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Development of a bi-modal directional wave spectrum
Ocean Engineering 105 (2015) 104–111
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Development of a bi-modal directional wave spectrum R. Panahi, A. Ghasemi K., M. Shafieefar Civil and Environmental Engineering Department, Tarbiat Modares University, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 17 May 2014 Accepted 12 June 2015 Available online 3 July 2015
Addressing sea state in a coastal region of Gulf of Oman by a directional wave spectrum for the very first time is the main focus of this study. The region encounters wind-sea as well as swell. So, proper modeling of the sea state requires in general a double peak spectral model. The research is firstly conducted to calibrate nondirectional Torsethaugen double peak spectrum for the region by entering the separation frequency; a frequency in which wind-sea and swell parts could be divided. This novel calibration procedure is simple while results in much better outcome. Besides, one has to decide about an appropriate Directional Spreading Function (DSF) for the wind-sea and swell components. Then, nine possible combinations of three well-known DSFs have been investigated and calibrated to provide maximum conformity between observed and modeled directional wave spectra. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nondirectional wave spectrum Directional Spreading Function Calibration Separation frequency
1. Introduction A trustworthy statistical modeling of wave directional properties at a specific location is a necessary prerequisite to ensure design precision and accuracy in coastal and offshore regions. Here, the spectral formulation, stemming from the early work of Phillips (Phillips, 1958), is among practical tools in ocean engineering. For the case of directional spectral formulation it actually addresses wave energy and its distribution over different wave frequencies and directions. Various methods like maximum entropy method, maximum likelihood method and Bayesian method have been developed over years to estimate a directional spectrum from measurements (Longuet-Higgins et al., 1963; Borgman, 1969; Isobe et al., 1984; Kobune and Hashimoto, 1986; Kuik et al., 1988). Such methods have been reviewed and categorized by Benoit et al. (1997). Among them is a common practical approach in which a directional spectrum Sðf ; θÞ is expressed as follows: ð1Þ S f ; θ ¼ Sðf ÞD f ; θ in which Sðf Þ is nondirectional spectrum and Dðf ; θÞ is Directional Spreading Function (DSF). Now, for coexistence of local wind generated wave and distant swell wave; from now on we briefly call them wind-sea and swell, respectively; one expects at least a double peak nondirectional spectrum. It is commonly assumed that nondirectional wave spectrum is of a single mode form, and can be well modeled by a well-known standard spectrum such as JONSWAP or Pierson– Moskowitz spectrum. This approach is totally reasonable for E-mail address: shafi[email protected] (M. Shafieefar). http://dx.doi.org/10.1016/j.oceaneng.2015.06.017 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
severe sea states. However, moderated and low sea states; as one encounters in most Iranian coastal regions at Gulf of Oman; are often of a mixed nature, consisting of wind-sea as well as swell. Then, such sea states should be addressed by a double peak nondirectional spectrum. To characterize this issue, an approach was pioneered by Strekalov and Massel (1971) to sum two individual spectra as follows: X Sðf Þ ¼ Si ðf Þ ¼ SW ðf Þ þ SS ðf Þ ð2Þ i ¼ wind sea; swell
here, SW ðf Þ and SS ðf Þ stand for wind-sea and swell components of the spectrum, respectively. This approach was used later by researchers to combine and modify available standard nondirectional spectra e.g. Pierson– Moskowitz or JONSWAP resulting in the well-known double peak spectra like Ochi–Hubble model and Torsethaugen model (Ochi and Hubble, 1976; Guedes Soares, 1984; Torsethaugen, 1993; Moon and Oh, 1998; Violante-Carvalho et al., 2004; Torsethaugen and Haver, 2004; Mackay, 2011). For DSF, there are some standard uni-modal forms as cosinepower distribution (Longuet-Higgins et al., 1963; Mitsuyasu et al., 1975; Hasselmann et al., 1980), wrapped normal distribution (Borgman, 1969; Briggs et al., 1995), wrapped-around Gaussian distribution (Mardia, 1972), hyperbolic distribution (Donelan et al., 1985), von Mises distribution (Abramowitz and Stegun, 1975; Hashimoto and Konube, 1986), Poisson distribution (Lygre and Krogstad, 1986) and Boxcar or Steklov distribution (Venugopal et al., 2005) which have been checked occasionally by other researchers based on observations (Ewans, 1998). However, no single model is universally accepted due to site specificity associated with particular formulations. In order to distribute a nondirectional spectrum over directions, it has been a common
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
practice to use distinct DSFs for distribution of wind-sea and swell parts as also recommended by rules and regulations (EM 1110-21100, 2006; DNV-RP-C205, 2010). So, for this situation it would be appropriate to rewrite Eq. (1) as follows: S f ; θ ¼ SW ðf ÞDW f ; θ þ SS ðf ÞDS f ; θ ð3Þ Wind part
Swell part
To this end, the paper aims at developing a directional spectrum Sðf ; θÞ for Chabahar coastal regions together with an introduction to a general practical approach for calibration of double peak spectra. Measured wave data in Chabahar bay in the northern coasts of Gulf of Oman are used to verify the proposed approach. The area is experiencing both wind-sea and swell regimes throughout the year especially in Monsoon seasons (Rashmi et al., 2013). As the border of Indian Ocean and Gulf of Oman, it is under construction and development activities and its field observations are recently released. Available data are briefly reviewed in the next section. Based on field observed spectra, nondirectional Torsethaugen double peak spectrum is firstly calibrated in Section 3 using a simple novel approach. Assessing some standard DSFs in Section 4 together with their calibration to maximize conformity, they finally resulted in developing a calibrated directional spectrum for the region.
2. Local measurements In order to monitor regional coastal waters a mega project had been initiated by Port and Maritime Organization of Iran since 2006. Fig. 1 shows the focal point of this paper together with measurement stations as stars for the Chabahar bay, also called Khalij-e Chabahar, which is around 251200 53″N and 601300 40″E in DMS (Degrees Minutes Seconds). Nortek Acoustic Wave and Current profiler (AWAC) had been used to gather such directional field data. Stations specifications are summarized in Table 1. Such raw data are processed using Nortek STORM software to recognize extreme events. The raw data come from direct measurements with no filter for anomalies e.g. gaps or spikes. Therefore, a simple code is developed to reject any records which do not pass certain quality requirements.
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calibrated by finding and tuning appropriate DSFs as presented in Section 4. For the purpose, Torsethaugen spectrum is nominated as the most recent well-known attempt to deal with double peak spectral presentation as ST ðf Þ ¼ ST W ðf Þ þ ST S ðf Þ in which T stands for Torsethaugen (Torsethaugen, 1993). Torsethaugen spectrum originally developed by combining two JONSWAP spectra. The spectrum is presented as a sum of wind-sea and swell components for j ¼ 1; 2 as follows (Torsethaugen, 1993): Sðf Þ ¼
2 X
Ej Snj ðf U T Pj Þ ¼
j¼1
2 X
Ej Snj ðf nj Þ
ð4Þ
j¼1
where: Ej ¼
1 2 H UT 16 Sj Pj
Snj ¼ G0 Aγ j Γ Sj γ Fj
ð5Þ ð6Þ
here, it should be noted that Torsethaugen only requires that the user provides two input parameters H s and T p . Its other main parameters are all expressed in terms of the significant wave height ðH s Þ and spectral peak period T p . At first, T f ¼ af H s1=3 should be calculated as the peak period in a fully developed sea in which af is slightly sensitive to the fetch length. Then, one could decide if the sea state is wind-sea dominated T P oT f or swell dominated T P 4 T f . It was shown later by Torsethaugen and Haver (2004) that some of the parameters for the general form of Torsethaugen spectrum, considering its complex formulation, are of marginal importance for design purpose. So, they introduced the simplified form as presented in Tables 2 and 3 for wind-sea dominated and swell dominated sea states. Here, bold parameters are those empirical coefficients tuned in this research together with af as it has the main rule in switching between formulations. 4 4 Besides, in Eq. (6) G0 ¼ 3:26, Γ Sj ¼ f nj exp½ f nj and Aγ j as well as γ Fj are functions of peak enhancement factor γ as follows: 2 2 ð7Þ γ F1 ¼ γ exp 1=2σ ðf n1 1Þ
γ F2 ¼ 1 (
0:07 f nj o 1 0:09 f nj Z 1
ð8Þ
3. Calibration of the nondirectional spectrum
σ¼
Nondirectional spectrum has always its own importance and application irrespective of its distribution over different angles. This motivated the authors to firstly focus on calibration of nondirectional spectrum as the sum of SW ðf Þ and SS ðf Þ based on the form introduced by Eq. (2). Then, the directional spectrum is
1:19 Aγ 1 ¼ 1 þ 1:1 ln γ =γ
ð10Þ
Aγ 2 ¼ 1
ð11Þ
measurement station
25°20'53" N and 60°30'40" E
Fig. 1. Chabahr bay in Gulf of Oman and measurement stations.
ð9Þ
here, γ is calculated using Tables 2 and 3. However, such a comprehensive parameterization of the spectrum has been originally shaped based on measurements at deep sea; Statfjord field in Norwegian waters (Torsethaugen, 1993); and later supported by other data close to that location (Ewans et al., 2006; Bitner-Gregersen and Toffoli, 2009). So, it has to be implemented for other locations like Gulf of Oman with care. Here, the spectrum is examined in three forms of standard version ðST ðf ÞÞ, calibrated version ðSCal T ðf ÞÞ and Separation Frequency Implemented (SFI) calibrated version ðSSFI Cal T ðf ÞÞ in order to model the sea states at the region. For ST ðf Þ, original values have been exactly used as the coefficients (Torsethaugen, 1993). To catch SCal T ðf Þ coefficients, minimizing Root Mean Square Error (RMSE) of modeled spectrum values, ΔA=A and Δf p =f p have been set as calibration targets using nonlinear Generalized Reduced Gradient (GRG) algorithm (Lasdon et al., 1973). Here, A and f p are the area under the spectrum and its peak frequency, while Δ
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Table 1 Wave measurement stations at the edge of Chabahar bay. Station no.
1 2 3
Local name
AbAmigh Behesthi Bargin
Coordination in DMS
Measurement interval
N
E
Begin
End
251150 40″ 251170 22″ 251170 30″
601390 03″ 601350 40″ 601280 30″
Aug., 27,2006 Aug., 30,2006 Aug., 25,2006
June,14, 2007 Sep.,2, 2007 Sep.,20, 2007
Table 2 Torsethaugen spectrum-wind-sea (Torsethaugen and Haver, 2004).
Secondary peak
HW1 ¼ RW U HS
HW2 ¼ ð1 RW 2 Þ0:5 U H S T p2 ¼ T f þ b1 γ2 ¼ 1
RW ¼ ð1 a10 Þe
εl a1
Distance to nearest coast (km)
30 11 10
2 0.3 1.5
dominated
Primary peak
T p1 ¼ T p γ 1 ¼ kg W U Sp 6=7 2
Depth (m)
þ a10
T T
εl ¼ Tff Tpl T l ¼ ae H0:5 S H W1 Sp ¼ 2π g T 2 p1
Fig. 2. Definition of different frequencies on a typical double peak spectrum (red line) based on separated wind-sea and swell spectra (blue lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 Torsethaugen spectrum-swell (Torsethaugen and Haver, 2004).
dominated
Primary peak
Secondary peak
HS1 ¼ RS U H S
H S2 ¼ ð1 RS 2 Þ0:5 U H S T p2 ¼ T f 1 γ2 ¼ 1
T p1 ¼ T p γ 1 ¼ γ f ð1 þ a3 εu Þ 2 RS ¼ ð1 a20 Þe
εu a2
þ a20
T T
εu ¼ T pu T ff T u ¼ au γ f ¼ kg S U Sf 6=7 HS Sf ¼ 2π g T 2 f
stands for the absolute difference between modeled and observed values. It should be noted that while f p is among inputs of Torsethaugen spectrum itself, spectrum coefficients change during the calibration procedure. So, the calibrated spectrum does not necessarily keep its peak at f p and considering this constraint is important. This is a common methodology as conducted in TMA spectrum calibration (Moon and Oh, 1998) or in combination of two JONSWAP spectra (Violante-Carvalho et al., 2004; Feld and Mørk, 2004). Finally, for SSFI Cal T ðf Þ, a separation frequency f s is calculated for observed double peak spectrum using the method proposed by Hwang (2012). Then, it is entered into the procedure to calibrate SW ðf Þ and SS ðf Þ parts separately, as follows (see Fig. 2 for definitions): First step: 1. Divide observed spectrum into S1 0-f s and S2 f s -f max . 2. Calibrate ST S by S1 0-f s introducing f S max 4 f s . Here, f S max is the frequency in which ST S diminishes. 3. Modify S2 f s -f max to S02 f s -f max by subtracting ST S between f s and f S max . 4. Calibrate ST W by S02 f s -f max introducing f W min o f s . Here f W min is the frequency in which ST W diminishes.
Fig. 3. Scatter diagrams of Hs vs Tp for wind-sea and swell spectral components.
So, it should be noted that he main output of this step would be f W min and f S max as minimum frequency of wind-sea part and maximum frequency of swell part, respectively (see Fig. 2). Second step: 1. Extract S1 0-f S max and S2 f W min -f max from field spectrum with an overlap interval. 2. Modify S1 0-f S max to S01 0-f S max by subtracting ST W between f W min and f S max . 3. Calibrate ST S by S01 0-fS max . 4. Modify S2 f W min -f max to S02 f W min -f max by subtracting ST S between f W min and f S max . 5. Calibrate ST W by S02 f S min -f max . The main output of this step would be SSFI Cal T as the sum of two separately calibrated wind-sea and swell spectra with the Torsethaugen base as ST W and ST S , respectively. It also results in updated f W min and f S max .
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Table 4 Performance and coefficients of S T , S Cal Torsethaugen spectrum. Coefficients
4. Calibration of directional spectrum T
and SSFI Cal
T
as three versions of
Wind-sea dominated First peak
Standard Calibrated SFI calibrated
kg W
af
a10
ae
a1
b1
35.00 34.90 36.38
6.60 4.24 5.62
0.70 0.71 0.82
2.00 1.99 1.95
0.50 0.54 0.53
2.00 3.46 2.50
Swell dominated First peak
Standard Calibrated SFI calibrated performance Standard Calibrated SFI calibrated
107
Second peak
kg S
af
a20
au
a2
a3
35.00 34.9 20.33
6.60 4.24 5.62
0.60 0.53 0.80
25.00 33.00 36.65
0.30 0.10 0.10
6.00 1.00 1.00
RMSE
ΔA=A
1.57 1.05 0.64
7% 4% 3%
Δf p =f p 0.00% 0.00% 0.00%
After the second step, RMSE, ΔA and Δf p =f p are calculated by comparing observed spectrum with SSFI Cal T and this step is repeated to reach an acceptable convergence in statistical measures. It should be remembered that, the very first output of the procedure is separation of field wind-sea and swell components. So, in order to provide an outlook into measurement data H s is plotted against T p in Fig. 3 for wind-sea and swell spectral components, separately. It is evident that the peak period of wind-sea component varies in a narrower interval when compared with that of swell component. Besides, significant wave height is of the same order for two spectral components although it is slightly larger for the swell part. All in all, the calibration procedure effect on 10 coefficients of Torsethaugen spectrum is presented in Table 4. However, by reviewing the performance of three aforementioned versions in last rows of Table 4, it is obvious that the second attempt in calibration has come up with a better adjustment when compared with the first calibration output. Here, RMSEof spectrum values decrease meaningfully up to 40%; from 1.57 in case of standard spectrum into 0.61 in case of SFI calibrated spectrum. The ability to predict two distinct peaks of observed spectra is also assessed in terms of peak spectral value and its frequency as well as significant wave height of swell component as well as wind-sea component in Fig. 4. When talking about maximum values, it is obvious that the second try in calibration improved data modeling to a great extent. It should be noted that SCal T performs better in modeling peak points of swell part than those of wind-sea part. But, when looking at performance of SSFI Cal T it presents peak values of wind-sea as well as swell with almost a similar acceptable accuracy level as it is reported by correlation factor R2 . However, there are still some difficulties for SSFI Cal T to adjust with peak spectral values of wind-sea. This is exactly the case when assessing such spectra in terms of significant wave height. Torsethaugen performance is generally much better in presenting swell component Hs rather than wind-sea component Hs. The Ochi–Hubble spectrum was also used in the analysis reported above. The procedure was completely useful in statistical terms when the SFI calibrated version compared with two other ones. However, the result was not as successful as Torsethaugen SFI calibrated version.
Calibration of double peak directional spectrum while its nondirectional part has been already calibrated is all about finding appropriate DSF and its calibration for sea and swell parts, see Eq. (3). To do this, three well-known forms of DSF have been examined to find out which combination of such DSFs provides the best directional description of the wave spectrum. Here, it should be noted that calculated wave directions are based on the first pair of Fourier coefficients describing the mean direction at a given frequency. This method can be considered inadequate when wave trains from two independent directions are present. This typically occurs along coastal areas where structures transform incident wave energy and create reflected waves that are measured at the same location leading to a weighted average of the two directions. Since the wave measurement stations in this research are located in deep water and far from coastal structure, there is no such condition that wave trains from two independent directions are presented at the same time. 4.1. Cosine-power model The very first attempt to model DSF is getting back to (Pierson et al., 1955) in which they proposed a cosine type function as: ( 2 cos 2 θ π2 þ θ0 o θ o π2 þ θ0 D θ ¼ π ð12Þ 0 otherwise in which θ is wave direction and θ0 is main peak frequency direction also called mean wave direction. The model did not have any frequency input. So, LonguetHiggins et al. (1963) took another step forward which led later to lots of efforts in this area. They introduced a more complicated model which could be written in a generic form as: θ θ0 Dðθ; sÞ ¼ IðsÞ cos 2s ; π rθ rπ ð13Þ 2 in which: pffiffiffiffi π Γ ðs þ 1Þ IðsÞ ¼ i 2π Γ ðs þ 12 Þ
ð14Þ
here, Γ is Gamma function, i is a constant, equal to one for almost all standard cases, and s is spreading parameter which controls spreading around θ0 as an empirical function. There have been lots of efforts to propose different versions based on this model from Krylov (1966) in which he entered mean wave frequency into account to Mitsuyasu (1975), Goda and Suzuki (1976) and Holthuijsen (1983) who clearly showed that s varies with wave frequency and it could be related to the stage of wave development as a combination of wind speed ðU Þ and fetch length ðF Þ. So μa 8 > f of p < smax f =f p μb s¼ ð15Þ > : smax f =f p f 4f p here, it has been shown that especially for deep water, smax is 10 for wind waves, 25 for swells present with wind waves of relatively large steepness, and 75 for swells with wind waves of small steepness in case of μa ¼ 5 and μb ¼ 2:5(Goda and Suzuki, 1976). Similar dependence of s to wave frequency introduced by Hasselmann et al. (1980) in which they proposed other values for smax , μa and μb working on JONSWAP project. In their proposal, there is a weak dependence between μb and the ratio between wind speed and phase velocity i.e. wave age.
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R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Fig. 4. Modeling of spectral peak values and significant wave height. (a) first peak and (b) second peak.
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
It is also discussed that s could be related to the wave length associated with peak frequency of the spectrum (LP ) (Wang, 1992). Such studies continued later (Krogstad et al., 1997; Kumar et al., 2000; Kumar, 2006).
Table 7 Best combination of DSFs in representing sea states at the region.
Wind-sea dominated
4.2. Hyperbolic model A cosh function proposed by Donelan et al. (1985) to calculate DSF as follows: 2 D θ; β ¼ αβ cos h β θ θ0 ð16Þ in which α ¼ 0:5 and 8 γ 1 > β1 ffp for > > > > < γ 2 for β ¼ β2 ffp > > γ 3 > > > : β 3 ff otherwise
109
Swell dominated
Wind-sea part
Swell part
Cosine-power model Smax ¼ 20:2 μa ¼ 7:3 μb ¼ 8 i ¼ 0:05 Cosine-power model Smax ¼ 13:5 μa ¼ 1:5 μb ¼ 6:5 i ¼ 0:06
Poisson model xmax ¼ 0:6 υa ¼ 0:5 υb ¼ 1:3 η ¼ 0:1 Cosine-power model Smax ¼ 71:2 μa ¼ 2 μb ¼ 3:4 i ¼ 0:02
0:56 o ωωp o 0:95 0:95 o ωωp o 1:60
ð17Þ
p
here, β 1 ¼ 2:61, β2 ¼ 2:28 and β 3 ¼ 1:24. Also, γ 1 ¼ 1:3, γ 2 ¼ 1:3 and γ 3 ¼ 0:0. To modify this model, Banner (1990) showed that β is not necessarily a constant for the third condition. So, he suggested an equation with a role for f p . 4.3. Poisson model Poisson distribution firstly proposed by Lygre and Krogstad (1986) is as follows: 1 1 x2 D θ; f ¼ η 2π 1 2x cos θ þ x2
ð18Þ
in which η is a constant; equal to one for the standard model. Besides υa 8 > f of p < xmax f =f p υ b x¼ ð19Þ > : xmax f =f p f 4f p here, one could find a relation between Poisson model and cosinepower model. However, for the Poisson distribution it has a
Table 5 Performance of calibrated DSFs to represent wind-sea dominated seat states.
DSF model Swell part
Cosine-power Hyperbolic Poisson
Wind part Cosine-power
Hyperbolic
Poisson
RMSE ¼ 2:41 ΔV =V ¼ 0:06 RMSE ¼ 2:31 ΔV =V ¼ 0:04 RMSE ¼ 2:29 ΔV =V ¼ 0:04
RMSE ¼ 2:33 ΔV =V ¼ 0:06 RMSE ¼ 2:39 ΔV =V ¼ 0:08 RMSE ¼ 2:37 ΔV =V ¼ 0:13
RMSE ¼ 2:42 ΔV=V ¼ 0:08 RMSE ¼ 2:40 ΔV=V ¼ 0:11 RMSE ¼ 2:38 ΔV=V ¼ 0:16
Table 6 Performance of calibrated DSFs to represent swell dominated seat states Wind part
Swell part
DSF model
Cosine-power
Hyperbolic
Poisson
Cosine-power
RMSE ¼ 3:49 ΔV =V ¼ 0:19 RMSE ¼ 3:56 ΔV =V ¼ 0:24 RMSE ¼ 3:75 ΔV =V ¼ 0:20
RMSE ¼ 3:84 ΔV =V ¼ 0:2 RMSE ¼ 3:89 ΔV =V ¼ 0:20 RMSE ¼ 4:08 ΔV =V ¼ 0:22
RMSE ¼ 3:65 ΔV=V ¼ 0:22 RMSE ¼ 3:68 ΔV=V ¼ 0:23 RMSE ¼ 3:78 ΔV=V ¼ 0:25
Hyperbolic Poisson
Fig. 5. Comparison of observed and calibrated directional spectrum in a wind-sea dominated sea state (swell part DSF: Poisson model and wind-sea part DSF: cosinepower model).
sharper peak but a longer tail than cosine-power distribution (ITTC, 2005). In addition, using 5-year directional wave data measurements from Haltenbanken (the Norwegian Sea), the model especially verified for swell component of wave spectrum by BitnerGregersen and Hagen (2002) with the numbers xmax ¼ 0:9 and νa ¼ 2:21 for f r f p and νb ¼ 0:35 for f 4 f p . Krogstad and Barstow (1999) later analyzed wave data collected in WADIC, WAVEMOD and SCAWVEX projects and concluded that general distributional shapes are between cosinepower and Poisson distributions; although they used limited number of storms. To address the issue of providing the best directional spreading of observed spectrum nine different combinations of three aforementioned DSFs have been examined in this paper. Final results of such combinations; considering calibration procedure; are presented in Tables 5 and 6 for wind-sea dominated sea states and
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Fig. 6. Comparison of observed and calibrated directional spectrum in a swell dominated sea state (swell part DSF: cosine power model and wind-sea part DSF: cosine power model).
swell dominated sea states, respectively. Here, minimizing RMSE as well as ΔV=Vhave been selected as calibration targets in which V stands for the volume under the frequency-directional spectrum. Calibrated coefficients of such nominated combination of DSFs; as shown by gray cells of Tables 5 and 6; in wind-sea dominated and swell dominated sea states are presented in Table 7. It is obvious that cosine-power model is always the most successful DSF in directional presentation of the wind-sea part, irrespective of having wind-sea dominated or swell dominated sea state. While, for the swell part, it is better to use the Poisson model under wind-sea dominated sea states and the cosine-power model under swell dominated conditions. To see how such a calibrated directional spectrum (Eq. (3)); including the calibrated DSFs as well as the SFI calibrated nondirectional spectrum; performs in the region, it has been drawn next to typical observed extreme events in Figs. 5 and 6. Such 3D presentations just provide a general view. So, to have a better outlook on performance of the final result, two cross sections of such directional spectra; as presented in Figs. 5 and 6; have been drawn in comparison with a standard recommendation at a typical wind-sea dominated as well as a typical swell dominated sea state in Figs. 7 and 8, respectively. Here, it should be noted that for the standard directional spectrum in these two figures, standard Torsethaugen (Torsethaugen, 1993) is used as the nondirectional spectrum together with standard DSF proposed by Goda and Suzuki (1976). So, the difference between the observed spectrum and the standard spectrum is partly from the facts which have been already discussed when presenting Fig. 4 and partly from the weak performance of implemented standard DSFs (Goda and Suzuki, (1976)) at this region. It is obvious that implementing available nondirectional spectra together with standard DSFs leads to totally unacceptable results either in peak frequency or in mean wave direction.
Fig. 7. Performance of calibrated directional spectrum in comparison with a standard recommendation (Goda and Suzuki, 1976) under a typical wind-sea dominated sea state. (a) at peak frequency and (b) at mean wave direction.
However, such error is apparently smaller in case of swell dominated sea states.
5. Conclusion While there are general standard spectral formulae to represent sea states as could be found in rules and regulations, it is always recommendable to calibrate them for a specific region while there are measurement data. For the case of double peak unidirectional spectrum, calibration of the whole spectrum has been the general approach but this research showed that separate calibration of sea and swell parts may result in better conformity. Here, the approach implemented on Torsethaugen spectrum with great success for the Chabahar bay, although it was originally developed for another region. Here it should be remembered that the simplified form of Torsethaugen spectrum used in this study which introduces a very limited number of empirical coefficients when compared with those of the original form. So, a more compatibility might be achievable by means of the original one. Besides, there are different DSFs which could be used for directional presentation of such spectrum. Here, it has been shown that cosine-power model is among the most successful DSFs for both sea and swell parts of wave field at Gulf of Oman. Here, Poisson model has a better performance just in case of swell part in wind-sea dominated sea states.
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Fig. 8. Performance of calibrated directional spectrum in comparison with a standard recommendation (Goda and Suzuki, 1976) under a typical swell dominated sea state. (a) at peak frequency and (b) at mean wave direction.
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Hasselmann, D.E., Dunckel, M., Ewing, J.A., 1980. Directional wave spectra observed during JONSWAP 1973. J. Phys. Oceanogr. 10, 1264–1280. Holthuijsen, L.H., 1983. Observation of the directional distribution of ocean-wave energy in fetch-limited conditions. J. Phys. Oceanogr. 13 (2), 191–207. Hwang, P., 2012. Wind-sea and swell separation of 1D wave spectrum by a spectrum integration method. J. Atmos. Ocean. Technol. 29, 116–128. Isobe, M., Kondo, K., Horikawa, K., 1984. Extension of MLM for estimating directional wave spectrum. In: Proceedings of the Symposium on Description and Modeling of Directional Seas. Denmark, pp. A-6-1–A-6-15. ITTC, 2005. Testing and Extrapolation Methods Loads and Responses, Ocean Engineering, Laboratory Modeling of Multidirectional Irregular Wave Spectra. Kobune, K., Hashimoto, N., 1986. Estimation of directional spectra from the maximum entropy principle. In: Proceedings of the 5th International Offshore and Arctic Engineering. Japan, pp. 80–85. 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A study of the characteristics of wave spectra over the seas around Korea by using a parametric spectrum method. Acta Oceanogr. Taiwanica 37 (1), 31–46. Ochi, M.K., Hubble, E.N., 1976. Six-parameter wave spectra. In: Proceedings of the 15th Coastal Engineering Conference, USA. Pierson, W.J., Neuman, G., James, R.W., 1955. Practical methods for observing and forecasting ocean waves by means of wave spectra and statistics. US Naval Hydrographic Office, USA (Pub. No. 603). Phillips, O.M., 1958. The equilibrium range in the spectrum of wind generated waves. J. Fluid Mech. 4, 426–434. Rashmi, R., Aboobacker, V.M., Vethamony, P., John, M.P., 2013. Co-existence of wind-seas and swells along the west coast of India during non-monsoon season. Ocean Sci. 9, 281–292. Strekalov, S., Massel, S., 1971. On the spectral analysis of wind waves. Arch. Hydrot. 18, 457–485. Torsethaugen, K., 1993. A two-peak wave spectral model. In: Proceedings of the 12th OMAE conference, U.S.A. Torsethaugen, K., Haver, S., 2004. Simplified double peak spectral model for ocean waves. In: Proceedings of the 14th International Offshore and Polar Engineering Conference, Toulon, France. EM 1110-2-1100, 2006. Coastal Engineering Manual, USACE. Venugopal, V., Wolfram, J., Linfoot, B.T., 2005. The Properties of Extreme Waves. Heriot-Watt University, Scotland (Research Report). Violante-Carvalho, N., Ocampo-Torres, F.J., Robinson, I.S., 2004. Buoy observations of the influence of swell on wind waves in the open ocean. Appl. Ocean Res. 26 (1-2), 49–60. Wang, D.W., 1992. Estimation of wave directional spreading in severe seas, 2nd International Offshore and Polar Engineering Conference. pp. 146–153.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Development of a bi-modal directional wave spectrum R. Panahi, A. Ghasemi K., M. Shafieefar Civil and Environmental Engineering Department, Tarbiat Modares University, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 17 May 2014 Accepted 12 June 2015 Available online 3 July 2015
Addressing sea state in a coastal region of Gulf of Oman by a directional wave spectrum for the very first time is the main focus of this study. The region encounters wind-sea as well as swell. So, proper modeling of the sea state requires in general a double peak spectral model. The research is firstly conducted to calibrate nondirectional Torsethaugen double peak spectrum for the region by entering the separation frequency; a frequency in which wind-sea and swell parts could be divided. This novel calibration procedure is simple while results in much better outcome. Besides, one has to decide about an appropriate Directional Spreading Function (DSF) for the wind-sea and swell components. Then, nine possible combinations of three well-known DSFs have been investigated and calibrated to provide maximum conformity between observed and modeled directional wave spectra. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nondirectional wave spectrum Directional Spreading Function Calibration Separation frequency
1. Introduction A trustworthy statistical modeling of wave directional properties at a specific location is a necessary prerequisite to ensure design precision and accuracy in coastal and offshore regions. Here, the spectral formulation, stemming from the early work of Phillips (Phillips, 1958), is among practical tools in ocean engineering. For the case of directional spectral formulation it actually addresses wave energy and its distribution over different wave frequencies and directions. Various methods like maximum entropy method, maximum likelihood method and Bayesian method have been developed over years to estimate a directional spectrum from measurements (Longuet-Higgins et al., 1963; Borgman, 1969; Isobe et al., 1984; Kobune and Hashimoto, 1986; Kuik et al., 1988). Such methods have been reviewed and categorized by Benoit et al. (1997). Among them is a common practical approach in which a directional spectrum Sðf ; θÞ is expressed as follows: ð1Þ S f ; θ ¼ Sðf ÞD f ; θ in which Sðf Þ is nondirectional spectrum and Dðf ; θÞ is Directional Spreading Function (DSF). Now, for coexistence of local wind generated wave and distant swell wave; from now on we briefly call them wind-sea and swell, respectively; one expects at least a double peak nondirectional spectrum. It is commonly assumed that nondirectional wave spectrum is of a single mode form, and can be well modeled by a well-known standard spectrum such as JONSWAP or Pierson– Moskowitz spectrum. This approach is totally reasonable for E-mail address: shafi[email protected] (M. Shafieefar). http://dx.doi.org/10.1016/j.oceaneng.2015.06.017 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
severe sea states. However, moderated and low sea states; as one encounters in most Iranian coastal regions at Gulf of Oman; are often of a mixed nature, consisting of wind-sea as well as swell. Then, such sea states should be addressed by a double peak nondirectional spectrum. To characterize this issue, an approach was pioneered by Strekalov and Massel (1971) to sum two individual spectra as follows: X Sðf Þ ¼ Si ðf Þ ¼ SW ðf Þ þ SS ðf Þ ð2Þ i ¼ wind sea; swell
here, SW ðf Þ and SS ðf Þ stand for wind-sea and swell components of the spectrum, respectively. This approach was used later by researchers to combine and modify available standard nondirectional spectra e.g. Pierson– Moskowitz or JONSWAP resulting in the well-known double peak spectra like Ochi–Hubble model and Torsethaugen model (Ochi and Hubble, 1976; Guedes Soares, 1984; Torsethaugen, 1993; Moon and Oh, 1998; Violante-Carvalho et al., 2004; Torsethaugen and Haver, 2004; Mackay, 2011). For DSF, there are some standard uni-modal forms as cosinepower distribution (Longuet-Higgins et al., 1963; Mitsuyasu et al., 1975; Hasselmann et al., 1980), wrapped normal distribution (Borgman, 1969; Briggs et al., 1995), wrapped-around Gaussian distribution (Mardia, 1972), hyperbolic distribution (Donelan et al., 1985), von Mises distribution (Abramowitz and Stegun, 1975; Hashimoto and Konube, 1986), Poisson distribution (Lygre and Krogstad, 1986) and Boxcar or Steklov distribution (Venugopal et al., 2005) which have been checked occasionally by other researchers based on observations (Ewans, 1998). However, no single model is universally accepted due to site specificity associated with particular formulations. In order to distribute a nondirectional spectrum over directions, it has been a common
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
practice to use distinct DSFs for distribution of wind-sea and swell parts as also recommended by rules and regulations (EM 1110-21100, 2006; DNV-RP-C205, 2010). So, for this situation it would be appropriate to rewrite Eq. (1) as follows: S f ; θ ¼ SW ðf ÞDW f ; θ þ SS ðf ÞDS f ; θ ð3Þ Wind part
Swell part
To this end, the paper aims at developing a directional spectrum Sðf ; θÞ for Chabahar coastal regions together with an introduction to a general practical approach for calibration of double peak spectra. Measured wave data in Chabahar bay in the northern coasts of Gulf of Oman are used to verify the proposed approach. The area is experiencing both wind-sea and swell regimes throughout the year especially in Monsoon seasons (Rashmi et al., 2013). As the border of Indian Ocean and Gulf of Oman, it is under construction and development activities and its field observations are recently released. Available data are briefly reviewed in the next section. Based on field observed spectra, nondirectional Torsethaugen double peak spectrum is firstly calibrated in Section 3 using a simple novel approach. Assessing some standard DSFs in Section 4 together with their calibration to maximize conformity, they finally resulted in developing a calibrated directional spectrum for the region.
2. Local measurements In order to monitor regional coastal waters a mega project had been initiated by Port and Maritime Organization of Iran since 2006. Fig. 1 shows the focal point of this paper together with measurement stations as stars for the Chabahar bay, also called Khalij-e Chabahar, which is around 251200 53″N and 601300 40″E in DMS (Degrees Minutes Seconds). Nortek Acoustic Wave and Current profiler (AWAC) had been used to gather such directional field data. Stations specifications are summarized in Table 1. Such raw data are processed using Nortek STORM software to recognize extreme events. The raw data come from direct measurements with no filter for anomalies e.g. gaps or spikes. Therefore, a simple code is developed to reject any records which do not pass certain quality requirements.
105
calibrated by finding and tuning appropriate DSFs as presented in Section 4. For the purpose, Torsethaugen spectrum is nominated as the most recent well-known attempt to deal with double peak spectral presentation as ST ðf Þ ¼ ST W ðf Þ þ ST S ðf Þ in which T stands for Torsethaugen (Torsethaugen, 1993). Torsethaugen spectrum originally developed by combining two JONSWAP spectra. The spectrum is presented as a sum of wind-sea and swell components for j ¼ 1; 2 as follows (Torsethaugen, 1993): Sðf Þ ¼
2 X
Ej Snj ðf U T Pj Þ ¼
j¼1
2 X
Ej Snj ðf nj Þ
ð4Þ
j¼1
where: Ej ¼
1 2 H UT 16 Sj Pj
Snj ¼ G0 Aγ j Γ Sj γ Fj
ð5Þ ð6Þ
here, it should be noted that Torsethaugen only requires that the user provides two input parameters H s and T p . Its other main parameters are all expressed in terms of the significant wave height ðH s Þ and spectral peak period T p . At first, T f ¼ af H s1=3 should be calculated as the peak period in a fully developed sea in which af is slightly sensitive to the fetch length. Then, one could decide if the sea state is wind-sea dominated T P oT f or swell dominated T P 4 T f . It was shown later by Torsethaugen and Haver (2004) that some of the parameters for the general form of Torsethaugen spectrum, considering its complex formulation, are of marginal importance for design purpose. So, they introduced the simplified form as presented in Tables 2 and 3 for wind-sea dominated and swell dominated sea states. Here, bold parameters are those empirical coefficients tuned in this research together with af as it has the main rule in switching between formulations. 4 4 Besides, in Eq. (6) G0 ¼ 3:26, Γ Sj ¼ f nj exp½ f nj and Aγ j as well as γ Fj are functions of peak enhancement factor γ as follows: 2 2 ð7Þ γ F1 ¼ γ exp 1=2σ ðf n1 1Þ
γ F2 ¼ 1 (
0:07 f nj o 1 0:09 f nj Z 1
ð8Þ
3. Calibration of the nondirectional spectrum
σ¼
Nondirectional spectrum has always its own importance and application irrespective of its distribution over different angles. This motivated the authors to firstly focus on calibration of nondirectional spectrum as the sum of SW ðf Þ and SS ðf Þ based on the form introduced by Eq. (2). Then, the directional spectrum is
1:19 Aγ 1 ¼ 1 þ 1:1 ln γ =γ
ð10Þ
Aγ 2 ¼ 1
ð11Þ
measurement station
25°20'53" N and 60°30'40" E
Fig. 1. Chabahr bay in Gulf of Oman and measurement stations.
ð9Þ
here, γ is calculated using Tables 2 and 3. However, such a comprehensive parameterization of the spectrum has been originally shaped based on measurements at deep sea; Statfjord field in Norwegian waters (Torsethaugen, 1993); and later supported by other data close to that location (Ewans et al., 2006; Bitner-Gregersen and Toffoli, 2009). So, it has to be implemented for other locations like Gulf of Oman with care. Here, the spectrum is examined in three forms of standard version ðST ðf ÞÞ, calibrated version ðSCal T ðf ÞÞ and Separation Frequency Implemented (SFI) calibrated version ðSSFI Cal T ðf ÞÞ in order to model the sea states at the region. For ST ðf Þ, original values have been exactly used as the coefficients (Torsethaugen, 1993). To catch SCal T ðf Þ coefficients, minimizing Root Mean Square Error (RMSE) of modeled spectrum values, ΔA=A and Δf p =f p have been set as calibration targets using nonlinear Generalized Reduced Gradient (GRG) algorithm (Lasdon et al., 1973). Here, A and f p are the area under the spectrum and its peak frequency, while Δ
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R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Table 1 Wave measurement stations at the edge of Chabahar bay. Station no.
1 2 3
Local name
AbAmigh Behesthi Bargin
Coordination in DMS
Measurement interval
N
E
Begin
End
251150 40″ 251170 22″ 251170 30″
601390 03″ 601350 40″ 601280 30″
Aug., 27,2006 Aug., 30,2006 Aug., 25,2006
June,14, 2007 Sep.,2, 2007 Sep.,20, 2007
Table 2 Torsethaugen spectrum-wind-sea (Torsethaugen and Haver, 2004).
Secondary peak
HW1 ¼ RW U HS
HW2 ¼ ð1 RW 2 Þ0:5 U H S T p2 ¼ T f þ b1 γ2 ¼ 1
RW ¼ ð1 a10 Þe
εl a1
Distance to nearest coast (km)
30 11 10
2 0.3 1.5
dominated
Primary peak
T p1 ¼ T p γ 1 ¼ kg W U Sp 6=7 2
Depth (m)
þ a10
T T
εl ¼ Tff Tpl T l ¼ ae H0:5 S H W1 Sp ¼ 2π g T 2 p1
Fig. 2. Definition of different frequencies on a typical double peak spectrum (red line) based on separated wind-sea and swell spectra (blue lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 Torsethaugen spectrum-swell (Torsethaugen and Haver, 2004).
dominated
Primary peak
Secondary peak
HS1 ¼ RS U H S
H S2 ¼ ð1 RS 2 Þ0:5 U H S T p2 ¼ T f 1 γ2 ¼ 1
T p1 ¼ T p γ 1 ¼ γ f ð1 þ a3 εu Þ 2 RS ¼ ð1 a20 Þe
εu a2
þ a20
T T
εu ¼ T pu T ff T u ¼ au γ f ¼ kg S U Sf 6=7 HS Sf ¼ 2π g T 2 f
stands for the absolute difference between modeled and observed values. It should be noted that while f p is among inputs of Torsethaugen spectrum itself, spectrum coefficients change during the calibration procedure. So, the calibrated spectrum does not necessarily keep its peak at f p and considering this constraint is important. This is a common methodology as conducted in TMA spectrum calibration (Moon and Oh, 1998) or in combination of two JONSWAP spectra (Violante-Carvalho et al., 2004; Feld and Mørk, 2004). Finally, for SSFI Cal T ðf Þ, a separation frequency f s is calculated for observed double peak spectrum using the method proposed by Hwang (2012). Then, it is entered into the procedure to calibrate SW ðf Þ and SS ðf Þ parts separately, as follows (see Fig. 2 for definitions): First step: 1. Divide observed spectrum into S1 0-f s and S2 f s -f max . 2. Calibrate ST S by S1 0-f s introducing f S max 4 f s . Here, f S max is the frequency in which ST S diminishes. 3. Modify S2 f s -f max to S02 f s -f max by subtracting ST S between f s and f S max . 4. Calibrate ST W by S02 f s -f max introducing f W min o f s . Here f W min is the frequency in which ST W diminishes.
Fig. 3. Scatter diagrams of Hs vs Tp for wind-sea and swell spectral components.
So, it should be noted that he main output of this step would be f W min and f S max as minimum frequency of wind-sea part and maximum frequency of swell part, respectively (see Fig. 2). Second step: 1. Extract S1 0-f S max and S2 f W min -f max from field spectrum with an overlap interval. 2. Modify S1 0-f S max to S01 0-f S max by subtracting ST W between f W min and f S max . 3. Calibrate ST S by S01 0-fS max . 4. Modify S2 f W min -f max to S02 f W min -f max by subtracting ST S between f W min and f S max . 5. Calibrate ST W by S02 f S min -f max . The main output of this step would be SSFI Cal T as the sum of two separately calibrated wind-sea and swell spectra with the Torsethaugen base as ST W and ST S , respectively. It also results in updated f W min and f S max .
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Table 4 Performance and coefficients of S T , S Cal Torsethaugen spectrum. Coefficients
4. Calibration of directional spectrum T
and SSFI Cal
T
as three versions of
Wind-sea dominated First peak
Standard Calibrated SFI calibrated
kg W
af
a10
ae
a1
b1
35.00 34.90 36.38
6.60 4.24 5.62
0.70 0.71 0.82
2.00 1.99 1.95
0.50 0.54 0.53
2.00 3.46 2.50
Swell dominated First peak
Standard Calibrated SFI calibrated performance Standard Calibrated SFI calibrated
107
Second peak
kg S
af
a20
au
a2
a3
35.00 34.9 20.33
6.60 4.24 5.62
0.60 0.53 0.80
25.00 33.00 36.65
0.30 0.10 0.10
6.00 1.00 1.00
RMSE
ΔA=A
1.57 1.05 0.64
7% 4% 3%
Δf p =f p 0.00% 0.00% 0.00%
After the second step, RMSE, ΔA and Δf p =f p are calculated by comparing observed spectrum with SSFI Cal T and this step is repeated to reach an acceptable convergence in statistical measures. It should be remembered that, the very first output of the procedure is separation of field wind-sea and swell components. So, in order to provide an outlook into measurement data H s is plotted against T p in Fig. 3 for wind-sea and swell spectral components, separately. It is evident that the peak period of wind-sea component varies in a narrower interval when compared with that of swell component. Besides, significant wave height is of the same order for two spectral components although it is slightly larger for the swell part. All in all, the calibration procedure effect on 10 coefficients of Torsethaugen spectrum is presented in Table 4. However, by reviewing the performance of three aforementioned versions in last rows of Table 4, it is obvious that the second attempt in calibration has come up with a better adjustment when compared with the first calibration output. Here, RMSEof spectrum values decrease meaningfully up to 40%; from 1.57 in case of standard spectrum into 0.61 in case of SFI calibrated spectrum. The ability to predict two distinct peaks of observed spectra is also assessed in terms of peak spectral value and its frequency as well as significant wave height of swell component as well as wind-sea component in Fig. 4. When talking about maximum values, it is obvious that the second try in calibration improved data modeling to a great extent. It should be noted that SCal T performs better in modeling peak points of swell part than those of wind-sea part. But, when looking at performance of SSFI Cal T it presents peak values of wind-sea as well as swell with almost a similar acceptable accuracy level as it is reported by correlation factor R2 . However, there are still some difficulties for SSFI Cal T to adjust with peak spectral values of wind-sea. This is exactly the case when assessing such spectra in terms of significant wave height. Torsethaugen performance is generally much better in presenting swell component Hs rather than wind-sea component Hs. The Ochi–Hubble spectrum was also used in the analysis reported above. The procedure was completely useful in statistical terms when the SFI calibrated version compared with two other ones. However, the result was not as successful as Torsethaugen SFI calibrated version.
Calibration of double peak directional spectrum while its nondirectional part has been already calibrated is all about finding appropriate DSF and its calibration for sea and swell parts, see Eq. (3). To do this, three well-known forms of DSF have been examined to find out which combination of such DSFs provides the best directional description of the wave spectrum. Here, it should be noted that calculated wave directions are based on the first pair of Fourier coefficients describing the mean direction at a given frequency. This method can be considered inadequate when wave trains from two independent directions are present. This typically occurs along coastal areas where structures transform incident wave energy and create reflected waves that are measured at the same location leading to a weighted average of the two directions. Since the wave measurement stations in this research are located in deep water and far from coastal structure, there is no such condition that wave trains from two independent directions are presented at the same time. 4.1. Cosine-power model The very first attempt to model DSF is getting back to (Pierson et al., 1955) in which they proposed a cosine type function as: ( 2 cos 2 θ π2 þ θ0 o θ o π2 þ θ0 D θ ¼ π ð12Þ 0 otherwise in which θ is wave direction and θ0 is main peak frequency direction also called mean wave direction. The model did not have any frequency input. So, LonguetHiggins et al. (1963) took another step forward which led later to lots of efforts in this area. They introduced a more complicated model which could be written in a generic form as: θ θ0 Dðθ; sÞ ¼ IðsÞ cos 2s ; π rθ rπ ð13Þ 2 in which: pffiffiffiffi π Γ ðs þ 1Þ IðsÞ ¼ i 2π Γ ðs þ 12 Þ
ð14Þ
here, Γ is Gamma function, i is a constant, equal to one for almost all standard cases, and s is spreading parameter which controls spreading around θ0 as an empirical function. There have been lots of efforts to propose different versions based on this model from Krylov (1966) in which he entered mean wave frequency into account to Mitsuyasu (1975), Goda and Suzuki (1976) and Holthuijsen (1983) who clearly showed that s varies with wave frequency and it could be related to the stage of wave development as a combination of wind speed ðU Þ and fetch length ðF Þ. So μa 8 > f of p < smax f =f p μb s¼ ð15Þ > : smax f =f p f 4f p here, it has been shown that especially for deep water, smax is 10 for wind waves, 25 for swells present with wind waves of relatively large steepness, and 75 for swells with wind waves of small steepness in case of μa ¼ 5 and μb ¼ 2:5(Goda and Suzuki, 1976). Similar dependence of s to wave frequency introduced by Hasselmann et al. (1980) in which they proposed other values for smax , μa and μb working on JONSWAP project. In their proposal, there is a weak dependence between μb and the ratio between wind speed and phase velocity i.e. wave age.
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R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
Fig. 4. Modeling of spectral peak values and significant wave height. (a) first peak and (b) second peak.
R. Panahi et al. / Ocean Engineering 105 (2015) 104–111
It is also discussed that s could be related to the wave length associated with peak frequency of the spectrum (LP ) (Wang, 1992). Such studies continued later (Krogstad et al., 1997; Kumar et al., 2000; Kumar, 2006).
Table 7 Best combination of DSFs in representing sea states at the region.
Wind-sea dominated
4.2. Hyperbolic model A cosh function proposed by Donelan et al. (1985) to calculate DSF as follows: 2 D θ; β ¼ αβ cos h β θ θ0 ð16Þ in which α ¼ 0:5 and 8 γ 1 > β1 ffp for > > > > < γ 2 for β ¼ β2 ffp > > γ 3 > > > : β 3 ff otherwise
109
Swell dominated
Wind-sea part
Swell part
Cosine-power model Smax ¼ 20:2 μa ¼ 7:3 μb ¼ 8 i ¼ 0:05 Cosine-power model Smax ¼ 13:5 μa ¼ 1:5 μb ¼ 6:5 i ¼ 0:06
Poisson model xmax ¼ 0:6 υa ¼ 0:5 υb ¼ 1:3 η ¼ 0:1 Cosine-power model Smax ¼ 71:2 μa ¼ 2 μb ¼ 3:4 i ¼ 0:02
0:56 o ωωp o 0:95 0:95 o ωωp o 1:60
ð17Þ
p
here, β 1 ¼ 2:61, β2 ¼ 2:28 and β 3 ¼ 1:24. Also, γ 1 ¼ 1:3, γ 2 ¼ 1:3 and γ 3 ¼ 0:0. To modify this model, Banner (1990) showed that β is not necessarily a constant for the third condition. So, he suggested an equation with a role for f p . 4.3. Poisson model Poisson distribution firstly proposed by Lygre and Krogstad (1986) is as follows: 1 1 x2 D θ; f ¼ η 2π 1 2x cos θ þ x2
ð18Þ
in which η is a constant; equal to one for the standard model. Besides υa 8 > f of p < xmax f =f p υ b x¼ ð19Þ > : xmax f =f p f 4f p here, one could find a relation between Poisson model and cosinepower model. However, for the Poisson distribution it has a
Table 5 Performance of calibrated DSFs to represent wind-sea dominated seat states.
DSF model Swell part
Cosine-power Hyperbolic Poisson
Wind part Cosine-power
Hyperbolic
Poisson
RMSE ¼ 2:41 ΔV =V ¼ 0:06 RMSE ¼ 2:31 ΔV =V ¼ 0:04 RMSE ¼ 2:29 ΔV =V ¼ 0:04
RMSE ¼ 2:33 ΔV =V ¼ 0:06 RMSE ¼ 2:39 ΔV =V ¼ 0:08 RMSE ¼ 2:37 ΔV =V ¼ 0:13
RMSE ¼ 2:42 ΔV=V ¼ 0:08 RMSE ¼ 2:40 ΔV=V ¼ 0:11 RMSE ¼ 2:38 ΔV=V ¼ 0:16
Table 6 Performance of calibrated DSFs to represent swell dominated seat states Wind part
Swell part
DSF model
Cosine-power
Hyperbolic
Poisson
Cosine-power
RMSE ¼ 3:49 ΔV =V ¼ 0:19 RMSE ¼ 3:56 ΔV =V ¼ 0:24 RMSE ¼ 3:75 ΔV =V ¼ 0:20
RMSE ¼ 3:84 ΔV =V ¼ 0:2 RMSE ¼ 3:89 ΔV =V ¼ 0:20 RMSE ¼ 4:08 ΔV =V ¼ 0:22
RMSE ¼ 3:65 ΔV=V ¼ 0:22 RMSE ¼ 3:68 ΔV=V ¼ 0:23 RMSE ¼ 3:78 ΔV=V ¼ 0:25
Hyperbolic Poisson
Fig. 5. Comparison of observed and calibrated directional spectrum in a wind-sea dominated sea state (swell part DSF: Poisson model and wind-sea part DSF: cosinepower model).
sharper peak but a longer tail than cosine-power distribution (ITTC, 2005). In addition, using 5-year directional wave data measurements from Haltenbanken (the Norwegian Sea), the model especially verified for swell component of wave spectrum by BitnerGregersen and Hagen (2002) with the numbers xmax ¼ 0:9 and νa ¼ 2:21 for f r f p and νb ¼ 0:35 for f 4 f p . Krogstad and Barstow (1999) later analyzed wave data collected in WADIC, WAVEMOD and SCAWVEX projects and concluded that general distributional shapes are between cosinepower and Poisson distributions; although they used limited number of storms. To address the issue of providing the best directional spreading of observed spectrum nine different combinations of three aforementioned DSFs have been examined in this paper. Final results of such combinations; considering calibration procedure; are presented in Tables 5 and 6 for wind-sea dominated sea states and
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Fig. 6. Comparison of observed and calibrated directional spectrum in a swell dominated sea state (swell part DSF: cosine power model and wind-sea part DSF: cosine power model).
swell dominated sea states, respectively. Here, minimizing RMSE as well as ΔV=Vhave been selected as calibration targets in which V stands for the volume under the frequency-directional spectrum. Calibrated coefficients of such nominated combination of DSFs; as shown by gray cells of Tables 5 and 6; in wind-sea dominated and swell dominated sea states are presented in Table 7. It is obvious that cosine-power model is always the most successful DSF in directional presentation of the wind-sea part, irrespective of having wind-sea dominated or swell dominated sea state. While, for the swell part, it is better to use the Poisson model under wind-sea dominated sea states and the cosine-power model under swell dominated conditions. To see how such a calibrated directional spectrum (Eq. (3)); including the calibrated DSFs as well as the SFI calibrated nondirectional spectrum; performs in the region, it has been drawn next to typical observed extreme events in Figs. 5 and 6. Such 3D presentations just provide a general view. So, to have a better outlook on performance of the final result, two cross sections of such directional spectra; as presented in Figs. 5 and 6; have been drawn in comparison with a standard recommendation at a typical wind-sea dominated as well as a typical swell dominated sea state in Figs. 7 and 8, respectively. Here, it should be noted that for the standard directional spectrum in these two figures, standard Torsethaugen (Torsethaugen, 1993) is used as the nondirectional spectrum together with standard DSF proposed by Goda and Suzuki (1976). So, the difference between the observed spectrum and the standard spectrum is partly from the facts which have been already discussed when presenting Fig. 4 and partly from the weak performance of implemented standard DSFs (Goda and Suzuki, (1976)) at this region. It is obvious that implementing available nondirectional spectra together with standard DSFs leads to totally unacceptable results either in peak frequency or in mean wave direction.
Fig. 7. Performance of calibrated directional spectrum in comparison with a standard recommendation (Goda and Suzuki, 1976) under a typical wind-sea dominated sea state. (a) at peak frequency and (b) at mean wave direction.
However, such error is apparently smaller in case of swell dominated sea states.
5. Conclusion While there are general standard spectral formulae to represent sea states as could be found in rules and regulations, it is always recommendable to calibrate them for a specific region while there are measurement data. For the case of double peak unidirectional spectrum, calibration of the whole spectrum has been the general approach but this research showed that separate calibration of sea and swell parts may result in better conformity. Here, the approach implemented on Torsethaugen spectrum with great success for the Chabahar bay, although it was originally developed for another region. Here it should be remembered that the simplified form of Torsethaugen spectrum used in this study which introduces a very limited number of empirical coefficients when compared with those of the original form. So, a more compatibility might be achievable by means of the original one. Besides, there are different DSFs which could be used for directional presentation of such spectrum. Here, it has been shown that cosine-power model is among the most successful DSFs for both sea and swell parts of wave field at Gulf of Oman. Here, Poisson model has a better performance just in case of swell part in wind-sea dominated sea states.
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Fig. 8. Performance of calibrated directional spectrum in comparison with a standard recommendation (Goda and Suzuki, 1976) under a typical swell dominated sea state. (a) at peak frequency and (b) at mean wave direction.
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