OverviewSplit 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 bandpass filters specifications.DownloadWith 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: Direct to the file: Matlab or at Matlab Central: DocumentationTheorie 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 righthandside 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.
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