculations and got help in
some evaluation by Barstow and Krogstad.

All the studies are ending
up with a significant 100-year-wave height in the order of 13.5 m and an
individual 100-year-wave height in the order of magnitude 24.3 - 25.7 m. 25 m
is probably the most likely (my feeling). The results are based on synoptic (3
hours) waverider and radar measurements starting in 1980 and/or continuous
storm recording since 1986.

For the crest evaluation
there is though a significant difference ranging from 14.2 m (DHI) to 18.8m
(NHL, 1994 a) above LAT. The differences in the results are based on use of
different instruments (radar/laser versus buoy) and different statistical
approaches. Lately wave staff and water pressure measurements have also been
brought into the discussion.

1. DHI (1993) based their
evaluations on 22 storm crest extremes from continuous records of storms from
1986-92. They used the maximum total crest elevation (crest and water level)
relative to the 3 hour peak significant wave height for the storm and
extrapolated to the design sea state.

Waverider wave height data
were combined with Plessey Radar water level data, and some allowance was made
for contributions from lower and higher sea states and for possible crest
underestimation by the Waverider.

The Plessey Radar data were
not used for crest assessments due to the influence of the nearby Ekofisk 2/4
Tank/Barrier Wall and 2/4 Hotel combined with the sensitivity of the sensor to
wave foam/sea spray.

The design sea state was
estimated using a POT/Gumbel distribution and MLE extrapolation. A design crest
elevation of 13.5 m above MSL or 13.2m above SWL was recommended.

DHI has made a new exercise
in which the distribution of the largest crest height during the most severe
storms was established (with some allowance for crest underestimation by the
Waverider). By integration over the storms to be expected in 100 years, the
central estimate for the design crest may be found. By combination with the
simultaneously recorded Plessey Radar water level data, a design crest value
above MSL was found.

Later the Waverider data
have been corrected for the transfer function of the buoy integration circuit
(resulting in increased sea states and reduced crest elevations).

Finally, DHI analysed data
from the WADIC experiment, including data from: SYMEX Wavestaff, Pressure
transducers, EMI Lasers and a Waverider Buoy. Based on the total amount of data
analysed, DHI has then recommended a revised design crest elevation of 14.0 m
above MSL (or Cr_{ max} = 13.7 m).

DHI maintains that EMI
Lasers when mounted on platform decks yield too high crest values due to a
combination of local effects around the structures and the sensitivity of the
laser sensors to sea spray and foam. In the view of DHI it is not possible to
correct these data to arrive at data useful for the accurate estimation of
design crest elevations for free field waves.

2. Dahl et al (1993) using the buoy data, a probabilistic
(PROBAN) method and log normal distributions end up with a 100-year value of
14,6 m above SWL and a standard deviation of 1.0 m, but they can accept 13,5 m
from MSL "for the time being". They found 1.4m from LAT to SWL, giving
16.0 m above LAT.

3. Haver (Statoil, 1994) analysed the crest values using
different approaches. He only used the radar data.

a) First he used the
formula of Jahns and Wheeler (1972) and the parameters from Haring and Heideman
(1986) approach and got 17 m above SWL,

b) He also used a method
based on skewness and kurtosis derided from Gullfaks C-data, with a depth
correction to Ekofisk after Vinje and Haver (1993), giving 16 m above SWL.

c) He used at heremite
expansion technique after Winterstein (1988) and got appr. 15 m above SWL.

d) He used a Stokes
expansion approach after Tayfun (1980) and Vinje (1989) and got about 15.5 m
above SWL.

Haver concluded the
100-year crest height to be in the range of 15.5 - 17.0 m above SWL.

4. Torsethaugen and Mathiesen (1993) used the Haring and
Heideman (1978) approach with their Gulf of Mexico parameters giving 17.2 m
above SWL. They also used a 18th order Fourier theory giving 15.7m, they
recommend 17.2m above SWL.

5. Mathiesen and Torsethaugen (1994) used the same approach,
but checked the Haring and Heideman parameters based on Ekofisk data. The
parametres were conservative and they recommended 17.2 m above SWL.

6. Myself I (1994) calculated extremes based on both buoy and
radar data. Using annual extremes (AE) and POT with variable threshold. I used
Gumbel for AE and the exponesiell distribution and a low threshold for POT. The
data was fitted using LLS. I got about 16 m for the buoy and 17 m for the radar
above LAT.

7. Samset and Krogstad
(1994) checked the short term distribution of DHI and say that it gave a small
underproduction because it reduce the natural variation. They evaluated the
underprediction to be about 0.5m.

8. Krogstad (1994)
calculated the expected crest height at Ekofisk to be 15.0m above SWL using
laser data and 13.3m above SWL using a Gaussian simulation. He used a short
term discription for the crest height for given significant wave height and
integrated over all the sea states from 1980-93.

All the differences in
different instruments is demonstrated in a figure made by Harald Krogstad. It
is based on a short term description of the sea states. The DHI values are
above the gausssian curve.

** **

**e) Text for
the ISO Standard**

I have the following
proposal for a text to the ISO standard:

*"For crest elevation calculations the approach of
Haring and Heideman (1978) can be used".*

Their approach and
parameters should be conservative also for North Sea conditions.

** **

**References:**

Barstow S F and Krogstad H
E: Analyses of full scale wave data for the Heidrun field development project;
final report, Oceanor report OCN-R-93029, Trondheim, 1993.

Barstow S, T-I Bern og
Harald Krogstad: Wave crest analysis for the Draugen field, Oceanor rapport OCN
R-92093, 7.10.1992.

Bjerken S and Barstow S F:
Preparation and quality control of wave data from the wavestaff used in the
Wadic experiment, Oceanor note OTNTN-94001, Trondheim, 5.5.1994.

Dahl F E, Gran S, Hagen Ø
and Skåtun H: Verification of environmental design criteria for the greater
Ekofisk Area, DnV-Defence report 036-93/Rev. 0, Oslo, 1.12.1993.

DHI: Greater Ekofisk
Environment design criteria study, Copenhagen, 1993

Haring and Heideman J: Gulf
of Mexico rare wave return periods, OTC 3230, Houston, 1978

Haver S and Anfindsen: On the
adequacy of the Gussian assumption regarding the sea surface elevation, Statoil
report F&U-ST 88010, Stavanger, Nov. 1988.

Haver S: A discussion of
the wave conditions in the Northern North Sea, Technical note, Shell, Troll
prosjektet, 18.12.1992.

Haver S: Extreme wave
conditions for the Ekofisk area, Statoil, Stavanger, 1993

Jahns H O and Wheeler J D:
Long-term wave probabilities based on hindcasting of severe storms OTC-1590,
Houston, 1972.

Kjeldsen S P: Dangerous
wave groups, Norwegian Maritime Research, no 2, 1984

Krogstad : Ekstreme bølgehøyder på Ekofisk, Sintef note 29.11.1994.

Kvitrud A: Bølgeforholdene på Ekofisk, Note 27.1.1994.

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London, 1986

Longuet-Higgins M S :
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depth, Journal of fluid mechanics, vol 186, pp 34-316, London, 1988.

Marthinsen T og S R
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international offshore and polar engineering conference, San Francisco, USA,
14-19. juni 1992.

Mathiesen M and K
Torsethaugen : Ekofisk Short-term Distribution of Crest Heights, Sintef, 1994

Mathiesen M and K
Torsethaugen : Some aspects of the wave climate at the Draugen field, Sintef,
1992

Myrhaug D og S P Kjeldsen :
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and asymmetry in deep water waves, Applied Ocean research, 1984, vol 6 no 4
side 207ff.

Myrhaug D og S P Kjeldsen :
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Ocean Engineering, vol 13, no 6 side 549ff, 1986.

Samset O and Krogstad H E:
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Sintef, Trondheim, 30.9.1994.

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Jacobsen : Analysis of wave heights measured from WADIC, DHI, Copenhagen,
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Mathiesen M: Ekofisk - extreme wave heights, Trondheim, 1993

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S Barstow : measurement of wave properties in extreme seas during the WADIC
experiment, Proceedings OTC, OTC 5964, Houston, 1989

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no 3, 1989.