Tayfun distribution: a distribution that doesn’t guess, it derives!

If you’re a coastal or ocean engineer, you’ve probably been taught that sea surface elevations are Gaussian — and that, as a result, wave heights follow a Rayleigh distribution. That’s the story we hear in every textbook and engineering manual. It’s tidy. It works. It’s convenient.

But is it true?

Well, like many things in engineering, it depends on your assumptions. And that’s where things get interesting.

The Rayleigh distribution comes from a beautiful mathematical argument: if the sea surface is made of many small, uncorrelated wave components with random phases, then their sum will be Gaussian (thanks to the central limit theorem). If the sea is “narrowband”, meaning all the energy is around a dominant frequency, then the heights of individual waves, defined as crest-to-trough, will follow a Rayleigh distribution.

This assumption works reasonably well in deep water under calm or average conditions. But real waves are not perfectly linear. They interact. They steepen. They deform. Crests become sharper and taller, troughs become wider and flatter. The sea becomes asymmetric.

So how do we model that?

In 1980, M. A. Tayfun proposed a new way to describe wave crests under weakly nonlinear conditions. Instead of assuming a purely linear Gaussian sea, he asked: what happens if we add second-order nonlinear effects? In other words, what if wave components actually interact with each other?

The answer starts with how we model the sea surface elevation, \( \eta(t) \), as a combination of a random amplitude \( a \), following a Rayleigh distribution, a dominant wave frequency \( \omega \), and a nonlinear correction proportional to \( a^2 \). This gives:

\[ \eta(t) = a \cos(\omega t) + \frac{1}{2}ka^2 \cos^2(\omega t) \]

Here, \( a \) is the amplitude of the wave envelope (Rayleigh-distributed) and \( k = \frac{2\pi}{L} \) is the wavenumber based on the mean wavelength \( L \). The second term is the nonlinear correction, a quadratic that introduces nonlinear interactions. This breaks the Gaussian symmetry and reshapes the wave.

Tayfun then asked: if \( a \) is Rayleigh-distributed, and \( \eta(t) \) follows the equation above, what is the distribution of the maximum wave crest?

He wrote the maximum crest height as:

\[ \eta_{\text{crest}} = a + \frac{1}{2}ka^2 \]

Using this, and assuming narrowband conditions, he derived a new PDF for the crest heights.

To make it usable, define the dimensionless crest height \( \xi = \eta_{\text{crest}} / \sigma \), where \( \sigma = H_s / (2\sqrt{2}) \). Also define the nonlinear parameter \( \mu = \frac{1}{2}k\sigma \).

The PDF becomes:

\[ f(\xi) = \frac{2}{\sqrt{1 + 4\mu \xi}} \cdot \exp\left( -\left[ \frac{-1 + \sqrt{1 + 4\mu \xi}}{2\mu} \right]^2 \right) \]

And the cumulative distribution function (CDF) is:

\[ F(\xi) = \int_0^{\xi} f(s) \, ds = \int_0^{\xi} \frac{2}{\sqrt{1 + 4\mu s}} \cdot \exp\left( -\left[ \frac{-1 + \sqrt{1 + 4\mu s}}{2\mu} \right]^2 \right) ds \]

This integral computes the cumulative probability of observing a normalised crest smaller than or equal to a given value \( \xi \), starting from 0 (the smallest possible crest) up to the crest of interest.

This is the Tayfun distribution. Unlike empirical fits, it comes directly from second-order wave theory.

Rayleigh works under linear assumptions. Tayfun works when nonlinearity starts to matter. It explains the skewness in the observed crest distribution, not as noise, but as a physical consequence of wave-wave interaction.

Back in the 1950s and 60s, Longuet-Higgins tried to correct the Gaussian assumption using the Gram-Charlier expansion, which modifies the Gaussian with Hermite polynomials:

\[ f(u) = \frac{1}{\sqrt{2\pi}} e^{-u^2/2} \left[ 1 + \frac{h_3}{6} H_3(u) + \frac{h_4}{24} H_4(u) + \cdots \right] \]

Where:

• \( H_3(u) = u^3 - 3u \)
• \( H_4(u) = u^4 - 6u^2 + 3 \)
• \( h_3 \), \( h_4 \) are normalised cumulants (skewness and kurtosis)

Think of them as special polynomials that, when combined with a Gaussian, let you add nonlinear effects to your model. Like a Taylor expansion, but custom-built for the Gaussian universe.

Instead of summing powers of \( x \), you add tailored shapes like \( H_3(x) \), \( H_4(x) \), and so on—designed to tweak the bell curve without breaking its core structure. It’s like tuning up a Gaussian with modular upgrades, each one mathematically orthogonal and ready to capture the messy truth of the real world. However, these work near the mean but misbehave in the tails, sometimes giving spurious results like negative probabilities.

Tayfun himself noted this limitation:
“It does not appear likely that the theoretical understanding based on Gram-Charlier can be extended to other wave field properties such as wave heights, crests, etc.”

His distribution is clean, bounded, and physically interpretable.

I remember attending a lecture by Prof. Miguel Onorato (a pioneer in non-linear wave research). He said something that stayed with me for years:
“The Tayfun distribution gives an upper bound for second-order crests. It’s not a rogue wave model, but it tells you what nonlinearity adds to the picture.” At the time, I was too junior to fully grasp it. But after digging into the classics, and working on my own simulations. I finally got it.

Finally, it is important to emphasise that most statistical courses are focused on teaching you how to fit distributions like GEV, GPD, or Weibull. These are great for design - say, if you’re building a breakwater or flood barrier. But they don’t explain the underlying physics. They describe data, they don’t derive it.

Tayfun’s approach is different. It doesn't guess. It derives. And it’s not alone in spirit. This idea, that nature follows probabilistic rules, but ones rooted in physical law, goes back to the 19th century. Ludwig Boltzmann was perhaps the first to try to explain thermodynamics with statistical distributions over microscopic states. This was a time when probability theory existed, but statistics had not yet emerged as a formal field of inference. The great figures of statistical science: Fisher, Neyman, Cramér, would only come later.

Tayfun’s work belongs to that older tradition: using probability to explain how nature works, not just to fit data. What happens beyond that, in the realm of truly rogue, multidirectional, modulationally unstable, or breaking waves, is a story for another time. But what Tayfun left us in 1980 is still, without doubt, a gem.

Catch you in the next post, where I’ll keep reflecting on the things I read, one idea at a time.
César

Reference
Tayfun, M. A. (1980). Narrow-band nonlinear sea waves. Journal of Geophysical Research: Oceans, 85(C3), 1548–1552. https://doi.org/10.1029/JC085iC03p01548