Wave scatter conversion: Theory

Notation

Let the list of sea state spectra and associated durations be denoted $Z_l$ and $D_l$ respectively, for $l = 1,\ldots,k$ where $k$ is the number of sea states.

Note that the sea state duration is only used to determine the distribution of sea states. The probability of occurrence of sea state $q$ is given by \begin{equation} P(Z_q) = \frac{D_q}{\displaystyle \sum_l D_l} \end{equation} The absolute values of $D_l$ are not important here, only their relative values.

The specified regular wave heights and periods are denoted $H_i,\ i=1,\ldots,m$ and $T_j,\ j=1,\ldots,n$, respectively. These wave heights and periods are ordered to be strictly increasing.

Regular wave bins

The wave height bins are denoted by $[H_i\low,H_i\upp]$ for $i=1,\ldots,m$. The first bin is taken to be \begin{equation} [H_1\low,H_1\upp] = [0,\ \tfrac12(H_1+H_2)] \end{equation} It is clearly important that 0 is contained in the bin. Doing so, however, means that $H_1$ might not fall in the middle of the bin. This should not matter provided that the bin is small enough. The last bin is taken to be \begin{equation} [H_m\low,H_m\upp] = [\tfrac12(H_{m-1}+H_m),\ H_m+\tfrac12(H_m-H_{m-1})] \end{equation} so that $H_m$ falls in the middle of the last bin.

Finally the intermediate bins are chosen to be \begin{equation} [H_i\low,H_i\upp] = [\tfrac12(H_{i-1}+H_i),\ \tfrac12(H_i+H_{i+1})] \end{equation} for $i = 2,\ldots,m-1$.

The wave period bins $[T_j\low, T_j\upp]$ for $j=1,\ldots,n$, are chosen in an analogous fashion.

We now have $m{\times}n$ individual wave height and period bins which we denote by $\Bij$ for $i=1,\ldots,m$ and $j=1,\ldots,n$. Each bin $\Bij$ represents individual waves whose heights and periods fall in the ranges $[H_i\low,H_i\upp]$ and $[T_j\low,T_j\upp]$ respectively.

Joint probability for individual waves

This section is based on Barltrop and Adams section 6.6.3. Following their lead, we use the formula of Longuet-Higgins, 1983, giving the joint probability density $p(R,S,Z)$ of individual waves in a given random sea state as \begin{equation} p(R,S,Z) = L(\nu)\ \exp\left[-R^2\left(1+\frac{(1-1/S)^2}{\nu^2}\right)\right]\ \frac{2R^2}{S^2\nu\sqrt\pi} \end{equation} where

$Z =$ spectrum of given random sea state

$m_i =\ i$^th spectral moment of $Z$

$\Tm = m_0/m_1$

$\Tz = (m_0/m_2)^{1/2}$

$S = T/\Tm$

$\Hrms = \Hs/\sqrt{2}$

$R = H/\Hrms$

$\nu = (\Tm^2/\Tz^2 - 1)^{1/2}$

$L(\nu) = 2 / (1 + (1+\nu^2)^{-1/2})$

Individual wave scatter

For each sea state $Z_l$ we can calculate the probability of occurrence of each individual wave height and period bin, denoted by $P(\Bij\vert Z_l)$, as \begin{equation} P(\Bij\vert Z_l) = \iint p(R,S,Z_l)\,dS\,dR \end{equation} where

the double integral is evaluated over the region $R_i\low \leq R \leq R_i\upp$, $S_j\low \leq S \leq S_j\upp$

$R_i\low = H_i\low / \Hrms$

$R_i\upp = H_i\upp / \Hrms$

$S_j\low = T_j\low / \Tm$

$S_j\upp = T_j\upp / \Tm$

Thus, the probability of occurrence $P(\Bij)$ of each individual wave height and period bin is calculated as \begin{equation} P(\Bij) = \sum_{l=1}^k P(Z_l) P(\Bij\vert Z_l) \end{equation} OrcaFlex does not report these probabilities directly; rather, it reports the total number of occurrences $O_{ij}$ for each bin $\Bij$, given by \begin{equation} O_{ij} = P(\Bij)\ \frac{D_\textrm{total}}{T_j} \end{equation} where $D_\textrm{total}$ is the total duration as given on the regular waves page.