Stationary/Propagative components decomposition of a Longitude/Time signal with Fourier harmonics

Overview

Split a signal S(SPACE,TIME) into its downward/upward space propagating and stationary components via a 2D Fourier decomposition. S is a (SPACE,TIME) matrix. We eventually proceed to a space and/or time filtering. DT,DX are temporal and space step. PERIOD=[Tmin Tmax] and WAVELENGTH=[Xmin Xmax] are time and space band-pass filters specifications.

Download

With SVN:
The fortran code is here:
svn checkout http://code.google.com/p/guillaumemaze/source/browse/trunk/fortran/WSEdecomp gmaze_WSEdecomp
The Matlab version here:
svn checkout http://code.google.com/p/guillaumemaze/source/browse/trunk/matlab/WSEdecomp gmaze_WSEdecomp

Direct to the file:
Matlab or at Matlab Central:

Documentation

Theorie of the 2D Fourier analysis

We can express a variable $ \eta(x,t)$ as a sum of Fourier harmonics: $ \sum_n\sum_m \eta_{n,m}$ and from them we can separate standing, eastward and westward propagating parts of a (longitude,time) signal, Park (1990).

First, for each time of the record we compute cosine and sine coefficients from Fourier decomposition of the variable, ie we express $ \eta(x,t)$ as:

$\displaystyle \eta(x,t)$   $\displaystyle = \sum_{n} A(n,t)\cos(k_nx) + B(n,t)\sin(k_nx)$ (1)

Next, for each point we compute cosine and sine coefficients from the Fourier decomposition of A and B:

$\displaystyle A(n,t)$   $\displaystyle = \sum_{m} a(n,m)\cos(\omega_mt) + b(n,m)\sin(\omega_mt)$ (2)
$\displaystyle B(n,t)$   $\displaystyle = \sum_{m} c(n,m)\cos(\omega_mt) + d(n,m)\sin(\omega_mt)$ (3)

Last, inserting 2 and 3 into 1 we obtain the $ \eta_{n,m}$ 2D Fourier decomposition:

\begin{displaymath}\begin{split}\eta_{n,m} &= a(n,m)\cos(k_nx)\cos(\omega_mt) + ... ...x)\cos(\omega_mt) + d(n,m)\sin(k_nx)\sin(\omega_mt) \end{split}\end{displaymath} (4)

and following the notation from Park et al. (2004):

\begin{displaymath}\begin{split}\eta_{n,m} &= C_{ws}^{n,m}\sin(\theta_w^{n,m}) +... ...(\theta_e^{n,m}) + C_{ec}^{n,m}\cos(\theta_e^{n,m}) \end{split}\end{displaymath} (5)

with the westward/eastward sine and cosine coefficients:
$\displaystyle C_{ws} = (b+c)/2$ $\displaystyle ;$ $\displaystyle C_{wc} = (a-d)/2$  
$\displaystyle C_{es} = -(b+c)/2$ $\displaystyle ;$ $\displaystyle C_{ec} = (a+d)/2$  

and phases of East and West waves: $ \theta_e=k_nx-\omega_mt$, $ \theta_w=k_nx+\omega_mt$.

We can rewrite equation 5 in the more significative form:

$\displaystyle \eta_{n,m}$   $\displaystyle = A_w\cos(\theta_w-\varphi_w) + A_e\cos(\theta_e-\varphi_e)$ (6)

where we can identify West/East waves having amplitude $ A_w=\sqrt{C_{ws}^2+C_{wc}^2}$, $ A_e=\sqrt{C_{es}^2+C_{ec}^2}$ and phase lags $ \varphi_w=arctan(C_{ws}/C_{wc})$, $ \varphi_e=arctan(C_{es}/C_{ec})$.

We state that a stationnary wave is the sum of an eastward and a westward wave having the same amplitude. This allow us to rewrite expression 6 separating standing from propagating (eastward or westward, it depends on the relative amplitude of each part) wave components.

If $ A_e>A_w$:

\begin{displaymath}\begin{split}\eta_{n,m} &= A_w\cos(\theta_e-\varphi_e) + A_w\... ... &\quad + (A_e-A_w)\cos(k_nx-\omega_mt-\varphi_e) \end{split}\end{displaymath} (7)

and if $ A_w>A_e$:

\begin{displaymath}\begin{split}\eta_{n,m} &= 2A_e\cos\left(k_nx-\frac{\varphi_w... ... &\quad + (A_w-A_e)\cos(k_nx-\omega_mt-\varphi_w) \end{split}\end{displaymath} (8)

First terms on the right-hand-side of equations 7 and 8 are standing wave parts of the harmonic while second term is an Eastward or Westward wave.

We reconstruct the signal by summing harmonics over the two directions $ n$ and $ m$ but doing it separetly for each wave components (according to equation 7 and 8 discrimination) allow separation of the signal into its 3 wave parts. Remark that restricting the summation over a selected range in space and time allow 2D filtering of the signal.

Finally, this method allow to plot a very powerfull diagram: a 2D power spectral density. Contours of amplitudes $ A_w$ and $ A_e$ in the ($ k_n$,$ \omega_m$) plan can show waves properties. Pics in only one part of the plan (westward side or eastward side) stands for a propagating wave and identical pics in both sides stand for a stationnary wave.

Bibliography

Park, Y.-H. (1990).
Mise en évidence d'ondes planétaires semi-annuelles baroclines au sud de l'océan indien par altimétrie satellitaire.
C. R. Acad. Sci. Paris, 310(2):919-926.

Park, Y.-H., Roquet, F., and Vivier, F. (2004).
Quasi-stationary enso wave signals versus the antarctic circumpolar wave scenario.
Geophys. Res. Lett., 31(L09315).



Comments