In The Heights PDF Free Download

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In The Heights PDF Free Download
The statistical distribution of wave heights in a random seaway TOR VINJE* VERITEC, 1322-Hovik, Norway. Three theoretical approaches to the statistical distribution of the wave height in a random seaway are compared with the results of analysis of about 80 months of wave records from the Statfjord field in the Northern North Sea. The Rayleigh distribution with a Rayleigh coefficient equal to: I-72 = 4(1 + p)mo, where 0 < p < 1.0 fit the data very well. Determined on the basis of the theory of Gaussian processes, p was found to be slightly less than 0.7. F r o m direct fitting to the wave data, in the range of noticeable non-linear effects, it was found to be about 0.8. It is shown that the asymptotic expansion of the probability density function for higher waves in a narrow-banded seaway is given as this Rayleigh distribution with the factor (1 + p/2p) 1/2 > 1.0 in front. Introduction of this factor in the theoretical predictions showed an improved fitting to the wave data. It is shown that this correction factor has a negligible influence in connection with extreme value prediction when the number of members is greater than, typically, 103 . The wave data showed a clear tendency for the wave heights to be less influenced by nonGaussian/non-linear effects than the crests and the troughs. The crest heights and the trough depths showed a behaviour corresponding to positive skewness and excess. All these observations are in accordance with qualitative, theoretical predictions. A further quantification of these effects was not possible, based on the Statfjord wave data.
1. I N T R O D U C T I O N The problem of determination of wave heights in irregular ocean waves has occupied scientists (on and off) for a period of several years. The importance of the determination of the wave heights is debatable, but the concept is widely used in oceanography and ocean engineering, and thus of considerable interest. The research has been of theoretical nature, as well as based on measurements, and, of course, on both. A review of the work done before 1978 is found in Ref 3. Here a short s u m m a r y of what is regarded as the key references will be given. In 1952 Longuet-Higgins 1 pointed out that if the wave spectrum became extremely narrow, the wave height would become Rayleigh distributed:
P ( H > h) = exp( - h2/h 2)
(1)
where the author has been careful not to identify h closer than being the 'root-mean-square' wave height. On the basis of the theory for extremely narrow banded Gaussian processes the value of h 2 has later been identified to be equal to 8too where m0 is the variance of the free surface elevation, yielding:
P ( H > h) = e x p ( - h2/8mo).
(2)
Several attempts have been made to verify the expression Equation (2) by comparison with measured Accepted November 1988. Discussion closes December 1989. * Present address: Norwegian Contractors, 1320-Stabekk, Norway.
© 1989 Computational Mechanics Publications
data. According to Ref 3 the trend is that Equation (2) over-predicts the wave heights, but the results (up to 1978) have not been conclusive regarding the rate of over-prediction. Haring et al. 2 and Forristall3 have fitted other distributions than the Rayleigh distribution to measured data, measured over a period of 116 hours (amounting to about 55,000 individual waves). The measurements were made in the Gulf of Mexico, off the coast of Louisiana, during five hurricanes occurring in the period of August 1969 through September 1975. Forristall's fitting (which is regarded as the more successful) was in terms of a Weibull distribution:
P ( H > h) = exp( - h~/~mo),
(3)
yielding tx = 2.126 and ~ = 8.42. Forristall 3 also points out that the non-linear wave effects on the distribution of the wave height is expected to be small, which is confirmed by Longuet-Higgins 4. In Ref 4 Longuet-Higgins claims that introducing /~ = 1.85 • (2mo)1/2 into Equation (1) this expression fits the Gulf o f Mexico wave data just as well as Forristall's approximation does. By assuming that the wave train consists of a carrier wave, plus a perturbation on top of this, he showed that the expected value of h 2 would be less than 8m0. By means of a perturbation procedure, followed by a series expansion, he managed to quantify the decrease. More recently Boccotti5 and N~ess 6 have approached the problem, also from the theoretical point of view. For r/(t) being a Gaussian process, Boccotti computed numerically the conditional probability density function
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143
The statistical distribution of wave heights in a random seaway: Tor Vinje of ~(t) for t > 0, provided 7(0) = A forms a crest. The following was found to take place for all his numerical computations: -
The conditional mean value of ~(t) approached A .R,(t)/R~(O) for A approaching infinity.
-
r/(t) approached the conditional mean value of rt(t) with probability approaching one for A going to infinity. [I.e. the conditional standard deviation of ~(t) goes to zero for A going to infinity].
In the paper some examples of typical autocorrelation functions and time development of ~(t) were shown. N~ess '6 approach is different. He uses the general expression for the probability density function of the wave height as a basis (provided the half wave period is given), and develops the asymptotic expansion of the probability distribution of the wave height of narrow banded Gaussian processes. His result does in fact correspond to Equations (1) and (8), giving additional evidence that the wave heights are (approximately) Rayleigh distributed. The discussion in Chapter 2 is meant to supplement the results obtained in Refs 1, and 4 through 6, both with more details and additional information. The analysis involved is based on the restriction that the stochastic process is assumed to be 'nearly narrow banded' (in Rice' 7 sense, as specified later), in the same way as done by Tayfun 12. All practical experience shows that application of this method leads to results valid for relatively broad banded spectra, which is expected to be the case for the present problem as well. In Chapter 3 the non-linear wave effects will be discussed, giving some qualitative results concerning the effect on the statistical distribution of wave heights, as well as on the distribution of crests and of troughs. In Chapter 4 the wave data from the Statfjord field, used for verification, are presented and the quality of these data is discussed. Chapter 5 deals with the comparison of the theoretical approaches with the results from the analysis of the wave data. Finally, in Chapter 6 the conclusions are drawn. 2. THE P R O B A B I L I T Y D I S T R I B U T I O N OF T H E
WAVE HEIGHT O17 A GAUSSIAN STOCHASTIC PROCESS; SOME S U P P L E M E N T I N G RESULTS
In the first part of this discussion Longuet-Higgins' claim that the distribution of the wave heights of narrow banded processes is Rayleigh distributed will be verified. Assuming, in line with Longuet-Higgins 4, that the Gaussian wave elevation can be written as: (t) = A (et)cos [fi (et)t + 0 (et)] + ~1 (t)
(4)
where the process ~l(t) is regarded as a small perturbation on top of the carrier wave, A(et)cos[fi(et)t+ O(et)], having a slowly varying amplitude, frequency and phase. This means that the maxima and minima of ~(t) will, as Longuet-Higgins observed, coincide approximately in time with those of A cos(fit + O ). Thus defining the wave height as the difference in elevation at time t--tmax and at time t = (tmax + 71'/fi), we get the following expression for the wave height: H = r/(tmax) -- ~(tmax + 71'/~) 144
(5)
Applied Ocean Research, 1989, Iioi. 11, No. 3
By defining the Gaussian stochastic process:
X(t)=~(t)-~(t+~)
(6)
one can identify the wave height, H, as the maximum of the Gaussian process, X(t), formed as the difference between two Gaussian processes. H will therefore be (approximately) Rayleigh distributed with a Rayleigh coefficient:
[t2 = 2Rx(O) = 4 [R~ (O) - R~ (~) ]
(7)
where R, (r) is the autocorrelation function of ~(t) and Rx(r) that of X(t). As will be shown later [see Equations (I I) and (23) that r = ~r/fl coincides, for narrow banded processes, well with the time for which R,(r) has its global minimum. For small variations of fi/~z will thus not vary significantly, and R, (~r/fi) can, as a good approximation, be replaced by [Rn(~')]min. This supplies a simple, and rather convincing, proof that Longuet-Higgins' claim is actually true (which, of course, also was shown by N~ess 6, but by a more complex method). Defining p = - R, (Tr/fi)/R, (0) = - [R, (r)/R, (0)] m a x and mo = R, (0), and introducing this into Equation (7) one gets: h 2 = 4mo(1 + p).
(8)
Longuet-Higgins' approximation h = 1.85 (2mo) 1/2, from the curve-fitting to the data from the Gulf of Mexico, corresponds to p = 0.711. In Ref 4 Longuet-Higgins quantified the value for h2 by means of the first terms of a series expansion. For the Pierson-Moskowitz spectrum he estimated the value to be p = 0.734, versus the correct value: p = 0.653. This indicates that his series expansion yields qualitatively correct results, but are not accurate enough for detailed analysis. In the following the assumption of Rayleigh distributed wave heights with a Rayleigh coeffÉcient given from Equation (8) will be denoted the 'LonguetHiggins/N~ess approach'. Now the problem stated by Boccotti: to determine the probability distribution of depth of a trough, following a crest at a given height, will be regarded. In this discussion the waves will be assumed to be given from a 'nearly narrow banded' process (in Rice' 7 sense), specified as a 'harmonic' wave with slowly varying amplitude, frequency and phase: ~(t) = A (ct)cos [fi(et)t + ~I,(et)],
(9)
where e ~ 1 is introduced to indicate that A, fl and xI, are slowly varying with t. The joint probability distribution of (A, fi) at two different times, assuming n(t) to be a stationary Gaussian stochastic process is given by Sveshnikov s, from which the following conditional probability density function is deduced:
p(a2, al) = ( o ~ )2 exp(
2(~-q)~} a2 '~exp(-(1-2(aq) 2q2)az~]
I°( 'ala2(1-(oq~q2) 1/2,~] (10) Io(Z) being the modified Bessel function of order zero. The conditional distribution for the frequency is likewise given as:
The statistical distribution o f wave heights in a random seaway: Tor Vinje al [ p(¢o, I a,) = (27r)l/ZoA expL
(¢Ol - h i ) 2] ~-~A/~)2J
and (11)
where (al, cox) represents [A (et), fl(et)] at t = tl = 0 and a2 represents A (et) at t = tl + r = r. a 2 is the variance of ~(t), (a2 = R ( 0 ) ) , and q 2 = 1 - r 2 - k 2. Here r = R ( r ) / R(0) is the correlation coefficient and k=K(T)/R(O), where:
R(r) = 2
0
S(~o)cos(~0r) d~
VAR(X I al) = E(X2[ a,) - [ E ( X I al)] 2 where
E(XZlal)=I+
~-1 al
A2 al(1 - q 2 ) 1/2,
(14)
leading to the conditional probability density function:
p ( x l a l ) = olEx e x p [ - ½aE(x2 + 1)] I0(a2x)
(15)
where:
(19)
-
• 1 +~2 + O ( 1~z )]
K(r) = 2
X-
2 Ot2'
The asymptotic expansions of these expressions are:
(12)
S(~o)sin(~or) d~0 (13) 0 S(~o) being the one-sided spectrum of ~(t). Further more, A is given as: A = ( f l ~ - f ~ ) 1/2, where: ~, = (mn/mo) 1/', m n = 2 ~o ~o~S(~o) d~o, being the n-th moment of the spectrum. Assume in the following that ~(t) has a maximum, al, for T = tl := 0. In this case ~(t) will have its global minimum for t = t2 ~ 7r/c01 with the value ( - A 2 ) . The conditional distribution of (A2 [ al) is given above, provided ¢01 is given as well. Wl supplies the input, r, to r(r) and k(r). To reduce the number of parameters in the problem the following new variable is introduced:
(18)
'q 8a4
~ (1 _ q 2 ) 1 / 2
(20)
and
w ( ,a0 as a--, ~ . Which, again, corresponds to Boccotti's results. Notice that [E(A2/aI [ al) - (1 - q2)1/21 approaches zero as 1/a 2, while [VAR(AE/al l al) 1/2] approaches zero as 1/~. This means that one will experience a variability in the result for A2 for relatively high values of al, even though the expected value is rather close to its asymptote. According to Equation (11) (~011 al) is Gaussian with expected value fh and variance (1 - q2)A2/(qa)2. This, in turn, means that Wl approaches f~l as al ~ oo at the same rate as A2 approaches (1 - q 2 ) 1 / 2 . Accordingly: (~(r) I A(0) = al -~ oo) ~ (1 - q ( 7 ) 2 ) 1/2' al COS(~IT) (22) as al -~ oo (and r being finite), which corresponds to
al(1 - q 2 ) 1/2 oq If the asymptotic expansion of Io(z) for large z is introduced into Equation (15) one gets:
p(xl
al) ~ 2zc]
o~' exp - - ~ ( x - 1) z • [1
Since
L
1 1 +8--a--zx+O(-~X2) ] • (16)
°2
this means that x will approach one with probability approaching one as a ~ oo. This result is quite similar to Boccotti's observation: the E(~(t)) approaches r(r) for al --' 0% when 7 ( 0 ) = al is a maximum, meaning that the conditionally expected value of the trough depth will be al [ r(7')min I ' The only exception is that I r(r) I is replaced by (1 - q2)1/2 = (r 2 + k2)1/2, which is greater or equal to I r ( r ) [ . Boccotti made his conclusions on the basis of numerical computations of E(A2/al [ a l ) and V A R ( A 2 / a l I a l ) . In the present case these parameters can be found in closed form through Equation (14) as E(XI al) =
(
(°41
1 / 2 lot exp -
Io ~ -
+~
+ I1
Io
(17)
Boccotti's general result, with the exception that I r(r) [ is again replaced by (1 - q ( T ) 2 ) 1/2 = (rE(T) + k2('r)) 1/2. For a narrow banded process, as defined by Equation (9), the r(*) and k(*) functions are given as:
r(r) = F(er)cos(fhr) - G(er)sin(f~ir) k(r) = F(cr)sin(fllr) + G(er)cos(fllr)
(23) (24)
where e is proportional to the spectrum width (and must be small compared with ~1) and is introduced to indicate that F ( , ) and G(*) are slowly varying in time. For (er)-~ 0 we have that F(er)-~ 1 + O((er) 2) and G(er) = O((er)3). Further more dr(r)/dr -~ - ~21k(r) +
O(eZr). This implies that k ( r ) ~ - 0 when r(r) has an extremum, and accordingly: (1 - q ( r ) 2 ) 1/2 ~ IT(r)] at this point, corresponding exactly to Boccotti's observation. This is not very surprising, since most experience so far shows that the Gaussian stochastic process tends to behave like a narrow-banded process when the 'amplitudes' go to infinity. In any case, it is clear that the depth of a trough, which follows a high crest, is strongly correlated to this height, showing a tendency to be slightly less than the crest height (of the order of 0.5-0.8 (in absolute value) for most wave spectra observed in nature). From the analytic expressions for the expected value and the variance, the following results, which have a certain bearing on the discussion in the following section, can be deduced: -
The expected value of A2 for A I given, A1 = al, will be independent of al as al -~ 0. This expected
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The statistical distribution of wave heights in a random seaway: Tor Vinje value is obviously given as the expected value of A (t), E(A (t)) = (7r/2) 1/2. a. - For some threshold value of al, depending on ~, E(A21 al) is equal to al. Below this threshold the expected value is greater than al; above the threshold it is less than al. This means that low crests have a tendency to be followed by relatively deeper troughs, and high crests by relatively shallower troughs• These effects tend to make the probability distribution of the wave heights more 'compressed' than that of the crest heights (or the trough depths). In the last section of this chapter, the probability distribution of the wave height, H = (A1 + A2) will be discussed. As a basis for this discussion the joint probability density function of A1 and A2 is given:
( (al~p~-~ + ) p(al, az)= aala -~' exp.-
fala4
exp~)
• io(ala2(l~q2q2)l/2 •)
(25)
Introducing H = A1 -F A2 we get: p(h)=
SI
o)
corresponding to Equation (13) of Tayfun 12. Introducing the new variable: Y = H[a and the dummy variable: x = a/h we get:
P(Y) - •q2'
Si (1
x,
• exp(- (y/q)Z(x- ½)2). I0 [ (y/q)2(1 - q2)1/2
• x(1-x)]
dx.
From this expression one notices that y3 . P ( Y ) ' ~ q e [1 + O((y/q)2)]
(27)
3. D I S C U S S I O N OF NON-LINEAR EFFECTS ON THE STATISTICAL DISTRIBUTION OF WAVE HEIGHTS As mentioned in the introduction, the non-linear wave effects on the probability distribution of the wave height has been discussed by both Forristall3 and by LonguetHiggins 4. Both concluding that this effect is weak. In the following some qualitative indications that this
(28)
as y --' 0, which means that p(y) approaches zero, for small y, much quicker than the Rayleigh distribution does. For (y/q)2. ( 1 - q2)1/2 _._~oo the following asymptotic expansion can be found, when introducing the asymptotic expansion of Io(z): 1 . y__22 1 ( y2 P(Y)~(27r) 1/2 q ( l - q 2 ) 1/4'exp 4q 2 • (1-(1-q2)1/2)
Here 0 has replaced (1 - q2)1/2 and the original variable and H = a . Y has been reintroduced. This expression resembles that of the LonguetHiggins/N~ess approach, but deviates from this by the multiplicative factor ((l+p)]2p)l/2> 1.0. The influence of this term will, for extreme value predictions, be of minor importance, creating an increase of the 'effective' number of waves involved in the prediction. Equation (27) has been integrated numerically, applying Romberg integration and rational approximations for I0(*). The result for p = 0.7 is shown on Fig. 1. In the figure the asymptotic approximations, Equation (28) and (30), are plotted as well as Equation (2) and the Longuet-Higgins/N~ess approach. The figure shows a remarkably good correspondence between the asymptotic expansion, Equation (30), and the numerical results for surprisingly low values of the wave height. The correspondence for the asymptotic expansion for low values of h, Equation (28), is not that good, but seems to catch the trend of the probability density function to have a horizontal tangent and a positive curvature for smaller values of the wave height. The comparison with the Rayleigh distribution (with a Rayleigh parameter equal to 8m0) only confirms Tayfun's 12claim that this approximation is not an acceptable one. The probability density function corresponding to the Longuet-Higgins/Nzess approach shows a significantly better fit than the Rayleigh distribution, as should be expected.
' o ( X ( l - x ) ) 1/2'exp
The numerical,solution Asymptotic expansion, h ---Asymptotic expansion, h ---0
J . ~
The Longuet-Higgins/N~ss approch. The Rayleigh distribution
.L
0.5
~: 0.7
q2
• (1 +(1 -q2)l/2)(x-½)21 dx
(29)
In the limit, (y/q)2. (1 - q2)1/2 _~ ~ , Laplace's method can be applied to the integral (Erdelyi9), leading to the following expression, after an integration from y to infinity of p(y) has been introduced:
(l+fi~ I/2
P ( H > h ) - Q(h)-* 20 ,/
/' 'expt
h2
)
4(1+0)o ~ ' (30)
146 Applied Ocean Research, 1989, Vol. 11, No. 3
1
2
3
h
2-'~
Figure 1. The probability density function of the wave height, according to the four different approaches
The stat&tical distribution o f wave heights in a random seaway: Tor Vinje effect is considerably weaker for the distribution of the wave heights for the crests (and troughs) will be given. If we consider the waves to be long crested and narrow banded we may assume that the wave elevation is given f r o m a Stokes expansion, where the first order term is assumed to be a Gaussian process. The third order Stokes expansion for the free surface elevation at infinite water depth is given as follows: ~/(t) = ~7o(t) + k ' r/o(/) 2 + ~ k 2 r / 0 ( t ) 3
(31)
assuming k . 070 -- (5 to be 'small'. A similar expansion, directly in terms of ~o(t), can not be brought through for higher order expansions, or at finite water depth. The Stokes expansion is based on long-crested seas and does therefore, most probably, tend to over-predict the non-linear effects. The results developed on the basis of Equation (31) must therefore only be regarded as indications of trends. For the crest height Equation (31) yields:
kC = kat + ½(kal) 2 + 3(kal)3,
(32)
and for the trough:
k] T I = ka2 - ½(ka2) 2 + 3(ka2) 3,
(33)
where k stands for the actual wave number (corresponding to ~1) and ( a l , a2) are the amplitudes at times t = t, and t = t2. If we assume that we can put al = a2 = a (which is rather restrictive) we get the following expression for the wave height:
k H = k(C + I TI) = 2ka + 3(ka) 3.
(34)
It is clear f r o m this small exercise that the probability distribution of the wave height is expected to be less influenced by non-linear wave effects than the distribution for the crest and for the trough. F r o m our previous analyses we expect the coupling between al and a2 to be somewhat weaker than indicated above. This means that a non-linear effect of second order m a y be found for the wave height as well. (It is a question if the formulation in terms of the Stokes expansion is proper if we allow a(t) to change within a wave period). By assuming that ~o(t) is given according to Equation (4) we find similarly that the wave heights are determined as the maxima of the stochastic process
X ( t ) = ~(t) - n(t + ~)
(35)
For the stationary stochastic process X(t) the frequency of positive crossings through the level x can be found
as:
N* (x) =
+ -~ )k4 He4 (~X)1
0
(36)
2 . p x , ~ x , 2) d2
N + (x) = C. px(X) where C =
2 . p2(2) d2 (37) o
This, in turn, means that in the limit x --, ~ one gets (see References Neess 6 or Lin ~° that:
N +(x) _ px(x)
• 1 - Fmax(X) = Qmax(X)
(38)
px(O)
where Fmax(X)is the probabilitydistribution function of the maxima of X(t).
(39)
where the skewness, ),3, is given as: k3=E(X-EX)3[a~, and the excess, k4, as: X4 [ E ( X - E X ) a / e 4 - 3]. nej(*) stands for the Hermitian polynomials. This expression is generally valid as an asymptotic expansion for small kn, making provision for the assumptions made about independence of the process and its time derivative, and for the convergence of Equation (39). The parameters EX, ox, k3 and ),4 have to be determined, either from wave measurements or from theoretical considerations. In the present case the expressions Equations (34) and (35) will be applied, assuming o0(t) to be a stationary Gaussian process; it can be shown that we can rewrite X(t) as:
21 +3k2x13, X(t)=xl[1 +kx2+~9k2XzJ
(40)
where the independent variables x,(t) and X E ( t ) a r e written:
x,(t) = ,lo(t) - oo(t + ~)
(41)
and
x2(t) = ~lo(t) + ~lo(t + ~)
(42)
This means that we have:
E(X2n-, ) ___0
(43)
+(11-2p)62+O(64)]
(44)
and E ( X 4 ) = 12(1 +p)2oo2[1 + 6 ( 6 - 0 ) 6 2 + 0 ( 6 4 ) ]
(45)
where p = - Rno(rr/fl)/R~o(O) as defined earlier, o~ = R,o(0) and (5 = kao. Accordingly the following is found: Xm = 0
(46)
and ~kH4= 6(7 - p)62 + 0(64)
By assuming that X ( t ) is statistically independent of X ( t ) (which is obviously the case for ~o(t)) we get:
N +(0)
By applying the Gram-Charlier expansion (see for instance Lin'°), px(X) can be approximated as follows:
(47)
where the index H indicates that the results are for the wave height, kn4 will be small, of order (52, and contributes by adding to a multiplicative factor in front of Equation (1), making it greater than 1, and not affecting the exponential order of Equation (39). The non-linear correction to the distribution of the wave height is thus expected to be relatively weak, corresponding to the conclusions drawn by Longuet-Higgins 4. Furthermore, )kn4 > 0 for reasonably small values of 6 2, leading to an increase of the wave heights when related to those determined from the appropriate Rayleigh distribution (i.e. based on Equation (44)). For the wave height it was shown that k m was basically zero. For the wave elevation Longuet-
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The statistical distribution o f wave heights in a random seaway: Tor Vinje Higgins 11 showed that for a second order Volterra expansion )`3 would in general be positive. The influence of k3 is then to increase the crest heights and reduce the trough depths, which is consistent with the influence of the second order term of Equations (32) and (33). The influence of ),3 on the crests and the troughs indicates that the wave height should not be seriously affected to this order, which corresponds to )`m = 0. A similar analysis as the one made by LonguetHiggins H for )`3 has not been made for )`4. To get an impression of the order and the sign of )`3 and )`4, Equation (31) has been considered, leading to: )`3 = 66 + O(63) > 0
(48)
X(t) = r ( t ) - -
r(t)
~(t) (50) I Y(t) l where a(t) is a stochastic process, 0 < a(t)~< A = 0.1 m, assumed to have a rectangular distribution over A, and to be independent of Y(t). From Equation (50) the expected value of Y(t) is found as:
E(Y)= E(X) +E[ . Y(t).]E(a)=EX +tzy A
and )`4 = 8462 + 0(64) > 0
investigate the effect on the two lowest moments of the process: the expected value and the variance. The original signal, before the truncation is introduced, is in the following denoted Y(t). The process, emerging from the truncation is denoted X(t). The two processes are connected through:
(49)
A positive value of )`4 tends to increase both the crest heights and the trough depths. The application of the Stokes expansion indicates therefore that the non-linear effects are significantly stronger for the crests and troughs than for the wave height. This is deduced from the fact that )`n3 = 0 while )`3 = 0(6). )`Ha and )`4 are both of order 6 2, and will in principle have an equivalent influence on the respective probability distributions, and first comes into effect for higher values of the argument. This means that the deviations from the Rayleigh distribution of wave heights will appear for higher values of the significant wave height than it will for the crests and the troughs. The consequences of )`3 > 0 and )`4 > 0 will be further commented on in Chapter 5. 4. THE WAVE D A T A USED FOR THE VERIFICATION OF THE THEORETICAL APPROACHES Wave data from the Statfjord field will be used for the verification of the probability distributions discussed in Chapters 1 and 2. The Statfjord field is situated in the Northern part of the North Sea, basically East-NorthEast of the Shetland Islands. Measured wave data from this field from New Year 1981 on, including those from July 1987, have been analyzed. The measuring device is a Wave Rider buoy. The time series are of 20 minutes duration, measured at three-hour intervals. The sampling frequency is 2 Hz. After a standard quality check of the data, 15,986 time series were regarded as acceptable. This corresponds to approximately 2.5 million individual waves. The vertical acceleration of the Wave Rider buoy is integrated aboard the buoy and transmitted to the platform at Statfjord for further analysis. The software at the platform transfers the analog data to digital form, and at the same time truncates the data to integer numbers of one-tenth of a meter. The truncation takes place relative to the artificial zero value of the analog signal. The analog signal is not stored. The truncation may at first sight seem quite innocent, and, as a matter of fact, does not influence the extreme value predictions at, say, the 50 years or 100 years levels, significantly. When looking into the finer details, such as: non Gaussian effects, comparison of theory and experiments and so forth, the influence can be shown to be considerable. In the following we will 148 A p p l i e d Ocean Research, 1989, Vol. 11, No. 3
L I Y(t) I J
(51)
where #y is equal to [ P ( Y > 0 ) - P ( Y < 0)]. This is related to the corresponding value for X through:
/Ly : /LX+ A2 dpy(0___~)+ O(A4 )
(52)
dy where py(y) is the probability density function of Y(t). In gx the probability of X = 0 is excluded, causing the correction. Assuming Y(t) to be Gaussian leads to: I£Y = ItX -- 2
(ly]
where O is given as: O : (2)'/2 e x p ( _ ~I(Ey~2~ --~y ] ] .
(54)
The correction of gy due to the truncation is in other terms of second order in (Afirv), and the error in E Y by replacing gv by gx is of third order in (A]oy). Computing the variance of X one gets: a } = E ( X 2) - (EX) 2 = a 2 - A [E( I YI)
-#yEY]-[~-I]A2.
(55)
Assuming Y(t) to be Gaussian then leads to: E(I YI) = a r O + #yE(Y)
(56)
and to: (57) after introduction of Equation (53). Solving for ay from Equation (57) gives:
= _ ~ + [ti2 + A2(~_~_ 0 2
1/2 (58)
Notice that O in this case is referred to the process Y(t), and thus contains higher order terms in (Ajax). The first order expansion of O r becomes:
+ (~xX) 2 ( ~ ) O x +
O((aA---x)2)] (59)
Based on the distribution function of the Gaussian variable, Y, the distribution of the discrete variable, X, emerging from the truncation, has been computed numerically, and the ratio ov/ox has been determined
The stat&tical distribution of wave heights in a random seaway: Tor Vinje Table 1. av/ax as a function o f E ( Y ) and or IEY[
0.25m
0.5m
0.75m
ay 1.0m
1.5m
2.0m
2.5m
1.0m 0.Vm 0.4m 0.0m
0.993 0.996 1.036 1.171
1.009 1.028 1.058 1.083
1.021 1.034 1.047 1.055
1.024 1.031 1.037 1.041
1.021 1.024 1.026 1.027
1.018 1.019 1.020 1.020
1.015 1.015 1.016 1.016
Here m0 is the variance of the wave elevation, 0 is the maximum value of [ - R~(r)[R~(O)] and Q(h) = P(H> h). These will form the basis for the verification of the theoretical probability distributions (in the broad sense). All the suggested probability distributions have the functional form: Q(h) = c~ e x p ( - ~ ) ,
Table 2. The ratio o f azv and the estimate o f 0 2 based on the first order correction, as a function E Y and ay
0.25m
0.5m
1.0m
2.5m
1.0m 0.Vm 0.4m 0.0m
0.987 0.982 0.965 0.997
0.993 0.992 0.995 0.999
0.998 0.999 1.000 1.000
1.000 1.000 1.000 1.000
1 - e x p [ - NQ(XN)]. (63) According to Equation (62) the function [ NQ(hN)] can P(XN>
for different values of EY and oy. The results are given in Table 1. The ratio is considerably different from 1.0 for most of the cases in the table. This has to be taken into account when comparing the theories with the results from the measurements. The first order approximations of Equation (58) read: cry
=
ax 1
AO + 0 +2-~x .
(61)
The ratio between the actual value, o]~, and the one estimated from Equation (61) is given in Table 2. The approximation shows to be rather good for all values, except for the extremely low values of or. Equation (51) (with /~y replaced by ~tx) and Equation (61) have been used for estimation of the expected value of the variance of the original signal when comparing the theories and the results from the measured data. BETWEEN
THEORY
AND
in Chapters 1 and 2 the following approaches to the probability distributions for the individual wave heights were given: The Rayleigh Distribution: Q(h) = e x p ( - 8-~o) -
The Longuet-Higgins/N~ess approach: h2
Q ( h ) = e x p ( - 4(l ~-o)mo ) -The present asymptotic modification of the Longuet-Higgins/N~ess approach:
(1 + O~~/z exp( Q ( h ) = 20 ] -
Forristall's approach:
Q(h) = exp(
=
1 - [1
-
Q(XN)] N
~-,
be written
NQ(hN)=czN.exp(-h~U)=N*exp(-h~).
(64)
where N* = c~N. The expected value of H N , given from Equation (62) is found by Forristall 3 to be:
E(HN) = [6.In(N*)] 1/~ 1 +/3 [
Y In(N*) ~
+ O ((ln(1.))z)]
(rx
-
XN)
(60)
and a 2=azx l + h O +
5. C O M P A R I S O N MEASUREMENTS
(62)
Following Longuet-Higgins ~ it can be shown that the probability distribution of the extreme of a population of N variables, equally distributed according to P(X > x) = Q(x), is found as:
oy ]EY I
h>0.
8 . ~ : moh/Z'~z6-
h2 4(1 +o)mo)
(65)
where -y = 0.577216... is Euler's constant. For the 15,986 time series from Statfjord, which form the basis for the verification, the maximum crest to trough wave height from each series (after adding A = 2 E ( a ) = 0.1 m), has been divided by its expected value, based on each of the theoretical approaches. The actual values of m0 have been corrected in accordance with Equation (61) and the number of waves for each individual time series has been used for the computation of the expected value. The value of 0 has been determined from the autocorrelation function computed on the basis of the truncated data. No correction has been applied to this, since the error can be shown to be of order (zX/o)z. The results are presented in Fig. 2 as the mean values of the respective ratios, computed within each 1 m class in Hmo = 4as. From this figure it is clearly seen that the Rayleigh distribution over-predicts the extreme values, but seems to give a constantly better fit for higher values of Hmo than for lower values. The reason for this is two-fold: the spectrum width tends to decrease for increasing Hmo partly due to contribution from swell for low Hmo, and the non-linear wave effects tend (at least for the moderate wave conditions) to increase the wave height. Forristall's approach seems to provide sensible results, even for the present North Sea data. This is a bit surprising in view of the fact that it has been developed on the basis of data from the Gulf of Mexico. On the other hand, this indicates that the results for the parameters involved in the statistical distribution are globally valid, and not restricted to special ocean areas (parameters like water depth and so forth may, of course, have a certain influence). The curve found for
Applied Ocean Research, 1989, Vol. 11, No. 3 149
The statistical distribution of wave heights in a random seaway: Tor Vinje Forristall's approach forms, more or less, a straight line in Fig. 2 which it has in c o m m o n with the curve related to the Rayleigh distribution. The main difference is that Forristall's approach provides a line passing through 1.0 for Hmo = 5m, while the curve for the Rayleigh aL THE RAYLEIGHDISTRIBUTION ---m-- FORRISTALL'SAPPROACH. • THE LONGUET-HIGfilNS/N/ESSAPPROACH. ' THE PRESENTASYMPTOTICEXPANSION
HI41AX E'(H~ x) 1.05
1.0
/
~e 0.95
0.9.
O.IE
I
I
I
I
',
S
:
'
',
;
:: 10
Table 3. The ratio (In N*/ln N ) 1/2 f o r some selected values o f N
[ml
Figure 2. The ratio of the maximum wave height within each 20 Minutes record and its expected value according to the four approaches. The plots are for the mean values within each lm class of Hmo. The data and the variances have been modified according to Chapter 4
¢J
distribution is situated relatively far below this value. C o m p a r e d with the two remaining approaches, it is the relatively strong slope of the curve for Forristall's approach, which makes it less attractive. The Longuet-Higgins/N~ess approach provides an estimate that forms a nearly horizontal line. The approach tends to under-predict the maxima. For time series of 20 minutes duration, and referred to the Statfjord field, the under-prediction is about 1%. F r o m Hmo = 7.5 m and upwards the non-linear wave effects seem to have a noticeable influence on the results. The increase of the ratio f r o m Hmo =7.5 m to 9.5 m is rather close to that of the Rayleigh distribution. Since the main effect of the spectrum width is taken into account in the Longuet-Higgins/N~ess approach, this correspondence indicates that the non-linear wave effects are responsible for the increase in the tail of the Rayleigh distribution as well. The present asymptotic modification of the LonguetHiggins/N~ess approach shows the same features as the unmodified solution does, with the only exception: it is situated closer to the value one. The distance between the two curves corresponds quite well to the leading term in the expression for the expected value: (ln(N*)/ln(N)) 1/2 = 1.012, where p = 0.65 and N = 150 have been introduced as representative for the data. Table (3) gives the ratio between the expected values for some representative values of N, and p chosen to be 0.7. From this it is clear that the modification is of marginal practical interest, specially for larger values of N. On the other hand it provides a means of improving the results and identifying the leading order correction to
N In N * ) 1/2 In N /
10 2
10 3
10 4
1.010
1.007
1.005
•L
S WITHIN EACH CLASS

'DESIGNVALUE' FOR S WITHIN EACH CLASS
T
sg, STANDARD DEVIATION FOR WITHIN EACH CLASS
1.0. Q Q
0.7
0.S
Hm 0
0.1.
S
10
Ira]
Figure 3. The value of p within each 20 Minutes record. The plots are for the mean values and the standard deviation o f p with each lm class of Hmo 150
Applied Ocean Research, 1989, Vol. 11, No. 3
The statistical distribution o f wave heights in a random seaway: Tor Vinje the Rayleigh distribution (provided the process is Gaussian). This, from an academic point of view, is satisfactory in itself. A key problem in relation to application of the Longuet-Higgins/N~ess approach, and the present modification as well, is determination of p. During the data analysis the value of P has (obviously) been computed for each time series. The mean values of P within each 1 m class of Hmo have been plotted on Fig. 3. The standard deviation within each class has been given as well. From the plot it is clear that ~ is increasing with Hmo. Since P is a measure of the spectrum width, this also indicates that the width decreases with increasing Hmo. The values of ~ seem to stabilize closely below 0.7 for the higher values of Hmo, which in turn indicates that the increasing slope of the curves for the higher values of Hmo in Fig. 2 is due to non-linear wave effects. The standard deviation of the individual p-s within each class does not change much with Hmo. For the higher values it is close to 0.07, indicating that p values as high as 0.85, and as low as 0.5, can be expected. On the figure a 'design value', po, of p is given. This is determined as the class mean of the p value which gives the maximum wave height when introduced into the Longuet-Higgins/N~ess approach, and is strictly speaking only valid for 20 Minutes duration in time. PD is given as: PD-
HZmax -
-
1.
Equation (1)) through /~ = 1.90' (2mo) x/2
(67)
which is about 2.5°70 higher than the value suggested by Longuet-Higgins 4. When predicting the wave height corresponding to the extreme crest height it makes sense choosing a value of P more in line with the value of ~. The reason is that the non-linear effects already should have been taken into account when estimating the crest height. These effects have a stronger influence on the crests than on the wave heights. To avoid taking these non-linear effects into account more than once, the value of p should be chosen to be lower than the 'design value'. The value suggested by Longuet-Higgins, P = 0.711, seems to be in line with what should be applied. A value of P = 0.66-0.68 might be a better estimate. This can only be confirmed by reanalyzing the data to determine the wave height corresponding to the maximum crest height during the time series. In Fig. 4 the ratios between the maximum crest heights and trough depths, divided by the expected value according to the Rayleigh distribution, are plotted in the same way as the wave height is in Fig. 2. The influence of the non-linear wave effects is obviously much stronger for the crests and troughs than for the wave heights. The effects of over-prediction by the theoretical estimates for low values of Hmo may be due to the relatively broad banded spectra in this domain. It is more likely, though, that it is due to the correction made to the data to account for the shift of the expected value and the effect of the truncation of the data. In addition to the possibility that the estimates of the corrections are too rough, the correction is based on all extreme crests to be positive (before truncation) and that all extreme
(66)
4mo ln(N)[1 + ln-~N)] If the aim is to predict the actual wave height for higher values of Hmo it seems to make sense to apply a value of PD which is of the order 0.8. This relates to /~ (see
CRESTMAx
1.05
E(CREST~,x} ~, TROUfiH~a~ E
(
T
/ R
~
1.0
0.95
Hmo
5
10
[m]
Figure 4. The maximum crest elevation and the minimum trough depression within each 20 Minutes record divided by their expected values according to the Rayleigh distribution. The values plotted are the mean values within each lm class of Hmo. The data and the variance have been modified according to Chapter 4 Applied Ocean Research, 1989, Vol. 11, No. 3
151
The stat&tical distribution o f wave heights in a random seaway: Tor Vinje troughs are negative. In both cases the errors in the results will show up the lower values o f Hmo. Both the curves in Fig. 4 show all features o f positive ~,3 > 0 and o f )k4 > 0. The estimates based on the Rayleigh distribution show a consistent, and steadily stronger, under-prediction o f the crests as H, n0 increases. For the troughs the estimates show a tendency to overpredict the measured values for intermediate values o f amo, in correspondence with )~3 > 0, and a tendency to tend towards under-prediction as nmo increases further, as would be the case for ~,4 > 0. The same effect o f )k4 being positive is reflected for the wave height in Fig. 2. Altogether this confirms the theoretical predictions o f the non-linear effects on the statistical distributions o f the wave heights, the crest heights and the trough depths. 6. C O N C L U S I O N S The conclusions f r o m the present investigations can be summarized as follows: -
The probability distribution o f the wave heights o f Gaussian waves can quite well be approximated by a Rayleigh distribution, as suggested by LonguetHiggins t and N~ess 6. This has been verified t h r o u g h a c o m p a r i s o n with measured data f r o m the Statfjord field over a period o f a b o u t 80 months. - The leading order asymptotic modification o f the probability density function for high waves is a multiplicative factor in front o f the Rayleigh distribution (Eq. (30)). This correction has a negligible influence on the results for extreme-value predictions when the n u m b e r o f waves exceeds, typically, 103 . -The Rayleigh coefficient was f o u n d to be: /~2 = 4m0(1 + p), where p = - [R(r)/R(O)] max. For higher values o f Hmo, 19 takes a value slightly below 0.7. Due to non-linear effects a value closer to 0.8 is r e c o m m e n d e d for design purposes. W h e n predicting the wave height corresponding to the highest crest, H = ( 1 +p)Cmax, a value slightly below 0.7 is recommended, due to the stronger non-linear wave effects on the crests than on the wave heights. - The wave data showed typical non-Gaussian behaviour, corresponding to both positive skewness and excess. The predicted weaker nonGaussian effects for the wave heights, than for the crests and troughs, were reflected in the data. - The truncation o f the Statfjord wave data to integer values o f 0.1 m introduces errors, both for the mean value and for the variance. This was insignificant for the severe sea-states, b u t made the results for calm sea (Hmo typically less than 1.0-1.5 m) unreliable. The skewness and excess,
152
Applied Ocean Research, 1989, Vol. 11, No. 3
c o m p u t e d directly f r o m the Statfjord data, would be completely erroneous. The results, concerning the applicability o f the Rayleigh distribution for the wave heights in irregular seas, are quite convincing. The present investigation will, hopefully, put an end to, what appears for an outsider to be, a controversy regarding this distribution. The result presented in this paper are clearly not valid for shallow water waves, which are strongly influenced by non-linear effects. Furthermore, the quantification o f the weak non-linear wave effects on the statistical distribution o f wave heights, crests and troughs in deep water will still be a subject for research in the years to come. For verification o f theoretical models for these effects carefully designed measurements, or experiments, have to be supplied. Standard wave measurements do not seem to seem fit for this purpose, at least not when the data are processed the way the Statfjord wave data were. 7. A C K N O W L E D G E M E N T S The paper has been prepared on the basis o f results f r o m Project No. FK-DC-87-078 for S A G A Petroleum, N o r w a y , and Project No. T-7574 for S T A T O I L , Norway, carried out by V E R I T E C on behalf o f the two clients. I would like to t h a n k the two clients for giving me the o p p o r t u n i t y to work on this interesting problem. I would also like to express m y thanks to Dr. Elzbieta Bitner-Gregersen and to Dr. Sverre H a y e r for interesting discussions and valuable suggestions made during the course o f the work.
.
REFERENCES
1 Longuet-Higgins, M. S. On the Statistical Distribution of the
Heights of Sea Waves, J. Marine Res., 1952, XI, 3 2 Hating, R. E., Osborne A. R. and Spencer L. P. Extreme wave parameters based on continental shelf storm wave records, Proc. 15th Conf. Coastal Eng., New York, 1976 3 Forristall, G. Z. On the statistical distribution of wave heights in a storm, J. Geophysical Res., 1978, 83, C5 4 Longuet-Higgins, M. S. On the distribution of the heights of sea waves: some effects of non-linearities and finite band width, J. Geophysical Res., 1980, C3 5 Boccotti, P. Some new results on statistical properties of wind waves, Appl. Ocean Res., 1983, 5, 3 6 N~ess,A. On the Statistical Distribution of Crest to Trough Wave Heights, Ocean Engineering, 1985, 12, 3 7 Rice, S. O. Mathematical analysis of random noise, Bell Tech. J., 1944, 23, and 1945, 24 8 Sveshnikov, A. A. Applied methods of the theory of random functions, Pergamon Press, 1966 9 Erdelyi, A. Asymptotic expansions, Dover, 1956 10 Lin, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill, N.Y., 1967 11 Longuet-Higgins, M. S. The effect of non-linearities on statistical distributions in the theory of sea waves, J. Fluid Mech., 1963, 17 12 Tayfun, M. A. Effects of spectrum band width on the distribution of wave heights and periods, Ocean Engineering, 1983, 10, 2

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