OTC to develop a profile with an n-year recurrence. We describe. how the analysis applies to real data.

Transcription

1 OTC 8267 Design Current Profiles Using Empirical Orthogonal Function (EOF) and Inverse FORM Methods George Z. Forristall, Shell International Exploration and Production, BV, and Cortis K. Cooper, Chevron Petroleum Technology Company Copyright 1997, Offshore Technology Conference This paper was prepared for presentation at the 1997 Offshore Technology Conference held in Houston, Texas, 5--8 May 1997 This paper was selected for presentation by the OTC Program Committee following review of information contained in an abstract submitted by the author(s), Contents of the paper, as presented, have not been reviewed by the Offshore Technology Conference and are subject to corredion by the author(s), The material, as presented, does not necessanly refled any position of the Offshore Technology Conference or its officers, Electronic reproduction, distribution, or storage of any pari of this paper for commercial purposes without the written consent of the Offshore Technology Conference is prohibited, Permission to reproduce in print is restrided 10 an abstract of not more than 300 words; illustrations may not be copied, The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented, Abstract In the past, the oil industry has used highly simplified design current profiles. The simplification process produces errors which are typically unimportant in shallow water but the errors can be substantial in deeper water where currents are more complex and some design concepts are sensitive to current. We suggest a new method to develop more accurate current profiles without significantly burdening the design engineer. The method consists of two steps. In the first step, we simplify the current data using Empirical Orthogonal Functions (EOF), a method that accurately expresses complex data with just a few energetic modes. To these modes, we then apply the inverse First Order Reliability Method (FORM) to develop a profile with an n-year recurrence. We describe the EOF and FORM methods and provide some examples of how the analysis applies to real data. IntrOduction Historically the oil industry has based the vertical variation of design current profiles on either simple theoretical formulas or piecewise linear profiles. The latter are usually derived by applying some simplistic vertical averaging to numerical model hindcast results. Figure I shows several examples of design profiles given in the codes of API (1993), DOE (1992), and DnV (1991). Note the simple shapes. The magnitudes of the profiles are not important because they reflect local forcing. These simple design profiles are reasonable for shallow water and more traditional structures like jackets where waves are a more important load factor then currents. For these cases, the extreme loads occur during extreme storms and the current profiles are relatively simple. Errors in the profiles are of little consequence because the waves dominate the load equation. In deeper water the situation can change, especially for newer concepts like spars and subsystems like risers. In these cases, currents can actually dominate the load equation so simplification of the profile can introduce substantial errors. In addition, the currents tend to be much more complex and less constant with depth. The extreme load may indeed occur during a storm but it may be accompanied by a persistent and strong non-storm generated current. A good example of this condition is found west of Shetlands where there is often a strong (I m/s) current which is largely independent oflocal wind forcing. Figure 2 shows some examples of strong, non-storm current profiles measured in various sites around the world. Note the complex profiles. This paper describes a technique to develop more realistic current profiles with two techniques used in sequence: empirical orthogonal functions (EOF) followed by the inverse First Order Reliability Method (FORM). EOFs are used to reduce a vertical profile into a small number of values, called modes. These are analyzed by the inverse FORM to develop design currents ofa specified recurrence interval. EOFs have been used by meteorologists and oceanographers for several decades to analyze complex time series. Wunsch (1996) gives a number of examples and references in the field of oceanography. In the case of currents, EOFs have been used to simplify time series ofocean currents into a series of modes. Just a few modes can replicate extremely complex current profiles. In addition, one can often gather substantial irtsigl!!jnto the physical processes driving the currents by examining the shape and frequency ofthe EOF components. Once the EOF procedure has been used to reduce the data to a few characteristic modes, we apply the inverse FORM to the modal components to derive currents at specified 11

2 2 GEORGE Z. FORRISTALL AND CORTIS K. COOPER OTC 8267 recurrence intervals. The inverse FORM is an elegant way to develop loads from multiple inputs that may be statistically dependent In our case, the inputs correspond to the dominant modes derived from the EOF. Winterstein et al (1993) describe the inverse FORM and give examples which include the determination of loads based on wave height and period. The next chapter outlines the basic EOF procedure. It is followed by a chapter that demonstrates the application of the EOF method to data collected west of Shetlands. This application illustrates how EOFs can dramatically reduce the amount of data needed to describe time series. It also illustrates how they can provide insight into the dominant physical processes driving the currents. The fourth chapter describes the inverse FORM, and applies it to hindcasted typhoon-generated current profiles from the South China Sea. We compare the 100-yr extremes to values derived from traditional means. The final chapter discusses some of the more important results and closes with conclusions. (U) and a spatial modulation (V) multiplied by a constant weight vector (W). Equation (I) is known as the Singular Value Decomposition (SVD) of A. The decomposition consists of finding the matrices W, U and V given A. SVD is a powerful technique for solving over- and under-determined systems of linear equations. Wunsch (1996) gives a comprehensive discussion of the relationship of the SVD to least squares estimation and its use in an oceanographic context Another good explanation of the use of the SVD for an oceanographic problem can be found in Davis (1986). Press et al (1986) give a FORTRAN routine that performs the SVD along with a simple explanation of its use. Alternatively, Equation (1) can be solved as a classical eigenvalue problem. Wunsch (1996) outlines the procedure. The first step is to calculate the covariance matrix The next step is to solve the eigenvalue problem (5) EOFs from Singular Value Decomposition Using matrix notation, any M x N matrix A can be written as which yields the values for V and W. given by (6) The U matrix is then U=AVW- I (7) and A l U where U has the same dimensions as A, and Wand V are N x N square matrices. W is a diagonal matrix and U and V are orthogonal so that Written out explicitly, the matrix multiplication in equation (1) is 1\/ Au =L WkU,kVjk k=l Equation (I) decomposes the data into a time modulation (2) (3) (4) The necessary calculations can be written in a few lines using languages such as MATLAB. The computational burden of these calculations is small. Finding the singular value decomposition of 1000, 30-level profiles takes only seconds on Pentium Pc. Each current profile is written as a row in A. Each row is a separately observed profile and each column represents the time series of the current at one depth. Neither the times nor the depths need to be evenly spaced, although the depths must be the same in every profile. The rows of V T (columns of V) are called the EOFs. Each EOF is a vector with a value at each depth in the original data, and there are the same number of functions as there are depths. As indicated in equation (3), the EOFs are orthogonal to each other. They play the same role as sine waves do in an ordinary Fourier spectral analysis. The current profiles are expressed as a sum of the EOFs just as they could be expressed as a sum of sine waves. The difference is that the EOF modes are functions which fit the data matrix most efficiently as opposed the predetermined functions used in a Fourier analysis. The diagonal elements of Ware called the magnitudes of the EOF modes. There is one non-negative magnitude per mode. The matrix U gives the amplitudes of the modes in each current profile; that is they modulate the fit in the time domain. There is one row in U for each profile, and the N 12

3 OTC 8267 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS 3 amplitudes in the row multiply the N modes which add to give the current profile at that time. The coefficient of mode k for profile i is thus wkuik, but since both the EOF and amplitude matrices are orthonormal, all of the information about the relative importance of the modes in explaining the data is contained in the magnitudes wk. We will use the notation convention that uk is the variable giving the amplitude of the kth mode. That is fit is generally excellent. Similar fits are found at other depth bins. Figure 5 quantifies the error by showing the percent of the variance in the data that is reproduced using the EOF fits as the finction of the number of modes used. Our measure of error is defined as uk = uk (i) = Uik The shapes of the EOF modes and their magnitudes can provide considerable insight into the physical processes that cause the currents. For example, if the mode with the largest magnitude is nearly constant with depth, barotropic processes such as tides are likely to be important in the dynamics. As it stands, equation (4) is not a more efficient representation of the data - there are the same number of amplitudes as there were data points. The gain in efficiency results from the fact that the magnitudes of some of the modes are generally much larger than the rest. A good approximation to the data can thus be made by summing over only a few modes: where Yg is the approximation to the current profile and K can be much smaller than N. EOFs of ADCP Profiles from West of Shetlands As an example of EOF decomposition we consider data taken by an ADCP (Acoustic Doppler Current Profiler) every 10 min in 450 m from west of Shetlands, north of the Wyville- Thompson ridge. Measurements were made during the summer of 1988 using an ADCP suspended from a drilling rig in a downward looking mode. For illustration we only consider the first seven days of the record. The ADCP recorded 16 m bins yielding 22 total bins. Therefore, a complete representation of the data requires 22 EOFs. We focused on the alongshelf component which runs 45' from hue North. We chose this data set because the EOF modes correspond closely to physical processes identified by Grant et al. (1995). Figure 3 shows three of the more complex individual profiles from the data (solid curves) and EOF fits (dashed curves) using the three most energetic modes. Note the complex shape of these profiles are matched quite well with just three EOF modes. Figure 4 compares the actual time series at the top bin (-28 m) to the EOF time series based on the three modes. The The error in the variance is a good measure for engineering problems since drag force is proportional to velocity squared. Figure 5 indicates that 96% of the variance can be explained with the first mode. Using the first two modes accounts for 98% of the variance. This figure shows how effective EOFs are at reducing the volume of data. Recall that we originally started with 22 depth bins and can now describe 99% of the variance in the original data using only three modes. This effectively reduces the data storage by a factor of seven. The figure also shows the mode magnitude for the first 10 modes. Modes are ranked so that the most energetic mode has the lowest index (number). In this case the first mode is an order of magnitude larger then any of the other modes. Mode 2 is roughly twice the magnitude of the higher modes. Figure 6 shows the shapes of the first three EOF modes. The first mode is nearly barotropic (constant with depth) and corresponds to the barotropic tide and slope current identified in Grant et al. (1995). The second mode suggests a diurnal baroclinic tide although at times this mode may also occur when the flow near the bottom is southerly and the flow near the surface is northerly corresponding to a northward-flowing warm Slope Current overlaying cold, southward-flowing Norwegian water. The physical interpretation of the third mode is uncertain in part because it contributes only 1% to the total variance. A closer look at the detailed time series suggests that mode 3 is fairly important during times of stronger, near-surface current events suggesting that it may be linked to local wind forcing. Figure 7 shows the EOF-based time series derived by multiplying the mode amplitude (W) by the mode magnitude (U). The barotropic tidal component is clearly seen in a 12.4 hr sinusoid in Mode 1. The low frequency variability of the Slope Current is evident in the trend of Mode I. Spectral analysis of Mode 2 shows a diurnal peak. Mode 3 shows no dominant periods.

4 4 GEORGE 2. FORRISTALL AND CORTIS K. COOPER OTC 8267 Extreme Current Profiles inversion Criteria from Inverse FORM. The inverse FORM method provides a general procedure for calculating joint environmental conditions for extreme responses. Contours of equal probability of the joint occurrence of environmental parameters, say wave height and period, are constructed. The contour is then searched for the point which maximizes some response function such as total force on a pile. The environmental parameters at that point then become the design criteria. Winterstein et al. (1993) give a good introduction to the method. Inverse FORM calculates environmental contours by mapping the joint probability of the environmental parameters to standard normal distributions. This process is most easily understood using a simple example. Let the standard normal probability distribution be denoted If the probabilities are expressed as annual extremes and the return period of interest is 100 yrs, then the probability of exceeding the 100-yr value is p = 11100, and the reliability index is From tables of the normal distribution, we find P = 2.32 in this example. The contour of the standard normal variables xi is given by the hypersphere defined by In two dimensions, the hypersphere is a circle. Each point on the hypersphere or circle has the same probability of exceedence. Now suppose that the distributions of the actual environmental parameters, say wave height H and period Tare given by FH and FT. We assume for the moment that these distributions are independent. Then for each point on the contour given by equation (1 I), the corresponding physical parameters are A plot of the resulting contour can then be made with H and T as the axes. The design point is then the point on this contour where the response is maximized. If T is correlated with H, then instead of the distribution FT, we must use the conditional distribution F ~ H The is generally no more difficult than when the parameters are independent. Hindcast Typhoon Currents As an illustration of estimating design current profiles using FORM, we consider typhoon current hindcasts in the South China Sea. The hindcasts were made with the one dimensional turbulence closure model developed by Kantha and Clayson (1994), an improved version of the familiar Mellor and Yamada (1982) model. These models describe the vertical mixing of momentum due to the wind stress at one point on the surface of the ocean. The output is a profile of the current speed and direction at each time step in the history of the storm. Wind stress histories for a point in the center of the South China Sea were taken from the SEAMOS hindcasts of Cardone and Grant (1994). These hindcasts include the 156 strongest typhoons to affect the South China Sea in the 47 yrs from 1946 through 1992, a long enough period for reasonably reliable estimates of 100-yr criteria. Figure 8 shows some representative current profiles from the typhoon hindcasts. The profiles have similar shapes but as the current speed near the surface increases, the current also extends deeper. Only larger current speeds are shown because these govern the extreme statistics. Since the current direction is nearly constant with depth, we focus on the current speed rather than velocity components. The first step to develop design profiles is to calculate the EOFs for the 3 19 current profiles discretized at 28 levels for all profiles with speeds greater than 1 m1s at the 5 m depth level. The shapes of the first four modes are shown in Figure 9. The first mode is a simple sheared profile from the surface to the deepest depth reached by the wind mixing. The higher modes are surface intensified and have large oscillating components at depth which can add to make the speed there near zero when needed to match the hindcast profiles. Figure 10 shows the magnitudes of the first 10 modes. The first three modes contain over 99% of the variance in the data. The second step is to apply the inverse FORM. The question is which variables from our EOF do we apply it to? A little thought suggests that inverse FORM should be done on the amplitudes of the EOF modes uk. since these are essentially the time series of each mode, e.g. Figure 7. We thus need to find the joint probability distribution of the EOF amplitudes. This is the most difficult stage of the inverse FORM calculations because it involves subjective judgments about what functional forms should be used to fit the data. Weibull distributions are often used to fit extreme value data. and we found that the amplitudes of the first mode could be fit

5 OTC 8267 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS 5 well using the two parameter Weibull distribution. The least squares fit gives F(u, ) = 1 - ( ) exp } (I5) where the factor in front of the exponential is based on 3 19 observations in 47 yrs and a decorrelation time for the currents of three hours, or eight samples per day. Equation (1 5) is thus the probability that the amplitude u, will not be exceeded in a randomly chosen three hour sample. Figure I I is a scatter plot of u, against u,. A trend is apparent, suggesting that these amplitudes are clearly not statistically independent. As discussed by Wunsch (1996), the orthogonality of the EOF amplitudes does not ensure statistical independence. It seems reasonable to fit the distribution of u2 given u, as a deterministic straight line plus normally distributed scatter. A least squares fit gives where E is normally distributed with zero mean and standard deviation of Expressed as a conditional probability, The reliability index for the 100-yr contours is The contours of the 100-yr amplitudes are shown as the bold curve in Figure 12. For the next step, we need to specify a response function. A reasonable function for a current profile is the total drag force on a cylinder, which is proportional to the integral of the square of the velocity. For each point on the 100-yr EOF contour, we can calculate the drag by summing the two modes to find a current profile and then integrating over depth to get the total drag. The design point is then the point on the EOF contour which gives the maximum drag. For illustrative purposes, we actually calculated the drag for many points over the plane in Figure 12 and contoured the drag as the thin curves in the figure. One advantage of inverse FORM is that this calculation is not really necessary since finding the contour with a given reliability index reduces the dimensions of the problem by one. The maximum drag occurs near the maximum of the first mode, with a drag of A 100-yr drag can also be found by calculating the drag for each hindcast profile and doing the extreme value analysis on those drags. The result of that calculation was The precise agreement is fortuitous given the vagaries of an extreme value fit. The advantage to finding the drag using inverse FORM is that it gives the most probable shape of the current profile which produces that extreme drag. For example, the first mode of the 100-yr profile is given by where x, and x, are the standard normal variables which plot as a circle. The EOF amplitudes are found by taking each pair of x, and x, points on the circle and converting them to the amplitudes using and Equation (20) is particularly easy to use since it can be rewritten where u, = 0.92 from Fig 12, W, and Vj, come from the EOF analysis of the data, and j=l..28 corresponding to the 28 depth levels used to discretize the original hindcast profiles. The 100-yr profile from the two-mode inverse FORM calculation is shown as the dashed curve in Figure 13. We also found a 100-yr profile using the first three EOF modes, and that profile is shown as the dotted line. The 100-yr drag from the three-mode solution was The three-mode profile is stronger at depth, agreeing more with the shape of the strongest hindcast profile. Conclusions The method we have developed provides more accurate and realistic design profiles than traditionally used yet requires no

6 6 GEORGE Z. FORRISTALL AND CORTIS K. COOPER OTC 8267 more effort from the design engineer. The improved accuracy should be especially important in deeper waters and with new design concepts like spars and deepwater risers that are sensitive to current. The method uses two steps. The first step applies the EOF technique to decompose the measured or hindcast time series of current profiles into spatially- and temporally- dependent parts multiplied by a constant vector. When applied to oceanographic data, EOFs usually yield a set of dominant modes. We demonstrated this for data from west of Shetlands which we found could be accurately described with just three modes, amounting to an order of magnitude decrease in data. EOF analysis can also give insight into the physical processes of importance in the data. In the case of the west of Shetlands, the first mode corresponds to the barotropic tide and slope current while the second mode is probably generated by an internal tide. The combination of modes which gives an extreme current profile is found using the inverse FORM method of Winterstein et al. (1993). Inverse FORM searches a hypersphere of a constant and specified recurrence level to find the amplitudes of the EOF modes that give the maximum response function. We provided an example for the case of typhoon-generated currents in the S. China Sea. We used a drag response function for our example, though clearly other response functions are possible. We found that the method produced estimates of the 100-yr response that were within 1% of a more traditional approach. The advantage of our approach is that one can quickly back out the profile that generated the 100-yr response. This is not so easy if inverse FORM is applied to the full 3-D current data. Our method retains the advantage of inverse FORM over the more straightforward simulation used by Forristall et al. (1991). That is, one can quickly find the design profile for another response function without rerunning the entire simulation. One only has to re-plot the load curves in Figure 12. A description of current profiles as a sum of EOFs should be ideal for use in fatigue analyses of risers. The joint probabilities of mode amplitudes given in scatter plots like Figure 11 can be combined into bins, producing a scatter diagram of mode amplitudes similar to the scatter diagrams of wave height and period used in the fatigue analysis of fixed platforms. A riser analysis would be performed for the current profile represented by each bin, and the fatigue damage added up taking into account the number of profiles in the bin. The number of modes which need to be considered will depend on how sensitive the riser stresses are to the higher modes as well as how energetic the modes are. We suspect that only a few modes will be necessary for an accurate calculation of fatigue damage. While the method appears to be a clear improvement over previous methods, there are some potential limitations which must be investigated hrther. For example, there is one subtle problem with extrapolating the amplitudes of the modes to produce a 100-yr typhoon current profile. Since all of the hindcast current profiles were zero below 130 m depth, all of the mode vectors also go to zero below that depth. It is, however, likely that the 100-yr storm would create enough mixing to produce a deeper current profile. Extrapolation of the modes of the profiles in the hindcast data can not duplicate such a profile. In the present example, it appears that this theoretical problem was not too important, since the 100-yr drag from the inverse FORM calculations agreed well with direct analysis of the hindcast profiles. We also suspect that the EOF would underestimate the profile in regions where high wave number currents occur infrequently. An example would be the northwest shelf of Australia where solitons add substantially to the extreme currents but because they occur for only a few minutes each day and represent a relatively small portion of the total variance, they will not be well captured by the EOF method. Keep in mind, the design profiles in present design codes are of course no better. Acknowledgments Rabi Dee gave us a beginner's guide to inverse FORM calculations. Lakshmi Kantha kindly sent us a copy of the code for his turbulence closure model. Colin Grant gave us permission to use the west of Shetlands ADCP data, and he, Chris Shaw, Kevin Ewans and Paul Taylor made helpful comments on a draft of the paper References 1. API( 1993), Recommended practice for planning, designing, and constructing fixed offshore platforms, American Petroleum Institute, API RP 2A, Washington. 2. Cardone, V.J. and C.K. Grant (1994), Southeast Asia meteorological and oceanographic hindcast study (SEAMOS). In Proceedings of the 10th Offshore South East Asia Conference and Exhibition, OSEA , Singapore, Davis, R.E. (1986), Predictability of sea surface temperature and sea level pressure anomalies over the North Pacific Ocean, Journal ofphysica1 Oceanography, 6, DOE (1992), Offshore Installations: Guidance on Design, Construction and Certification, UK Department of Energy, London. 5. DnV (Det norske Veritas) (1991), Environmental conditions and environmental loads. Hovik, Norway, March. 6. Forristall, G.Z.. R.D. Larrabee and R.S. Mercier (1991), Combined oceanographic criteria for deepwater structures in the Gulf of Mexico, Proceedings of the 23rd Annual Offshore Technology Conference, OTC Houston

7 OTC 8267 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS 7 7. Grant, C., R. Dyer. and I. Leggett (1995), Development of a new metocean design basis for the NW shelf of Europe, Proceedings of the 27th Annual Oflshore Technology Conference, OTC 7685, Houston. 8. Kantha, L.H. and C.A. Clayson (1994), An improved mixed layer model for geophysical applications, Journal o/ Geophysical Research, 99, 25,235-25, Mellor, G.L. and T. Yamada (1982), Development of a turbulence closure model for geophysical fluid problems, Reviews of Geophysics and Space Physics, 20, Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling (1 986), Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, London. I I. Winterstein, S.R., T.C. Ude. C.A. Cornell, P. Bjerager, and S. Haver (I 993), Environmental parameters for extreme response: Inverse FORM with omission factors. In ICOSSAR-93, Innsbruck. 12. Wunsch, C. (1996), The Ocean Circulation Inverse Problem, Cambridge University Press, London.

8 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS 0, I Fig. 1 Examples of profiles from various design codes used by the oil industry. DOE (L DnV are based on 37 wind and water depth. Fig. 3 Sample current profiles from west of Shetlands. The solid curves show the measured along-sbpe profiles and the dashed curves show the profiles using three EOF modes. Fig. 2 Examples of measured current profiles taken from various parts of the world. Fig. 4 Time series of the alongshelf current at 28 m depth from west of Shetlands.

9 OTC 8267 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS 9 Modes Used Fig. 5 The dashed curve shows the magnitudes of the EOF Fig. 7 Time series of the size of the first three EOF modes modes used to fit ADCP measurements west of Shetlands, and fined to the ADCP measurements from west of Shetlands. The the solid curve shows the percentage of the variance in the mode size is given by its amplitude multiplied by its magnitude. original data as a function of the number of modes used in the fit. Current Speed (m/s) L Fig. 6 Shapes of the first three EOF modes for the ADCP measurements from west of Shetlands. Fig. 8 Representative current profiles from the typhoon hindcasts.

10 10 GEORGE Z. FORRISTALL AND CORTIS K. COOPER OTC 8267 Mode ).. I... I. ' '... i' Current Speed (m/s) Fig. 9 The first four EOF modes for the typhoon hindcasts. Fig. 11 Scatter plot of the first two EOF modal amplitudes for the typhoon hindcasts. Mode Number Fig. 10 Magnitudes of the first 10 EOF modes for the typhoon hindcasts. U Fig. 12 The bold curve is the IOOyr contour of the amplitudes of the first two EOF modes for the typhoon profiles. The thin contours show equal drag on a single pile. The design current profile is given by the point on the bold curve with the maximum drag.

11 OTC 8267 DESIGN CURRENT PROFILES USING EOF AND INVERSE FORM METHODS Maximum in Hindcosts -, Yeor profile from 2 modes 1 I 100 Yeor profile from 3 modes 11 Current Speed (rn/s) Fig yr typhoon current profiles along with the hindcast profile which produced the largest drag.