Formulation of PDF of overtopping volume

The PDF of the wave height or wave crest distribution was assumed as the Weibull distribution by Eq.(9), and the overtopping volume $Q$ per wave was assumed to follow Eq.(14). Substituting Eq.(14) into Eq.(9) gives the PDF of overtopping as a function of the normalized overtopping volume $Q$

$$p_Q(Q,z_c)dQ = \frac{\alpha\beta}{2} \left( \frac{1}{2}... ...{1}{2}Q + z_c\right)^{\alpha} - z^\alpha \right] \right\}dQ$$ (15)

Integrating Eq.(15) between $[0, \infty)$, we have the exceedance probability of $Q$ on the deck given by

$$P_Q(Q,z_c) = \exp\left\{ -\beta\left[ \left(\frac{1}{2}Q + z_c\right)^{\alpha} - z^\alpha \right] \right\}$$ (16)

Eq.(15) and (16) represent the overtopping distribution based on a modified form of the Weibull wave amplitude distribution and is simply referred to as the ``modified Weibull overtopping distribution" hereafter for clarity. The truncated Weibull distribution for wave crest amplitude $A$ by Eq.(9) is normalized by the rms value of $A$, and the normalized wave amplitude $A$ has a linear relationship with the overtopping volume $Q$ by Eq.(13). As a result, the overtopping volume $Q$ in Eq.(15) and (16) is normalized by the rms value of $Q$. It is worth noting that the number of statistical events (overtopping events) depends on the deck level due to the transformation of variable from $A$ to $Q$ using Eq.(14). Physically, this means that the probability of overtopping volume $Q$ has zero value below $A<z_c$. Therefore, number of overtopping events is different from number of incident wave events.

The mean and rms values of the overtopping volume $Q$ in Eq.(15) are given by

$$\bar{Q} = \int_0^\infty Q\,p_Q(Q,z_c)\,dQ$$

$$= \frac{2}{ \alpha\beta^{ \frac{1}{\alpha}} } \exp( \beta z_c^\alpha ) \,\Gamma\left( \frac{ 1 }{ \alpha }, \beta z_c^\alpha \right)$$ (17)

and

$$Q_{rms}^2 = \int_0^\infty Q^2\,p_Q(Q,z_c)\,dQ$$

$$= \frac{8}{ \alpha\beta^{ \frac{2}{\alpha}} } \exp( \beta z_c^\alph... ...a \right) + \Gamma\left( \frac{ 2 }{ \alpha }, \beta z_c^\alpha \right) \right]$$ (18)

In the case of $z_c$=0, the deck level is equal to the still water level (SWL), and the PDF and exceedance probability can be simplified as

$$p_Q(Q,0)dQ = \frac{ \alpha\beta }{ 2^\alpha } Q^{\alpha-... ...exp\left( -\frac{ \beta }{ 2^\alpha } Q^{\alpha} \right)dQ$$ (19)

$$P_Q(Q,0) = \exp\left( -\frac{ \beta }{ 2^\alpha } Q^{\alpha} \right)$$ (20)

and

$$\bar{Q} = \frac{2}{ \alpha\beta^{ \frac{1}{\alpha}} } \,\Gamma \left( \frac{ 1 }{ \alpha } \right)$$ (21)

$$Q_{rms}^2 = \frac{8}{ \alpha\beta^{ \frac{2}{\alpha}} } \,\Gamma \left( \frac{ 2 }{ \alpha } \right)$$ (22)

Thus, the PDF of the modified Weibull overtopping distribution has a similar profile compared to the Rayleigh distribution for wave amplitude. This implies that wave nonlinearities (or spectrum band width) directly affects the overtopping volume via $\alpha$ in Eq.(15) to (22).

For the special case of $\alpha$=2 and $\beta$=1/2 which is the Rayleigh condition for the wave amplitude distribution, Eq.(15) and (16) are simplified as

$$p_Q^R(Q,z_c)dQ = \frac{1}{4} \left( Q+2z_c \right) \exp\left[ - \frac{1}{8} Q(Q+4z_c) \right] dQ$$ (23)

$$P_Q^R(Q,z_c) = \exp\left[ - \frac{1}{8} Q(Q+4z_c) \right]$$ (24)

This gives the mean and rms value of overtopping volume $Q$ as

$$\bar{Q} = \sqrt{2\pi} \exp\left(\beta z_c^2\right)$$ (25)

$$Q_{rms}^2 = 8 - 4\sqrt{\pi}\,z_c \exp\left(\beta z_c^2\right)$$ (26)

Following the same manner, Eq.(23) and (24) represent the overtopping distribution based on a modified form of the Rayleigh wave amplitude distribution and is simply referred to as the ``modified Rayleigh overtopping distribution" hereafter for clarity.

Figure 3: Probability density function of overtopping volume $Q$ as a function of deck level $z_c$ with (a) $\alpha$=2.0 and (b) $\alpha$=1.5.

=10cm

(a) $\alpha$=2.0

=10cm

(b) $\alpha$=1.5

Figure 4: Probability density function of overtopping volume $Q$ as a function of $\alpha$ with (a) $z_c$=0 and (b) $z_c$=1.

=10cm

(a) $z_c$=0

=10cm

(b) $z_c$=1

Having developed a theoretical framework for the statistical modeling of the overtopping via Eq.(15) to (26), the qualitative tendency of the distributions can now be explained. Fig.3 shows the PDF of the modified Weibull overtopping distribution as a function of deck level $z_c$ with $\alpha$=2.0 and 1.5, where the value of $\alpha$=2.0 indicates the Rayleigh (linear random wave) condition for the incident waves and $\alpha=1.5$ indicates extreme nonlinear condition. As the deck height $z_c$ increases, the peak of the PDF shifts to lower values of $Q$ until it eventually reaches a peak at $Q$=0. The PDF of $Q$ becomes a monotonically decreasing function for $z_c \ge 1$ for the case of $\alpha \ge 1$. Fig.4 shows the PDF of the modified Weibull overtopping distribution as a function of $\alpha$ with $z_c$=0 and 1.0. As the value of $\alpha$ decreases, the tail of the PDF increases. Thus, the exceedance probability of larger value of $Q$ increases for $\alpha < 2$ in comparison with the case of $\alpha=2$. The PDF of the wave amplitude $A$ and overtopping volume $Q$, Eq.(9) and (15), have a similar exponential decay for large values of $A$ and $Q$, respectively. Therefore, the PDF of the modified Weibull overtopping volume $Q$ shows a tendency similar to that of the PDF of wave amplitude $A$.

Fig.5 shows the sensitivity of the exceedance probability of the modified Weibull overtopping distribution on $\alpha$ for the case of $z_c$=1.0. With a change in $\alpha$ from $\alpha$=1.5 to $\alpha$=2, the exceedance probability of $Q$ decreases one order of magnitude at $Q \simeq 4.3$ and two orders of magnitude at $Q \simeq 6.4$. Fig.6 shows the sensitivity of the exceedance probability of $Q$ to the deck level $z_c$ for the case of $\alpha$=2.0. With a change of $z_c$ from $z_c$=0 to $z_c$=2, the exceedance probability decreases one order of magnitude at $Q \simeq 2.3$ and two orders of magnitude at $Q \simeq 4.6$.

Figure 5: Dependence of the exceedance probability of overtopping volume $Q$ on $\alpha$ with $z_c$=1.0.

=12cm

Figure 6: Dependence of exceedance probability of overtopping volume $Q$ on deck level $z_c$ with $\alpha$=2.0.

=12cm

Figure 7: Experimental setup.

=16cm

Figure 8: Experimental results of the relationship between zero-crossing wave height $H$ and wave period $T$.

=12cm

Figure 9: 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.

=12cm