Proceedings of the 30th International Conference on Ocean, Offshore and Arctic Engineering OMAE 2011 June 19–24, 2011, Rotterdam, Netherlands

OMAE2011-49866

TURKSTRA PROFILES OF NORTH SEA CURRENTS: WIDER THAN YOU’D THINK

Steven R. Winterstein [email protected]

Sverre Haver [email protected]

Borge Kvingedal [email protected]

Alok K. Jha [email protected] Einar Nygaard [email protected]

ABSTRACT To design marine structures in deep water, currents must be modelled accurately as a function of depth. These models often take the form of T -year profiles, which assume the T -year extreme current speed occurs simultaneously at each depth. To better reflect the spatial correlation in the current speeds versus depth, we have recently introduced Turkstra current profiles. These assign the T -year speed at one depth, and “associated” speeds expected to occur simultaneously at other depths. Two essentially decoupled steps are required: (1) marginal analysis to estimate T -year extremes, and (2) some type of regression to find associated values. The result is a set of current profiles, each of which coincides with the T -year profile at a single depth and is reduced elsewhere. Our previous work with Turkstra profiles suggested that, when applied in an unbiased fashion, they could produce unconservative estimates of extreme loads. This is in direct contrast to the findings of Statoil, whose similar (“CCA”) current profiles have generally been found to yield conservative load estimates. This paper addresses this contradiction. In the process, we find considerable differences can arise in precisely how one performs steps 1 and 2 above. The net finding is to favor methods that properly emphasize the upper tails of the data—e.g., using peak-over-threshold (“POT”) data, and regression based on class means—rather than standard analyses that weigh all data equally. By applying such tail-sensitive methods to our dataset, we find the unconservative trend in Turkstra profiles to essentially vanish. For our data, these tail-fit results yield profiles with both larger marginal

extremes, and broader profiles surrounding these extremes— hence the title of this paper.

INTRODUCTION To design marine structures in deep water, currents must be modelled accurately as a function of depth. For design against current loads, a simple profile choice combines the marginal T year extreme current speed at each depth. These are generally 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 generally yield increasingly inaccurate predictions of structural loads. To reflect this imperfect correlation, we have recently introduced Turkstra models of current profiles [1]. 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., [2], [3]). This results in a set of current profiles, each of which coincides with the T -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.

Modelling Current Loads With Turkstra’s Rule In applying Turkstra’s rule to North Sea currents, we follow a practice that has been commonly adopted by Statoil in recent years, a procedure known as Conditional Current Analysis or “CCA.” Indeed, one of the mandates of our work has 1

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been to apply a conceptually similar analysis to the North Sea current data set measured at Ormen Lange, which has been analyzed by Statoil’s in-house CCA procedure (e.g., [4]). Following their work, we focus here on total current speed irrespective of direction. Directional effects will be a topic of additional study; Appendix 1 shows some preliminary models that include directionality. We also hope to consider VIV limit states in follow-on work, studying whether empirical orthogonal functions (e.g., [5], [6], [7], [8])—or more general system identification methods—can isolate a limited number of profile shapes that govern for a particular riser geometry. If a limited number of shapes can be identified, the statistics of their weighting functions can be expressed through T -year contours (e.g., [9]). To establish Turkstra (or CCA) profiles, two distinct steps are required:

and broader profiles surrounding these extremes—hence the title of this paper.

Step 1: At each depth for which current speeds are available, one must estimate the marginal T -year current, xT . Step 2: Given that the T -year current occurs at a particular depth, one must estimate associated currents that are expected to occur simultaneously at other depths. These steps essentially decouple; the first requires marginal analysis of the current at a particular depth, while the second typically uses some type of regression. As noted above, the result is a set of current profiles, each of which assigns the T -year speed xT at a single depth and the reduced, “associated” values elsewhere. Our preliminary work on Turkstra profiles [1] led to the following findings: 1. A straightforward, unbiased analysis of the current data— with steps 1 and 2 above—led to unconservative estimates of 100-year loads. 2. This unconservatism could be avoided by choosing conservatively biased regressions in step 2. In particular, 84% fractile values of the associated currents were found to yield essentially unbiased loads.

Figure 1. Location of the Ormen Lange field. The red line shows an underwater pipeline, running to Norway (Nyhamna) and finally to the United Kindom (Easington) after passing through the Sleipner field. This is currently the world’s longest underwater pipeline.

THE CURRENT DATA AT ORMEN LANGE Ormen Lange is the largest natural gas field in development on the Norwegian contential shelf. Production began in September 2007. The field is situated 140 km northwest of Nyhamna (Fig. 1). It is the location of the underwater Storegga Slides, considered among the largest known landslides (“Storegga” is Norwegian for “Great Edge”). As a result, seabed depths along the field vary between 800 and 1100 meters. Statoil has an ongoing campaign measuring current speeds at the Ormen Lange field. Our previous study [1] utilized a current dataset of roughly 16 months duration. We consider here a longer dataset, which spans roughly 22 months. Specifically, average current velocities are reported every 10 minutes at 10 depths simultaneously: z=20, 35, 50, 75, 200, 300, 400, 500, 600, and 750m. The data set comprises N=97024 points at each depth, corresponding to a duration of N/(6 × 24 × 365)=1.85 years (about 22.1 months). These data have been extracted from

These findings are in direct opposition to those of Statoil, who generally find their CCA profiles to yield conservative estimates of 100-year loads (e.g., [4], [10]). This paper seeks to explain this contradiction. In the process, we find considerable differences can arise in precisely how one performs steps 1 and 2 above. The net result is to favor methods that properly emphasize the upper tails of the data—e.g., using peak-over-threshold (“POT”) data, and regression based on class means1 —rather than standard analyses that weigh all data equally. By applying such tail-sensitive methods to our dataset, we find the unconservative trend in Turkstra profiles to essentially vanish. For our data, these tail-fit results yield profiles with both larger marginal extremes,

1 The concept here is to divide (x, y) data into equally-spaced intervals over the x-axis, and report only the centroidal value (x, y) within each interval. This is discussed further in subsequent sections.

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measurements over the period 1997–2008, excluding intervals when measurements are missing at one or more depths. Marginal statistics of these data, and moment-based models, are shown in Appendix 1.

model is then rescaled and shifted (by κ and xmin , respectively) to preserve the mean and variance of the shifted data. For positive ε, Eqn. 1 results in a model that can capture a range of skewness values, between that of W and that of W 2 . Experience suggests that this includes most practical cases of interest. For cases where the data show a skewness less than that of W , the tails of W are diminished by interchanging X and W in Eqn. 1. The algorithm FITS has been established to automate this fitting, and accommodate an arbitrary threshold, in predicting extremes [11].

MARGINAL MODELS OF T-YEAR EXTREMES The first step in constructing curent profiles is to estimate the marginal, T -year extreme current speed at each depth. This is, in principal, a fairly straightforward problem. Indeed, in the offshore community it is fairly routine to estimate 100-year wave heights. Note, however, that for waves we generally have at least an order of magnitude more data than our T =1.85 year database of current speeds. The limited extent of our current data requires that we select carefully among different plausible models. For the Ormen Lange data, these models have included

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1. A 2-moment fit to monthly maxima [1] 2. A 3-moment fit to all data [10] 3. A 2- or 3-moment fit to peak-over-threshold (“POT”) data (to be shown here). Regarding monthly maxima, it was found [1] that there was a “quiet” period—of about 4 successive months—with unusually low maxima, and that biased results could be found if one included these in a single Gumbel model of all monthly maxima. Thus, even after all data other than monthly maxima are removed, there still appears to be a danger of a heterogeneous sample, and hence a mis-fitting when applying a simple, 2-parameter probability distribution model.

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Figure 2. Comparing the observed maxima in the T =1.85-year database with the predicted average maxima over many such T =1.85year intervals. Predictions use the Weibull and Quadratic Weibull models fit to POT data, and a random process model (Hermite) described further in Appendix 1.

PEAK-OVER-THRESHOLD MODELS The peak-over-threshold method is designed specifically to achieve a more homogeneous sample, by excluding points that are uninteresting (due to their low values) or show serial correlation (e.g., points neighboring a local peak). Here we choose a speed threshold, vT H , corresponding to the 99% fractile speed at each depth. Between every upcrossing of vT H and its subsequent downcrossing, the largest speed value, V , is identified for analysis. We have then applied two models:

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1. A Weibull model of W =V − vT H , based on its first two moments. 2. A Quadratic Weibull model, commonly related to the 2moment Weibull variable W as

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X = xmin + κ(W + εW 2 ) ; W = V − vT H

Weibull Quadratic Weibull Hermite Observed Max

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Figure 3. Predictions of the 100-year maximum current at various depths. Our predictions use POT data, while the Statoil results use 3moment fits to all data.

Here ε is first chosen so that the skewness of W + εW 2 matches that of the shifted data, V − vT H . The resulting 3

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Result 1: The T=1.85-Year Extreme To test results, it is useful to first use the various models to predict the (average) extreme over a duration of T =1.85 years, the duration of the database. This can then be compared with the observed maxima in the database. While the single observed T =1.85-year maximum at each depth is a rather noisy estimate of the mean extreme over many such intervals, we are concerned here with whether the models show any notable bias. Indeed, Fig. 2 shows that both the Weibull and Quadratic Weibull models agree well with the observed maxima in the dataset. Similar agreement is found for a random process (Hermite) model, which is discussed further in Appendix 1.

tary cumulative distribution function (CCDF), for all current data at this depth. Weibull and Quadratic Weibull fits to all data are also shown. It is first notable that based on these fits, Statoil’s 100-year estimate of V100 =120 cm/s (Fig. 3) becomes quite plausible. As Fig. 4 shows, however, one may reasonably question the accuracy of these fits in the upper tails. The observed CCDF appears to flatten in its upper tail, a phenomenon occurring at least above the speed v=90 cm/s. Relative to this population of all 10-minute values, however, the level v=90 cm/s is quite far into the tail; e.g., with exceedance probability less than .001 from Fig. 4. It is not surprising that 2and 3-moment fits do not capture information at this high-fractile level.

Result 2: The T=100-Year Extreme Figure 3 shows the predicted 100-year current speeds arising from both the Weibull (2-moment) and Quadratic Weibull (3-moment) fits to POT data. Also shown are 100-year predictions formed by applying the Statoil CCA method to the same dataset [10]. This is also a 3-moment fit, hence should yield comparable results to the Quadratic Weibull model. Agreement is indeed good except for the depth d=300m, where the Statoil result is lower by about 10%. Because the largest observed currents occur at this depth, this discrepancy is studied further below.

P [ Peak > x ] in POT Event

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MARGINAL MODELS FIT TO ALL DATA In Fig. 3, the main difference in the Statoil prediction is that it uses all data, not simply POT data. To better understand this method, we perform similar (2- and 3-moment) fits to all data. Because the significant difference arises in Fig. 3 at depth d=300m, we choose to study the current data at this depth. 1

P [ Value > x ]

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Weibull and Quadratic Weibull fits to POT data at depth 300m.

To better understand the upper-tail behavior of these data, Fig. 5 shows the corresponding POT data for this d=300m depth. (For this depth, the threshold speed is vT H =65 cm/s.) Also shown are the Weibull and Quadratic Weibull fits to these data, which have led to the 100-year estimates (132 and 145 cm/s) shown in Fig. 3. This figure shows that the apparent anomolous tail behavior shown in Fig. 4 essentially vanishes when POT data are considered. The data in Fig. 5 appear to support both the Weibull and Quadratic Weibull fits, and make it difficult to justify a 100year estimate as low as 120 cm/s. Note too that Fig. 5 reveals that there are no independent (peak) data between speeds of 90–95 cm/s. It is thus suggested that the apparent “flattening” anomoly, from the cumulative data of Fig. 4, is merely the deceptive view one may find when plotting all data of a continuously varying time function. Since all sample paths that upcross v=90 cm/s here will also soon upcross v=95 cm/s, it is not surprising that the CCDF of all data flattens (PDF notably reduces) in this region. This shows the potential danger even in visualizing the exceedance probability plot based on all data, not to mention the possibly misleading results that may arise from fitting to all data.

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Figure 4. Weibull and Quadratic Weibull fits to all current data; depth=300m.

Figure 4 shows the exceedance probability, or complemen4

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This is a conclusion we find repeated in offshore engineering: if (upper tail) extremes are of interest, one can be considerably misled by a global, moment-based fit to all data. This was the conclusion when modelling monthly maxima of these current data, where it was found that the lowest values came from a separate sub-population and should be excluded [1]. We therefore favor the use of POT data here, and in similar applications. A similar finding has been made with respect to predicting 100-year wave heights in the North Sea [12].

Observed max, observed associated values Observed max, 50%-regression on POT values Observed max, 84%-regression on POT values Observed max, 90%-regression on POT values

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REGRESSION ANALYSIS AND TURKSTRA PROFILES The foregoing results show that with POT models, design currents—e.g., at depth d=300m—can show higher speeds than those found from global fits to all data. We show here that tail-fit models may also yield wider design current profiles, compared to unbiased results that weigh all data equally. The general idea of Turkstra profiles is to combine the T year speed, xT , at a given depth with the “associated” speed, y, expected at other depths at the same time. The conventional model for this is linear regression. It is convenient here to use a general result for y p , the conditional p-fractile of y given x=xT , in terms of the corresponding p-fractile U p of a standard normal variable2 : y p = mY + ρX,Y σY (

q xT − mX ) +U p σY 1 − ρ2X,Y σX σ2X

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Figure 6. Estimated current profiles when the current at depth d =300m attains its maximum speed, 106 cm/s.

at the instant in the database when the speed at d=300m is maximum) is also wider than the unbiased (p=0.5) regression-based profile. An “inflated” profile, with p=0.84–0.90, appears needed to better model this observed behavior. This observation formed the basis for our previous work [1], which suggested use of the p=.84 fractile. To test the accuracy of these profiles, we have constructed the current-induced drag load

(2) L(t) = ∑ wi vi (t) p ; i

σY2

Here mX and mY are the means of X and Y , and their variances, and ρX,Y their correlation coefficient. The familiar (mean) regression result follows with p=.50 and U p =0, so that the last term in Eqn. 2 disappears. In contrast, the p=.84 and .90 fractiles use this result with U.84 =1 and U.90 =1.28, respectively.

∑ wi = 1

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in which p=2. The sums here are over the ten squared current speeds, v2i (t), with weights wi proportional to the vertical lengths over which they apply. (L(t) can then be viewed as a drag load based on constant Cd over the entire water column.) Because the weights wi are normalized to sum to one, we then report Veq (t)=[L(t)]1/p , the equivalent constant speed that would give the same load as L(t) at each time point t. For this drag load case, we find the maximum load in the T =1.85-year database is produced when the observed current speed at d=300m is maximum. This is the point in time when the observed profile is given by the upper curve shown in Fig. 6. This leads to several observations:

Which Regression Should We Use? As in the previous section, we find it convenient to first focus on predicting T =1.85-year extremes. Because this is the duration of the database, we can use the observed extremes at each depth, thereby avoiding the issue of extreme value estimation. Figure 6 shows the result of predicting the T =1.85-year profile associated with an extreme value at the reference depth d=300m. (This value is fixed at the maximum speed found at this depth in the database, 106 cm/s, as shown in Fig. 2.) Predicted profiles are found from regression on the POT data, using p=.50, .84, and .90 in Eqn. 2. Not surprisingly, these profiles become wider as p increases from its unbiased value of 0.5. What is surprising, though, is that the “true”, observed profile (found

1. The observed current profile in the extreme event—here, the instant that yields the largest drag load in the database—is wider than the average (p=0.5) regression result from our POT data. 2. It may be argued that higher fractiles should be used to compensate for errors in Turkstra’s rule. Because the maximum load occurs here precisely when one of the current histories (at d=300m) is maximum, Turkstra’s rule is exact. Therefore, this is not a factor in choosing higher fractiles in this case.

2 Strictly speaking, the conditional distribution of y given x should be Gaussian

in order for Eqn. 2 to be exact for all p. We believe the Gaussian assumption is not very critical for the results used here, which focus on p values at or near 0.5.

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Regression Data To explain these effects, it is useful to examine the POT data on which these regressions are based. Figure 7 shows the variation of y=current speed at d=50m, versus x=speed at d=20m. (Other depths show similar behavior.) Recall that our threshold is taken as the 99% fractile for each dataset; for current speeds at depth d=20m this yields the threshold xT H =68 cm/s. If we consider all data above this threshold, we find virtually no trend (ρX,Y =0.06). By excluding values other than peaks of x above this threshold we increase the trend slightly, now finding ρX,Y =0.16 and a slightly steeper regression slope.

These separate the data into 20 equally-spaced intervals between xT H and the maximum observed x, and, for each interval where we find data, we report the centroidal location (x, y). This is the method used in the Statoil CCA procedure [10]. Based on the observed trend in class means in Fig. 8, we find a dramatically larger correlation (ρX,Y =0.71) and hence a much steeper regression line. It is clear that the average regression (on class means) better follows these trends; we therefore choose it here as our recommended practice to construct Turkstra profiles. Figure 8 also shows clearly that if we consider regression based on all data, we need to shift our regressions upward, e.g., to p=.84–.90, to better match the trend in the upper tails of the class-mean data. (The shifted regressions in Fig. 8 use p=.90.)

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Figure 10. Load predictions from observed profiles, when the maximum current occurs at various depths, and from regression-based profiles.

It seems clear, however, that the largest speeds in Fig. 7 show a steeper trend than either of these regression results. This is revealed by considering the class means of these data (Fig. 8).

Figures 9–10 complete the picture. Figure 9 is a compa6

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Predicted 100-Year Drag -- Speed Veq [cm/s]

rable plot to Fig. 6, again considering the critical time in the T =1.85-year database when the d=300m current (and resulting drag load) is maximum. It repeats the earlier p=.50 and .84 regressions based on all POT data. In addition, it shows the preferred method: mean (p=.50) regression results based on classmean results. As might be expected, these are also broader than the p=.50 results based on POT data. They therefore obviate the need to inflate POT results to a higher-than-50% fractile. Figure 10 shows corresponding load predictions, based on profiles with observed maxima at d=300m (as in Fig. 9) and at all other depths as well. The red curve shows “true” results; i.e., the observed maximum load (in equivalent current speed terms) when each of the component currents attains its maximum speed in the database. As noted above, the worst load is shown to occur when the speed at d=300m is maximum (Veq =71.8 cm/s). This is found to be well-predicted by our preferred method (50% regressions on class means), as well as by our earlier suggestion (84% regressions on POT data). Most notably, if one uses an unbiased (p=.50) regression on all POT data in Fig. 7, one finds a misleadingly low estimate of associated current speeds—even though these data have already excluded 99% of the lowest x values. The remaining 1% of the data remains concentrated at relatively low x values in the figure, swamping out the true upper-tail behavior if all data are weighed equally. Thus, Turkstra profiles can be wider than you might think.

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Figure 11. Predicting 100-year drag loads, in terms of the equivalent 100-year current, Veq,100 . Time-Domain Prediction 50%-regression on POT values 84%-regression on POT values 50%-regression on class means

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RESULTS FOR 100-YEAR LOADS The foregoing results considered T =1.85-year loads, so that the predicted extremes could be compared directly with the observed extremes found in the T =1.85-year database. We now consider T =100-year loads. In this case we estimate an “exact” 100-year load in precisely the same way we estimate 100-year current speeds: we first construct a load history L(t) at each 10minute data point, and then fit a Quadratic Weibull model to POT data above the 99% fractile of L(t). This results here √ in a 100year estimate of the equivalent current speed, Veq,100 = L100 , of 85.0 cm/s. Figure 11 compares this estimate with those based on various regression-based Turkstra profiles. Again, our preferred method (based on class means) is rather accurate, yielding a mildly conservative estimate (Veq,100 =88.0 cm/s) in this case. In contrast, both p=.50 and p=.84-fractile regressions based on POT data are unconservative.

Figure 12.

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Predicting 100-year wave-current loads, in terms of the equiv-

alent 100-year current, Veq,100 .

velocity, u(t): L(t) = Cd [u(t) + v(t)]2 = Cd [u(t)2 + 2u(t)v(t) + v(t)2 ]

(4)

The first term in Eqn. 4 does not involve the current v(t), while the last term corresponds to our base case example. The remaining cross-term reflects a wave-current interaction effect. It is linearly proportional to v(t), with weights proportional to the variation in the wave-induced speed, u(t), as a function of depth z. Considering a discretized load Li (t) on vertical element i of height Hi , exposed to wave velocity ui and current velocity vi , this cross-term load is of the form

Wave-Current Interaction Loads The foregoing results have considered a drag load L(t) = Cd v(t)2 , proportional to the square of the current speed v(t). More generally, the fluid velocity passing by the structure is the sum of two terms: the current speed, v(t), and the wave-induced

Li (t) ∝ Hi ui (t)vi (t) ∝ Hi exp(−kzi )vi (t)

(5)

The latter form of this result uses linear wave theory, which predicts that for a regular wave with length λ (and hence wavenum7

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ber k=2π/λ), u decays with depth z like exp(−kz). For random waves, we should consider k as the wave number of an “equivalent” regular wave. Because we are concerned with 100-year conditions, we will assume the wave length λ=300m, which the linear dispersion relation associates with a regular wave of duration T ≈14s. This is meant to be roughly suggestive of extreme North Sea conditions. This leads again to a total load of the form of Eqn. 3, now with p=1 and weights wi ∝ Hi exp(−kzi ). Compared to the pure drag case, we now assign higher weights to near-surface currents. In particular, with λ=300m (k=0.021 m−1 ), we find the normalized weights wi =.32, .14, .17, .34, ..., so that 97% of the total weight is associated with current speeds at the first four depths (z=20, 30, 50, and 75m). This implies that the critical profiles are now those with peaks nearer the surface. Specifically, Turkstra profiles with peaks at d=50m are found to govern in all cases (Fig. 12). As this figure shows, our preferred method (based on class means) remains rather accurate in this case as well.

is to avoid global fits (e.g., moment fitting) to the entire population of current speeds, as these may not focus sufficiently on the extreme current speeds of most interest (Fig. 4). Step 2: Estimation of Associated Values at Other Depths. For each depth, the T -year values found in step 1 are combined with “associated” values expected to occur simultaneously at other depths. Here we recommend regression on class-mean data, again to properly emphasize events with extreme current speeds. With this approach, no need has been found to inflate the regression results; average regression results are therefore suggested.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of Statoil, through research contracts 4501838847 and 4501975985 for 2009 and 2010 respectively.

REFERENCES [1] Winterstein, S. R., Haver, S., and Nygaard, E., 2009. “Turkstra models of current profiles”. In Proceedings, 28th Intl. Conf. on Offshore Mech. and Arctic Eng. Paper number OMAE2009-79691. [2] Turkstra, C., 1970. Theory of Structural Safety. Technical report, University of Waterloo, Waterloo, Ontario, Canada. SMStudy No. 2. [3] Madsen, H., Krenk, S., and Lind, N., 1986. Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] Haver, S., and Winterstein, S. R., 2008. “Environmental contour lines: A method for estimating extremes by a short term analysis”. In Proceedings, 2008 SNAME Annual

SUMMARY In comparing Statoil’s CCA method and our original Turkstra profiles (the latter as defined in [1]), we generally find CCA to yield broader profiles—that is, profiles that decay more slowly as one moves away from the location with maximum speed. We show here that these differences are largely due to the difference in regression approaches: Original Turkstra Profiles: Regressions have been based on monthly maxima [1], or on peak values above a threshold corresponding to the 99% fractile (Fig. 6). CCA Profiles: Regressions have been based on class-mean values in 20 speed intervals, ranging from the 99% fractile level to the maximum observed speed. We find here that regressions not using class-means—i.e., those of the original Turkstra method—fail to capture the important trends in large-current events. The inflation recommended in [1]—e.g., to the 84% fractile—is valid within the context it is given. This need for inflation, however, is tied to the use of these (generally flawed) regression schemes. In contrast, by regressing on class-mean data, more accurate profiles are found from average regressions; no need for inflation is found. Our recommendations—what we define as the “best” definition of Turkstra profiles—are then as follows: Step 1: Marginal T -year Predictions. T -year current speeds are fit at each depth, with a method that emphasizes the extreme largest values of the current data. Here we use a 3moment fit to POT data, with a threshold that retains only the upper 1% of the original data. The main suggestion here 8

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Meeting and Ship Production Symposium. Paper number B3-067. [10] Kvingedal, B., 2009. Ormen Lange field—CCA profiles. Statoil memo, Statoil, Stavanger, Norway. Document No. MGE NO 209. [11] Manuel, L., Kashef, T., and Winterstein, S. R., 1999. Moment-Based Probability Modelling and Extreme Response Estimation: The FITS Routine (Version 1.2). Technical report, Civil Engineering Department, Stanford University. Report RMS–38. [12] Winterstein, S. R., Kleiven, G., and Hagen, O., 2001. “Comparing extreme wave estimates from hourly and annual data”. In Proceedings, ISOPE 2001. Stavanger, Norway.

or weakly non-Gaussian models (e.g., Hermite models) are natural candidates for the current speed components x(t) and y(t). This is shown below. Figure 13 shows the correlation between x(t) and y(t) as a function of water depth. Interestingly, there is little correlation shown, especially at shallow water depths. The correlation coefficients are found to be less than ρxy =0.3 for depths up to 600m, and no more than ρxy =0.5 even at the deepest measurement site. (Recall that for correlation ρ, the fraction of variance explained by a linear regression is ρ2 . Thus, ρxy =0.5 remains fairly insubstantial, as only 25% of the variability is explained by a regression line.) Note that one can remove these non-zero correlations, if desired, by rotation of axes. For a given depth z and hence correlation coefficient ρxy , one may define a rotation angle φ which will lead to (1) uncorrelated velocities (x0 , y0 ), and (2) maximum variance explained by x0 (t). The benefit of this rotation is that we no longer need model correlations, and modelling sensitivity to the details of y0 (t) will be lessened because its variance will be minimized. The corresponding drawback is that it adds a layer of modelling complexity: we need transform back from the (x0 , y0 ) coordinates to the original (x, y) coordinates, rotating by a different angle φ at each depth. Because the correlations ρxy are relatively low in this case, we choose not to perform this rotation in this early phase of the data analysis. It should be noted, however, as a possible extension for future study.

APPENDIX 1: DIRECTIONAL INFORMATION AND HERMITE MODELS Unlike some other results reported for this site, our current data set includes the direction of the current velocity, θ(t), as well as its magnitude v(t). Rather than model the direction itself as a random process, we believe it more convenient to model the current components, x(t) and y(t), in the x- and y-directions as well as the total speed v(t): x(t) = v(t) cos θ(t) ; y(t) = v(t) sin θ(t)

(6)

35

Total Speed: Mean X-Component: Mean Y-Component: Mean Total Speed: Std Dev X-Component: Std Dev Y-Component: Std Dev

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Figure 14. Means and standard deviations, total current speed and xand y components.

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Figure 13. Correlation between current components in direction, as a function of water depth.

x- and y-

Marginal Statistics Figure 14 shows the means p and standard deviations of x(t), y(t), and the total speed v(t)= x2 (t) + y2 (t). Note that the mean of the x(t) component dominates that of y(t), although their standard deviations are comparable. The mean of v(t) is therefore

By choosing to model x(t) and y(t), we avoid the obvious “wrap-around” problems associated with θ(t). In addition, probabilistic models may often be more conveniently applied to x(t) and y(t), because they are two-sided and unbounded. Gaussian 9

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Hermite Models of Component Speeds The near-Gaussian behavior of x(t) and y(t) suggest they may be well-represented by Hermite models, based on their respective α3 and α4 values. The expected maximum xmax can then be estimated from umax , a similar estimate of the mean maximum of a standard Gaussian process:

driven by the mean of x(t); note especially the similar shape of its dependence on water depth. 6

Total Speed: Kurtosis X-Comp: Kurtosis Y-Comp: Kurtosis Total Speed: Skewness X-Comp: Skewness Y-Comp: Skewness

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Unitless Statistic

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xmax = mx + κσ[umax + c3 (u2max − 1) + (u3max − 3umax )] (7) √ .577 (8) umax = 2 ln N + √ 2 ln N

3 2 1 0

p Here c4 =( 1 + 1.5(α4 − 3) − 1)/18, c3 =α3 /(6 + 36c4 ), κ=(1 + 2c23 + 6c24 )−1/2 , and N is the expected number of upcrossings of the mean, mx , in the interval of interest. (Since we seek here to predict the average maxima over the database duration, T =1.85 years, we use for N the observed number of upcrossings of mx .) Figure 16 compares the observed extremes, of x(t) and y(t), with the average behavior expected over many such intervals, as predicted from the Hermite model. Agreement is generally quite good, and deviations can for the most part be attributed to the “noise” in the observed maxima, which use only a single observation to predict an average value. Greatest deviations occur for x(t) at depths 100 and 200m, where the Hermite predictions exceed the observed extremes.

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Figure 15. Skewness and kurtosis of total speed components.

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Turkstra Models of Total Speed Finally, we consider how the foregoing results p can be used to predict the maximum of the total speed, v(t)= x2 (t) + y2 (t). In view of our rather accurate predictions of xmax and ymax , it may be natural to apply a Turkstra model to estimate vmax . Formally, the logic of Turkstra’s rule suggests an estimate of the form

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vmax = max v[x(t), y(t)] ≈ max{v[x(t1 ), y(t1 )]; v[x(t2 ), y(t2 )]}

Figure 16. Comparing observed maxima of speed components vx and vy , in x- and y-directions, with average results predicted from a Hermite

0≤t≤T

(9) Here t1 and t2 are the respective times when the individual components, x(t) and y(t), attain their maximum values. If applied “in the time domain,” as Eqn. 9 suggests, the approximation should be a lower bound to vmax . This is because the result neglects all other times at which v(t) could attain its maximum value. However, Eqn. 9 is almost never applied in this fashion. It is common to define the “associated terms”—e.g., y(t1 ) and x(t2 )— in some statistical sense; e.g., as the mean value conditional on a known (maximum) value of the other process. Here, because the directional components show such low correlation (Fig. 13), we simply replace y(t1 ) and x(t2 ) by their marginal mean values. Further, we replace the observed maxima, xmax and ymax , by

model fit to observed moments. (Maximum values of vx and vy are alternatively denoted xmax and ymax in the text.)

Figure 15 shows the skewness and kurtosis, α3 and α4 , of the total speed, v(t), as well as of the components x(t) and y(t). Note the nearly symmetric behavior (skewness near zero) of x(t) and y(t). The positive skewness of v(t) reflects, in large part, its onesided probability distribution. Kurtosis values are weakly nonGaussian (slightly above 3), at least for moderate water depths. Larger kurtosis values than 3 would be expected here even if the histories were locally Gaussian, due to nonstationary effects over the period of measurement. 10

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Hermite estimates formed as in Eqns. 7–8: vmax ≈ max{v[E[xmax ], my ]; v[mx , E[ymax ]]}

(10)

Equation 10 results in the predictions labelled “Hermite” in Fig. 2. Agreement is strikingly good, perhaps somewhat by happenstance: at shallow depths the conservatism of the Hermite model—compare its estimate of E[xmax ] with the observed xmax in Fig. 16—serves to somewhat offset the non-conservatism of Turkstra’s rule. While this may be fortuitous, it suggests the general value of this combined approach of applying the Hermite model to the individual x- and y-components of current speeds, and estimating extremes of the net speed, if this is desired, by Turkstra’s rule.

11

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