3.1. First order

When truncated at leading (first, linear) order (3.2) is simply

(3.3) \begin{equation} \phi (s)=\mbox{i}\zeta s-\frac {s^2}{2}, \end{equation}

such that the lone saddle point is determined from (2.4):

(3.4) \begin{equation} \phi ^\prime (\tilde {s}) = \mbox{i}\zeta -\tilde {s} = 0, \end{equation}

yielding $\tilde {s}=\mbox{i}\zeta$ . Invoking this and that $\phi ^{\prime \prime }(\tilde {s})=-1$ within (2.6) then leads to

(3.5) \begin{equation} p(\zeta ) = \frac {1}{\sqrt {2\pi }}\exp \left (-\frac {\zeta ^2}{2}\right ), \end{equation}

which corresponds to the Gaussian (normal) distribution. In this case, the saddle-point approximation yields the well-known exact result, first derived by LH63. Hence, in the present case, the equals sign is invoked above.

3.2. Second order

Truncated at second order, (3.2) incorporates the skewness:

(3.6) \begin{equation} \phi (s)=\mbox{i}\zeta s -\frac {s^2}{2} + \frac {\mbox{i}\kappa _3}{6} s^3. \end{equation}

The saddle points now correspond to those where

(3.7) \begin{equation} \phi ^\prime (\tilde {s}) = \mbox{i}\zeta -\tilde {s} + \frac {\mbox{i}\kappa _3}{2}\tilde {s}^2=0, \end{equation}

yielding

(3.8) \begin{equation} \{\tilde {s}_1,\tilde {s}_2\}=\frac {\mbox{i}(-1 \pm \sqrt {\lambda })}{\kappa _3}, \end{equation}

where

(3.9) \begin{equation} \lambda =1+2\zeta \kappa _3. \end{equation}

The second derivative at the saddle point is

(3.10) \begin{equation} \phi ^{\prime \prime }(\tilde {s})=\mbox{i}\kappa _3 \tilde {s} - 1. \end{equation}

Invoking all of the above into (2.6) leads to the two options

(3.11) \begin{equation} \{p_1(\zeta ),p_2(\zeta )\}\approx \frac {1}{\sqrt {2\pi }\sqrt {\pm \lambda ^{1/2}}}\exp \left (\frac {(\sqrt {\lambda } \mp 1)(\sqrt {\lambda }\mp 1\mp 4\zeta \kappa _3)}{6\kappa _3^2} \right ). \end{equation}

Figure 1.

Plots showing the second-order saddle-point positions (black circles) in the complex plane, contours where ${\rm Im} (\phi (s))=0$ (full black lines) as well as ${\rm Re} (\phi (s))$ (colours). Results use fixed $\kappa _3=0.3$ with ( $a$ ) $\zeta =-2$ , ( $b$ ) $\zeta =\zeta _{crit}=-1.\overline {6}$ , ( $c$ ) $\zeta =0$ and ( $d$ ) $\zeta =2$ . The dashed red lines in ( $a$ ) depict additional contours where ${\rm Im} (\phi (s))={\rm Im} (\phi (\tilde {s}))\neq 0$ .

Note that both contain singularities where $\lambda =0$ , corresponding to $\zeta =\zeta _{crit}= -1/(2\kappa _3)$ .

To determine which saddle points contribute we will inspect their position in relation to the behaviour of the truncated $\phi (s)$ in the complex plane. Figure 1 exemplifies the complex plane with fixed $\kappa _3=0.3$ and various $\zeta$ values, as indicated in the caption. Colour density plots for ${\rm Re} (\phi (s))$ are shown, as well as the contours where ${\rm Im} (\phi (s))=0$ (full lines), with the saddle points shown as filled (contributing) or open (non-contributing) circles, as will be explained further below. For $\zeta \gt \zeta _{crit}$ (figure 1 c,d) the saddle points are purely imaginary and lie directly on these contours. It is clear that the uppermost saddle point ( $\tilde {s}_1$ ) lies on the contour representing the path of steepest descent ( ${\rm Re} (\phi (s))$ decays very rapidly along this contour), and will thus be the only contributor; the contour passing through the lower saddle point ( $\tilde {s}_2$ ) actually represents the path of steepest ascent. At $\zeta =\zeta _{crit}$ (figure 1 b) the two saddle points coalesce. For $\zeta \lt \zeta _{crit}$ (figure 1 a), the saddle points bifurcate, and no longer lie on a ${\rm Im} (\phi (s))=0$ contour. Both must contribute in order for the integration path to span the entire real axis, but with the integration made along the contours where ${\rm Im} (\phi (\tilde {s}))= \mathrm{constant}$ , additionally depicted as the red dashed lines. As $\phi (s)$ has non-zero imaginary part along these contours, oscillations will inevitably occur in the negative tail. For $\zeta \lt \zeta _{crit}$ , $p_1(\zeta )$ and $p_2(\zeta )$ , in fact, form complex conjugates, and hence once added together as in (2.7) form a real-valued approximation of the PDF. The saddle-point approximation to the PDF at second order is thus

(3.12) \begin{equation} p(\zeta ) \approx \begin{cases} p_1(\zeta ) + \text{c.c.}, \hspace {0.5cm} \zeta \lt \zeta _{crit},\\ p_1(\zeta ), \hspace {1.4cm} \zeta \geqslant \zeta _{crit}, \end{cases} \end{equation}

where $\text{c.c.}$ represents the complex conjugate. This approximation is compared with the second-order FKZ distribution (full line), $ {Ai}$ being the Airy function,

(3.13) \begin{equation} p(\zeta )=\left (\frac {2}{\kappa _3}\right )^{1/3} \exp \left (\frac {1}{3\kappa _3^2} + \frac {\zeta }{\kappa _3}\right ) \mbox{Ai}(\chi ), \quad \chi = \left (\frac {2}{\kappa _3}\right )^{1/3}\left (\frac {1}{2\kappa _3}+\zeta \right ), \end{equation}

in figure 2 for an example with $\kappa _3=0.3$ . Apart from the previously discussed singularity, the saddle-point approximation provides an excellent match with the full theoretical distribution. This is consistent with the rapid decay of ${\rm Re} (\phi (s))$ along the contours where ${\rm Im} (\phi (s))={\rm Im} (\phi (\tilde {s}))$ observed in figure 1, indicating that the approximation introduced via the Taylor series (2.5) should be very accurate.

Figure 2.

Comparison of the second-order saddle-point approximation with the FKZ distribution, with $\kappa _3=0.3$ . Note the singularity occurring at $\zeta =\zeta _{crit}=-1/(2\kappa _3)=-1.\overline {6}$ .

3.3. Third order

When truncated at third order (3.2) incorporates both skewness and kurtosis:

(3.14) \begin{equation} \phi (s)=\mbox{i}\zeta s-\frac {s^2}{2} + \frac {\mbox{i}\kappa _3}{6} s^3 + \frac {\kappa _4}{24}s^4. \end{equation}

The saddle points now correspond to those where

(3.15) \begin{equation} \phi ^\prime (\tilde {s}) = \mbox{i}\zeta - \tilde {s} + \frac {\mbox{i}\kappa _3}{2}\tilde {s}^2 + \frac {\kappa _4}{6}\tilde {s}^3=0. \end{equation}

Being a third-order polynomial in $\tilde {s}$ , there are three saddle points, all of which are given by rather lengthy expressions.

To determine which of these contribute, we will again perform an inspection of their positions in the complex plane. These are exemplified in figure 3 for cases with fixed $\kappa _3=0.3$ and $\kappa _4=0.2$ and variable $\zeta$ , in a fashion similar to that in figure 1. A critical point occurs at

(3.16) \begin{equation} \zeta _{crit}=\frac {\kappa _3^3-3\kappa _3\kappa _4}{3\kappa _4^2} \end{equation}

where all three saddle points lie on ${\rm Im} (\phi (s))=0$ contours (full lines), as depicted in figure 3(b). For $\zeta \gt \zeta _{crit}$ (figure 3 c,d), as well as $\zeta \lt \zeta _{crit}$ (figure 3 a), it is clear that there is a lone (purely imaginary) contributing saddle point, which lies directly on a ${\rm Im} (\phi (s))=0$ contour, which clearly represents the path of steepest descent. This structure implies that both positive and negative tails should be well behaved at third order. This is in stark contrast to the structurally inevitable oscillations in the negative tail observed at second order, which were explained through similar inspection of figure 1.

Figure 3.

Plots showing the third-order saddle-point positions (black circles) in the complex plane, contours where ${\rm Im} (\phi (s))=0$ (full black lines) as well as ${\rm Re} (\phi (s))$ (colours). Results use fixed $\kappa _3=0.3$ and $\kappa _4=0.2$ with ( $a$ ) $\zeta =-2$ , ( $b$ ) $\zeta =\zeta _{crit}=-1.275$ from (3.16), ( $c$ ) $\zeta =0$ and ( $d$ ) $\zeta =2$ .

Since for $\zeta \neq \zeta _{crit}$ there is only a single contributing saddle point, we will utilise it to formulate an approximate PDF at third order. The (purely imaginary) saddle point, stemming from a solution to (3.15), may be written reasonably compactly as

(3.17) \begin{equation} \tilde {s}=\mbox{i}\gamma ,\end{equation}

where

(3.18) \begin{equation} \gamma = -\frac {1}{\kappa _4}\left ( \kappa _3 + \frac {\kappa _3^2-2\kappa _4}{\alpha ^{1/3}} + \alpha ^{1/3} \right ), \end{equation}

(3.19) \begin{equation} \alpha = \kappa _3^3 - 3\kappa _3\kappa _4 + \kappa _4(\sqrt {\beta } - 3\zeta \kappa _4), \end{equation}

(3.20) \begin{equation} \beta = -3\kappa _3^2\lambda + 2\kappa _4(4+9\zeta \kappa _3) + 9\zeta ^2\kappa _4^2 \end{equation}

and $\lambda$ is from (3.9). To arrive at the above it has been assumed that

(3.21) \begin{equation} \kappa _4\gt \frac {\kappa _3^2}{2}. \end{equation}

This condition likewise ensures that the cumulant generating function truncated at third order is convex for all real $s$ . The second derivative at the saddle point is

(3.22) \begin{equation} \phi ^{\prime \prime }(\tilde {s})=-1 - \gamma \kappa _3 - \frac {\gamma ^2\kappa _4}{2}. \end{equation}

Inserting all of the above into (2.6) finally leads to the third-order saddle-point approximation of the PDF:

(3.23) \begin{equation} p(\zeta ) \approx \tilde {p}(\zeta ) = \frac {1}{\sqrt {-2\pi \phi ^{\prime \prime }(\tilde {s})}} \exp \left (- \gamma \zeta + \frac {\gamma ^2}{2} + \frac {\gamma ^3\kappa _3}{6} + \frac {\gamma ^4\kappa _4}{24} \right ). \end{equation}

We will denote this approximation $\tilde {p}(\zeta )$ , as indicated above. This new result is compared with the Gaussian (first-order exact solution) distribution (3.5), those of LH63 (second- and third-order approximations) and FKZ (second-order exact solution) (3.13) in figure 4, for an example with $\kappa _3=0.3$ and $\kappa _4=0.2$ . It is seen that the present third-order saddle-point approximation has a heavier positive tail than all of these prior distributions, important, for example, in the prediction of large surface elevations typical of rogue waves. Moreover, apart from the Gaussian distribution, it is the only one of those shown to be free of unphysical oscillations in the negative tail. This behaviour is consistent with the expectations based on figure 3.

Figure 4.

Comparison of the present third-order saddle-point approximation $\tilde {p}$ from (3.23), with the Gaussian (normal) distribution (3.5), the (second- and third-order) Gram–Charlier approximations of LH63 and the second-order FKZ distribution (3.13). All results use $\kappa _3=0.3$ and $\kappa _4=0.2$ .

4.1. First order

The first-order asymptotics are fully governed by the Gaussian distribution (3.5). From this, it is seen that both positive and negative tails decay with a term proportional to $\zeta ^2$ in the exponential function.

4.2. Second order

Let us now consider the asymptotics of the second-order saddle-point approximation from (3.12). In the positive-tail limit where $\zeta \rightarrow \infty$ this tends to

(4.1) \begin{equation} p(\zeta )\sim \frac {C}{\zeta ^{1/4}}\exp \left (a_1\zeta ^{3/2} + a_2\zeta + a_3\zeta ^{1/2} + a_4 + a_5\zeta ^{-1/2} + \cdots \right ). \end{equation}

For completeness we note that

(4.2) \begin{equation} a_1 = -\frac {2}{3}\left (\frac {2}{\kappa _3}\right )^{1/2}, \end{equation}

i.e. the dominant term in the exponential tail function is governed entirely by the skewness. The other $a_i$ coefficients can be inferred from (4.1) of FKZ, or alternatively are (up to $a_3$ ) as provided in table 1 of Klahn et al. (Reference Klahn, Zhai and Fuhrman2024). The prefactor is

(4.3) \begin{equation} C = \frac {1}{2^{3/4}\sqrt {\pi }\kappa _3^{1/4}}. \end{equation}

The positive-tail asymptotics (4.1) from the saddle-point approximation thus coincide with those known from the second-order FKZ distribution (3.13). As emphasised by FKZ, with dominant term proportional to $\zeta ^{3/2}$ in the exponential function, the positive tail is fundamentally heavier than predicted by the Gaussian distribution (3.5), as well as both second- and third-order Gram–Charlier approximations of LH63, whose asymptotics are both similar to Gaussian (see § 4 of FKZ).

In the negative-tail limit $\zeta \rightarrow -\infty$ (3.12) tends to

(4.4) \begin{equation} p(\zeta )\sim \frac {1+\mbox{i}}{\sqrt {2}}\frac {C}{|\zeta |^{1/4}}\exp \left (\mbox{i}a_1|\zeta |^{3/2} + a_2|\zeta | + \mbox{i}a_3|\zeta |^{1/2} + a_4 + \mbox{i}a_5|\zeta |^{-1/2} + \cdots \right ) + \text{c.c.} \end{equation}

The (unphysical) oscillations in the negative tail are obviously due to the imaginary unit appearing in front of the $a_1$ , $a_3$ and $a_5$ terms. Although this is the first time the second-order negative-tail asymptotics have been formally shown, these similarly match those of the FKZ distribution (3.13).

Comparison of these positive and negative tail asymptotics with the full second-order FKZ distribution is shown in figure 5, where an excellent match is indeed seen at both extrema. At second order, there are again no non-oscillatory solutions for the negative tail.

Figure 5.

Comparison of the second-order FKZ distribution with the positive and negative asymptotic tail solutions with $\kappa _3=0.3$ .

4.3. Third order

We will now consider the tail asymptotics of the third-order saddle-point approximation $\tilde {p}(\zeta )$ from (3.23). In the positive-tail limit where $\zeta \rightarrow \infty$ (3.23) tends to

(4.5) \begin{equation} \tilde {p}(\zeta )\sim \frac {C}{\zeta ^{1/3}}\exp \left (a_1\zeta ^{4/3} + a_2\zeta + a_3\zeta ^{2/3} + a_4\zeta ^{1/3} + a_5 + a_6\zeta ^{-1/3} + a_7\zeta ^{-2/3} + \cdots \right ). \end{equation}

For completeness we note that

(4.6) \begin{equation} a_1 = -\frac {3}{2}\left (\frac {3}{4\kappa _4}\right )^{1/3}, \end{equation}

i.e. the dominant term in the exponential tail function is now governed entirely by the kurtosis. The other $a_i$ coefficients (up to $a_4$ ) are as given in table 1 of Klahn et al. (Reference Klahn, Zhai and Fuhrman2024), with

(4.7) \begin{align} a_5=-\frac {3\kappa _3^4-12\kappa _3^2\kappa _4 + 8\kappa _4^2}{8\kappa _4^3}, \\[-28pt] \nonumber \end{align}

(4.8) \begin{align} a_6=\frac {\kappa _3^5-5\kappa _3^3\kappa _4+6\kappa _3\kappa _4^2}{3\times 6^{1/3}\kappa _4^{11/3}} \\[0pt] \nonumber \end{align}

and

(4.9) \begin{equation} a_7 = \frac {-7\kappa _3^6 + 42\kappa _3^4\kappa _4 - 72\kappa _3^2\kappa _4^2 + 24\kappa _4^3}{36\times 6^{2/3}\kappa _4^{13/3}}. \end{equation}

The (previously unknown) prefactor is

(4.10) \begin{equation} C = \frac {1}{6^{1/3}\sqrt {\pi }\kappa _4^{1/6}}. \end{equation}

The asymptotic form (4.5) stemming from the saddle-point approximation (3.23) reproduces the exact exponential scaling obtained previously via the method of dominant balance by Klahn et al. (Reference Klahn, Zhai and Fuhrman2024). Analysis of the saddle-point approximation additionally yields the prefactor $C$ , thereby newly providing the full formal asymptotic expression, accounting for both skewness and kurtosis. Determination of $a_5$ , $a_6$ and $a_7$ above is likewise new. As emphasised by Klahn et al. (Reference Klahn, Zhai and Fuhrman2024), with dominant term proportional to $\zeta ^{4/3}$ in the exponential function, the positive third-order tail is fundamentally even heavier than at second order, consistent with what is shown in figure 4.

In the negative-tail limit where $\zeta \rightarrow -\infty$ (3.23) tends to

(4.11) \begin{equation} \tilde {p}(\zeta )\sim \frac {C}{|\zeta |^{1/3}}\exp \left (a_1|\zeta |^{4/3} - a_2|\zeta | + a_3|\zeta |^{2/3} - a_4|\zeta |^{1/3} + a_5 - a_6|\zeta |^{-1/3} + a_7|\zeta |^{-2/3} + \cdots \right ). \end{equation}

This result is entirely new. Interestingly, the theoretical negative tail is formally just as heavy as the positive tail, with dominant term proportional to $|\zeta |^{4/3}$ in the exponential. The more rapid decay in the negative, relative to the positive, tail in practice is due to the sign reversal on the subdominant $a_2$ , $a_4$ and $a_6$ terms.

Comparison of both positive- and negative-tail asymptotics with the third-order saddle-point approximation (3.23) is provided in figure 6 for a case with $\kappa _3=0.3$ and $\kappa _4=0.2$ , where an excellent match is again observed. In contrast to the second-order solution, the third-order saddle-point approximation clearly yields a negative tail free of unphysical oscillations, as discussed previously. This thus resolves a problem common to both second- and third-order approximations of LH63, the exact second-order FKZ distribution (3.13) as well as the third- and higher-order numerical solutions obtained by Klahn et al. (Reference Klahn, Zhai and Fuhrman2024) and Zhai et al. (Reference Zhai, Klahn, Onorato and Fuhrman2025).

Figure 6.

Comparison of the third-order tail asymptotics with the saddle-point approximation $\tilde {p}(\zeta )$ from (3.23), with $\kappa _3=0.3$ and $\kappa _4=0.2$ .

4.4. Higher orders

A novel perturbation method is developed in Appendix A, which is utilised to derive the positive- and negative-tail asymptotics at fourth and fifth orders. These are again generally in line with the form arrived at by Klahn et al. (Reference Klahn, Zhai and Fuhrman2024), with the previously unknown prefactors again newly determined. Similar to second order, the fourth-order negative tail is oscillatory. Similar to third order, at fifth order a non-oscillatory, decaying solution for the negative tail is found. Hence, it seems that unphysical, oscillatory negative tails are structural artefacts common to the LH63 integral truncated at even orders, which can be remedied by going to the next odd order.

5.1. Un-normalised

To analyse the accuracy of $\tilde {p}(\zeta )$ from (3.23), we will consider its fundamental integral

(5.1) \begin{equation} \tilde {\mathcal{I}} = \int _{-\infty }^\infty \tilde {p}(\zeta )\, \mbox{d}\zeta , \end{equation}

which should, of course, ideally give unity. We will additionally consider the mean (assumed zero)

(5.2) \begin{equation} \tilde {\mu }=\int _{-\infty }^\infty \zeta \tilde {p}(\zeta )\, \mbox{d}\zeta , \end{equation}

variance (assumed to be unity)

(5.3) \begin{equation} \tilde {\sigma }^2=\int _{-\infty }^\infty \zeta ^2 \tilde {p}(\zeta )\, \mbox{d}\zeta , \end{equation}

as well as its skewness (in principle, input as $\kappa _3$ )

(5.4) \begin{equation} \tilde {\mathcal{S}}=\int _{-\infty }^\infty \zeta ^3 \tilde {p}(\zeta )\, \mbox{d}\zeta \end{equation}

and excess kurtosis $\tilde {\mathcal{K}}-3$ (in principle, input as $\kappa _4$ ), where

(5.5) \begin{equation} \tilde {\mathcal{K}}=\int _{-\infty }^\infty \zeta ^4 \tilde {p}(\zeta )\, \mbox{d}\zeta . \end{equation}

Figure 7.

Contour plots depicting the variation of ( $a$ ) $\tilde {\mathcal{I}}$ , ( $b$ ) $\tilde {\mu }$ , ( $c$ ) $\tilde {\sigma }^2$ , ( $d$ ) $\tilde {\mathcal{S}}$ and ( $e$ ) $\tilde {\mathcal{K}}-3$ for input $\kappa _3$ and $\kappa _4$ based on the third-order saddle-point approximation $\tilde {p}(\zeta )$ from (3.23). Note that the red lines correspond to $\kappa _4=\kappa _3^2/2$ , below which $\tilde {p}(\zeta )$ becomes complex.

Contour plots for each of these quantities are provided in figure 7. Because $\tilde {p}(\zeta )$ from (3.23) is only an approximation of $p(\zeta )$ at third order, it is seen that each of the integrated quantities above do not result in exactly the idealised values. They are, however, reasonably close, especially for moderate $\kappa _3$ and $\kappa _4$ . The contours for $\tilde {\mathcal{S}}$ and $\tilde {\mathcal{K}}-3$ are, respectively, nearly vertical and horizontal in figures 7(d) and 7(e), such that (3.23) gives quite a reasonable first approximation. Note that in the uncontoured region in the lower right of each subfigure in figure 7, (3.23) yields complex numbers, and cannot be used. This region corresponds to where assumption (3.21) is violated, and this threshold is indicated by the red lines (similarly in forthcoming figures 8–10).

5.2. Normalised

A proper PDF can be formed from $\tilde {p}(\zeta )$ via the following normalisation:

(5.6) \begin{equation} p(\zeta ) = \frac {\hat {\sigma }}{\tilde {\mathcal{I}}}\tilde {p}(\hat {\sigma }\zeta +\hat {\mu }), \end{equation}

where

(5.7) \begin{equation} \hat {\mu }=\frac {1}{\tilde {\mathcal{I}}}\int _{-\infty }^\infty \zeta \tilde {p}(\zeta )\,\mbox{d}\zeta \end{equation}

and

(5.8) \begin{equation} \hat {\sigma }^2=\frac {1}{\tilde {\mathcal{I}}}\int _{-\infty }^\infty (\zeta -\hat {\mu })^2\tilde {p}(\zeta )\,\mbox{d}\zeta . \end{equation}

Figure 8.

Contour plots depicting the variation of ( $a$ ) $\mathcal{S}$ and ( $b$ ) $\mathcal{K}-3$ for input cumulants $\kappa _3$ and $\kappa _4$ , based on the normalised PDF $p(\zeta )$ from (5.6). The red lines correspond to $\kappa _4=\kappa _3^2/2$ , the lower bound from assumption (3.21).

This normalisation shifts $\kappa _1$ and scales $\kappa _2$ relative to their raw values in the original truncated cumulant generating function. It is here motivated by pragmatism, as it is required to enforce a proper PDF, $p(\zeta )$ , having

(5.9) \begin{equation} \mathcal{I}=\int _{-\infty }^\infty p(\zeta )\,\mbox{d}\zeta =1, \end{equation}

(5.10) \begin{equation} \mu =\int _{-\infty }^\infty \zeta p(\zeta )\,\mbox{d}\zeta =0 \end{equation}

and

(5.11) \begin{equation} \sigma ^2=\int _{-\infty }^\infty \zeta ^2p(\zeta )\,\mbox{d}\zeta =1, \end{equation}

i.e. unit integral, zero mean and unit variance, fully consistent with the true statistical properties of $\zeta$ .

Contour plots depicting the resulting integral skewness

(5.12) \begin{equation} \mathcal{S}=\int _{-\infty }^\infty \zeta ^3p(\zeta )\,\mbox{d}\zeta \end{equation}

and excess kurtosis $\mathcal{K}-3$ , where

(5.13) \begin{equation} \mathcal{K}=\int _{-\infty }^\infty \zeta ^4p(\zeta )\,\mbox{d}\zeta , \end{equation}

for various input $\kappa _3$ and $\kappa _4$ are provided in figure 8. Since the first two cumulants are already manipulated to enforce zero mean and unit variance, it is natural to treat the third and fourth cumulants with similar freedom. Inspection of figure 8 shows that the input cumulants $\kappa _3$ and $\kappa _4$ can likewise be adjusted (typically slightly, especially for moderate values) to yield PDFs matching desired skewness and kurtosis combinations in practice. The adjusted inputs thus correspond to the $(\kappa _3,\kappa _4)$ coordinates at the intersection of the desired $\mathcal{S}$ and $\mathcal{K}-3$ contours in figure 8. Finding these is easily automated using a root-finding function, which has been used to produce the inverse of figure 8 presented in figure 9, i.e. contour plots for $\kappa _3$ and $\kappa _4$ required to yield a prescribed $\mathcal{S}$ and $\mathcal{K}-3$ combination. This procedure essentially involves simultaneous iterative solution of (5.12) and (5.13) (with integrals evaluated numerically) to determine the required $(\kappa _3,\kappa _4)$ combination, using $(\mathcal{S},\mathcal{K}-3)$ as initial guesses. It is emphasised that the resulting tail behaviour in such normalised PDFs will be governed by the adjusted $\kappa _3$ and $\kappa _4$ values.

Figure 9.

Contour plots depicting the variation of input cumulants ( $a$ ) $\kappa _3$ and ( $b$ ) $\kappa _4$ required to yield a desired combination of $\mathcal{S}$ and $\mathcal{K}-3$ using the normalised PDF $p(\zeta )$ from (5.6). The red lines correspond to $\mathcal{K}-3=\mathcal{S}^2/2$ , the lower bound based on (3.21).

Figure 10.

Possible parameter space for $p(\zeta )$ in terms of $|\mathcal{S}|$ and $\mathcal{K}-3$ , achievable through normalisation of $\tilde {p}(\zeta )$ as in (5.6) and modification of input cumulants $\kappa _3$ and $\kappa _4$ . Cases considered in § 6 are indicated as circles.

The possible parameter space in terms of $|\mathcal{S}|$ and $\mathcal{K}-3$ combinations is presented as the shaded region in figure 10, there extending up to 1.2 on both axes. (Note that the possible space is symmetric about $\mathcal{S}=0$ , i.e. solutions admit both positive and negative skewness, and hence $|\mathcal{S}|$ is used on the abscissa.) As seen there, as well as from the contoured regions in figure 9, the possible parameter space essentially corresponds to

(5.14) \begin{equation} \mathcal{K}-3 \gt \frac {\mathcal{S}^2}{2}, \end{equation}

with this threshold presented as the full red line, for $|\mathcal{S}|\lt 0.6$ , consistent with the assumption (3.21). For $|\mathcal{S}|\gt 0.6$ the possible space is more restrictive, as is evident from figure 10. Also shown as a reference in figure 10 (dashed black line) is the relation implied by Socquet-Juglard et al. (Reference Socquet-Juglard, Dysthe, Trulsen, Krogstad and Liu2005):

(5.15) \begin{equation} \mathcal{K}-3 = \frac {8\mathcal{S}^2}{3}, \end{equation}

for uni-directional, narrow-banded, deep-water wave fields, as well as the test cases to be considered in § 6 ( $\mathcal{S}\gt 0$ : filled circles, $\mathcal{S}\lt 0$ : open circles), which loosely follow this trend. This reference line thus serves as a rough indication of $\mathcal{S}$ and $\mathcal{K}-3$ combinations likely to be encountered in practice, as skewness and kurtosis are not independent in irregular seas.

Normalised PDFs, as described above, will be presented exclusively in what follows, ensuring that all third-order PDFs shown have (numerically) integrated statistics ( $\mathcal{I}=1$ , $\mu =0$ , $\sigma ^2=1$ , prescribed $\mathcal{S}$ and $\mathcal{K}$ ) matching those of $\zeta$ from the various datasets against which they are compared in the next section.