Proceedings of the 28th International Conference on Ocean, Offshore and Arctic Engineering OMAE 2009 May 31–June 5, 2009, Honolulu, Hawaii, USA

OMAE2009-79691

TURKSTRA MODELS OF CURRENT PROFILES

Steven R. Winterstein [email protected]

Sverre Haver [email protected]

ABSTRACT Design of deep-water structures requires accurate models of currents versus water depth. Common models are N-year profiles, which conservatively assume the N-year extreme current speed occurs simultaneously at each depth. To address this conservatism, we introduce Turkstra models of current profiles here. These yield a set of current profiles, each of which coincides with the N-year profile at a single depth and is reduced elsewhere. The degree of reduction is a direct function of the spatial correlation structure of the current process. Results are shown for a deep-water North Sea site, and compared with time-domain prediction of extreme loads for linear and drag load mechanisms. Extensions are suggested to combine these methods with procedures such as Empirical Orthogonal Functions, permitting the data to define the most economical set of basis vectors upon which the Turkstra logic is applied.

Einar Nygaard [email protected]

profile at a single depth and is reduced elsewhere. The degree of reduction is a direct function of the spatial correlation structure of the current process. Figure 1 shows an example of such profiles for a deep-water North Sea site. We show here how such profiles are produced, and test their abilities to predict extreme loads in both linear and quadratic (drag-dominated) cases. Extensions are suggested to combine these methods with procedures such as Empirical Orthogonal Functions (e.g., [3], [4], [5], [6]), to allow the data to choose the most economical set of basis vectors upon which the Turkstra logic is applied.

Modelling Current Loads With Turkstra’s Rule To illustrate concepts, we first consider only two current velocities, x1 (t) and x2 (t), measured simultaneously at different water depths. Interest then focuses on the maximum value of a load, L(t), which varies with both x1 (t) and x2 (t):

INTRODUCTION To design marine structures in deep water, currents must be modelled accurately as a function of depth. These models often take the form of N-year profiles, combining the marginal Nyear extreme current speed at each depth. These are essentially worst-case profiles, which assume the worst current speed occurs simultaneously at each depth. As correlations among currents at various depths lessen, these worst-case profiles yield increasingly conservative predictions of structural loads. To address this conservatism, we introduce Turkstra models of current profiles here. These use the logic of Turkstra’s load combination rule, introducing a set of load scenarios where one component is at its extreme level, and all others are at their expected “associated” values (e.g., [1], [2]). In our case the result is a set of current profiles, each of which coincides with the N-year

Lmax = max {L(t)} = max {L[x1 (t), x2 (t)]} 0≤t≤T

0≤t≤T

(1)

The duration T is a reference interval of interest, e.g., T =100 years. Turkstra’s rule suggests that Lmax be estimated by the larger of L(t) values at two times only: the times when each of the current time histories xi (t) is maximized. Formally, define these times t1 and t2 as x1 (t1 ) = max {x1 (t)} = x1,max ; x2 (t2 ) = max {x2 (t)} = x2,max 0≤t≤T

0≤t≤T

(2) 1

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100-Year Profiles 1.6

Normalized Current Speed

1.4 1.2 1

Worst-Case Turkstra@70m

0.8

Turkstra@100m Turkstra@200m Turkstra@400m

0.6 0.4 0.2 0 0

200

400

600

800

1000

1200

Depth [m]

Figure 1. Traditional (worst-case) N =100-year profile, with set of 100-year Turkstra current profiles. Current speeds are normalized by the largest observed speed in the 16-month data set.

The Turkstra’s rule estimate of Lmax is then Lmax ≈ max{L[x1 (t1 ), x2 (t1 )]; L[x1 (t2 ), x2 (t2 )]}

responding conservative (upper-bound) estimate of Lmax is easily constructed by setting each xi (t) to its maximum value. For two processes, written in terms analogous to Eqn. 3:

(3)

Lmax ≤ L(x1,max , x2,max ) = L[x1 (t1 ), x2 (t2 )]

This result is particularly convenient for structural design. It suggests that the load need be evaluated for only two current profiles: (1) [x1 (t1 ), x2 (t1 )]= maximum value of x1 and associated x2 value when x1 is maximum; and (2) [x1 (t2 ), x2 (t2 )]= maximum value of x2 and associated x1 value when x2 is maximum. Equation 3 extends easily to the general case of n currents, x1 (t), ..., xn (t), whose maxima occur at times t1 , ...,tn respectively. In this case, we need consider the worst load caused among n current profiles, each of which reflecting a time ti when one of the current histories is maximum, and the other n − 1 are at associated values. In principle, Turkstra’s rule estimates such as Eqn. 3 should be unconservative; i.e., yield lower bounds on the desired maximum load Lmax . This is conceptually clear, since these estimates ignore all other times—when none of the component processes xi (t) is at its global maximum—at which Lmax may occur. A cor-

(4)

This “worst-case” result becomes correct only if t1 =t2 ; more precisely, if the maximum values of x1 (t), x2 (t), and L(t) occur simultaneously. As correlation among the processes xi (t) lessens, this worst case profile will likely yield increasingly conservative results, while the Turkstra rule estimates may become increasingly unconservative. We investigate here the consequence of these approximations—worst case profile vs. Turkstra profiles— for a particular North Sea location where current data are available.

THE CURRENT DATA Current velocities have been measured simultaneously at 6 depths: d=70, 100, 200, 400, 600, and 1000m. Measurements are 2

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Monthly Maxima 1.2

Normalized Current Speed

1

0.8

d=70 d=100 d=200

0.6

d=400 d=600 d=1000

0.4

0.2

0 1

2

3

4

5

6

7

8

9

10

11

12

13

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15

16

Month n (n=1...16)

Figure 2.

Variations in monthly maxima over 16 month data set.

Annual Maxima and Worst-Case Profile Due to our focus on extreme values, we model monthly maxima at each current elevation. Practical interest focuses, however, on extremes with longer return periods; e.g., 1-, 10- and 100-year maxima. To estimate these, we must fit a probabilistic model to the monthly maxima, and extrapolate it accordingly. Since our population comprises maxima X1 , ..., X16 over each month, it is natural to adopt a Gumbel (Extreme Type I) probability model of X. Details of this model are included in an appendix that follows. For our purposes, the critical result is that if monthly maxima follow a Gumbel model, annual maxima Y =max(X1 ...XN ) also follow a (shifted) Gumbel model. The annual maximum fractile yq (with exceedance probability q) can then be directly related to mX and σ2X , the mean and variance of the monthly maxima:

reported every 10 minutes, and should be viewed as 10-minute average current speeds at each depth. There are thus 6 × 24=144 measurements per day. The entire data set includes 68661 observations, reflecting a duration of 68661/144=476.8 days, or 15.9 30-day periods defined here as “months.” Also, to not lose the information in the last 0.9 month of this short data set, we treat it here as if it were a full (30 day) month, and hence assume a population of 16 monthly maxima.

Figure 2 shows these 16 monthly maxima for each of the 6 current histories. (Here and throughout, current speeds are normalized by the largest observed speed in the data set, which occurs in month 2 at depth d=70m.) Qualitatively similar behavior is found in adjacent currents; e.g., the 3 “shallow” currents (at d ≤ 200m) vary similarly, although their global maxima do not coincide. However, many of the local oscillations in these shallow currents are not reflected at the lower depths (especially d ≥ 600m). It is precisely these imperfect correlations that Turkstra models seek to capture.

y q = mX +

σX [− ln(− ln(1 − q)) + ln N − 0.577] 1.28

(5)

In the square brackets, the first term reflects the desired fractile level q, the second term adjusts from X to Y , and the last term 3

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Values at 100m | Peaks at 70m

corrects for the difference between the mean and mode. In this result, N reflects the number of months modelled per year; e.g., N=12 if all data are used to fit the Gumbel model. Equation 5 has been used to predict 100-year values (q=.01), with mX and σX estimated from the monthly maxima at each elevation. Two fits have been performed: (1) based on all 16 data, and (2) excluding months 9–12, which may be argued to come from a separate statistical population (e.g., a “quiet” season). As Fig. 2 shows, this seems particularly plausible for the upper currents (d ≤ 200m). With approach (2) we retain only 16-4=12 months during which larger currents are encountered on average, but this is offset by noting that our model now applies to only an 8-month year (hence N=8 is used here in Eqn. 5). We believe approach (2) to more accurately predict extreme currents, and we generally show results from only this approach (e.g., the 100-year profiles in Fig. 1). If seasonal effects are indeed significant, approach (1) may likely overestimate extremes, assigning a single, over-broad Gumbel model to what is perhaps instead a mixture of different seasonal models. (Appendix I demonstrates this effect.) Note, however, that relative results— such as biases in load estimates from various profiles—are shown below to be relatively insensitive to this treatment of seasonal effects.

1.2

Associated Value at 100m

1

0.8

Observed

0.6

Regression (rho=.73)

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

Monthly Maximum at 70m

Figure 3.

Normalized current speed at

d =100m when topmost current

(at d =70m) reaches a monthly maximum. Also shown is regression fit. Values at 200m | Peaks at 70m 1 0.9

Associated Value at 200m

0.8 0.7 0.6 Observed

0.5

Regression (rho=.71)

0.4 0.3 0.2

TURKSTRA PROFILES Turkstra profiles are constructed in two basic steps: (1) currents at each depth are analyzed separately, to predict marginal extremes such as 100-year values; and (2) given current values at a particular reference depth, associated currents at all other depths are analyzed. The marginal analysis in step (1) has been described above. This analysis is not unique to Turkstra profiles; indeed it is commonplace. It is therefore not our main focus here. We note, however, that when based on such short data sets as those available here, step (1) can involve considerable subjectivity and hence statistical uncertainty. Our main purpose here is to implement the conditional analysis of Step 2. It generally relies on standard linear regression models, of the form E[Y |X = x] = mY + ρX,Y σY (

x − mX ) σX

0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

Monthly Maximum at 70m

Figure 4. Normalized current speed at d =200m when topmost current (at d =70m) reaches a monthly maximum. Also shown is regression fit.

Turkstra Profiles Based on Monthly Maxima Consider first the Turkstra profile associated with maximum x70 , the current at depth d=70m. Figures 3–6 show the values at 4 lower depths (z=100, 200, 400, and 600m) that occur when x70 reaches each of its 16 monthly maxima. Also shown are regression fits to these data. (These fits are virtually unchanged if months 9–12 are removed.) As may be expected, highest correlation is shown at d=100m, the measured depth nearest to d=70m. Correlation decreases quickly as depth increases, and the lowest depths (d ≥ 600m) appear nearly uncorrelated. Finally, the Turkstra profile associated with maximum x70 again uses Eqn. 6. The means and variances of X and Y , together with their correlation coefficient ρX,Y , are estimated from the data in Figs. 3–6. Equation 6 is then used to estimate the N-year profile, with appropriate x=maxx70 value for the desired return period (Eqn. 5 with q=1/N). Figure 1 shows the resulting Turkstra profile associated with maximum x70 , labelled “Turkstra@70m.” Because it assumes maximum current speed at depth d=70m, it coincides with the

(6)

Here x is the current at the reference location, assumed equal to its N-year extreme value. The y values in Eqn. 6 are the estimated currents at other depths. These will decrease as the correlation coefficient, ρX,Y , is reduced; in the limit as ρX,Y → 0, knowledge of x becomes immaterial and we will estimate the current simply by mY , its marginal mean. We will use Eqn. 6 to produce Turkstra profiles in two distinct ways: based on the monthly maxima data, and based on the statistics of the entire current processes. 4

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Values at 400m | Peaks at 70m

pure drag load, with constant Cd over the entire structural length.) Extreme value analysis of L(t) parallels that of the current velocities: time histories of L(t) are constructed at each time t, monthly maxima found, and their means and variances used with a Gumbel model to estimate N-year values as in Eqn. 5. For our purposes these are viewed as “exact” results, to which approximations based on N-year current profiles are compared.

0.7

Associated Value at 400m

0.6

0.5

0.4 Observed Regression (rho=.47)

0.3

0.2

0.1

Bias = Estimated / Exact L100 0 0

0.2

0.4

0.6

0.8

1

1.2

Load Model:

Monthly Maximum at 70m

Figure 5.

Normalized current speed at

d =400m when topmost current

(at d =70m) reaches a monthly maximum. Also shown is regression fit.

Linear

Drag

1.28

1.42

1.33

1.49

0.89

0.92

0.88

0.87

Worst-Case Profile

Values at 600m | Peaks at 70m

(all months)

0.3

Worst-Case Profile Associated Value at 600m

0.25

(excludes months 9–12) 0.2

Turkstra Profiles Observed

0.15

Regression (rho=.38)

(all months)

0.1

Turkstra Profiles

0.05

(excludes months 9–12)

0 0

0.2

0.4

0.6

0.8

1

Table 1.

1.2

Biases in 100-year load estimates from various profiles.

Monthly Maximum at 70m

Figure 6. Normalized current speed at d =600m when topmost current (at d =70m) reaches a monthly maximum. Also shown is regression fit.

Table 1 confirms that applying the worst-case current profile yields a conservative estimate of the 100-year load. As may be expected, this conservatism increases with p; e.g., biases are roughly 30% in the linear (p=1) case, while they increase to above 40% in the case of pure drag (p=2). In contrast, and also as expected, the Turkstra profiles yield a mildly unconservative estimate, underestimating loads by 8–13% across the two cases. As noted above, while absolute load levels are affected if months 9– 12 are excluded, these bias values are relatively insensitive to this treatment of seasonal effects. Figures 7–8 show that these biases remain relatively constant over a range of return periods. They also show that the Turkstra profile with maximum at d=100m governs in these cases, although the profile with maximum at d=400m is roughly comparable in the linear case (Fig. 7).

worst-case profile at that depth. At lower depths it predicts lower than worst-case velocities; specifically, the associated values expected when the maximum at depth d=70m occurs. Compared with the worst-case profile, the Turkstra profile falls off more sharply with depth, in roughly exponential fashion. This is a direct result of the similar exponential decay shown by the spatial correlation of the monthly maxima (ρ values shown in Figs. 3– 6). The other Turkstra profiles in Fig. 1 are found similarly. Each profile conditions on monthly maxima at a single reference depth, and 5 regressions are performed to predict the conditional mean (Eqn. 6) at the other 5 depths. To test the accuracy of these profiles, load histories of the form L(t)=∑i wi xi (t) p have been constructed for p=1 (linear) and p=2 (quadratic) load cases. The sum is over the six current speeds xi (t), with weights wi proportional to the vertical lengths over which they apply1 . (The quadratic case can be viewed as a

Turkstra Profiles Based on Random Process Models In view of our interest in extreme values, it is logical to use monthly maxima to fit regressions such as Eqn. 6. It is also consistent with our marginal extreme analysis (step (1) above), which also is based here on monthly maxima. This focus on monthly maxima does not come without cost, however. It drastically reduces the amount of data—here, 4320

1 Specifically, we first infer current speeds x =1.1x 0 70 and x1500 =0.75x1000 at depths z=0 and z=1500m, and then assume the resulting 8 currents apply over lengths 40, 40, 70, 150, 200, 300, 400, and 300m (e.g., [7]).

5

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Linear Load Case Current Correlation Functions

1.4 1.2

1.2

1 Worst-Case Turkstra@70

0.8

0.8

Turkstra@100 Turkstra@200

Correlation

Estimated / Exact

1

Turkstra@400

0.6

Turkstra@600 Turkstra@1000

0.4

Fit d=70 d=100

0.6

d=200 d=400 d=600

0.4 0.2 0.2 0 1

10

100 0

Return Period [yrs]

0

200

400

600

800

1000

Vertical Separation [m]

Figure 7. Bias associated with estimating linear load. Results exclude quiet season in months 9–12. (Note that Turkstra@100 results here are

Figure 9. Correlation between current speed at depth d and depth d + ∆, viewed as a function of the vertical separation ∆. Also shown is the exponential fit: ρ(∆)=exp(−∆/∆char ), where the characteristic correlation length is ∆char =800m. Results here use the entire data set.

obscured by virtually identical results of Turkstra@400.) Drag Load Case 1.6 1.4

depths z1 =d and z2 =z1 +∆, as a function of the vertical separation ∆. For example, the current at depth z1 =d=70m has correlation coefficients (.96, .86, .64, .18, .20) with the current at depths z2 =(100, 200, 400, 600, 1000m) respectively. These correlation values are plotted in the figure at separation lengths ∆=(10070=30, 130, 330, 530, 930m). Several features are notable:

Estimated / Exact

1.2 Worst-Case

1

Turkstra@70 Turkstra@100

0.8

Turkstra@200 Turkstra@400 Turkstra@600

0.6

Turkstra@1000 0.4 0.2

1. There are clearly two different current regimes: an upper regime for depths d ≤ 400m, and a lower regime with d ≥ 600m. There is essentially no correlation between currents in these two regimes (computed ρ values of .25 at most). 2. Within either the lower or upper regime, current speeds appear spatially homogeneous. In other words, the correlation between currents at depths d and d + ∆ appear to vary only with ∆. Moreover, in either regime this correlation is found to be well estimated by a simple exponential model:

0 1

10

100

Return Period [yrs]

Figure 8.

Bias associated with estimating pure drag load. Results ex-

clude quiet season in months 9–12.

time points (per 30-day month) are reduced to a single outcome. This can greatly increase statistical uncertainty. It can also enhance the effects of subjective modelling choices, such as how one deals with seasonal variations. For these reasons, it is of interest to consider how the results would change if we instead use the entire current process to establish the regression results in Eqn. 6, rather than merely its monthly peak values. To use Eqn. 6 in this context, the means, variances, and correlations should be estimated from the entire current histories. Specifically, letting x(t, z) denote the current speed at time t and depth z, we seek the correlation ρx (z1 , z2 )=Corr[x(t, z1 ), x(t, z2 )]—physically, the correlation between the current experienced at two different depths, z1 and z2 , at the same time t. (This is sometimes called a “lag-zero” correlation, to distinguish it from correlation between process values viewed at two different times.) If the current process is spatially homogeneous, ρx (z1 , z2 ) will depend on only the vertical separation, ∆=|z1 −z2 |, between the two depths. Figure 9 shows the correlation between current speeds at

ρx (z, z + ∆) = exp(−

∆ ) ; ∆char = 800m ∆char

(7)

3. If the quiet period in months 9–12 is excluded, these correlation functions are found to remain virtually unchanged. For example, the current at depth z1 =d=70m now yields correlation coefficients (.95, .86, .61, .15, .17) with the current at lower depths, as opposed to the values (.96, .86, .64, .18, .20) from Fig. 9 which use all data. Thus, we confirm a main virtue of the random process modelling approach: it focuses on statistics, such as marginal moments and lag-zero correlations, that are most robustly estimated from a limited time series. It follows that these statistics are relatively insensitive to whether we include or exclude a relatively quiet period during months 9–12. We also note that the correlations among currents at an arbitrary point in time—as described 6

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100-Year Profiles, Max at D=200m

by Eqn. 7—are larger than the correlations found when monthly maxima occur (see ρ values in Figs. 3–6).

1.4

Normalized Current Speed

1.2

100-Year Profiles, Max at D=70m 1.6

Normalized Current Speed

1.4 1.2

1

0.8 Turkstra_Peaks Turkstra_Process

0.6

0.4

1 0.2

Turkstra_Peaks

0.8

Turkstra_Process 0

0.6

0

200

400

600

800

1000

1200

Depth [m] 0.4

Figure 12.

0.2

0

200

400

600

800

1000

Comparing Turkstra profiles using regression from (a)

monthly peaks and (b) entire current process. Results shown here for reference depth d =200m.

0 1200

Depth [m]

Figure 10. Comparing Turkstra profiles using regression from (a) monthly peaks and (b) entire current process. Results shown here for reference depth d =70m.

100-Year Profiles, Max at D=400m 1.2

Normalized Current Speed

1

100-Year Profiles, Max at D=100m 1.6

Normalized Current Speed

1.4

0.8

Turkstra_Peaks

0.6

Turkstra_Process

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1.2 0 1

0

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200

400

600

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1200

Depth [m]

Turkstra_Peaks Turkstra_Process

Figure 13. Comparing Turkstra profiles using regression from (a) monthly peaks and (b) entire current process. Results shown here for reference depth d =400m.

0.6 0.4 0.2 0 0

200

400

600

800

1000

1200

(upper or lower), and (2) lower correlations between currents in the two different regimes. This causes process-based Turkstra profiles to fall off more slowly near their peak values, and more quickly at remote spacings (which span the two regimes). Table 2 shows that in this case, at least, process- and peakbased models show quite similar biases. Further, their unconservatism is effectively removed here by using not the mean but rather the “mean plus one sigma” value. With the Gaussian model, this is equivalent to the p=84% fractile of the associated current level y:

Depth [m]

Figure 11. Comparing Turkstra profiles using regression from (a) monthly peaks and (b) entire current process. Results shown here for reference depth d =100m.

Finally, we consider the Turkstra profiles that result when the random process statistics are used in Eqn. 6. Figures 10–13 show the resulting profiles with maxima at depths d=70, 100, 200, and 400m. For reference, these figures also show the previous Turkstra profiles based on monthly maxima. From Figs. 10– 13, it seems that little accuracy, if any, is lost when proceeding from a peak-based to a process-based Turkstra model. The main difference comes from their respective correlations: compared with the peak-based model, the random process model estimates (1) higher correlations among currents within the same regime

y.84 = mY |X + σY |X

(8)

To apply this result, the conditional mean, mY |X , is given by Eqn. 6. Because ρ2X,Y is the fraction of variance “explained” by the regression, the linear model predicts the remaining (condi7

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tional) variance to be σY2 |X = σY2 (1 − ρ2X,Y )

use more robust statistics and seem to introduce little additional bias, it seems useful to construct Turkstra profiles from random process statistics. Moreover, Table 2 shows that to compensate for the basic unconservatism of Turkstra’s rule, load bias is removed here by using the 84% fractile of the associated currents (Eqn. 10 rather than Eqn. 6).

(9)

Substituting Eqn. 6 and Eqn. 9 into Eqn. 8, we find the 84% fractile to be y.84 = mY + ρX,Y σY (

q x − mX ) + σY 1 − ρ2X,Y σX

DISCUSSION AND EXTENSIONS Turkstra models have been introduced here to account for the imperfect correlation among currents at different depths. This correlation will become less important as loads become more “localized” in space. In the limit, if the load depends only on a single measured current component, correlation becomes immaterial and both worst-case and Turkstra profiles will produce the same (marginal) N-year extreme. Conversely, cases that most challenge these methods are those in which many or all current components, xi (t), contribute comparably to the structural load L(t). In that sense, our test cases—which use constant weighting of either a linear or quadratic load over the vertical length of the structure—are a good challenge to these methods. In our limited testing of cases with non-constant weights, we find various profiles—either Turkstra or worst-case—to be at least as accurate (i.e., to have equal or lesser bias) as the results shown here for equal weighting. Finally, as presented here these models are intended only for loads that are non-decreasing in all current components xi ; i.e., ∂L/∂xi ≥ 0. To describe more complex phenomena (e.g., VIV) within the Turkstra framework, it may prove useful to change the vector basis. Collecting the current histories xi (t) into an ndimensional vector x(t), we may write these as a weighted sum over a new set of time-independent basis vectors, Vj :

(10)

Preliminary study suggests that this simple “mean plus one sigma” choice of associated currents suffices across a range of load cases. This follows the common use of inflated response fractiles to compensate for omitted uncertainty (e.g., [8]). Of course, the precise choice of p=.84 is based here on a single data set, and should be tested across a wider range of cases.

Bias = Estimated / Exact L100 Load Model: Linear

Drag

Worst-Case Profile

1.33

1.49

Mean Turkstra Profiles (Peak-Based)

0.88

0.87

Mean Turkstra Profiles (Process-Based)

0.87

0.87

p=84% Turkstra Profiles (Peak-Based)

1.00

0.99

p=84% Turkstra Profiles (Process-Based)

0.98

1.00

Table 2.

Biases in 100-year load estimates from various profiles.

x(t) = w1 (t)V1 + ... + wq (t)Vq SUMMARY We have introduced Turkstra models of current profiles here. These are constructed in two basic steps: (1) currents at each depth are analyzed separately, to predict marginal extremes such as N-year values; and (2) given current values at a particular reference depth, associated currents at all other depths are analyzed. Steps (1) and (2) are essentially decoupled; any standard method of extreme value analysis may be used in step (1). Step (2) assumes that the marginal N-year extreme occurs at one current elevation, and asks what values occur simultaneously at other depths. We have addressed this with two types of models, one based on random peaks (monthly maxima) and another based on random process statistics (from the complete current histories). In our cases, these two models yield quite similar profiles (Figs. 10–13) and load predictions (Table 2). Because they

(11)

The vectors Vj can be chosen by the analyst, to reflect current shapes most “damaging” to the structure. Alternatively, the current data can be used to dictate the shapes: empirical orthogonalization formulates an eigenproblem whose solution yields shapes Vj that “best explain” the variance of x(t). In any case, once the Vj are specified, the time-varying functions to be analyzed are no longer the original current histories, xi (t), i = 1...n, but rather the weight functions w j (t), j = 1...q. If q < n the w j (t) histories can be estimated by least squares methods. The load can then be viewed as L[w(t)], a function of the time-varying weights w j , and the Turkstra logic would yield L100 ≈ max L[w j = w j,100 ; other wk at “associated” values] j=1...q

(12) 8

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The basic model we have used here can be seen as the trivial case in which all the Vj are unit vectors, and the weights w j (t) become simply the individual measured current components. If we can model the problem sufficiently with only q=2 components, a 100-year contour of [w1 (t), w2 (t)] can also be constructed and searched (e.g., [8]). But the basic Turkstra approach permits a rather general number of components, which may also include directional current components instead of speed values alone.

predicted from, the mean mX and standard deviation σX of X: mX = uX +

1.28 0.577 ; σX = αX αX

To find the fractile x p with a specified probability of nonexceedance, we set FX (x)=p in Equation 13 and solve for x: x p = uX +

REFERENCES [1] Turkstra, C., 1970. Theory of Structural Safety. Technical report, University of Waterloo, Waterloo, Ontario, Canada. SMStudy No. 2. [2] Madsen, H., Krenk, S., and Lind, N., 1986. Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ. [3] Forristall, G. Z., and Cooper, C. K., 1997. “Design current profiles using empirical orthogonal function (eof) and inverse form methods”. In Proceedings, 1997 Offshore Technology Conference, pp. 11–21. Paper number OTC 8267. [4] Kleiven, G., 2002. “Identifying viv vibration modes by use of the empirical orthogonal functions technique”. In Proceedings, Vol. 1, 21st Intl. Conf. on Offshore Mech. and Arctic Eng., pp. 711–719. [5] Meling, T., and Eik, K., 2002. “An assessment of eof current scatter diagrams with respect to riser viv fatigue damage”. In Proceedings, Vol. 1, 21st Intl. Conf. on Offshore Mech. and Arctic Eng., pp. 85–93. [6] Srivilairit, T., and Manuel, L., 2007. “Vortex-induced vibration and coincident current velocity profiles for a deepwater drilling riser”. In Proceedings, 26th Intl. Conf. on Offshore Mech. and Arctic Eng. Paper number OMAE2007-29596. [7] Nygaard, E., and Eik, K. J., 2004. Deep Water Current Profile in the Norwegian Seas. Technical report, Statoil, Stavanger, Norway. Document No. PTT-NKG-RA 00063. [8] Haver, S., and Winterstein, S., 2008. “Environmental contour lines: A method for estimating extremes by a short term analysis”. In Proceedings, 2008 SNAME Annual Meeting and Ship Production Symposium. Paper number B3-067.

1 [− ln(− ln p)] αX

(15)

It is also useful to note that if X has Gumbel distribution, the maximum Y =max(X1 , ..., XN ) of N such Xi again has Gumbel distribution, with the same shape parameter, αY =αX , and the (upward) shifted mode:

uY = uX +

ln N αX

(16)

For example, if the Xi represent monthly maxima, using this result with N=12 yields the mode of the annual maxima under the Gumbel model. Finally, combining Equations 14–16, the fractile of y p can be expressed as y p = mX +

σX [− ln(− ln p) + ln N − 0.577] 1.28

(17)

This is the result cited in Eqn. 5 of the main text, with q=1-p. Seasonal Effects Finally, we consider briefly the extent to which our analysis may be influenced by seasonal effects. The foregoing Gumbel modelling approach assumes that monthly maxima of current histories form a homogeneous statistical population. This may be questioned if, for example, these currents show significant seasonal variations. Ideally, this issue could be avoided by studying annual rather than monthly maxima. Unfortunately, the limited duration of our data set precludes this option here. In discussing the maxima in Fig. 2, we have noted an apparent seasonal variation and, in particular, a “quiet” period producing relatively low maxima. This is observed particularly in the uppermost currents; e.g., at depths 70–200m. In these cases we may question whether the lower maxima in fact follow a different distribution, and a different fit would be found if these uninteresting, lower maxima were excluded. Figures 14–15 explore this for the current at depth d=200m. Figure 14 shows the full population of 16 monthly maxima observed at this depth, On the Gumbel scale shown, samples from a

APPENDIX I: GUMBEL FIT TO EXTREMES It is common to model extreme values with the Gumbel (Extreme Type I) probability distribution. A Gumbel random variable X has the following cumulative distribution function: FX (x) = P[X ≤ x] = exp[−e−αX (x−uX ) ]

(14)

(13)

Here uX is the mode of X, and αX is an inverse measure of dispersion. The parameters uX and αX can be related to, and hence 9

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Monthly maxima at d=200m

current speed, and hence 1.172 =1.37 in the associated drag force. This suggests the original model in Figure 14 somewhat overestimates extremes, by assigning a single, over-broad Gumbel model to what is perhaps instead a mixture of different seasonal models. Of course, we may still question the adequacy of this particular Gumbel model, and of the precise numerical extreme values it generates. (Why not exclude still more of the lower tail of the data, for example? We stop here at the lowest 4 points because they occur contiguously in time, and thus are most plausibly the result of seasonal variation.) It would seem difficult, in summary, to exclude this model—or, indeed, quite a number of different variations—given the limited, 12-month duration on which it is based. Most importantly, one should recognize the considerable statistical uncertainty inherent in predicting 100year results from such short data sets. Finally, we note that the foregoing fits are moment-based, and hence weigh all of the data equally. Maximum likelihood methods also treat all data equally, and generally yield comparable fits to those based on moments. These are in marked contrast to least-square fits that may be applied, for example, to data on the Gumbel scale shown in Figure 14 or Figure 15. While leastsquares fits to these figures can be visually pleasing, they weigh the data unequally, giving greater import to the expanded, uppertail region in this case. If one begins with the assumption of a homogeneous population, equally weighted fitting (based on moments or maximum likelihood) is generally preferred. Probability scale figures, such as those shown here, are intended mainly as a diagnostic tool, to test the quality of these “global” fits more locally, e.g., in the upper-tail region. One should not expect, nor require, visually optimal fits when viewed on these distorted scales.

8 7 6

-ln(-ln(F))

5 4 Observed

3

Gumbel Model

2 1 0 0

0.5

1

1.5

2

-1 -2 Speed x / x_max,obs

Figure 14. Gumbel fit to all N =16 monthly maxima of the current at depth d =200m. Extrapolation shown to predicted 100-year level.

Monthly maxima at d=200m 7 6 5

-ln(-ln(F))

4 3 Observed

2

Gumbel Model

1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 -2 -3 Speed x / x_max,obs

Figure 15.

Gumbel fit to the largest

N =12 monthly maxima of the cur-

rent at depth d =200m. The “quiet” season during months 9–12 has been excluded. Extrapolation shown to predicted 100-year level.

single homogeneous Gumbel sample should appear as a straight line (e.g., Eqn. 15). The figure shows the lowest points to be in some disagreement. It also reminds us of how far we need extrapolate to predict 100-year values—and hence the limitations of having only a 16-month data set. For reference, this model predicts the normalized current to have 100-year value x100 =1.54— i.e., 1.54 times the largest observed current at this depth. (This value follows by setting the monthly exceedance probability 1-F to 0.01/12, and hence − ln(− ln(F))=7.09 in the figure.) Figure 15 shows the result of removing the lowest 4 maxima, which occur during months 9–12, and performing a new moment-based Gumbel fit. Not surprisingly, the remaining data appear to better follow the new Gumbel line. By removing lowertail values we find a Gumbel model with smaller variance, and hence a lower 100-year prediction. Specifically, using Figure 15 with 1-F=0.01/8 for an 8-month year, we now find x100 =1.32 rather than 1.54—the difference is a factor of 1.54/1.32=1.17 in 10

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