The general data analysis techniques described below are the same as those employed in the zonal wave analysis of \citet{IrvingSimmonds2015}. The following text is derived from there with minor modifications.
All anomaly data discussed in the paper represent the daily anomaly. For instance, in preparing the 30-day running mean surface air temperature anomaly data series, a 30-day running mean was first applied to the daily surface air temperature data. The mean value for each day in this 30-day running mean data series (over the entire 1979–2014 study period) was then calculated to produce a daily climatology (i.e. the multi-year daily mean). The corresponding climatological daily mean value was then subtracted at each data time to obtain the anomaly.
Composite mean fields are presented throughout the paper for various temporal subsets (e.g. all data times corresponding to the positive or negative phase of the PSA pattern). For the composite mean anomalies of surface temperature, precipitation and sea ice, two-sided, one sample t-tests were applied at each grid point to examine the null hypothesis that the composite mean anomaly had been drawn from a population centered on zero. In order to account for autocorrelation in the data (which was substantial due to the 30-day running mean applied to the daily timescale data), the sample size (i.e. the number of data times used in calculating the composite; denoted \(n\)) was reduced to an effective sample size (\(n_{eff}\)) according to,
\[\label{eq:effective_sample_size} n_{eff} = \frac{n}{1 + 2\displaystyle\sum_{k=1}^{n-1} \frac{n-k}{n}\rho_k}\]
where \(\rho_k\) represents the autocorrelation for a given time lag \(k\) \citep{Zieba2010}.
The characteristics of data series that have been Fourier-transformed are often summarized using a plot known as a periodogram or Fourier line spectrum \citep{Wilks2011}. These plots are also referred to as a power or density spectrum, and most commonly display the squared amplitudes (\(C_k^2\)) of the Fourier transform coefficients as a function of their corresponding frequencies (\(\omega_k\)). As an alternative to the squared amplitude, we have chosen to rescale the vertical axis and instead use the \(R^2\) statistic commonly computed in regression analysis. The \(R^2\) for the \(k\)th harmonic is,
\[\label{eq:variance_explained} R_k^2 = \frac{(n/2)C_k^2}{(n-1)s_y^2}\]
where \(s_y^2\) is the sample variance and \(n\) the length of the data series. This rescaling is particularly useful as it shows the proportion of variance in the original data series accounted for by each harmonic \citep{Wilks2011}.
Two of the major modes of SH climate variability are the SAM and ENSO. In order to assess their relationship with the PSA pattern, the Antarctic Oscillation Index \citep{Gong1999} and Niño 3.4 index \citep{Trenberth2001} were calculated from 30-day running mean data (i.e. the same timescale that was used for the PSA pattern analysis). The former represents the normalized difference of zonal mean sea level pressure between 40\(^{\circ}\)S and 65\(^{\circ}\)S, while the latter is the sea surface temperature anomaly (relative to the 1981–2000 base period) for the region in the central tropical Pacific Ocean bounded by 5\(^{\circ}\)S–5\(^{\circ}\)N and 190–240\(^{\circ}\)E.