Introduction

Many offshore platforms have suffered from significant wave loading on their lower deck (Bea et al., 1999), and floating production storage and offloading systems (FPSOs) have also suffered topside damage from `green water' on their decks during storms (e.g. Leonhardsen et al., 2001). However, our understanding of the wave kinematics such as crest velocities, accelerations, and overtopping rates for extreme events remains rudimentary.

Many recent advances have been made in predicting the extreme wave and crest height distributions in relation to the deck clearance or air gap problem. Determining the deck elevation above the calm water level is one of the most important aspects in the design of an offshore platform. For example, Haring et al. (1976) shows that large wave heights observed in storms are on the order of 10 percent less than those predicted by the Rayleigh distribution. Forristall (1984), and Myrhaug and Kjeldsen (1987) reported that occurrence probabilities of large wave heights in the field are smaller than the values predicted by the Rayleigh distribution. Recently, Forristall (2000) investigated the influence of sensor type in field measurements of total wave height and crest elevation. He indicated that a surface buoy cannot accurately measure the distribution of wave crests, and he reported that occurrence probabilities of large wave crest heights measured in the field by laser are larger than the Rayleigh distribution. On the other hand, numerical investigations by Yasuda et al. (Yasuda and Mori, 1994; Mori and Yasuda, 2002a) found that the third order nonlinear interactions enhance occurrence probabilities of extreme wave heights compared to the Rayleigh distribution. There are also several empirical or semi-empirical formulations for the distribution of wave crest/trough amplitudes for weakly nonlinear deep-water waves (e.g. Mori and Yasuda, 2002b).

Although the overtopping of shallow water coastal structures has been well studied, there have been very few studies on overtopping of a fixed deck in deep water with high wave steepness theoretically (Mori and Cox, 2003). For the case of extreme waves on fixed and floating structures, particularly for wave loading on lower decks (e.g. Bea et al., 1999; Hellan et al., 2001) as well as for vane breakwaters to deflect the water on the deck (e.g. Buchner, 1995), it is important to be able to estimate overtopping volume, overtopping rate and velocity distribution on the deck. Although one of these parameters may be more important than another for a particular design problem, these events are linked together. The overtopping volume is not only important for engineering purpose but also the dynamic properties, maximum velocity and overtopping rate, are significantly required for designing offshore structures. Many recent studies have been made to measure wave kinematics near the wave crest (e.g. Kway et al., 1998; Mathew et al., 2001). Gudmestad and Spids (1990) and Gudmestad (1998) summarized recent efforts to predict measured deep water wave kinematics in regular and irregular seas. However, the wave velocity components on the deck is difficult to estimate analytically. Because the wave velocity components depend on both of randomness and nonlinearity of wave field, and it also influences of geometrical setup of the deck. Moreover, the maximum value of velocity component is very sensitive to measure in the experiments. Generally, velocity components of random wave field are calculated or estimated by i) nonlinear regular wave theory, ii) stream function theory, iii) stretching method using the Fourier transform, iv) local approximation using polynomials, or v) nonlinear kernel function. Many studies (e.g. Gudmestad and Spids, 1990; Delft-Hydraulic-Laboratory, 1982) have been reported that the linear random wave theory overestimates the horizontal velocity above SWL and the nonlinear regular wave theory underestimate it near the crest. There is no general theory to predict the velocity distributions for random waves near the wave crest. Moreover, few studies have been investigated statistical characteristics of the maximum velocity for random waves (Cieslikiewicz and Gudmestad, 1995).

This paper presents the formulations for a statistical model developed to predict wave overtopping volume, maximum horizontal velocity velocity and maximum overtopping rate of extreme waves on a fixed deck as a extension of our previous paper (Mori and Cox, 2003). The objective of this work are (1) to extend existing theories developed for extreme crest amplitude statistics to the problem of wave overtopping and (2) to compare the theoretical model to experimentally determined overtopping probability distributions. `Physical Model Study' section 2 of this paper presents the experimental setup and procedures of the small-scale hydraulic model test. `Mathematical Formulation and Model/Data Comparisons' section 3 presents the statistical formulation of the problem and the validity of the the formulation under several assumptions is examined in comparison with the experimental results step by step, and presents a detailed discussion of the model-data comparisons. The last section summarizes and concludes this paper.

Figure 1: Experimental setup.

=16cm

Figure 2: Comparison of free surface elevation measured at cross-tank location corresponding to leading edge of deck for case without (solid) and with (chain) deck (Cox and Ortega 2002). Deck level is $z_c$=6.25 cm.

=11cm