- Email: [email protected]
Estimating wave crest distributions using the method of L-moments
Applied Ocean Research 31 (2009) 37–43
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Estimating wave crest distributions using the method of L-moments Amir H. Izadparast, John M. Niedzwecki ∗ Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, United States
article
info
Article history: Received 1 September 2008 Received in revised form 2 April 2009 Accepted 12 April 2009 Available online 22 May 2009 Keywords: Method of L-moments Ocean waves Wave-structure interaction Mini-TLP model test data Wave crest estimates Parameter estimation
abstract The design of fixed or floating offshore structures requires accurate information of the met-ocean data at the intended offshore site. In the design process it is recognized that this environmental data is modified in the near-field by the interaction with the particular geometrical configuration of the offshore structure. This transformation of the incident wave field around and beneath an offshore structure presents a challenge for ocean engineers when specifying the wave gap elevation to avoid impact loads on the underside of the deck and inundation of the topsides. Thus, the accurate estimation of the wave crest distributions from measurements at various locations near and under the offshore structure during model test studies is essential. A semi-empirical approach is presented herein that builds upon the findings of previous studies and introduces the Method of L-moments. A three parameter model for a wave crest probability distribution function is presented and explicit relationships between the parameters of the distribution and its’ first three L-Moments are established. Furthermore, three narrow-band models from earlier research studies are reviewed and compared with the new model. Wave measurements from a mini-TLP model test program are used as the basis for comparison of the four distributions. The rootmean-square error is used as a metric to quantify the overall fit of the data and its accuracy in the high end tail of the data. The L-Moment model is shown to be more robust in representing the data in both the far-field and beneath the deck of the mini-TLP where the wave field demonstrates increased non-linear behavior. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The distribution of wave crests and in particular the amplification of those wave crests beneath deepwater platforms presents a significant design problem for an ocean engineer considering the air gap elevation needed to avoid potential impact loads on the underside of the deck and inundation of the topsides. For these platforms, vertical columns, submerged pontoons, diagonal bracings and exposed riser arrays can greatly influence the non-linear nature of the design seas through diffraction and radiation effects. Due to the complex nature of this interaction, designers routinely rely on calibrated numerical models and semi-empirical probabilistic models to provide guidance regarding the general behavior and amplification of wave field for use in the design process. The specification of probability density and cumulative distribution functions for non-linear wave crests has been the focus of previous research studies. In this study, a wave crest is defined as the maximum positive wave elevation between two successive zero-upcrossings. Assuming that the wave surface elevation could be modeled as a narrow-band process, Tayfun [1,2] developed a theoretical model that described the transformation of a
∗
Corresponding author. Tel.: +1 979 862 1463. E-mail address: [email protected] (J.M. Niedzwecki).
0141-1187/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2009.04.002
narrow-band linear crest distribution to a narrow-band non-linear crest distribution. This model was subsequently modified by other researchers and the results predicted using the various models yield nearly identical results for the deepwater limit [4–6]. The assumption that the wave surface elevations are narrow-band processes is a fairly robust assumption for the analysis of offshore structures [7]. This assumption however, appears to be somewhat constraining for the development of parametric models as the analysis of field data suggests that the wave elevation tends to be more broad banded, that is the time series often contain multiple peaks and troughs in a single wave cycle. With the development of each new model there was an improvement in the parametric representation of the wave data especially for the region of large wave crests. In 1994 Tayfun [8] developed a non-linear crest distribution function using a Gram–Charlier type of distribution, resulting in a model that was less restrictive with regards to the spectral properties of ocean wave elevations. Later Al-Humoud et al. [9] examined this distribution for ocean surface waves and showed that their improved model was better than the linear model but that the qualitative accuracy of the predictions remained somewhat elusive. Earlier Forristall [10] used a two-parameter Weibull distribution model as the basis to estimate the general probability distribution of wave heigths. He later related his Weibull model parameters to wave steepness and the Ursell
38
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
parameter providing a different perspective for estimating the probability distribution of wave crests [11,12]. That improved model was accurate for large wave crests but was shown to be less so for small waves. More recently, Tayfun [2] updated the quadratic transformation of linear to non-linear wave crests in order to address wave crest height distribution for transitional water depths. In that study, Tayfun used the Weibull model proposed by Forristall to estimate the parameters of the quadratic transformation, and this model was shown to have better accuracy even in deepwater when compared to other non-linear models but again this model deviated for smaller wave amplitudes. It appears that the various two paramter models developed in earlier studies are unable to capture the complete distribution structure of the observed data and that perhaps an unconstrained three parameter model would yield better results. This idea was the foucs of this research study and has led to the development of a three parameter model that utilized the method of L-moments to evaluate each parameter from the meaured data. The model incorporates the second order transfomation of linear to non-linear crests based upon the research work of Tayfun [1], but here a third empircal parmeter is introduced and no parameter is specified as a constant. In this new formulation, each model parameter was explicitly derived in terms of L-moments [13]. The motivation for introducing the method of L-moments into the evaluation of the distribution model parameters was that L-moments are computed from a linear combination of ordered data and that consequently estimates of the parameters are less influenced by outliers and the bias associated with small sample estimates remains fairly small. The method of L-moments is also effective in estimating high-order moments such as skewness and kurtosis whose evaluations are quite sensitive and can be unduly influenced by exceptionally large values [2,14]. Further the method of L-moments has been widely used for extreme value analysis and its efficiency has been studied on both simulated and real data sets [15–18]. In the discussion to follow, the mathematical formulation of the three-parameter model is presented and its performance evaluated using data from an earlier model test program investigating the rigid and compliant response of a mini-TLP designed for West Africa [19,20]. 2. Mathematical formulation 2.1. Basic model description The quadratic transformation of linear to non-linear wave crests introduced by Tayfun [1] is used here with an additional parameter γ which provides the flexibility to estimate the shift between the linear narrow-band and non-linear wave crests. The equation for the normalized non-linear wave amplitude ζn then can be expresses as
ζn = ζ + αζ 2 + γ
(1)
where, ζn = an /σ and similarly the normalized wave crest amplitude is ζ = a/σ with an and a being the non-linear and linear crests elevations respectively. Here the variance is expressed in terms of the single sided wave amplitude spectrum Gηη (ω) as R∞ σ 2 = 0 Gηη (ω)dω which can be specified or developed from the time series η(t ) of the wave amplitude. Solving Eq. (1) to obtain an expression for the normalized linear wave amplitude yields the following expression
ζ =
−1 +
√
1 + 4α (ζn − γ )
. (2) 2α It is assumed that the probability density function of ζ is the Rayleigh law, specifically f (ζ ) =
ζ R
exp −
ζ2 2R
(3)
where, the parameter R is determined from the appropriate data set. Substituting Eq. (2) into Eq. (3), one obtains f (ζn ) =
1 − [1 + 4α (ζn − γ )]−1/2 2α R
−1 + [1 + 4α (ζn − γ )]1/2 × exp − 8Rα 2
2 ! .
(4)
It follows then that the cumulative distribution function (CDF) and quantile function of the non-linear wave amplitude can then be expressed respectively as
2 ! −1 + [1 + 4α (ζn − γ )]1/2 F (ζn ) = 1 − exp − 8Rα 2
(5)
and,
ζn (u) = γ − 2α R ln (1 − u) + [−2R ln (1 − u)]1/2 , 0 ≤ u < 1.
(6)
Upon examination of these last three equations, it is observed that the probability distribution of ζn has three unknown empirical constants, specifically α , γ and R. In the discussion that follows approximations for these three parameters from earlier studies will be discussed. This will be followed by the presentation of a derivation that relates these three parameters directly to Lmoments and consequently directly to the appropriate time series measurements. 2.1.1. Earlier parametric models Longuet-Higgins [21] derived an expression for the Rayleigh parameter; specifically he showed that R = (a2rms /σ 2 )/2 where arms is the root-mean-square of the crest-height. For the case of √ linear narrow-band waves arms = 2σ and subsequently the Rayleigh parameter has a value of unity [22]. Later, LonguetHiggins [23] showed that the non-linear effects tend to increase the ratio of a2rms /σ 2 /2 while spectrum width effects may either increase or decrease the ratio. The effect of the combination of non-linearity and spectrum width on the value of R is not clear from a theoretical perspective, but what is important is that the value of the Rayleigh parameter can be influenced to some extent by these effects. There also is the issue that in the transformation of the narrow-band linear model, the effects of spectrum width are modeled using the quadratic term which is not physically correct and this may cause overestimation of large crests. In later studies, the processes were assumed to be narrow banded and it was assumed that two of these parameters namely γ and R had the following values, γ = 0 and R = 1 [1–3,5,6]. In addition, the linear assumption requires the parameter α to have a value of zero in this model. A Rayleigh model using these values is referred to in this study as the narrow-band linear model. Assuming that individual waves are adequately described by Stokes second-order wave theory, it follows that
α=
kσ 2
−
1 sinh 2kd
+
cosh kd (2 + cosh 2kd) 2 sinh3 kd
(7)
where, k = 2π /L is the wave number, L is the wave length associated with the mean period T¯ calculated from the ratio of the first two moments of the wave elevation spectrum m0 /m1 , and d is the water depth. For the deep water limit of interest in this study, Eq. (7) reduces to α = k2σ [1]. In this study the statistics estimated using γ = 0, R = 1 and α estimated from Eq. (7) will be referred to as the narrow-band non-linear model. More recently Tayfun [2] showed that the same quadratic transformation defined in Eq. (1) could be used for transitional water depths where the parameter α represents wave steepness
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
and is specified to be consistent with the two parameter Weibull distribution model used by Forristall [11] specifically
α=
8c 3 b
0
r
3
−
b
(8)
128
where b and c are the shape and scale parameters of the two parameter Weibull distribution respectively, and 0 is the gamma function. The parameters b and c were approximated by Forristall [11] based upon fitting Weibull distribution to 2-D and 3-D computer simulations of second order non-linear waves. The computer simulations used in Forristall’s study covered the practical range of wave peak period, wave steepness, and water depth. In order to define parameters b and c as functions of the water depth and wave spectrum characteristics, Forristall [11] used wave steepness S and Ursell number Ur defined as below S=
8π σ
Ur =
(9)
gT 4σ k2 d3
and the notation in brackets denotes binomial coefficients, for example
π
.
(10)
Analyses of the 2-D and 3-D simulations resulted in relations given in Eqs. (11) and (12) respectively
39
r j
=
r!
r ≥ j.
j! (r − j)!
(16)
The probability distribution parameter estimation using method of L-moments is based on the probability weighted moments βr defined as
βr =
1
Z
x (u) ur du.
(17)
0
Note that br defined in Eq. (15) is an unbiased estimator of probability weighted moment βr . The difference between conventional moments and probability weighted moments is that conventional moments involve successively higher powers of the quantile function x (u), whereas probability weighted moments involve successively higher power of u [14]. The L-moments λr of a random variable X with cumulative distribution function F (x) and quantile function x(u) are basically the linear combination of the probability weighted moments. The L-moments λr of X are then defined as,
λr =
Z
1
x (u) Pr∗−1 (u) du.
(18)
0
b2 = 2 − 2.1597 S + 0.0968 Ur 2 1 c2 = √ + 0.2892 S + 0.1060 Ur 2 2 b3 = 2 − 1.7912 S − 0.5302 Ur + 0.2848 Ur 2 1 c3 = √ + 0.2568 S + 0.0800 Ur 2 2
(11)
These linear combinations arise from integrals of x (u) weighted by a set of orthogonal polynomials called shifted Legendre polynomials Pr∗ (u) [14]
(12)
Pr∗ (u) =
2.1.2. L-moment model and parameter estimation The use of L-moments for parameter estimation was developed by Hosking [13] and these moments represent the expectations of certain linear combinations of order statistics. The main advantage of L-moments is that they are less influenced by outliers since they are developed from a linear combination of data and consequently the bias of their small sample estimates remains fairly small. This makes the method of L-moments an efficient means for estimating the extreme values when the data is limited and the existence of outliers is expected. In order to estimate the m parameters of a distribution with the method of L-moments, its first m L-moments, expressed as functions of the distribution parameters, would be equated to their corresponding sample values [14]. The sample L-moment lr of an ordered sample x1:n ≤ x2:n ≤ · · · ≤ xn:n of size n is defined as r X
r = 0, 1, . . . , n − 1
p∗r ,j bj
(13)
j=0
−1
br = n
r j
n−1 r
r +j j
(14)
−1 X n j−1 j =r +1
(19)
In terms of weighted moments this can be expressed as
λ r +1 =
r X
p∗r ,j βj .
(20)
j =0
In order to estimate the parameters R, α, and γ the relation between these parameters and the first three L-moments of ζn need to be established. It can be shown that the first three distribution L-moments are
λ1 = β 0 λ2 = 2β1 − β0 λ3 = 6β2 − 6β1 + β0
(21)
where, λ1 , λ2 , and λ3 are the measures of location, scale, and skewness of the probability distribution. Substituting the quantile function of ζn , given in Eq. (6), into Eq. (17) and using the relations given in Eq. (21) one obtains after integration the following equations
λ1 = γ + 2α R + 0
3
√ 2R
2
√ 3 √ λ2 = α R + 2−1 0 R
(22)
2
√ 3 rR 1 3/2 3/2 λ3 = α R + 6+2 −3 0 . 3
2
3
Then upon equating the foregoing L-moments with their corresponding sample values and solving for the distribution parameters one obtains the following equations
where, p∗r ,j = (−1)r −j
p∗r ,j uk .
j =0
where, the subscripts ‘‘2’’ and ‘‘3’’ refer to the 2-D and 3-D estimates. The largest value of α estimated by using either (b2 , c2 ) or (b3 , c3 ) in Eq. (8) will be used in the non-linear crests probability distribution [2]. This model is basically designed to address large crests and will be referred to as the modified non-linear model in this study. Note that the parameter γ = 0 in this model.
lr +1 =
r X
r
xj:n
(15)
R = (4.139l2 − 12.418l3 )2
√ α = l2 /R − 0.367/ R
(23)
√
γ = l1 − 2α R − 1.253 R.
40
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Table 1 The main particulars of the mini-TLP. Draft Column diameter Column spacing Pontoon height Pontoon width Water depth
Table 2 Basic characteristics of the observed waves from the model test program. 28.50 m 8.75 m 28.50 m 6.25 m 6.25 m 668 m
Description
Undisturbed waves
Waves, center of TLP
Wave count
978 11.97 m2 0.60 11.03 s
1089 13.03 m2 0.76 9.90 s
σ2 ε Tz
Fig. 1. A plan view of the tested mini-TLP and the probe locations, values are in model scale (cm).
Thus, the parameters α , γ and R are estimated directly from the data using the L-moments and in order to estimate l1 , l2 and l3 one needs to first estimate b0 , b1 and b2 from the ordered data using Eq. (15). The relations between the first three sample L-moments l1 , l2 , and l3 and b0 , b1 , and b2 are the same as the relations between distribution L-moments λ1 , λ2 , and λ3 and β0 , β1 , and β2 defined in Eq. (21). This model constitutes what herein is referred to as the L-moments non-linear model. 3. Description of the model test data Data from a model study investigating the response behavior of a mini-TLP was selected as the basis to evaluate the Lmoments non-linear model for capturing the extreme behavior and accurately modeling a probability distribution of wave crest data. Selected particulars of the mini-TLP model are presented in Table 1 and additional details can be found in the articles by Teigen et al. [19] and Niedzwecki et al. [20]. The model tests were performed at a model scale of 1:40 and the prototype water depth was 668 m. The unidirectional design seas were generated using the JONSWAP wave amplitude spectrum model with a significant wave height Hs = 13.1 m, a peak period Tp = 14.0 s, a peakedness factor of γs = 2.2, and each time series realization corresponded to a 3-h time series at the prototype scale. A schematic of the wave probe locations used in the miniTLP experiments is presented in Fig. 1. For comparative purposes waves measured at location 6, representing the nearly undisturbed waves far from the structure, and the waves measured at location 3, beneath the center of the offshore structure were analyzed. Each of the measured time series were initially low-pass filtered with a cutoff-frequency of 1.0 Hz assuming that the waves with periods smaller than 1 s were not physically important for this analysis. This assumption was investigated and verified by comparing the unfiltered and filtered auto-spectrum of the data to insure that the difference between the variance of filtered and raw data was negligible. Basic summary information on the time series measured at these two locations is presented in Table 2, and includes the number of wave in the time series, the variances, the spectral width, ε 2 = 1 − (Tc /Tz )2 and mean zero crossing period. Note that
Fig. 2. Frequency spectrum of normalized wave elevations from the mini-TLP model tests.
Tc is the mean crest period and Tz is the mean zero-crossing period. The effects of the incident waves passing over the pontoons and being amplified within the confines of the pontoon boundaries beneath the mini-TLP is presented in Table 2 in terms of the four metrics. It is observed that the spectral broadness of the incident wave field increases suggesting the development of a more non-linear wave field. This is consistent with the shortening of the mean zero-crossing period, the slight increase in wave cycles and the increased variance suggesting the interactions of incident waves modified with diffracted and radiated waves. The frequency spectrum of the normalized wave elevation for waves at these two wave probe locations is presented in Fig. 2. The spectral estimates show that waves at the center of TLP have just slightly more energy off the peak frequency than undisturbed far field waves. This is consistent with the earlier conclusions based upon Table 2 regarding the modification of the wave field. The amplification of the wave field beneath the platform deck exhibits higher non-linearity as a consequence of the platform geometrical considerations and the resulting diffraction and radiation of the incident wave field. 4. Analysis of the model test data Accurate estimates of crest heights with specified probabilities of exceedance are needed for the design of offshore structures. As an indicator of the accuracy of the four methods addressed in this study, the root-mean-square error (RMSE) of the estimated quantiles will be used that will allow one to quantify the difference between the measured data and model estimates. Thus, for a random process
v u N u1 X [x (ui )data − x (ui )estimated ]2 RMSE = t N i=1
(24)
where, N is the number of waves in the time series, x (ui ) is the quantile function of X with a probability of ui = ni / (N + 1), and ni is the number of data points for X ≤ xi . As ocean
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Fig. 3. Probability density function estimates and histogram of the normalized crests of the far-field waves measurements.
41
Fig. 4. Normalized crests of the undisturbed waves as a function of probability of exceedance.
Table 3 The estimated model parameters and RMSE estimates for the far-field wave crests. Method
R
α
γ
Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
1.000 1.000
0.000 0.057
1.000 1.280
0.048 0.031
−0.108
RMSE%
Tail RMSE%
0.000 0.000
5.3 1.8
12.0 2.6
0.000
2.2 0.9
4.0 1.1
engineers are typically interested in the distribution of large crests and the largest crests possible with specified small probability of exceedance, it will be important to examine the overall model fit to the data as well as the accuracy of the model fit to the tail of the distribution. For this purpose, two RMSEs, one estimated considering all the crests and the other one considering only large crests with probability ui ≥ 0.9 are used in this study. The RMSE estimates were normalized using the significant wave height, Hs = 4σ , allowing this error estimate to be interpreted as a percentage of error relative to the significant wave height. 4.1. Analysis of the far-field waves A histogram of the wave crest elevations at wave probe location 6 is presented in Fig. 3. In that figure the probability density function (pdf) obtained using each of the four models are overlaid. The narrow-band linear model overshoots the peak and underestimates the distribution in the high end tail. The narrowband non-linear and the modified non-linear models are similar to each other in that they have nearly equal peak values and that they better represent the high-end tail of the data. The L-moment nonlinear model with its three parameters underestimates the peak value of the distribution but appears to better address the high end tail of the distribution. Table 3 presents a summary of the values used in each of the models and the normalized RMSE errors. For the tail RMSE estimate there were 98 largest crests in the data set. The L-moment non-linear model performs the best and surprisingly, is followed by the narrow-band non-linear model. As expected, the narrowband linear model has the highest RSME values. Actually, none of the four models adequately models the smaller wave crests and interestingly although the location of the each model peak is similar they do not match the peak location of the data. Although, it can be observed that the L-moments model does the best of the four models in the region of small wave heights. In his research study Tayfun [2] concludes that the narrow-band non-linear model and modified non-linear models are not appropriate for small
Fig. 5. Contribution of linear and non-linear terms for the far-field waves as a function of probability of exceedance.
crests, ζn < 0.75 and recommends that these models are suitable for ζn > 1.2. Another perspective of the normalized wave crest distributions is presented in Figs. 4 and 5. It can be observed from these figures that L-moments non-linear method is the most accurate method for capturing the global structure of the probability distribution and for estimating the distribution of large crests. Referring to Table 3 for the L-moments non-linear model, the value of parameter R = 1.280 indicates that this model has captured larger contribution of linear term in quadratic transformation than narrow-band and modified non-linear models. On the other hand, the contribution of the non-linear term is smaller for the Lmoments non-linear model in comparison with the other two nonlinear models but its performance most likely lies in the balance of the two remaining parameters. Fig. 5 shows the contribution of linear and non-linear terms of the transformation as functions of probability exceedance. In that figure the vertical axis is ζ − γ where γ is the shifting constant estimated by the L-moments non-linear model and provides another perspective from which to compare the results of different models more clearly. While the L-moments non-linear model most closely follows the data in Fig. 5 there is still a discrepancy as the probability of exceedance becomes quite small, but the data is also quite sparse in that region. The RMSE values for the L-moments non-linear method as shown in Table 3 are basically half or less than half of the smallest values
42
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Table 4 The observed and estimates of crest heights for the far-field wave crests. Method
E [a] (m)
a (P = 0.1) (m)
a (P = 0.01) (m)
a (P = 0.001) (m)
Observed Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
4.81 4.34 4.73
8.67 7.43 8.34
12.47 10.50 12.33
15.70 12.86 15.60
4.67 4.81
8.20 8.66
12.04 12.78
15.17 16.08
Table 5 The estimated model parameters and RSME estimates for the wave crests beneath the center of the mini-TLP deck. Method
R
α
γ
RMSE%
Tail RMSE%
Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
1.000 1.000 1.000 1.124
0.000 0.074 0.066 0.051
0.000 0.000 0.000 −0.207
4.9 4.3 3.9 1.0
11.2 4.2 2.8 1.9
Fig. 7. Normalized crests of the waves beneath the center of the mini-TLP deck as a function of probability of exceedance. Fig. 6. Probability density function estimates and histogram of the normalized crests of waves beneath the center of the mini-TLP deck.
of the other models which suggests higher relative accuracy of this model comparing to other methods used in this study. The sensitivity of the extreme estimates for the various models is investigated for probabilities of exceedance of P = 0.10, 0.01, 0.001 for the far-field wave data and the results are presented in Table 4. As expected, the extreme estimates of narrowband linear model are significantly smaller than the estimates of non-linear models, especially for crest heights with small exceeding probability. The extreme estimates of the narrowband and L-moments non-linear models maintain a relatively constant difference where as the modified non-linear model predicts values consistently smaller. In all cases the L-moments non-linear model has a slightly larger mean and predicts higher wave crest elevations. Returning to the RMSE values shown in Table 3, it is concluded that the L-moments non-linear model not only captures the overall probability distribution of the data set more accurately, but also has relatively smaller error in estimating the tail distribution of the data which is consistent with the results presented in Table 4. 4.2. Analysis of the waves beneath the mini-TLP As noted earlier the waves at the center of the region bounded by the pontoons have a larger spectral broadness and the wave train contains more non-linearity than undisturbed incident waves. A histogram of the wave crest elevations at wave probe location 3 is presented in Fig. 6. In that figure the probability density function (pdf) obtained using each of the four models are overlaid. The histogram clearly shows significant number of small crests in this data set which is consistent with Table 2. The narrowband linear model overshoots the peak and underestimates the behavior in the critical high end tail. The narrow-band non-linear and modified non-linear models perform quite similarly and their peaks match the peak of the data histogram. The L-moments
non-linear model has a lower peak value which is slightly shifted to the left of the other models and qualitatively seems to fit the histogram better than the other models. In a format similar to Table 3, Table 5 presents the coefficients of the four models and the RMSE errors. Once again the L-moments non-linear model performs quite well with the modified nonlinear model being the best of the remaining three models. It is also confirmed that the narrow-band non-linear and the modified non-linear models perform with about the same RSME values. The overall RSME value is smallest for the L-moments non-linear model indicating a good overall fit for the full range of waves observed in the experiment. It can be observed in Fig. 7 that the three non-linear models follow the data reasonably well with the major discrepancy being in the region of small probability of exceedance where the data is quite limited. Here again the L-moments non-linear method is seen to be a better overall fit to the data. In trying to better understand the role of the three model parameters, α , γ and R, Fig. 8 separates out the linear and non-linear contributions of the three non-linear models. For the L-moments non-linear model the linear contribution is the larger than that of the other two models and its non-linear contribution is smaller than either of the other models. However, the L-moments non-linear model also has the third parameter that provide a slight shift effect which can be verified from the values of the three parameters presented in Table 5. A comparison with the previous example, see Fig. 5, shows that the data actually deviates further from the linear contribution of the L-moments non-linear model and that the corresponding parameter values in Table 5 are noticeably larger than those in Table 3. While the RMSE value for the overall fit remains nearly identical for both examples, the RMSE of the tail increases but remains better than the other non-linear models. Similar to the previous sensitivity investigation for the farfield wave measurements, predictions by the four models of mean and extreme wave crest elevations below the center of the miniTLP deck are presented in Table 6. With the exception of the mean, the predications based upon the narrow-band linear model
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
43
contrasting view of the four models investigated in this research study. Basically, the extremal prediction based upon the Lmoments nonlinear model indicated that the estimates of the other three models underestimated the wave crest elevations in the far-field where the influence of the offshore platform was negligible and overestimated the wave conditions where the offshore platform significantly increased the nonlinearity of the wave field. In previous research studies the Method of L-moments has been applied to interpret large and diverse data sets. However, in this study it has been demonstrated that for limited data the method can be used quite effectively, in this case for model testing associated with the design of an offshore structure. Acknowledgements
Fig. 8. Contribution of linear and non-linear terms in waves beneath the center of the mini-TLP deck as a function of probability of exceedance.
Table 6 The observed and estimates of crest heights for the waves beneath the center of the mini-TLP deck. Method
E [a] (m)
a (P = 0.1) (m)
a (P = 0.01) (m)
a (P = 0.001) (m)
Observed Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
4.46 4.52 5.06
8.51 7.75 8.98
12.75 10.96 13.42
14.90 13.42 17.11
5.00 4.47
8.84 8.42
13.14 12.77
16.69 16.33
are significantly lower than the three non-linear models. Unlike the results presented in Table 4, the narrow-band non-linear and the modified nonl-inear models predict higher values than the Lmoments non-linear model for the mean and each probability of exceedance estimate. This can be related back to the increased non-linearity of the wave field beneath the mini-TLP deck as characterized by the increased spectral broadness and changes in the other parameters used to characterize the wave field as shown in Table 2. Returning to Table 5, it can be confirmed that again the L-moments nonlinear model provides the smallest RMSE estimates for both the overall fit and fit of the high end tail of the measured data, suggesting that the model predictions are more accurate. 5. Summary and conclusions This study introduced the L-moments non-linear model as alternative to the narrow-band linear, the narrow-band nonlinear and the modified non-linear models for developing more robust probability distribution functions of both undisturbed and disturbed wave field measurements. An advantage of the L-moments approach is that their evaluation is less influenced by outliers since they are developed from a linear combination of the data and consequently the bias associated with small samples remains small. In addition, the formulation took advantage of using three parameters in defining the probability distribution functions. The evaluation of the three parameters released the constraint on the linear term and thus all three parameters could be used in fitting the data and the consequence was a better overall fit to the data and in the tail needed for extreme values estimates. This was verified using the root-mean-square error function as a measure of accuracy. The sensitivity analyses of the far-field data and the data measured below the center of the mini-TLP deck provided a
The writers would like to gratefully acknowledge the partial financial support of the Texas Engineering Experiment Station and the R.P. Gregory ’32 Chair endowment during this research investigation. In addition, we would like to recognize Dr. Per Teigen of StatoilHydro and the Offshore Technology Research Center for permission to utilize the experimental data presented in this study. References [1] Tayfun MA. Narrow-band nonlinear sea waves. Journal of Geophysical Research 1980;85(C3):1548–52. [2] Tayfun MA. Statistics of nonlinear wave crests and groups. Journal of Ocean Engineering 2006;33:1589–622. [3] Huang EN, Lung SR, Tung CC, Yuan L, Bliven F. A non-Gaussian model for surface elevation of nonlinear random wave fields. Journal of Geophysical Research 1983;88:7597–606. [4] Arhan M, Plaisted RO. Non-linear deformation of sea wave profiles in intermediate and shallow water. Oceanologica Acta 1981;4(2):107–15. [5] Kriebel DL, Dawson TH. Nonlinearity in wave crest statistics. In: Proc. 2nd int. symp. on ocean wave measurement and analysis. 1993. p. 61–75. [6] Kriebel DL, Dawson TH. Nonlinear effects on wave groups in random seas. Journal of Offshore Mechanics and Arctic Engineering 1991;113:142–7. [7] Moe G, Niedzwecki JM. Frequency of maxima of non-narrow banded stochastic processes. Applied Ocean Research 2006;27(6):265–72. [8] Tayfun MA. Distribution of envelope and phase in weakly nonlinear random waves. Journal of Engineering Mechanics (ASCE) 1994;120(4):1009–25. [9] Al-Humoud J, Tayfun MA, Askar H. Distribution of nonlinear wave crests. Ocean Engineering 2002;29(15):1929–43. [10] Forristall ZG. On the statistical distribution of wave heights in a storm. Journal of Geophysical Research 1978;83(C5):2353–8. [11] Forristall ZG. Wave crest distribution: Observations and second-order theory. Journal of Physical Oceanography 2000;30(8):1931–43. [12] Prevosto M, Forristalll GZ. Statistics of wave crests from models vs. measurements. In: Proceedings, 21st international offshore conference on offshore mechanics and arctic engineering. ASME; 2002. p. 1–11. OMAE200228443. [13] Hosking JRM. L-moments: Analysis and estimation of distributions using linear combinations. Journal of the Royal Statistical Society. Series B (Methodological) 1990;52(1):105–24. [14] Hosking JRM, Wallis JR. Regional frequency analysis an approach based on Lmoments. Cambridge University Press; 1997. [15] Hosking JRM, Wallis JR, Wood EF. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 1985;27(3):251–61. [16] Hosking JRM, Wallis JR. Parameter and quantile estimation for the generalized pareto distribution. Technometrics 1987;29(3):339–49. [17] Pandey MD, Van Gelder PHAJM, Vrijling JK. The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach. Structural Safety 2001;23:179–92. [18] Park HW, Sohn H. Parameter estimation of the generalized extreme value distribution for structural health monitoring. Probabilistic Engineering Mechanics 2006;21:366–76. [19] Teigen P, Niedzwecki JM, Winterstein SR. Wave interaction effects for noncompliant TLP. In: Proc. 11th international offshore and polar engineering conference, vol. 1. 2001. p. 453–61. [20] Niedzwecki JM, Liagre P-YJ, Roesset JM, Kim MH, Teigen P. An experimental research study of a mini-TLP. In: Proc. 11th international offshore and polar engineering conference, vol. 1. 2001. p. 631–4. [21] Longuet-Higgins MS. On the statistical distribution of the heights of sea waves. Journal of Marine Research 1952;11(3):245–66. [22] Cartwright DE, Longuet-Higgins MS. The statistical distribution of the maxima of a random function. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1956;237(1209):212–32. [23] Longuet-Higgins MS. On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width. Journal of Geophysical Research 1980;85(C3):1519–23.
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Estimating wave crest distributions using the method of L-moments Amir H. Izadparast, John M. Niedzwecki ∗ Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, United States
article
info
Article history: Received 1 September 2008 Received in revised form 2 April 2009 Accepted 12 April 2009 Available online 22 May 2009 Keywords: Method of L-moments Ocean waves Wave-structure interaction Mini-TLP model test data Wave crest estimates Parameter estimation
abstract The design of fixed or floating offshore structures requires accurate information of the met-ocean data at the intended offshore site. In the design process it is recognized that this environmental data is modified in the near-field by the interaction with the particular geometrical configuration of the offshore structure. This transformation of the incident wave field around and beneath an offshore structure presents a challenge for ocean engineers when specifying the wave gap elevation to avoid impact loads on the underside of the deck and inundation of the topsides. Thus, the accurate estimation of the wave crest distributions from measurements at various locations near and under the offshore structure during model test studies is essential. A semi-empirical approach is presented herein that builds upon the findings of previous studies and introduces the Method of L-moments. A three parameter model for a wave crest probability distribution function is presented and explicit relationships between the parameters of the distribution and its’ first three L-Moments are established. Furthermore, three narrow-band models from earlier research studies are reviewed and compared with the new model. Wave measurements from a mini-TLP model test program are used as the basis for comparison of the four distributions. The rootmean-square error is used as a metric to quantify the overall fit of the data and its accuracy in the high end tail of the data. The L-Moment model is shown to be more robust in representing the data in both the far-field and beneath the deck of the mini-TLP where the wave field demonstrates increased non-linear behavior. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The distribution of wave crests and in particular the amplification of those wave crests beneath deepwater platforms presents a significant design problem for an ocean engineer considering the air gap elevation needed to avoid potential impact loads on the underside of the deck and inundation of the topsides. For these platforms, vertical columns, submerged pontoons, diagonal bracings and exposed riser arrays can greatly influence the non-linear nature of the design seas through diffraction and radiation effects. Due to the complex nature of this interaction, designers routinely rely on calibrated numerical models and semi-empirical probabilistic models to provide guidance regarding the general behavior and amplification of wave field for use in the design process. The specification of probability density and cumulative distribution functions for non-linear wave crests has been the focus of previous research studies. In this study, a wave crest is defined as the maximum positive wave elevation between two successive zero-upcrossings. Assuming that the wave surface elevation could be modeled as a narrow-band process, Tayfun [1,2] developed a theoretical model that described the transformation of a
∗
Corresponding author. Tel.: +1 979 862 1463. E-mail address: [email protected] (J.M. Niedzwecki).
0141-1187/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2009.04.002
narrow-band linear crest distribution to a narrow-band non-linear crest distribution. This model was subsequently modified by other researchers and the results predicted using the various models yield nearly identical results for the deepwater limit [4–6]. The assumption that the wave surface elevations are narrow-band processes is a fairly robust assumption for the analysis of offshore structures [7]. This assumption however, appears to be somewhat constraining for the development of parametric models as the analysis of field data suggests that the wave elevation tends to be more broad banded, that is the time series often contain multiple peaks and troughs in a single wave cycle. With the development of each new model there was an improvement in the parametric representation of the wave data especially for the region of large wave crests. In 1994 Tayfun [8] developed a non-linear crest distribution function using a Gram–Charlier type of distribution, resulting in a model that was less restrictive with regards to the spectral properties of ocean wave elevations. Later Al-Humoud et al. [9] examined this distribution for ocean surface waves and showed that their improved model was better than the linear model but that the qualitative accuracy of the predictions remained somewhat elusive. Earlier Forristall [10] used a two-parameter Weibull distribution model as the basis to estimate the general probability distribution of wave heigths. He later related his Weibull model parameters to wave steepness and the Ursell
38
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
parameter providing a different perspective for estimating the probability distribution of wave crests [11,12]. That improved model was accurate for large wave crests but was shown to be less so for small waves. More recently, Tayfun [2] updated the quadratic transformation of linear to non-linear wave crests in order to address wave crest height distribution for transitional water depths. In that study, Tayfun used the Weibull model proposed by Forristall to estimate the parameters of the quadratic transformation, and this model was shown to have better accuracy even in deepwater when compared to other non-linear models but again this model deviated for smaller wave amplitudes. It appears that the various two paramter models developed in earlier studies are unable to capture the complete distribution structure of the observed data and that perhaps an unconstrained three parameter model would yield better results. This idea was the foucs of this research study and has led to the development of a three parameter model that utilized the method of L-moments to evaluate each parameter from the meaured data. The model incorporates the second order transfomation of linear to non-linear crests based upon the research work of Tayfun [1], but here a third empircal parmeter is introduced and no parameter is specified as a constant. In this new formulation, each model parameter was explicitly derived in terms of L-moments [13]. The motivation for introducing the method of L-moments into the evaluation of the distribution model parameters was that L-moments are computed from a linear combination of ordered data and that consequently estimates of the parameters are less influenced by outliers and the bias associated with small sample estimates remains fairly small. The method of L-moments is also effective in estimating high-order moments such as skewness and kurtosis whose evaluations are quite sensitive and can be unduly influenced by exceptionally large values [2,14]. Further the method of L-moments has been widely used for extreme value analysis and its efficiency has been studied on both simulated and real data sets [15–18]. In the discussion to follow, the mathematical formulation of the three-parameter model is presented and its performance evaluated using data from an earlier model test program investigating the rigid and compliant response of a mini-TLP designed for West Africa [19,20]. 2. Mathematical formulation 2.1. Basic model description The quadratic transformation of linear to non-linear wave crests introduced by Tayfun [1] is used here with an additional parameter γ which provides the flexibility to estimate the shift between the linear narrow-band and non-linear wave crests. The equation for the normalized non-linear wave amplitude ζn then can be expresses as
ζn = ζ + αζ 2 + γ
(1)
where, ζn = an /σ and similarly the normalized wave crest amplitude is ζ = a/σ with an and a being the non-linear and linear crests elevations respectively. Here the variance is expressed in terms of the single sided wave amplitude spectrum Gηη (ω) as R∞ σ 2 = 0 Gηη (ω)dω which can be specified or developed from the time series η(t ) of the wave amplitude. Solving Eq. (1) to obtain an expression for the normalized linear wave amplitude yields the following expression
ζ =
−1 +
√
1 + 4α (ζn − γ )
. (2) 2α It is assumed that the probability density function of ζ is the Rayleigh law, specifically f (ζ ) =
ζ R
exp −
ζ2 2R
(3)
where, the parameter R is determined from the appropriate data set. Substituting Eq. (2) into Eq. (3), one obtains f (ζn ) =
1 − [1 + 4α (ζn − γ )]−1/2 2α R
−1 + [1 + 4α (ζn − γ )]1/2 × exp − 8Rα 2
2 ! .
(4)
It follows then that the cumulative distribution function (CDF) and quantile function of the non-linear wave amplitude can then be expressed respectively as
2 ! −1 + [1 + 4α (ζn − γ )]1/2 F (ζn ) = 1 − exp − 8Rα 2
(5)
and,
ζn (u) = γ − 2α R ln (1 − u) + [−2R ln (1 − u)]1/2 , 0 ≤ u < 1.
(6)
Upon examination of these last three equations, it is observed that the probability distribution of ζn has three unknown empirical constants, specifically α , γ and R. In the discussion that follows approximations for these three parameters from earlier studies will be discussed. This will be followed by the presentation of a derivation that relates these three parameters directly to Lmoments and consequently directly to the appropriate time series measurements. 2.1.1. Earlier parametric models Longuet-Higgins [21] derived an expression for the Rayleigh parameter; specifically he showed that R = (a2rms /σ 2 )/2 where arms is the root-mean-square of the crest-height. For the case of √ linear narrow-band waves arms = 2σ and subsequently the Rayleigh parameter has a value of unity [22]. Later, LonguetHiggins [23] showed that the non-linear effects tend to increase the ratio of a2rms /σ 2 /2 while spectrum width effects may either increase or decrease the ratio. The effect of the combination of non-linearity and spectrum width on the value of R is not clear from a theoretical perspective, but what is important is that the value of the Rayleigh parameter can be influenced to some extent by these effects. There also is the issue that in the transformation of the narrow-band linear model, the effects of spectrum width are modeled using the quadratic term which is not physically correct and this may cause overestimation of large crests. In later studies, the processes were assumed to be narrow banded and it was assumed that two of these parameters namely γ and R had the following values, γ = 0 and R = 1 [1–3,5,6]. In addition, the linear assumption requires the parameter α to have a value of zero in this model. A Rayleigh model using these values is referred to in this study as the narrow-band linear model. Assuming that individual waves are adequately described by Stokes second-order wave theory, it follows that
α=
kσ 2
−
1 sinh 2kd
+
cosh kd (2 + cosh 2kd) 2 sinh3 kd
(7)
where, k = 2π /L is the wave number, L is the wave length associated with the mean period T¯ calculated from the ratio of the first two moments of the wave elevation spectrum m0 /m1 , and d is the water depth. For the deep water limit of interest in this study, Eq. (7) reduces to α = k2σ [1]. In this study the statistics estimated using γ = 0, R = 1 and α estimated from Eq. (7) will be referred to as the narrow-band non-linear model. More recently Tayfun [2] showed that the same quadratic transformation defined in Eq. (1) could be used for transitional water depths where the parameter α represents wave steepness
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
and is specified to be consistent with the two parameter Weibull distribution model used by Forristall [11] specifically
α=
8c 3 b
0
r
3
−
b
(8)
128
where b and c are the shape and scale parameters of the two parameter Weibull distribution respectively, and 0 is the gamma function. The parameters b and c were approximated by Forristall [11] based upon fitting Weibull distribution to 2-D and 3-D computer simulations of second order non-linear waves. The computer simulations used in Forristall’s study covered the practical range of wave peak period, wave steepness, and water depth. In order to define parameters b and c as functions of the water depth and wave spectrum characteristics, Forristall [11] used wave steepness S and Ursell number Ur defined as below S=
8π σ
Ur =
(9)
gT 4σ k2 d3
and the notation in brackets denotes binomial coefficients, for example
π
.
(10)
Analyses of the 2-D and 3-D simulations resulted in relations given in Eqs. (11) and (12) respectively
39
r j
=
r!
r ≥ j.
j! (r − j)!
(16)
The probability distribution parameter estimation using method of L-moments is based on the probability weighted moments βr defined as
βr =
1
Z
x (u) ur du.
(17)
0
Note that br defined in Eq. (15) is an unbiased estimator of probability weighted moment βr . The difference between conventional moments and probability weighted moments is that conventional moments involve successively higher powers of the quantile function x (u), whereas probability weighted moments involve successively higher power of u [14]. The L-moments λr of a random variable X with cumulative distribution function F (x) and quantile function x(u) are basically the linear combination of the probability weighted moments. The L-moments λr of X are then defined as,
λr =
Z
1
x (u) Pr∗−1 (u) du.
(18)
0
b2 = 2 − 2.1597 S + 0.0968 Ur 2 1 c2 = √ + 0.2892 S + 0.1060 Ur 2 2 b3 = 2 − 1.7912 S − 0.5302 Ur + 0.2848 Ur 2 1 c3 = √ + 0.2568 S + 0.0800 Ur 2 2
(11)
These linear combinations arise from integrals of x (u) weighted by a set of orthogonal polynomials called shifted Legendre polynomials Pr∗ (u) [14]
(12)
Pr∗ (u) =
2.1.2. L-moment model and parameter estimation The use of L-moments for parameter estimation was developed by Hosking [13] and these moments represent the expectations of certain linear combinations of order statistics. The main advantage of L-moments is that they are less influenced by outliers since they are developed from a linear combination of data and consequently the bias of their small sample estimates remains fairly small. This makes the method of L-moments an efficient means for estimating the extreme values when the data is limited and the existence of outliers is expected. In order to estimate the m parameters of a distribution with the method of L-moments, its first m L-moments, expressed as functions of the distribution parameters, would be equated to their corresponding sample values [14]. The sample L-moment lr of an ordered sample x1:n ≤ x2:n ≤ · · · ≤ xn:n of size n is defined as r X
r = 0, 1, . . . , n − 1
p∗r ,j bj
(13)
j=0
−1
br = n
r j
n−1 r
r +j j
(14)
−1 X n j−1 j =r +1
(19)
In terms of weighted moments this can be expressed as
λ r +1 =
r X
p∗r ,j βj .
(20)
j =0
In order to estimate the parameters R, α, and γ the relation between these parameters and the first three L-moments of ζn need to be established. It can be shown that the first three distribution L-moments are
λ1 = β 0 λ2 = 2β1 − β0 λ3 = 6β2 − 6β1 + β0
(21)
where, λ1 , λ2 , and λ3 are the measures of location, scale, and skewness of the probability distribution. Substituting the quantile function of ζn , given in Eq. (6), into Eq. (17) and using the relations given in Eq. (21) one obtains after integration the following equations
λ1 = γ + 2α R + 0
3
√ 2R
2
√ 3 √ λ2 = α R + 2−1 0 R
(22)
2
√ 3 rR 1 3/2 3/2 λ3 = α R + 6+2 −3 0 . 3
2
3
Then upon equating the foregoing L-moments with their corresponding sample values and solving for the distribution parameters one obtains the following equations
where, p∗r ,j = (−1)r −j
p∗r ,j uk .
j =0
where, the subscripts ‘‘2’’ and ‘‘3’’ refer to the 2-D and 3-D estimates. The largest value of α estimated by using either (b2 , c2 ) or (b3 , c3 ) in Eq. (8) will be used in the non-linear crests probability distribution [2]. This model is basically designed to address large crests and will be referred to as the modified non-linear model in this study. Note that the parameter γ = 0 in this model.
lr +1 =
r X
r
xj:n
(15)
R = (4.139l2 − 12.418l3 )2
√ α = l2 /R − 0.367/ R
(23)
√
γ = l1 − 2α R − 1.253 R.
40
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Table 1 The main particulars of the mini-TLP. Draft Column diameter Column spacing Pontoon height Pontoon width Water depth
Table 2 Basic characteristics of the observed waves from the model test program. 28.50 m 8.75 m 28.50 m 6.25 m 6.25 m 668 m
Description
Undisturbed waves
Waves, center of TLP
Wave count
978 11.97 m2 0.60 11.03 s
1089 13.03 m2 0.76 9.90 s
σ2 ε Tz
Fig. 1. A plan view of the tested mini-TLP and the probe locations, values are in model scale (cm).
Thus, the parameters α , γ and R are estimated directly from the data using the L-moments and in order to estimate l1 , l2 and l3 one needs to first estimate b0 , b1 and b2 from the ordered data using Eq. (15). The relations between the first three sample L-moments l1 , l2 , and l3 and b0 , b1 , and b2 are the same as the relations between distribution L-moments λ1 , λ2 , and λ3 and β0 , β1 , and β2 defined in Eq. (21). This model constitutes what herein is referred to as the L-moments non-linear model. 3. Description of the model test data Data from a model study investigating the response behavior of a mini-TLP was selected as the basis to evaluate the Lmoments non-linear model for capturing the extreme behavior and accurately modeling a probability distribution of wave crest data. Selected particulars of the mini-TLP model are presented in Table 1 and additional details can be found in the articles by Teigen et al. [19] and Niedzwecki et al. [20]. The model tests were performed at a model scale of 1:40 and the prototype water depth was 668 m. The unidirectional design seas were generated using the JONSWAP wave amplitude spectrum model with a significant wave height Hs = 13.1 m, a peak period Tp = 14.0 s, a peakedness factor of γs = 2.2, and each time series realization corresponded to a 3-h time series at the prototype scale. A schematic of the wave probe locations used in the miniTLP experiments is presented in Fig. 1. For comparative purposes waves measured at location 6, representing the nearly undisturbed waves far from the structure, and the waves measured at location 3, beneath the center of the offshore structure were analyzed. Each of the measured time series were initially low-pass filtered with a cutoff-frequency of 1.0 Hz assuming that the waves with periods smaller than 1 s were not physically important for this analysis. This assumption was investigated and verified by comparing the unfiltered and filtered auto-spectrum of the data to insure that the difference between the variance of filtered and raw data was negligible. Basic summary information on the time series measured at these two locations is presented in Table 2, and includes the number of wave in the time series, the variances, the spectral width, ε 2 = 1 − (Tc /Tz )2 and mean zero crossing period. Note that
Fig. 2. Frequency spectrum of normalized wave elevations from the mini-TLP model tests.
Tc is the mean crest period and Tz is the mean zero-crossing period. The effects of the incident waves passing over the pontoons and being amplified within the confines of the pontoon boundaries beneath the mini-TLP is presented in Table 2 in terms of the four metrics. It is observed that the spectral broadness of the incident wave field increases suggesting the development of a more non-linear wave field. This is consistent with the shortening of the mean zero-crossing period, the slight increase in wave cycles and the increased variance suggesting the interactions of incident waves modified with diffracted and radiated waves. The frequency spectrum of the normalized wave elevation for waves at these two wave probe locations is presented in Fig. 2. The spectral estimates show that waves at the center of TLP have just slightly more energy off the peak frequency than undisturbed far field waves. This is consistent with the earlier conclusions based upon Table 2 regarding the modification of the wave field. The amplification of the wave field beneath the platform deck exhibits higher non-linearity as a consequence of the platform geometrical considerations and the resulting diffraction and radiation of the incident wave field. 4. Analysis of the model test data Accurate estimates of crest heights with specified probabilities of exceedance are needed for the design of offshore structures. As an indicator of the accuracy of the four methods addressed in this study, the root-mean-square error (RMSE) of the estimated quantiles will be used that will allow one to quantify the difference between the measured data and model estimates. Thus, for a random process
v u N u1 X [x (ui )data − x (ui )estimated ]2 RMSE = t N i=1
(24)
where, N is the number of waves in the time series, x (ui ) is the quantile function of X with a probability of ui = ni / (N + 1), and ni is the number of data points for X ≤ xi . As ocean
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Fig. 3. Probability density function estimates and histogram of the normalized crests of the far-field waves measurements.
41
Fig. 4. Normalized crests of the undisturbed waves as a function of probability of exceedance.
Table 3 The estimated model parameters and RMSE estimates for the far-field wave crests. Method
R
α
γ
Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
1.000 1.000
0.000 0.057
1.000 1.280
0.048 0.031
−0.108
RMSE%
Tail RMSE%
0.000 0.000
5.3 1.8
12.0 2.6
0.000
2.2 0.9
4.0 1.1
engineers are typically interested in the distribution of large crests and the largest crests possible with specified small probability of exceedance, it will be important to examine the overall model fit to the data as well as the accuracy of the model fit to the tail of the distribution. For this purpose, two RMSEs, one estimated considering all the crests and the other one considering only large crests with probability ui ≥ 0.9 are used in this study. The RMSE estimates were normalized using the significant wave height, Hs = 4σ , allowing this error estimate to be interpreted as a percentage of error relative to the significant wave height. 4.1. Analysis of the far-field waves A histogram of the wave crest elevations at wave probe location 6 is presented in Fig. 3. In that figure the probability density function (pdf) obtained using each of the four models are overlaid. The narrow-band linear model overshoots the peak and underestimates the distribution in the high end tail. The narrowband non-linear and the modified non-linear models are similar to each other in that they have nearly equal peak values and that they better represent the high-end tail of the data. The L-moment nonlinear model with its three parameters underestimates the peak value of the distribution but appears to better address the high end tail of the distribution. Table 3 presents a summary of the values used in each of the models and the normalized RMSE errors. For the tail RMSE estimate there were 98 largest crests in the data set. The L-moment non-linear model performs the best and surprisingly, is followed by the narrow-band non-linear model. As expected, the narrowband linear model has the highest RSME values. Actually, none of the four models adequately models the smaller wave crests and interestingly although the location of the each model peak is similar they do not match the peak location of the data. Although, it can be observed that the L-moments model does the best of the four models in the region of small wave heights. In his research study Tayfun [2] concludes that the narrow-band non-linear model and modified non-linear models are not appropriate for small
Fig. 5. Contribution of linear and non-linear terms for the far-field waves as a function of probability of exceedance.
crests, ζn < 0.75 and recommends that these models are suitable for ζn > 1.2. Another perspective of the normalized wave crest distributions is presented in Figs. 4 and 5. It can be observed from these figures that L-moments non-linear method is the most accurate method for capturing the global structure of the probability distribution and for estimating the distribution of large crests. Referring to Table 3 for the L-moments non-linear model, the value of parameter R = 1.280 indicates that this model has captured larger contribution of linear term in quadratic transformation than narrow-band and modified non-linear models. On the other hand, the contribution of the non-linear term is smaller for the Lmoments non-linear model in comparison with the other two nonlinear models but its performance most likely lies in the balance of the two remaining parameters. Fig. 5 shows the contribution of linear and non-linear terms of the transformation as functions of probability exceedance. In that figure the vertical axis is ζ − γ where γ is the shifting constant estimated by the L-moments non-linear model and provides another perspective from which to compare the results of different models more clearly. While the L-moments non-linear model most closely follows the data in Fig. 5 there is still a discrepancy as the probability of exceedance becomes quite small, but the data is also quite sparse in that region. The RMSE values for the L-moments non-linear method as shown in Table 3 are basically half or less than half of the smallest values
42
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
Table 4 The observed and estimates of crest heights for the far-field wave crests. Method
E [a] (m)
a (P = 0.1) (m)
a (P = 0.01) (m)
a (P = 0.001) (m)
Observed Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
4.81 4.34 4.73
8.67 7.43 8.34
12.47 10.50 12.33
15.70 12.86 15.60
4.67 4.81
8.20 8.66
12.04 12.78
15.17 16.08
Table 5 The estimated model parameters and RSME estimates for the wave crests beneath the center of the mini-TLP deck. Method
R
α
γ
RMSE%
Tail RMSE%
Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
1.000 1.000 1.000 1.124
0.000 0.074 0.066 0.051
0.000 0.000 0.000 −0.207
4.9 4.3 3.9 1.0
11.2 4.2 2.8 1.9
Fig. 7. Normalized crests of the waves beneath the center of the mini-TLP deck as a function of probability of exceedance. Fig. 6. Probability density function estimates and histogram of the normalized crests of waves beneath the center of the mini-TLP deck.
of the other models which suggests higher relative accuracy of this model comparing to other methods used in this study. The sensitivity of the extreme estimates for the various models is investigated for probabilities of exceedance of P = 0.10, 0.01, 0.001 for the far-field wave data and the results are presented in Table 4. As expected, the extreme estimates of narrowband linear model are significantly smaller than the estimates of non-linear models, especially for crest heights with small exceeding probability. The extreme estimates of the narrowband and L-moments non-linear models maintain a relatively constant difference where as the modified non-linear model predicts values consistently smaller. In all cases the L-moments non-linear model has a slightly larger mean and predicts higher wave crest elevations. Returning to the RMSE values shown in Table 3, it is concluded that the L-moments non-linear model not only captures the overall probability distribution of the data set more accurately, but also has relatively smaller error in estimating the tail distribution of the data which is consistent with the results presented in Table 4. 4.2. Analysis of the waves beneath the mini-TLP As noted earlier the waves at the center of the region bounded by the pontoons have a larger spectral broadness and the wave train contains more non-linearity than undisturbed incident waves. A histogram of the wave crest elevations at wave probe location 3 is presented in Fig. 6. In that figure the probability density function (pdf) obtained using each of the four models are overlaid. The histogram clearly shows significant number of small crests in this data set which is consistent with Table 2. The narrowband linear model overshoots the peak and underestimates the behavior in the critical high end tail. The narrow-band non-linear and modified non-linear models perform quite similarly and their peaks match the peak of the data histogram. The L-moments
non-linear model has a lower peak value which is slightly shifted to the left of the other models and qualitatively seems to fit the histogram better than the other models. In a format similar to Table 3, Table 5 presents the coefficients of the four models and the RMSE errors. Once again the L-moments non-linear model performs quite well with the modified nonlinear model being the best of the remaining three models. It is also confirmed that the narrow-band non-linear and the modified non-linear models perform with about the same RSME values. The overall RSME value is smallest for the L-moments non-linear model indicating a good overall fit for the full range of waves observed in the experiment. It can be observed in Fig. 7 that the three non-linear models follow the data reasonably well with the major discrepancy being in the region of small probability of exceedance where the data is quite limited. Here again the L-moments non-linear method is seen to be a better overall fit to the data. In trying to better understand the role of the three model parameters, α , γ and R, Fig. 8 separates out the linear and non-linear contributions of the three non-linear models. For the L-moments non-linear model the linear contribution is the larger than that of the other two models and its non-linear contribution is smaller than either of the other models. However, the L-moments non-linear model also has the third parameter that provide a slight shift effect which can be verified from the values of the three parameters presented in Table 5. A comparison with the previous example, see Fig. 5, shows that the data actually deviates further from the linear contribution of the L-moments non-linear model and that the corresponding parameter values in Table 5 are noticeably larger than those in Table 3. While the RMSE value for the overall fit remains nearly identical for both examples, the RMSE of the tail increases but remains better than the other non-linear models. Similar to the previous sensitivity investigation for the farfield wave measurements, predictions by the four models of mean and extreme wave crest elevations below the center of the miniTLP deck are presented in Table 6. With the exception of the mean, the predications based upon the narrow-band linear model
A.H. Izadparast, J.M. Niedzwecki / Applied Ocean Research 31 (2009) 37–43
43
contrasting view of the four models investigated in this research study. Basically, the extremal prediction based upon the Lmoments nonlinear model indicated that the estimates of the other three models underestimated the wave crest elevations in the far-field where the influence of the offshore platform was negligible and overestimated the wave conditions where the offshore platform significantly increased the nonlinearity of the wave field. In previous research studies the Method of L-moments has been applied to interpret large and diverse data sets. However, in this study it has been demonstrated that for limited data the method can be used quite effectively, in this case for model testing associated with the design of an offshore structure. Acknowledgements
Fig. 8. Contribution of linear and non-linear terms in waves beneath the center of the mini-TLP deck as a function of probability of exceedance.
Table 6 The observed and estimates of crest heights for the waves beneath the center of the mini-TLP deck. Method
E [a] (m)
a (P = 0.1) (m)
a (P = 0.01) (m)
a (P = 0.001) (m)
Observed Narrow-band linear Narrow-band non-linear Modified non-linear L-moments non-linear
4.46 4.52 5.06
8.51 7.75 8.98
12.75 10.96 13.42
14.90 13.42 17.11
5.00 4.47
8.84 8.42
13.14 12.77
16.69 16.33
are significantly lower than the three non-linear models. Unlike the results presented in Table 4, the narrow-band non-linear and the modified nonl-inear models predict higher values than the Lmoments non-linear model for the mean and each probability of exceedance estimate. This can be related back to the increased non-linearity of the wave field beneath the mini-TLP deck as characterized by the increased spectral broadness and changes in the other parameters used to characterize the wave field as shown in Table 2. Returning to Table 5, it can be confirmed that again the L-moments nonlinear model provides the smallest RMSE estimates for both the overall fit and fit of the high end tail of the measured data, suggesting that the model predictions are more accurate. 5. Summary and conclusions This study introduced the L-moments non-linear model as alternative to the narrow-band linear, the narrow-band nonlinear and the modified non-linear models for developing more robust probability distribution functions of both undisturbed and disturbed wave field measurements. An advantage of the L-moments approach is that their evaluation is less influenced by outliers since they are developed from a linear combination of the data and consequently the bias associated with small samples remains small. In addition, the formulation took advantage of using three parameters in defining the probability distribution functions. The evaluation of the three parameters released the constraint on the linear term and thus all three parameters could be used in fitting the data and the consequence was a better overall fit to the data and in the tail needed for extreme values estimates. This was verified using the root-mean-square error function as a measure of accuracy. The sensitivity analyses of the far-field data and the data measured below the center of the mini-TLP deck provided a
The writers would like to gratefully acknowledge the partial financial support of the Texas Engineering Experiment Station and the R.P. Gregory ’32 Chair endowment during this research investigation. In addition, we would like to recognize Dr. Per Teigen of StatoilHydro and the Offshore Technology Research Center for permission to utilize the experimental data presented in this study. References [1] Tayfun MA. Narrow-band nonlinear sea waves. Journal of Geophysical Research 1980;85(C3):1548–52. [2] Tayfun MA. Statistics of nonlinear wave crests and groups. Journal of Ocean Engineering 2006;33:1589–622. [3] Huang EN, Lung SR, Tung CC, Yuan L, Bliven F. A non-Gaussian model for surface elevation of nonlinear random wave fields. Journal of Geophysical Research 1983;88:7597–606. [4] Arhan M, Plaisted RO. Non-linear deformation of sea wave profiles in intermediate and shallow water. Oceanologica Acta 1981;4(2):107–15. [5] Kriebel DL, Dawson TH. Nonlinearity in wave crest statistics. In: Proc. 2nd int. symp. on ocean wave measurement and analysis. 1993. p. 61–75. [6] Kriebel DL, Dawson TH. Nonlinear effects on wave groups in random seas. Journal of Offshore Mechanics and Arctic Engineering 1991;113:142–7. [7] Moe G, Niedzwecki JM. Frequency of maxima of non-narrow banded stochastic processes. Applied Ocean Research 2006;27(6):265–72. [8] Tayfun MA. Distribution of envelope and phase in weakly nonlinear random waves. Journal of Engineering Mechanics (ASCE) 1994;120(4):1009–25. [9] Al-Humoud J, Tayfun MA, Askar H. Distribution of nonlinear wave crests. Ocean Engineering 2002;29(15):1929–43. [10] Forristall ZG. On the statistical distribution of wave heights in a storm. Journal of Geophysical Research 1978;83(C5):2353–8. [11] Forristall ZG. Wave crest distribution: Observations and second-order theory. Journal of Physical Oceanography 2000;30(8):1931–43. [12] Prevosto M, Forristalll GZ. Statistics of wave crests from models vs. measurements. In: Proceedings, 21st international offshore conference on offshore mechanics and arctic engineering. ASME; 2002. p. 1–11. OMAE200228443. [13] Hosking JRM. L-moments: Analysis and estimation of distributions using linear combinations. Journal of the Royal Statistical Society. Series B (Methodological) 1990;52(1):105–24. [14] Hosking JRM, Wallis JR. Regional frequency analysis an approach based on Lmoments. Cambridge University Press; 1997. [15] Hosking JRM, Wallis JR, Wood EF. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 1985;27(3):251–61. [16] Hosking JRM, Wallis JR. Parameter and quantile estimation for the generalized pareto distribution. Technometrics 1987;29(3):339–49. [17] Pandey MD, Van Gelder PHAJM, Vrijling JK. The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach. Structural Safety 2001;23:179–92. [18] Park HW, Sohn H. Parameter estimation of the generalized extreme value distribution for structural health monitoring. Probabilistic Engineering Mechanics 2006;21:366–76. [19] Teigen P, Niedzwecki JM, Winterstein SR. Wave interaction effects for noncompliant TLP. In: Proc. 11th international offshore and polar engineering conference, vol. 1. 2001. p. 453–61. [20] Niedzwecki JM, Liagre P-YJ, Roesset JM, Kim MH, Teigen P. An experimental research study of a mini-TLP. In: Proc. 11th international offshore and polar engineering conference, vol. 1. 2001. p. 631–4. [21] Longuet-Higgins MS. On the statistical distribution of the heights of sea waves. Journal of Marine Research 1952;11(3):245–66. [22] Cartwright DE, Longuet-Higgins MS. The statistical distribution of the maxima of a random function. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1956;237(1209):212–32. [23] Longuet-Higgins MS. On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width. Journal of Geophysical Research 1980;85(C3):1519–23.