Page 88 - Marine Structural Design

64 Part I Structural Design Principles

Gnmbel Fitting
Gumbel Fitting is based on the assumption that for a wide class of parent distributions whose
tail is of the form:
(3.32)
F(X) = 1 - exp(-g(x))
where g(x) is a monotonically increasing function of x, the distribution of extreme values is
Gumbel (or Type I, maximum) with the form:
(3.33)

The MPME typically corresponds to an exceedance probability of 1/1000 in a distribution
function of individual peaks or to 0.63 in an extreme distribution function. The MPME of the
response can therefore be calculated as:
x,,,, = ry - K . In(- MF(XA.fP,E 1)) (3.34)
Now the key is to estimate the parameters land K based on the response signal records
obtained fiom time-domain simulations. The SNAME Bulletin recommends to extract
maximum simulated value for each of the ten 3-hour response signal records, and to compute
the parameters by maximum likelihood estimation. Similar calculations are also to be
performed using the ten 3-hour minimum values. Although it is always possible to apply the
maximum likelihood fit numerically, the method of moments (as explained below) may be
preferred by designers for computing the Gumbel parameters in light of the analytical
difficulty involving the type-I distribution in connection with the maximum likelihood
procedure.
For the type-I distribution, the mean and variance are given by
Mean: p = v+y. K, where y= Euler constant (0.5772.. .)
Variance: c2 =Z~K~I~
By which means the parameters y and K can be directly obtained using the moment fitting
method:
(3.35)

WintersteinIJensen Method
The basic premise of the analysis according to Winterstein (1988) or Jensen (1994) is that a
non-Gaussian process can be expressed as a polynomial (e.g., a power series or an orthogonal
polynomial) of a zero mean, narrow-banded Gaussian process (represented here by the symbol
v). In particular, the orthogonal polynomial employed by Winterstein is the Hermite
polynomial. In both cases, the series is truncated after the cubic terms as follows:
Winterstein:
R(u)=~, +OR -K[u+A,(u~ -1)+h,(u3 -w)] (3.36)
Jensen:
R(U)=C, +C,U+C2U2 +C,U3 (3.37)

Within this framework, the solution is essentially separated into two phases. First, the
coefficients of the expansions, i.e., K, h3, and in Winterstein’s formulation and & to C3 in