We employ MEMD to objectively identify intrinsic modes of variability in nonlinear and nonstationary spatio-temporal data. To understand MEMD, it is easier to first consider the simpler implementation of the 1-D version, empirical mode decomposition (EMD), as outlined by : i.

Local minima and maxima of the input (1-D) time series (e.g. see black solid line in Fig. ) are identified.

Additionally, each IMF has to satisfy two criteria: (a) the number of extrema and the number of zero crossings differs at most by one and (b) the mean value of the envelope of the IMF is zero. Note that the procedure from i to iv does not necessarily satisfy (a)–(b) immediately; thus, an additional sifting process (typically iterative) is used, which requires a stopping criteria to ensure the physical meaning of the IMFs. The stopping criteria can be based on the standard deviation of each IMF and on the maximum number of iterations, among others, which set tolerance and confidence limits for the IMF for details see. Ultimately, this process extracts modes of variability that consist of (1) the typical timescale of the mode and (2) its time series. The modes are automatically ordered from the highest (shortest) in the first mode to lowest (longest) frequency (period) in the last mode. Note that modes that emerge are largely independent with only small correlations between them.

MEMD is a generalisation of the EMD to multivariate datasets of more than two time series (for bivariate and trivariate data separate methods exist; ). The method solves a similar problem as in i–vi, but the mean envelope is computed as “an integral of all the envelopes along multiple directions in an N-dimensional space” (i.e. on an N sphere). This is much more complex, but the basic idea remains similar to the 1-D method, and the method retains similar stopping criteria for the sifting process. For further details on and visualisations of the method the reader is referred to and .

The MEMD ultimately extracts timescales common to all input time series (i.e. synchronises signals; ) and provides multivariate IMFs (i.e. the outputs of MEMD method) describing those timescales. The timescales of IMFs are then consistent across the input time series – to visualise this, see Supplement Figs. S6–S10, Tables S1–S2, and Sect. S.3 for an idealised example.

As with all statistical methods, it is important to be aware of the drawbacks associated with the (M)EMD e.g.. Similar to other time-series-filtering methods, (M)EMD can encounter issues at the edges of the time series, which can result in travelling waves and thus unrealistic peaks in the time series e.g.. Another common challenge with (M)EMD is the mixing of modes (see below) where a single genuine mode may be split into multiple modes if inappropriate parameters are chosen e.g.. However, this issue can also arise when there is no clear timescale separation. To address these challenges, it is crucial to test the physical relevance of the modes of variability identified using (M)EMD and to ensure convergence and stability of the modes through different parameter sweeps that are related to the stopping criteria (see Sect. ). Note that different parameter sweeps may be relevant for different applications.

The code for the MEMD is freely available on GitHub (https://github.com/mariogrune/MEMD-Python- (last access: 1 December 2020); similarly for the EMD discussed above: https://github.com/laszukdawid/PyEMD, last access: 4 December 2023), and the user ultimately only decides about the stopping criteria. These are set to the “fix_h” parameter following , who suggest that limiting iterations yields better-behaved IMFs than other stopping criteria. We limit the number of iterations to 15 (parameter “n_iter” is 15), though other values were tested and a range for “n_iter” around 5–30 yielded similar results, suggesting some convergence for the significant modes of variability (see below).

Note that at higher/lower frequencies we find mode mixing in our MEMD analysis where timescales are not clear (here, this occurs on timescales shorter than about 8 months and longer than about 700 months), as well as with a larger number of iterations. These modes are not detected as different from red noise (see Sects.  and for more details). However, the significant modes of variability on interannual timescales that are of interest here are largely unaffected by this (Sect. ).

As mentioned in Sect. , we use 3-D data relevant for ENSO dynamics, i.e. A(Lt,Ly,Lx) (with Lt the length of the time dimension, Ly the length of the latitude dimension, Lx the length of the longitude dimension, and A the selected variable(s)/field(s); bold letters represent two- or three-dimensional arrays) as the basis for the MEMD analysis, which is done the following way. i.

First, we remove the smooth trend/seasonal cycle described above; see also from A to get a temporal anomaly, A′.

MEMD analysis identifies 21 IMFs that are ordered by frequency from the highest (IMF 1) to the lowest (IMF 21) with the last 21st mode typically representing a trend, which in our case was already removed (see above). Namely, we find 21 potential intrinsic timescales within the tropical Pacific, i.e. common to all input PCs. This means that we obtain 21 IMFs for each PC time series, i.e. PCm(Lt)=∑sIMFs(PCm(Lt)), with s the IMF number and m the PC number, where the sth IMF of each PCm has the same timescale (see Table S2). Since we initially obtained PCs via SVD analysis and we thus have corresponding EOF patterns, we can then reconstruct spatio-temporal patterns of variability for each field (i.e. SST, wind stress, and thermocline depth) from PCs and EOFs. The interest here is in the variability on different timescales (i.e. for each (sth) IMF separately, we can perform this reconstruction for each IMF across all 20 PCs/EOFs): 1IMFsspatial(Lt,Ly,Lx)≃∑mIMFs(PCm(Lt))EOFm(Lx,Ly). Please recall that input data for MEMD analysis were divided by σ; thus, for the variability of a field in its original units we need to multiply spatio-temporal IMFs by σ.

To compute an index, such as the eastern Pacific SST (Niño3), we can average over a latitude–longitude region (Table ) from spatio-temporal IMFs (Eq. ) to obtain time series of, for example, the eastern Pacific SST (Niño3) for each IMF separately – see Figs. S6–S10 for time series of different variables from Table  and Table S1 for their timescales. This yields an index for each IMF (e.g. IMFs(SST (Niño3)), with s the IMF number) corresponding to an equivalent index computed from input data (A′). The latter is approximately the same as the sum of indices computed from all IMFs, e.g. SST(Niño3)(Lt)≃∑sIMFs(SST(Niño3)(Lt)).

We identify two significant modes of variability in the eastern Pacific: the 11th IMF (IMF 11) and 12th IMF (IMF 12) (shown with blue arrows in Fig. ) with average timescales of ∼ 23 and ∼ 39 months (∼ 2–3 years; see also Table S1), respectively. Their timescales fall well within the typical ENSO timescale range (2–8 years), as can be identified via a typical power spectrum analysis (Appendix , Fig. ). These timescales (with their uncertainty ranges; see below) are consistent with previously identified quasi-biennial (QB; ∼ 2 years) and low-frequency/quasi-quadrennial (LF/QQ; ∼ 4 years) ENSO modes e.g..

The quasi-periodic mode of variability with a 39-month average period (IMF 12) is more statistically significant (than IMF 11). This can be seen in different ways. First, considering the Niño3 index (i.e. the eastern Pacific SST in the Niño3 region), IMF 11 (blue dot) lies only slightly above the 95 % red noise threshold (red dashed line), whereas IMF 12 lies well above the threshold (Fig. ). Second, considering the PC1's IMFs (i.e. before we do spatio-temporal reconstruction of IMF data; Fig. S2), only IMF 12 is found to be significant, despite the generally good agreement between the Niño3 index and PC1 of the tropical Pacific SST e.g.. Finally, we can perform a red noise test on each grid point of the tropical Pacific (similarly as for specific time series shown above) and plot the spatial structure of SST together with the significance test (Sect. , Figs. , S3). This shows that none of the grid points of spatio-temporal IMF 11 pass the red noise threshold (Fig. S3), but many grid points of the spatio-temporal IMF 12 are significant (not shaded in grey in Fig. ).

Nevertheless, even though IMF 11 is marginally statistically significant, the MEMD analysis suggests that there may be two quasi-periodic modes of interannual SST variability in the tropical Pacific region. This agrees with previous results that have identified two statistically significant modes of variability with specific frequencies and distinguished by their dynamics. Note that we also find that the two modes of variability are well separated (in terms of timescale) from the other modes and from each other, as shown by Fig. . This ensures no mode mixing on the quasi-periodic timescales.

On longer timescales, we do not find any behaviour that would be discernible from red noise, suggesting that the lower-frequency range of ENSO (timescales longer than ∼ 4.5 years) e.g. is better represented by red noise. Recall, however, that any potential oscillatory behaviour on timescales longer than 30 years was removed via detrending. Similarly, we do not find any quasi-periodic modes of variability on shorter timescales. Also, on these longer (> 700 months) and shorter (< 8 months) timescales we find some mode mixing. This is likely a consequence of inputting several different variables (i.e. SST, τx, thermocline depth) into the MEMD algorithm, where each variable can have different timescales represented. For example, wind stress can be much noisier than SST (or different parts of the tropical Pacific have different variabilities) and can thus lead to identification of several high-frequency modes of variability, resulting in mode mixing (see somewhat overlapping blue dots in Fig. ).

Above we have established that the Niño3 index in the tropical Pacific exhibits two quasi-periodic modes of variability with average periods ∼ 2–3 years (via MEMD analysis). We now analyse the associated time series of the two IMFs (IMF 11 and IMF 12) and compare them with a band-pass-filtered (16–53 months) Niño3 index (Fig. ).

The period/frequency of the two modes is not constant (i.e. varies in time). Thus, we also specify a range of periods/frequencies for the two modes. The mean periods of IMF 11 and 12 with their uncertainty ranges (in square brackets) are: 23 [16, 33] (IMF 11), and 39 [29, 53] (IMF 12) months. These ranges are defined based on the 6.7th and 93.3rd percentiles of IMF 11 and IMF 12's instantaneous period/frequency values, which roughly correspond to ±1.5σ(instantaneous periods). Figure S1 shows power spectra of IMF 11 and IMF 12, which visualise the range of periods the two modes have. Recall that the instantaneous period/frequency can be obtained via Hilbert transform (see Eq.  in Appendix ). This range was chosen as it captures most of the variability in a given mode. Additionally, this range yields a good agreement with other methods (e.g. band-pass filtering), although other (reasonable) percentile ranges give qualitatively similar results. We then use these period/frequency ranges to perform a band-pass filter (via fifth-order Butterworth filter) of the Niño3 index that is consistent with the individual modes (e.g. 29–53-month band of IMF 12 is used to construct band-pass-filtered Niño3 index composites in Fig. b; more below). We also use a band-pass filter of the Niño3 index consistent with the sum of the two modes where the band-pass range encompasses periods of both significant modes, i.e. 16–53 months (e.g. used to construct black solid line in Fig. ).

Time series of the sum of IMFs (black dashed line) and the band-pass index (black solid line) largely agree (i.e. their correlation is 0.95; Fig. ). Also, the individual modes show very good agreement with the band-passed index, with correlations of 0.68 (IMF 11) and 0.83 (IMF 12), which can be increased further if we consider only the specific IMF's timescale range (specified above) when band-pass filtering the Niño3 index. This merely confirms that MEMD results are consistent with other filtering methods.

Additionally, time series of the Niño3 index extracted from IMF 11 and 12 (Fig. ) are largely consistent with the modes identified in (their Fig. 9a), who used MSSA analysis, and with (their Fig. 3), who used EEMD on the Niño3.4 index. IMF 12 is also similar to the 4-year mode (their Fig. 10), which used an operator-theoretic approach. The average periods of the significant modes of variability (IMF 11 and 12) in this study are typically lower than in other studies; however, please recall that we have used much longer time series and that the time series of IMFs are nonstationary. Thus, within the overlapping time periods (e.g. recent decades), the timescale (and corresponding time series) is generally consistent across different studies. These similarities provide further confidence in the results from the MEMD. Note that these studies have focused on different ENSO timescales and associated different spatial patterns during QB and LF/QQ ENSO events. However, IMFs are nonstationary and can thus capture changing patterns over time (Sect. ). Also, previous studies have not necessarily considered relationships between different variables that are relevant for the ENSO dynamics (Sect. ) and related conceptual oscillator models on specific timescales.

Overall, MEMD is consistent with other filtering methods (i.e. it acts like an effective band-pass filter) and can thus be used for further analysis of ENSO dynamics and its nonstationarity to determine if IMFs hold any physical significance. If IMFs have physical meaning, the information inferred from them can be valuable for enhancing climate models, for long-term predictions, for understanding teleconnections, and for exploring the underlying physics and variability of specific fields of interest, such as ENSO. In the following sections we address some of these aspects.

The period ranges provided above (see also Fig. S1) suggest that there is some overlap between the period/frequency bands of IMF 11 and IMF 12, which is a result of nonlinear and nonstationary evolution of the modes (period is not constant as seen in Fig. ). Indeed, there is a low (yet statistically significant) correlation between the two modes (∼ 0.28). Thus, in some time windows the two IMFs can describe the variability of a similar timescale (e.g. similar temporal evolution in Fig.  around years 1923–1927, 1951–1955, and 1997), but in other time windows they describe variability on different timescales. Despite changing periods, the two IMFs pass the red noise threshold in the Niño3 region and are therefore still considered significant and thus quasi-periodic (Sects.  and ).

In some decades the band-pass-filtered Niño3 index (black solid line) is more consistent with the lower-frequency IMF 12 (red dashed line; approximately 1870–1917 and 1968–2000), and in other periods it is more consistent with the higher-frequency IMF 11 (red dotted line; approximately 1917–1968 and 2000–2010) (Fig. ). This is consistent with the interdecadal shifts in the frequency of ENSO that have occurred around years 1970 (from higher frequency to lower frequency) and 2000 (from lower frequency to higher frequency). Similar behaviour can also be seen earlier in the record. We can also see interdecadal changes in the amplitude of the ENSO modes, i.e. middle panel (period 1920–1965) in Fig.  compared with top and bottom panels (periods 1870–1920 and 1965–2010). This is somewhat consistent with , who found reduced amplitude of ENSO during the 1901–1931 and 1935–1965 periods relative to the post-1970 period.

Similarly, IMFs can capture different propagation directions of SST anomalies (Fig. ). Previous studies have highlighted that SST anomaly propagation changed from westward to stationary or eastward around 1970 . Indeed, the two IMFs show westward propagation of SST anomalies prior to 1970. However, after 1970 it is slightly more complex. SST anomaly propagation becomes stationary or eastward in IMF 11 (except for a period just before 1990; Fig. a), but IMF 12 (Fig. b) still indicates either stationary or westward propagation with some rare (eastward) exceptions. This is then reflected in the sum of the two modes (Fig. c), which generally shows variability in propagation of SST anomalies post-1970. Furthermore, some of the characteristics of SST anomaly propagation from the sum of IMF 11 and IMF 12 carry over to 1-year low-passed SST anomalies (e.g. in the 1960s and 1980s; Fig. d).

ENSO events have also been characterised as east Pacific (EP) or central Pacific (CP) depending on the longitude where SST anomalies maximise e.g.. The two IMFs are less able to reproduce this feature of ENSO diversity (Fig. ). Most EP events are captured by IMF 11 and 12. However, some CP events (e.g. early 1990s; Fig. d) are not captured by quasi-periodic modes of variability (Fig. a, b, c). This suggests that other processes are likely causing them, such as sub-ENSO variability . Similarly, some persistent La Niña events (e.g. mid 1970s, mid-1980s, and around year 2000; Fig. d) are not necessarily captured by the quasi-periodic modes of variability alone (potentially related to ENSO asymmetry; e.g. ). These results could also reflect that some aspects of ENSO nonlinearity are not represented by these two IMFs (e.g. ).

Note that the magnitude of ENSO ultimately depends on all underlying modes of variability in the tropical Pacific (not just on the IMFs discussed here) – as also seen in Fig. . In fact, we find ∼ 5 modes (i.e. IMFs 10–14 here) with timescales ranging from ∼ 1 to ∼ 11 years (assessed via the 6.7th and 93.3rd percentiles as above) that can reproduce the majority of the ENSO variability (not shown), but the rest of these modes (i.e. IMF 10, 13, 14 here) are consistent with red noise (Fig. ).

Figures  and also show that weak Niño3 events have either small amplitudes (e.g. 1930–1940) of both IMFs or opposite amplitudes (e.g. 1985–1987). However, strong Niño3 events generally show a constructive interference or mode combination (e.g. 1997, a super-El Niño event), which is consistent with , , , and .

This section has clearly demonstrated that MEMD together with a red noise test is suitable for identifying nonstationary quasi-periodic multivariate signals. This is a clear advantage of MEMD over other multivariate signal processing methods (e.g. PCA, MSSA). Below we now turn to ENSO dynamics to show that IMFs are also physical.

As ENSO dynamics is primarily related to the evolution of the ocean surface zonal wind stress (τx), thermocline depth, and the SST in the tropical Pacific e.g., we demonstrate their relationship with phase composite maps of IMF 12 (Fig. ). Shading in Fig.  represents SST anomalies, contours represent the thermocline depth anomalies (solid contours represent positive values and deeper thermocline), yellow arrows represent τx anomalies, and grey shading represents grid points of the SST that do not pass the red noise threshold. All values are standardised (i.e. divided by standard deviation).

The phase composites are computed using the instantaneous phase of the IMF 12's eastern Pacific SST (Niño3) time series (i.e. IMF 12 (SST (Niño3))) that we can obtain via Hilbert transform (Appendix , Eq. ). This assigns every point in the eastern Pacific SST (Niño3) time series a phase between 0 and 360∘, which can then be split into 12 phases (marked phase 0 through 11; e.g. phase 0 is 0–30∘ and phase 11 is 330–360∘), and all points in the time series (of 1-D or 3-D fields) belonging to a specific phase are then averaged to form map phase composites shown in Fig.  (see also Fig. S3 for IMF 11). Line phase composites in Fig.  (see also Fig. S4 for IMF 11) are constructed similarly. The eastern Pacific SST (Niño3) is chosen as an index here to focus on east Pacific (EP) ENSO events and related dynamics.

Figure  shows a typical cycle of EP ENSO in the tropical Pacific (on a ∼ 3-year timescale) as composited over the IMF 12's Niño3 index. This cycle can also be summarised with line phase composites (Fig. a, c) by averaging over specific regions of the tropical Pacific (as labelled; see also Table ). Here, we analyse the eastern Pacific SST (Niño3), central Pacific τx (Niño4), western Pacific τx (Niño5), off-equatorial western Pacific thermocline depth (Niño6), and Pacific mean thermocline depth as they have historically been considered important for ENSO dynamics and have been used in ENSO conceptual oscillator models e.g..

Together the two figures (Figs. , a, c) suggest the following sequence of events:

during La Niña (phases 5–7) we have negative SST anomalies and a shallower thermocline in the Niño3 region, stronger easterly wind stress in the Niño4 region, and deeper thermocline in the western Pacific (including the Niño6 region);

The evolution described above is also seen in the band-pass-filtered (29–53 months; 2.5–4.5 years) data (Fig. b). Note that the values in Fig. b are slightly larger than in Fig. a, because slightly different frequency ranges are ultimately represented in the two panels, but they remain qualitatively similar.

While Fig.  describes an average evolution of the variables considered, not all individual events have this exact behaviour. This is partly reflected in the reduced amplitudes of τx and thermocline depth in Fig. a compared with the SST amplitude. The full time series of these parameters from IMF 12 (Fig. ; line types/colours are the same as in Fig. ) show that the relationships from Fig.  occur often in the analysed period, especially for strong events. However, for weak events (middle panel in Fig. ) the relationships are harder to establish – every event seems to be slightly different. This is somewhat consistent with , who noted that the dynamics of ENSO was different prior to 1970 relative to after 1970, with a dominant recharge–discharge oscillator in the latter period.

Similar results (phase composites) can also be obtained for the 16–33-month band-passed data and IMF 11 (Figs. S3, S4). This suggests that on average the QB and LF/QQ ENSO events have similar evolution and associated dynamics. However, the frequency of events (see full time series in Fig. S5) that follow the dynamics identified in the phase composites (Fig. S4) is lower in the IMF 11 case than in the IMF 12 case. This may be indicative of other processes that could be relevant for the QB ENSO.

The results presented here and summarised in the line phase composites (Figs. , S4) have some interesting implications for the dynamics of ENSO and conceptual oscillator models. The out-of-phase relationship between the Pacific mean thermocline depth (blue dotted line) and eastern Pacific SST (Niño3; black solid line) is a typical feature of the recharge–discharge oscillator . This suggests that IMFs that emerge from MEMD analysis can capture physical processes in the tropical Pacific. Furthermore, co-variability of the Pacific mean thermocline depth and the western Pacific wind stress (Niño5; grey solid line) suggests that wind forcing in the western Pacific may play a role in the recharge–discharge process or in the onset of ENSO in general as it precedes ENSO events see also.

On the other hand, the co-variability between the central Pacific wind stress (Niño4; grey dashed line), off-equatorial western Pacific thermocline depth (Niño6; black dashed line), and eastern Pacific SST (Niño3; black solid line) suggests that the unified oscillator proposed by may need to be revised see also. This is because (i) in the unified oscillator model these three quantities should all be somewhat out of phase, but here we show that only the western Pacific wind stress and Pacific mean thermocline depth exhibit an out-of-phase relationship with the eastern Pacific SST (on average) and (ii) the Pacific mean thermocline depth does not co-vary with the off-equatorial western Pacific thermocline depth (Niño6), rendering the unified model's recharge–discharge oscillator simplification, which uses the off-equatorial western Pacific thermocline depth (Niño6), questionable.

That the central Pacific wind stress may be omitted in the unified model (due to co-variability with the eastern Pacific SST) was also mentioned in . However, they suggested that western Pacific wind stress (Niño5) could also be omitted from the unified model, but Fig.  suggests that on 1.5–4.5-year timescale this variable should be kept. This may be because used 1-year low-pass-filtered data, which could have obscured the signal on a 1.5–4.5-year timescale (see Sect. ). Additionally, suggested using the thermocline depth in the western Pacific averaged over a region that lies on the Equator rather than off-Equator. From Fig.  we can see that this would likely yield similar results to the off-equatorial western Pacific thermocline depth (Niño6) (to ±1 phase).

ENSO is a phenomenon that occurs on timescales of 2–8 years, and previous work has often used a 1-year low-pass filter to obtain ENSO. Thus, we now test the relationships between the eastern Pacific SST (Niño3), central Pacific τx (Niño4), western Pacific τx (Niño5), western Pacific off-equatorial thermocline depth (Niño6), and Pacific mean thermocline depth on slightly shorter and slightly longer timescales. We do this to test how relationships between different variables change across timescales that are still somewhat within the ENSO range.

Figure  shows phase composites similar to those in Fig. b but for band-passed data over a 12–19-month range (shorter periods; Fig. a) and over a 42–135-month range (longer periods; Fig. b). These timescale bands are consistent with IMF 10 (shorter periods) and IMFs 13 and 14 (longer periods), which are the modes that together with IMF 11 and 12 explain the majority of the variance in the eastern Pacific SST (Niño3).

Interestingly, on these shorter and longer timescales the evolution is different than on quasi-oscillatory timescales (1.5–4.5-year periods of IMF 11 and 12). Namely, the western Pacific wind stress (Niño5) closely follows the western Pacific off-equatorial thermocline depth (Niño6) (Fig. a, b). This suggests a very different role of the western Pacific wind stress on different timescales, which may be relevant for ENSO diversity and asymmetry. However, other variables remain similar across timescales (to ±1 phase). Thus, recharge–discharge processes operate throughout the ENSO timescale range (2–8 years), since the Pacific mean thermocline depth and eastern Pacific SST (Niño3) remain out of phase also on these shorter and longer timescales. Also, the results from these shorter and longer timescales indicate that the western Pacific wind stress (Niño5) may be omitted from conceptual oscillator models (on these timescales) as it is anti-correlated with the eastern Pacific SST (Niño3). This is consistent with , who suggested this revision of the unified oscillator, but using 1-year low-pass-filtered data, which may have obscured the different behaviour on quasi-periodic timescales (Sect. ; Fig. ).

The above analysis shows that it is important to filter the data to correct frequency bands as there may be different behaviour present on different timescales, even within the ENSO range of 2–8 years.