Paper notes

Central finding

Two reconstruction methods can reproduce the same measured directional moments but produce substantially different directional energy distributions.

Issue- Four moments cannot uniquely determine the entire directional distribution. There are infinitely many possible distributions that reproduce those four moments. The reconstruction methods determines how we fill in the missing directional information.

Methods compared

${\cos }^{2s}$

• prescribes a simple unimodal directional distribution.
• Limitation: cannot reproduce all four moments

MEM-I (Burg)

• Uses a maximum-entropy formulation with an analytical solution.
• Limitation: Often produces excessively sharp peaks and artificial bimodality.

MEM-II (Shannon)

• Maximizes Shannon entropy subject to the measured moments
• Limitation: historically computationally expensive

This paper establishes a much faster algorithm for MEM-II

Previous methods - Truncated Fourier Expansion

$D(\theta )=\frac{1}{2\pi }\left[1+2a_1\cos \theta +2b_1\sin \theta +2a_2\cos 2\theta +2b_2\sin 2\theta \right]$

MEM-I (Burg)

This is the Lygre and Krogstad (1986) method. It uses the complex directional moments: $c_1=a_1+i{b}_1,c_2=a_2+i{b}_2$ These are transformed into two complex parameters, conventionally written as ${\varphi }_1$ and ${\varphi }_2$.

The directional distribution then has a rational form: $D_{\mathrm{MEM}-\mathrm{I}}(\theta )\propto \frac{1}{{|1-{\varphi }_1{e}^{-i\theta }-{\varphi }_2{e}^{-2i\theta }|}^2}$ The sign of the angular exponent depends on the moment convention.

Derivation

It origonates in spectral estimation and autoregressive modeling.

Using the conventional maximization sign, its entropy functional is: $H_B[D]={\int }_{-\pi }^{\pi }\ln D(\theta )d\theta$ subject to the measured constraints.

Some presentations use the negative of this quanitity and formulate an equivalent minimization problem (why?)

The logarithm is integrated uniformly over direction, rather than weighted by the amount of wave energy in each direction.

This means the optimization treats every direction as part of the domain over which spectral uncertainty is being regularized.

To understand why this produces the Burg reconstruction, introduce Lagrance multipliers for the constrains: $\begin{array}{rl}\mathcal{L}[D]=& \int \ln D(\theta )d\theta \\ & -{\lambda }_0\left(\int D(\theta )d\theta -1\right)\\ & -\sum _{n=1}^2{\lambda }_{a_n}\left(\int D(\theta )\cos n\theta d\theta -a_n\right)\\ & -\sum _{n=1}^2{\lambda }_{b_n}\left(\int D(\theta )\sin n\theta d\theta -b_n\right)\end{array}$ To find the optimal distribution, require that small changes in $D(\theta )$ do not change $\mathcal{L}$ to first order.

The function derivative gives: $\frac{1}{D(\theta )}-{\lambda }_0-{\sum }_{n=1}^2\left({\lambda }_{a_n}\cos n\theta +{\lambda }_{b_n}\sin n\theta \right)=0$ Rearranging: $\boxed{D_B(\theta )=\frac{1}{{\lambda }_0+{\lambda }_{a_1}\cos \theta +{\lambda }_{b_1}\sin \theta +{\lambda }_{a_2}\cos 2\theta +{\lambda }_{b_2}\sin 2\theta }}$ This is the defining mathematical property of Burg reconstruction: The directional distribution is the reciprocal of a trigonometric polynomial.

This then becomes the Lygre-Krogstad formula via: a positive trigonometric polynomial can be factored into the squared magnitude of a complex polynomial.

Lygre and Kogstad exploit this to obtain a closed-form reconstruction. First combine the measured moments into complex coefficients: $c_1=a_1+i{b}_1,c_2=a_2+i{b}_2$ Then calculate ${\varphi }_1=\frac{c_1-c_2{c}_1^{\ast }}{1-|c_1{|}^2},{\varphi }_2=c_2-{\varphi }_1{c}_1$ where the * denotes the complex conjugate.

Then the reconstruction becomes $D_B(\theta )=\frac{{\sigma }^2}{2\pi {|1-{\varphi }_1{e}^{-i\theta }-{\varphi }_2{e}^{-2i\theta }|}^2}$ With normalization: ${\sigma }^2=1-{\varphi }_1{c}_1^{\ast }-{\varphi }_2{c}_2^{\ast }$ The issue is that whenever the denominator becomes small, the directional energy becomes large, which can produce very narrow peaks.

In autoregressive terminology, these resemble spectral resonances associated with poles close to the unit circle.

If the polynomial has two angular minima, the reconstruction can develop two directional maxima.

MEM-II (Shannon)

Rather than a rational function, the directional distribution takes the form $D_{\mathrm{MEM}-\mathrm{II}}(\theta )=\frac{\exp \left[{\kappa }_1\cos (\theta -{\mu }_1)+{\kappa }_2\cos 2(\theta -{\mu }_2)\right]}{A}$ where A normalizes the distribution. what are other variables?

This is a generalized Von Mises distribution. The parameters must be determined numerically so that the reconstruction reproduces the measured directional moments.

Github : https://github.com/davidcchristie/MaxEntropySPT

Chistie’s approach

He uses a different entropy functional: $H_S[D]=-{\int }_{-\pi }^{\pi }D(\theta )\ln D(\theta )d\theta$ This has an additional weighting by $D(\theta )$

In Shannon entropy, the contribution of a particular direction depends on how much probability, or normalized wave energy, is assigned to it.

Maximizing Shannon entropy therefore selects the distribution with the greatest uncertainty about wave direction while respecting everything measured by the buoy. (why is that advantageous?)

Derivation

Starting with the same moment constraints, the Lagrangian is now: $\begin{array}{rl}\mathcal{L}[D]=& -\int D(\theta )\ln D(\theta )d\theta \\ & -{\lambda }_0\left(\int D(\theta )d\theta -1\right)\\ & -\sum _{n=1}^2{\lambda }_{a_n}\left(\int D(\theta )\cos n\theta d\theta -a_n\right)\\ & -\sum _{n=1}^2{\lambda }_{b_n}\left(\int D(\theta )\sin n\theta d\theta -b_n\right)\end{array}$ Again, take the functional derivative with respect to $D$: $-\ln D(\theta )-1-{\lambda }_0-{\sum }_{n=1}^2\left({\lambda }_{a_n}\cos n\theta +{\lambda }_{b_n}\sin n\theta \right)=0$ Rearranging gives: $\ln D(\theta )={\tilde{\lambda }}_0+{\sum }_{n=1}^2\left({\tilde{\lambda }}_{a_n}\cos n\theta +{\tilde{\lambda }}_{b_n}\sin n\theta \right)$ Exponentiating: $D_S(\theta )\propto \exp \left[{\sum }_{n=1}^2\left({\tilde{\lambda }}_{a_n}\cos n\theta +{\tilde{\lambda }}_{b_n}\sin n\theta \right)\right]$ Each sin/cos pair can be written as one phase-shifted cosine.

This gives Christi’s Eq. (14): $D_S(\theta )=\frac{\exp \left[{\kappa }_1\cos (\theta -{\mu }_1)+{\kappa }_2\cos 2(\theta -{\mu }_2)\right]}{A}$ Where A ensures normalization. This is the generalized von Mises distribution.

The first harmonic ${\kappa }_1\cos (\theta -{\mu }_1)$, primarily controls directional concentration and orientation.

The second harmonic ${\kappa }_2\cos 2(\theta -{\mu }_2)$ , allows additional structure, including asymmetry and bimodality.

The interactions between these terms determine the actual distribution, not necessarily identical to the measured first-moment mean direction.

To determine the four parameters, we require th reconstructed moments to match the observed moments: $\int D_S(\theta )\left[\begin{array}{c}\cos \theta \\ \sin \theta \\ \cos 2\theta \\ \sin 2\theta \end{array}\right]d\theta =\left[\begin{array}{c}{a}_1\\ b_1\\ a_2\\ b_2\end{array}\right]$ Unlike Burg, there is no simple analytical solution for these parameters. We must solve a nonlinear optimization or root-finding problem.

This is exactly what makes Shannon MEM computationally more demanding.

Christie makes this more efficient by expressing the distribution relative to the measured mean direction, therefore reducing the numerical problem from four unknowns to three $({\kappa }_1,{\kappa }_2,\Psi )$ where $\Psi ={\mu }_2-{\mu }_1$

He then uses modified Bessel-function expansions to evaluat ethe reuired moments and their derivatives efficiently (whats a Bessel function?)

The result is an implementation of MEM-II that is substantially more practical for large observational datasets.

Comparison

The fundamental mathematical distinction between methods is: $\begin{array}{rl}\text{Burg:}& D_B(\theta )=\frac{1}{P(\theta )}\\ \text{Shannon:}& D_S(\theta )=\frac{e^{Q(\theta )}}{A}\end{array}$ This has important consequences.

Burg: sharp peaks and comparatively heavy tails

Because Burg takes the reciprocal of a polynomial, small values of the denominator produce disproportionately large directional peaks.

Its characteristic shape can involve:

• very narrow principal peaks
• relatively heavy directional tails
• secondary peaks that may be artificially amplified These are not inevitable, but are known tendencies of the method.

Shannon: a smoother distribution under the same constraints

The exponentiation can produce sharply peaked or bimodal distributions when the constrains support them, but it does not have Burg’s rational-function behavior.

It is also the maximum-Shannon_entropy solution: among distributions satisfying the available moment constraints, it introduces the least additional structure according to that particular information-theoretic criterion.

This doesn’t mean that Shannon necessarily reproduces the true wave field in every case. Four moments remain insufficient to determine the true distribution uniquely.

However, the validation studies discussed by Christie generally find MEM-II more representative of known directional spectra.

Analysis worth revisiting

• Directionally resolved Doppler corrections angular peak widths
• directional peak maxima
• interpretations involving the detailed shape of high-frequency directional distributions.

-> If the results remain consistent when you replace Burg with Shannon, that would provide stronger evidence that we’re observing a physical relationship between forcing history and directional wave structure, rather than a dependence on the assumed reconstruction.

-> I would be particularly interested in whether HFDS–wind-curl relationship remains stable across both methods after Doppler correction.