\section{Methodology}

While EOF analysis has been useful in highlighting the existence of the PSA pattern, it may not be the best tool for detailed climatological investigation. A particularly problematic shortcoming is that EOF analysis allows for only a crude representation of variations in wave phase (via the PSA-1 and PSA-2 modes, which are 90$^{\circ}$ out of phase), which makes it difficult to interpret characteristics such as the propagation of the pattern. This issue is further compounded by the degenerate \citep{North1982} nature of the PSA-2 mode \citep[e.g. Figure 1;][]{Mo2000}, which means the true second mode is some random combination of the apparent second and third modes.

In order to obtain more detailed information on variations in wave phase, existing studies of Rossby wave activity have tended to apply Fourier analysis along lines of constant latitude \citep[e.g.][]{Glatt2014}. More sophisticated methods have recently been considered for identifying/tracking non-zonal waves \citep[e.g.][]{Zimin2006,Souders2014}, however a key insight of the method developed here is that unlike the generalized case of all possible non-zonal propagation, analysis of the PSA pattern can make use of the fact that the waveform follows an approximate great circle path \citep{Hoskins1981}. By rotating the global coordinate system such that the equator (itself a great circle) traces the approximate path of the PSA pattern, we were able to simply apply Fourier analysis along the `equator' in the new zonal direction. Such grid rotation is commonly used in ocean modeling to avoid coordinate singularities caused by the convergence of meridians at the poles \citep[i.e. the grid is rotated to place the north pole over land; e.g.][]{Bonaventura2012}, but has not previously been applied in the context of tropospheric planetary wave activity. This new approach to PSA pattern identification is described below, along with the other more general data analysis techniques used in the study.

\subsection{Identification algorithm}

\subsubsection{Grid rotation}

In order to align the new equator with the approximate path of the PSA pattern, a global 0.75$^{\circ}$ latitude by 0.75$^{\circ}$ longitude grid was defined (i.e. the same resolution as the original ERA-Interim data) with the north pole located at 20$^{\circ}$N, 260$^{\circ}$E. The 500 hPa zonal and meridional wind data were used to calculate the meridional wind relative to the new north, and then the temporal anomaly of this new meridional wind was linearly interpolated to the rotated grid for use in the Fourier analysis (e.g. Figure \ref{fig:rotation}). It should be noted that existing zonal wave studies \citep[e.g.][]{IrvingSimmonds2015} tend to skip this final step of calculating the anomaly, because in the case of zonal waves the temporal mean of the meridional wind is typically close to zero (and hence waveforms defined by the meridional wind already oscillate about zero). 

On this rotated grid, the search region of interest was defined as the area bounded by 10$^{\circ}$S to 10$^{\circ}$N and 115$^{\circ}$E to 235$^{\circ}$E (this approximate area is referred to as the PSA sector at times throughout the paper). This region was selected via visual comparison with existing definitions of the PSA pattern (e.g. Figure \ref{fig:eof}), however the final results were not sensitive to small changes in pole location or search region bounds.

\subsubsection{Fourier analysis}

To prepare the meridional wind anomaly for Fourier analysis, the meridional mean was calculated over 10$^{\circ}$S to 10$^{\circ}$N (in order to eliminate the latitudinal dimension) and then values outside of 115$^{\circ}$E to 235$^{\circ}$E were set to zero. Zero padding is a commonly used technique in signal processing when the waveform of interest does not complete an integer number of cycles in a given domain, and is equivalent to multiplying the original signal (in this case the meridional mean meridional wind anomaly) by a square window function. This multiplication (or convolution) of two waves has consequences in frequency space, such that even a perfectly sinusoidal signal that would repeat exactly six times (for example) over the zero padded domain would show power at more than one frequency. This phenomenon is known as spectral leakage (into the side lobes of the frequency spectrum) and arises due to the fact that a square window function is not square in frequency space. In analyses where excessive leakage is undesirable, a Hanning or Hamming window can be used instead. In the frequency space these windows do not display as much spread into the side lobes, however this comes at the expense of the magnitude of the main lobes. Since our selection process (see below) focuses on identifying the main lobes, a square window function was considered most appropriate.

\subsubsection{Identification and characterization of PSA-like variability}

Given that the PSA pattern completes approximately 1.6 to 2.0 cycles (depending on the specific EOF mode) over the 120$^{\circ}$ search area (see Figure \ref{fig:eof}), our analysis focused on data times where a Fourier transform revealed wavenumber 5 and 6 as dominant frequencies over the zero padded 360$^{\circ}$ domain. In particular, a data time was said to display PSA-like variability (and hence was selected for further analysis) if the amplitude of the wavenumber 5 and 6 components of the Fourier transform were ranked in the top three of all frequencies. The vague `PSA-like' descriptor is used because a number of features besides the PSA pattern (e.g. Antarctic Dipole, Amundsen Sea Low) can exhibit wavenumber 5--6 variability in the PSA sector.

Once these data times were selected, additional information from the Fourier transform was used to characterize the phase and amplitude of the PSA-like variability. With respect to the former, it can be seen from Figure \ref{fig:transform} that within the search area the phase of the wavenumber 5 and 6 components of the transform (and usually also adjacent frequencies like wavenumber 4 and 7) tend to align both with each other and also with the phase of the actual signal. The phase of the wavenumber 6 component of the Fourier transform was therefore used as a proxy for the phase of the signal as a whole, and this information was used to separate data times displaying the actual PSA pattern from the larger population of PSA-like variability (note that similar results were obtained using wavenumber 5). The details of this separation process (e.g. the phase ranges used to define the PSA pattern) are discussed below. In order to quantify the amplitude of PSA-like variability, the wave envelope construct pioneered (in the atmospheric sciences) by \citet{Zimin2003} and recently applied by \citet{IrvingSimmonds2015} was used. The envelope is obtained by performing a Fourier transform, followed by an inverse Fourier transform for only the wavenumbers of interest (this is known as a Hilbert transform in digital signal processing). The complex number amplitude of the resulting waveform represents the envelope. Since the envelope of the complete signal (i.e. with all wavenumbers retained) can be quite noisy, the amplitude of PSA-like variability was defined as the maximum value of the envelope when only wavenumbers 4 to 7 are retained (see Figure \ref{fig:transform} for an example envelope).

\subsubsection{Timescale considerations}

In applying the identification algorithm to the ERA-Interim dataset, we focus on monthly timescale data at 500 hPa. This represents a mid-to-upper tropospheric level that is below the tropopause in all seasons and at all latitudes of interest. Given the equivalent barotropic nature of the PSA pattern (i.e. the wave amplitude increases with height but phase lines tend to be vertical) the results do not differ substantially for other levels of the troposphere. Monthly mean data were used for consistency with most previous studies and were obtained by applying a 30-day running mean to the daily (i.e. diurnally averaged) ERA-Interim data, in order to maximize the available monthly timescale information. As noted by previous authors \citep[e.g.][]{Kidson1988}, potentially useful information may be lost if only twelve (i.e. calendar month) samples are taken every year. Dates were labeled as the middle day of the 30-day period and this middle day was used to determine which season a given data time belonged to (e.g. the labeled date 1979-02-16 spans the period 1979-02-01 to 1979-03-02 and belongs to DJF). 

To explore the implications of this timescale selection, the Fourier transform used in the identification process was applied to the 500 hPa rotated meridional wind anomaly data (Figure \ref{fig:periodogram}). That analysis revealed wavenumber 7 as the most dominant frequency for daily timescale data in the PSA sector, with wavenumber 6 dominating the frequency spectrum for a 10--90 day running window. Given that the PSA pattern is itself characterized by wavenumber 5--6 variability in the PSA sector, this result suggests that (a) the PSA pattern is a dominant regional feature on weekly through to seasonal timescales, and (b) by extension the climatological results obtained from 30-day running mean data may also be relevant at those timescales.

