\label{fig:example_hilbert}
Wave envelope of the 30-day running mean, 500 hPa meridional wind for 22 May 1986 and 29 July 2006 at 54.75$^{\circ}$S. The original meridional wind (solid green), individual Fourier components for wavenumbers 1-9 (as defined in Equation \ref{eq:fourier_transform}; grey dashed), reconstructed signal from the inverse Fourier transform (Equation \ref{eq:inverse_transform} with wavenumbers 1-9 retained; dashed orange), and wave envelope (Equation \ref{eq:wave_envelope}; solid orange) are all shown.
    
    
    