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.
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.
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.
Longuet-Higgins M S : Eulerian and Lagrangian aspects of surface waves, Journal of Fluid Mechanics, London, 1986
Longuet-Higgins M S : Lagrangian moments and mass transport in Stoke waves, part 2. Water of finite depth, Journal of fluid mechanics, vol 186, pp 34-316, London, 1988.
Marthinsen T og S R Winterstein: On the skewness of random waves, Proceedings of the second (1992) 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 : Parametric modelling of joint probability density distributions for steepness and asymmetry in deep water waves, Applied Ocean research, 1984, vol 6 no 4 side 207ff.
Myrhaug D og S P Kjeldsen : Prediction of occurrences of steep and high waves in deep water, Journal of waterway, port and ocean engineering, volum 113, nr 2, 1987.
Myrhaug D og S P Kjeldsen : Steepness and asymmetry of extreme waves and the highest waves in deep water, Ocean Engineering, vol 13, no 6 side 549ff, 1986.
Samset O and Krogstad H E: Analysis of the Danish Hydraulic Institutes method for design Wave Conditions, Sintef, Trondheim, 30.9.1994.
Smed P F, C Dahl and V Jacobsen : Analysis of wave heights measured from WADIC, DHI, Copenhagen, 7.11.1994.
Torsethaugen K and Mathiesen M: Ekofisk - extreme wave heights, Trondheim, 1993
Vartdal L, H E Krogstad og S Barstow : measurement of wave properties in extreme seas during the WADIC experiment, Proceedings OTC, OTC 5964, Houston, 1989
Vinje T: The statistical distribution of wave heights in a random seaway, Applied ocean research, vol 11 no 3, 1989.