The annual variation of rainfall in Southern Africa, directly and indirectly, affect human livelihoods and ecosystem through droughts, temperature changes, water supply problems and reduced agricultural production. Most of Southern Africa experience Austral summer rainfall (October to March) and north-eastern to south-western regions experience austral winter rainfall (May to August) (Dieppois et al., 2016). The summer and winter rainfall variability has been shown to be influenced by El Niño-Southern Oscillation (ENSO) and becomes nonlinear, with extreme weather conditions (Phillippon et al., 2012; Fauchereau et al., 2009; Kane, 2009). The temperature in South Africa has increased by over one and half times more than the globally observed temperature increases (Kruger & Shongwe, 2004; DEA, 2011; Mackellar et al., 2014).

El Niño describes the warming of sea surface temperature that occurs periodically, typically concentrated in the central-east equatorial Pacific (Lehodey et al., 1997). La Niña is the term adopted which describes the opposite side of the fluctuations. The Niño 3.4 index is one of the key atmospheric indices used to gauge the strength of El Niño and La Niña. The other driver that has been shown to have an impact on weather patterns is the Quasi-Biennial Oscillation (QBO) (Begue et al., 2010). QBO is a regular variation of the winds that blow high above the equator. Strong winds in the stratosphere travel in a belt around the planet, and in about 14 months these winds completely change direction (Kane, 2009). In the study by Kane (2009), it was shown that the warming of climate and sea surface temperatures has an impact on the rainfall patterns. Due to the influences of these climatic drivers and others, the weather patterns become nonlinear and non-stationary such that linear models are sometimes inconclusive (Huang et al., 1999; Molla et al., 2006). Fourier based methods assume that the data is linear and the data must be strictly periodic which is not the case with climate data (Schulte, 2016).

Empirical mode decomposition (EMD), was introduced by Huang et al. (1998), which does not make assumptions about linearity and stationarity of the time series and it is best suitable to analyse climate data. The time series is decomposed into different time scales called intrinsic mode functions (IMFs), which can reveal intrinsic changes in the climate system (Huang et al., 1998; Wu et al., 2009). EMD has a challenge of mixing the signals from one IMF to another, therefore, to cater for this, a noise assisted method named Ensemble Empirical Mode Decomposition (EEMD) was introduced (Wu et al., 2009).

EEMD has been applied widely in the hydrological and atmospheric studies. Chiew et al. (2005) applied EMD to annual streamflow to find significant oscillations in the data. Twenty unimpaired catchments from different parts of the world were used and 3, 6–7, 11–15 and 20–25 year oscillations in the stream flows were identified. In another study, the climate variability was observed in the case study of East and Central Africa, which was demonstrated by mapping the coupling between precipitation variables, inter-annual vegetation changes and the ENSO and Indian Ocean Dipole (IOD). EEMD was adopted because of its ability to breakdown normalised vegetation index (NDVI) series into multiple time scales components and its generic ability to be used with other time series analysis tools (Hawinkel et al., 2015, 2016). The Pacific Decadal Oscillation (PDO) is known to have an influence over East China’s annual and summer rainfall. Using EEMD the monthly rainfall influence of PDO was identified (Yang et al., 2017). The intrinsic oscillations in the land surface temperature of Wuhan, China were revealed using EEMD by decomposing the data into annual, inter-annual, noise and trend (Liu et al., 2019). In South Korea several climate drivers have been shown to have an impact on the precipitation, however many studies did not take into consideration the inherent cycles in the long term precipitation. Using EEMD, cross-correlation and multiple linear regression on the influence of ENSO, QBO, Arctic Oscillation, Atlantic Meridional Mode and others was shown at the monthly level (Kim et al., 2018). Some notable applications of EEMD include finding the effects of temperature and precipitation trends in Plateau droughts (Sun & Ma, 2015) and identifying the variability of monthly precipitation in Iran (Alizadeh et al., 2019).

In this study EEMD is used to decompose a 38-year rainfall and temperature data for a selected region in Western Cape, South Africa to reveal underlying physical signals. The influence of ENSO, QBO and temperature on the rainfall pattern is identified. The selected region chosen has been facing a lot of water challenges recently. There has been growing general interest in the winter rainfall mainly due to a threat of “day zero” in 2018 (Kruger et al., 2017; Maxmen, 2018; Wolski, 2018). However, very few studies have used statistical analysis to investigate the rainfall variability during winter. This is the first study that uses EEMD and synchronisation to investigate the influence of ENSO and QBO on the rainfall pattern for the region. This paper is organized as follows: in section 2, the data set and methodology are described; in section 3, the analysis and discussion of the results is done and conclusion is done in section 4.

The average monthly rainfall for the region located between 18.2–19.2ºE and 33.5–34.5ºS was standardised for easy computation and comparison. The multivariate imputation by chained equations is used to impute the missing rainfall and temperature data. The data was decomposed until when there is at most one maximum and one minimum in the residual. From the standardised rainfall data, 7 IMFs were found and are shown in Figure 2. IMF 3 has a period of about 12 months which corresponds to an annual (seasonal) oscillation, IMF 4 has a period of about 26 months which approximated a 2 year oscillation, IMF 5 has an oscillation of about 54 months (4.5 years) and IMF 6 captures a quasi-decadal oscillation (7 year period). Previous studies have used Wavelet Analysis, identified similar oscillations in the South African rainfall that were also found in this study (Kane, 2009; Dieppois et al., 2016). In these previous studies, winter rainfall was found to be having significant 2–3 year period and 3–4 year period which contributed to the rainfall variability. As compared to Wavelet Analysis, EEMD is adaptive, intuitive and does not use basis functions. Additionally, the impact of different climate drivers at different time scales can be shown. The graph on the bottom right of Figure 2 illustrates the residual plot, which shows the general trend of the rainfall. A study by Maúre et al. (2018) used several climatic models to predict the rainfall trend over Southern Africa under global warming and the study pointed to a decreasing daily rainfall for Western Cape, which is in agreement with the obtained residual plot.

The probability density function for each IMF is approximately normally distributed. The IMFs and residual are added together to reconstruct the data. The reconstructed data approximates the original data with Root Mean Square Error of order 10–14. This clearly shows that EMD is lossless decomposition with minimal data being lost in the decomposition and managing to capture most of the oscillations in the data.

It is noted that the maximum value of orthogonality between the IMFs is found to be approximately equal to 0.001 and it is way below the acceptable value of less than 0.1. The index of orthogonality for the IMFs is 0.594 × 10–4 for rainfall, 0.154 × 10–6 for Niño 3.4 and 0.235 × 10–5 for QBO. It confirms that there is less amount of information leakage.

The cross-correlation between rainfall and QBO and Niño 3.4 shows that there is no correlation as shown in the auto-correlation function (ACF) plot in Figure 3. However, when the time series were decomposed correlation is identified for IMF 3 for both Niño 3.4 and QBO as shown in Figure 4. There is a correlation of rainfall’s IMF3 and Niño 3.4 index at lag –4, –5, –6, 2 and 3. These results confirm the influence of ENSO on the seasonal rainfall and also the quasi-biennial oscillation which is consistent with results found by Philippon et al. (2012) and Kane (2009). Additionally, the general pattern of the rainfall at different time scales is identified up to quasi-decadal oscillation.

Cross-correlation can be used only when the time series is stationary, the Augmented Dicky Fuller Test (ADF) and Phillips-Perron Unit Root Test shows that IMF 4 to 7 are not stationary therefore cross-correlation cannot be used. The synchronisation does not require that the time series to be stationary hence it was used to identify any relationship for those IMFs. These IMFs found are further synchronised with Niño 3.4 and QBO and the results are shown in Figure 5 below. The results show that there is weak coupling between the original rainfall time series and Niño 3.4 or QBO. However, there is phase-locking identified for IMF 5 for Niño 3.4, since there is a clear peak. These results for Niño 3.4 shows that there may be an influence of ENSO on the rainfall pattern. This is in agreement with the results that were found by Philippon et al. (2012), which found that there was a significant association between ENSO and winter rainfall. They showed that there is a strong correlation of the May-June-July season with ENSO than any other period of the year. In this study using EEMD, the correlation is further done at a monthly level than seasonal level. The clear peak on the histogram of rainfall’s IMF6 and QBO (Figure 5) shows that there is phase locking. This QBO signal identified is in agreement with a study by Kane (2009) which also identified the presence of QBO and it was shown that it contributed significantly to the variability of the winter rainfall for the same region.

Temperature for the selected region was also decomposed into 7 IMFs and a residual. There is weak coupling between the original rainfall time series and temperature as shown in Figure 6 below. However, phase locking is observed for IMF 1 and IMF5, since there is a clear peak for both of them. IMF1 captures the noise found in the signal and IMF5 is a 4.5-year oscillation. The phase-locking in IMF1 suggest that the temperature variability may have an impact on the rainfall patterns. The synchronisation on IMF5 shows that the temperature changes may have a long term impact on the rainfall variability. This is consistent with other studies that have used different climatic models to find the impact of increasing temperature (Maúre et al., 2018; Nangombe et al., 2018; Nikulin et al., 2018). These models predict that there will be an increase in extreme rainfall patterns. In our study, we have managed to show the direct impact of increasing temperature using historical data.

The effectiveness of EEMD to analyse nonlinear and non-stationary data was demonstrated. The rainfall and temperature data were decomposed into IMFs and residual data, which summed up to the original data. The decomposed IMFs found can be used with other methods such as regression and neural networks to predict the impact of climate drivers in the future. EEMD was effective in isolating the data into different timescales and therefore the variability of the rainfall pattern was identified, in the end, evidence of the effect of ENSO and QBO was provided. Cross-correlation and phase synchronisation was used to find the relationship of the IMFs from the different time series under study. It will be of interest for future studies to carry out a study for a longer period to find the pattern of the rainfall over decades.

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