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.

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svn checkout http://code.google.com/p/guillaumemaze/source/browse/trunk/fortran/WSEdecomp gmaze_WSEdecomp

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svn checkout http://code.google.com/p/guillaumemaze/source/browse/trunk/matlab/WSEdecomp gmaze_WSEdecomp

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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:

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

Last, inserting 2 and 3 into 1 we obtain the

2D Fourier decomposition:

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

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:

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

:

and if

:

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 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).