Swell Spectral Transform For understanding the quality of surf in general, I find that directional spectral plots, like this also below, are the best way for me to get a quick mental picture. I want to make a kind of video of these images so I can visually see surf evolution. CDIP offers XYZ displacement data, like this, but I don't know exactly what the transform is to go from XYZ displacement to a directional spectrum.
My naive intuition would be that the functions spanning the space might look something like,
\begin{align}
\vec{\psi}(t; \omega, \theta) &= \hat{x} \sin \omega t \sin \theta + \hat{y} \sin \omega t \cos \theta + \hat{z} \cos \omega t
\end{align}
I imagine that this might work because I have read that swell move in circles vertically and in the direction the swell is propagating. Thus the displacement on the X-Y plane will be handled by the $ \sin \theta, \cos \theta $ terms while the time cyclic terms handle the swell movement.
I would then decompose (numerically) the waves by doing the integral,
$$
F(\omega, \theta) = N^{-1} \int_\mathbb{R} dt\ \ \vec{f}(t) \cdot \vec{\psi}(t; \omega, \theta)
$$
where $ N $ is the normalization I would need to work out.
Is this the right way to do the directional spectral decomposition?

 A: It's called "Directional Spectrum of Ocean/Sea Waves".
Theory
The credit for this method goes to Longuet-Higgens. His paper is miserable to find, but I linked a scan of it below.
Using some relations from deep water Airy Wave Theory, we get the particle displacement field elements,
\begin{align}
\xi_i(\vec{x}, t) &= \mathbb{Re}\left[ \int d^2k\ p_i A(k)\ e^{i(k\cdot x - \omega t)} \right] \\
p_i &= (i \cos \phi, i \sin \phi, 1)_i
\end{align}
with $ \phi $ being the angle of the momentum vector $ k $.
The energy density is,
\begin{align}
E(k)d^2k &= \frac{1}{2}A(k)\overline{A(k)} d^2k \\
E(\omega, \phi) d\omega d\phi &= \frac{1}{2}A(\omega, \phi)\overline{A(\omega, \phi)} \frac{d^2k}{d\omega d\phi} d\omega d\phi \\
  &= \frac{1}{2}A(\omega, \phi)\overline{A(\omega, \phi)} \frac{k dk}{d\omega} d\omega d\phi \\
E(\omega, \phi) &= \frac{1}{2}A(\omega, \phi)\overline{A(\omega, \phi)} \frac{k dk}{d\omega}
\end{align}
Lets define,
\begin{align}
C_{ij} &= \frac{1}{2} \int_{[0, 2\pi]} d\phi \frac{d^2k}{d\omega d\phi} (p_i A(k)) \overline{(p_j A(k))} \\
  &= \int_{[0, 2\pi]} d\phi\ p_i \overline{p_j} E(\omega, \phi)
\end{align}
Note the symmetric tensor,
\begin{align}
p_i \overline{p_j} &= 
\begin{bmatrix}
\cos^2 \phi & \cos \phi \sin \phi & \cos \phi \\
\cdot & \sin^2 \phi & \sin \phi \\
\cdot & \cdot & 1
\end{bmatrix} \\
  &= 
\begin{bmatrix}
\frac{1}{4} \left(e^{i2\phi} + e^{-i2\phi} + 2 \right) & \frac{1}{4i} \left(e^{i2\phi} - e^{-i2\phi} \right) & \frac{1}{2}\left(e^{i\phi} + e^{-i\phi}\right) \\
\cdot & \frac{1}{4} \left(2 - e^{i2\phi} - e^{-i2\phi} \right) & \frac{1}{2i}\left(e^{i\phi} - e^{-i\phi}\right) \\
\cdot & \cdot & 1
\end{bmatrix} \\
\end{align}
We note that the Fourier series on $ \phi $ is,
\begin{align}
E(\omega, \phi) &= \sum_n E_n e^{in\phi} \\
E_n &= \frac{1}{2\pi} \int_{[0, 2\pi]} E(\omega, \phi) e^{-in\phi} \\
\end{align}
Mixing and matching the $ C_{ij} $ we get,
\begin{align}
E_0 &= \frac{1}{2\pi} C_{00} \\
E_1 &= \overline{E_{-1}} = \frac{1}{2\pi} (C_{13} - iC_{23}) \\
E_2 &= \overline{E_{-2}} = 2(C_{11} - C_{22}) + 4i C_{12}
\end{align}
Part I can't figure out!!
In order to get the $ C_{ij} $, I would expect to evaluate the integral at $ x=0 $ because I only have one data source so I can't integrate over space,
\begin{align}
\int dt \xi_i(x=0) \overline{\xi}_j(x=0) &= \int d^2k\ d^2k'\ p_i \overline{p_j} A(k) \overline{A(k')} \delta(\omega - \omega') \\
  &= \int d\omega d\omega' d\phi d\phi' \left[k(\omega) \frac{dk}{d\omega} \right] \left[k(\omega') \frac{dk'}{d\omega'} \right] p_i(\phi) \overline{p_j(\phi')} A(\omega, \phi) \overline{A(\omega', \phi')} \delta(\omega - \omega') \\
  &= \int d\omega d\phi d\phi' \left[k(\omega) \frac{dk}{d\omega} \right]^2 p_i(\phi) \overline{p_j(\phi')} A(\omega, \phi) \overline{A(\omega, \phi')} \\
\end{align}
I may be off by an overall normalization, but I can't move the integral past this to get anything that looks like $ C_{ij} $ because of that integral over $ \phi' $. I don't have a delta function for $ \phi' $ and thus I get a kind of averaging over $ \phi' $ which is inappropriate for my goals.
Conclusion
Donelan et al. states that you need many points to collect data.
Sources:
Donelan MA, Hamilton J, Hui WH. Directional spectra of wind
generated waves. Phil Trans R Soc London 1985;A315:509–62.
Observaton of the Directional Spectrum of Sea Waves Using the Motion of a Floating Bouy by Longuet-Higgens
Wikipedia Airy Wave Theory
NOAA
Section 2.1 of this
