# 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

as a sum of Fourier harmonics: 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

as:

(1)

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

(2)

(3)

2D Fourier decomposition:

(4)

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

(5)

with the westward/eastward sine and cosine coefficients:

and phases of East and West waves:

, .

We can rewrite equation 5 in the more significative form:

(6)

where we can identify West/East waves having amplitude

, and phase lags ,

.

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

:

(7)

and if

:

(8)

We reconstruct the signal by summing harmonics over the two directions and 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

and in the (,) 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).