The Wigner distribution

The principle result of this section is to present the Wigner distribution which can be viewed as an extension to the variance density spectrum in the sense that cross-correlation contributions between non-collinear wave components in the wave field are included. Such a field thus deviates from homogeneous statistics. We begin with a short review of the spectral description of quasi-homogeneous wave fields and then consider the extension to inhomogeneous fields.


We consider the random sea surface $\eta(\vec{x})$ that is Gaussian distributed with a zero mean. Its Fourier transform is given by^2.4

  $${\hat \eta}(\vec{k}) = \frac{1}{4\pi^2} \int \eta(\vec{x})\,e^{-{\rm i}\vec{k}\cdot\vec{x}}d\vec{x}$$ (2.188)

where $\vec{k} = (k_x,k_y)$ is the wave number vector. The inverse of this Fourier transform reads

  $$\eta(\vec{x}) = \int {\hat \eta}(\vec{k})\,e^{{\rm i}\vec{k}\cdot\vec{x}}d\vec{k}$$ (2.189)

Note $\eta$ and ${\hat \eta}$ forms the conjugate pair.


Under the assumption of homogeneity in space, that is, the waves are statistically independent, the variance density spectrum can be found as the Fourier transform of the following auto-covariance function

  $$R(\vec{\xi}) = \left < \eta ( \vec{x} )\,\eta^\star (\vec{x}+\vec{\xi}) \right >$$ (2.190)

with $\vec{\xi}$ the separation distance and $\star$ denoting the complex conjugate. The variance density spectrum is given by

  $$E(\vec{k}) = \frac{1}{4\pi^2} \int R(\vec{\xi}) \, e^{-{\rm i}\vec{k}\cdot\vec{\xi}}d\vec{\xi}$$ (2.191)

and, conversely, we obtain

  $$R(\vec{\xi}) = \int E(\vec{k}) \, e^{{\rm i}\vec{k}\cdot\vec{\xi}}d\vec{k}$$ (2.192)

The total wave variance is defined through its marginal distribution

  $$\left < \eta \, \eta^\star \right > = R(\vec{0}) = \int E(\vec{k}) d\vec{k}$$ (2.193)

Clearly, $E(\vec{k}) \geq 0$ yields the distribution of variance among different wave numbers. The first order statistics of the wave field are then completely defined by this spectrum.


However, the situation becomes different if any two distinct wave components are correlated owing to medium variations (introduced by the depth and mean currents) that are rapid compared to the typical correlation length of the wave field. For instance, coastal waves can scatter into multiple directions when interacted with irregular small-scale seabed changes and can create interference patterns, such as refractive focusing of swells over shoals and wave diffraction around breakwaters. Additionally, the different wave components may remain correlated over many wave lengths. Another example is the scattering of waves induced by submesoscale currents in the open ocean. Statistically, the wave interferences are described by cross correlations between different components of the scattered wave field. In turn, this can cause relatively rapid variations in wave statistics (Smit and Janssen, 2013; Smit et al., 2015a; Akrish et al., 2020).


To properly describe the spectral representation of the correlation between crossing waves we consider the covariance function $\Gamma$ of surface elevation $\eta$ between two spatial points $\vec{x}+\vec{\xi}/2$ and $\vec{x}-\vec{\xi}/2$

  $$\Gamma (\vec{x},\vec{\xi}) = \biggl < \eta \Bigl (\vec{x}+\frac{\... ...i}}{2} \Bigr )\,\eta^\star \Bigl (\vec{x}-\frac{\vec{\xi}}{2} \Bigr ) \biggr >$$ (2.194)

Next, we employ the following joint distribution in phase space $(\vec{x},\vec{k})$, called the Wigner distribution,

  $$W(\vec{x},\vec{k}) = \frac{1}{4\pi^2} \int \Gamma (\vec{x},\vec{\xi}) \, e^{-{\rm i}\vec{k}\cdot\vec{\xi}}d\vec{\xi}$$ (2.195)

Note that $\Gamma$ is symmetric with respect to the spatial lag $\vec{\xi}$, so that $\Gamma (\vec{x},\vec{\xi}) = \Gamma^\star (\vec{x},-\vec{\xi})$, and hence $W(\vec{x},\vec{k})$ is a real-valued function. Furthermore, it can also be expressed in terms of the spectrum as follows (Bastiaans, 1979)

  $$W(\vec{x},\vec{k}) = \int {\hat \Gamma}(\vec{k},\vec{u}) \, e^{{\rm i}\vec{x}\cdot\vec{u}}d\vec{u}$$ (2.196)

with

  $${\hat \Gamma}(\vec{k},\vec{u}) = \biggl < {\hat \eta} \Bigl ( \ve... ... \Bigr )\,{\hat \eta}^\star \Bigl ( \vec{k}-\frac{\vec{u}}{2} \Bigr ) \biggr >$$ (2.197)

where $\vec{k} = (\vec{k}_1 + \vec{k}_2)/2$ and $\vec{u} = \vec{k}_1 - \vec{k}_2$ are the mean and difference of two interacting wave components, respectively.


From the above we have the following expressions of the correlation functions

  $$\Gamma(\vec{x},\vec{\xi}) = \int W(\vec{x},\vec{k}) e^{{\rm i}\vec{k}\cdot\vec{\xi}} d\vec{k}$$ (2.198)

and

  $${\hat \Gamma}(\vec{k},\vec{u}) = \frac{1}{4\pi^2} \int W(\vec{x},\vec{k})e^{-{\rm i}\vec{x}\cdot \vec{u}} d\vec{x}$$ (2.199)

Thus, the Wigner distribution $W(\vec{x},\vec{k})$ essentially describes the complete second order wave statistics, including the cross-variance contributions, by virtue of the separation in the wave number $\vec{u}$. Furthermore, the local wave variance can be expressed as

  $$0 \leq m_0 (\vec{x}) = \left < \eta \, \eta^\star \right > (\vec{x}) = \int W(\vec{x},\vec{k}) d\vec{k}$$ (2.200)

Although its marginal distribution exists^2.5, the Wigner distribution cannot be associated with a joint distribution in the strict sense because it can take negative values. (The Wigner distribution is commonly referred to as a quasi-distribution.)