A variety of statistical tools have been used in climate science to gain a better understanding of the climate system's variability on various temporal and spatial scales. However, these tools are mostly linear, stationary, or both. In this study, we use a recently developed nonlinear and nonstationary multivariate time series analysis tool – multivariate empirical mode decomposition (MEMD). MEMD is a powerful tool for objectively identifying (intrinsic) timescales of variability within a given spatio-temporal system without any timescale pre-selection. Additionally, a red noise significance test is developed to robustly extract quasi-periodic modes of variability. We apply these tools to reanalysis and observational data of the tropical Pacific. This reveals a quasi-periodic variability in the tropical Pacific on timescales

The climate system is a highly complex system consisting of variability across many different timescales

The following are typical statistical tools used for exploring the patterns of variability on different temporal and/or spatial scales

Multivariate empirical mode decomposition (MEMD) addresses these drawbacks as it is an analysis tool that is entirely data adaptive and is designed to extract nonlinear and nonstationary signals. MEMD is a generalisation of the empirical mode decomposition

Despite their appeal, MEMD and EMD have hardly been used in climate research. EMD and its 1-D extension ensemble EMD (EEMD;

A major challenge in applying MEMD in climate analysis is that no statistical null hypothesis test for red noise has been developed. When applied to climate data, MEMD can reveal many modes that are consistent with red (or white) noise. In particular, sea surface temperature (SST) exhibits a red spectrum because it represents the integral response of the ocean to stochastic higher-frequency atmospheric (e.g. weather or white noise) variability

In this study, we combine the MEMD method with a red noise test (Sects.

ENSO is a quasi-periodic phenomenon occurring on (interannual) timescales of 2–8 years

The paper is structured as follows. Section

We focus on the intrinsic variability of the tropical Pacific and analyse monthly mean data of three different variables relevant for atmosphere–ocean exchange: sea surface temperature (SST) from the HadISST observational dataset

The MEMD analysis (Sect.

Tropical Pacific regions used for computing time series (see text for details). The rightmost column lists the variables that are averaged over specified regions.

While the SST, thermocline depth, and

Before performing the analysis, we detrend the data and remove their seasonal cycle, which is done the following way

This is done to avoid domination of the seasonal cycle or trend in the statistical analysis below, even though the MEMD can generally extract nonlinear trends by itself. Note that this means that we cannot assess the impact of long-term variability or seasonal cycle on ENSO variability in this study, but the latter may still be present indirectly, as ENSO is phase-locked to the seasonal cycle

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

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

Envelopes are created by interpolating between subsequent maxima (upper envelope, shown as grey dotted line in Fig.

An average envelope is obtained by taking the mean of the upper and lower envelopes (depicted by the red solid line in Fig.

The average envelope is subtracted from the original time series data.

The subtracted data, i.e. the original data minus the average envelope, represent the first mode of variability and typically correspond to the highest-frequency signal in the dataset. However, the average envelope can be further analysed.

Steps i–vi are repeated on the average envelope until only a trend or residual component remains. This occurs when we can no longer find at least two extrema in the dataset, which is a condition that needs to be satisfied by EMD's modes of variability.

Schematic for obtaining the average envelope during the EMD process. The black line shows a simple input signal that is a sum of two sine waves (a low-frequency and a high-frequency wave). Grey dotted and grey dashed lines show the upper and lower envelope, respectively. The red line shows the average envelope, which represents the low frequency of the input signal. If we remove the average envelope from our input data, we obtain the high-frequency signal (e.g. the first mode of EMD).

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

MEMD

The MEMD ultimately extracts timescales common to all input time series (i.e. synchronises signals;

As with all statistical methods, it is important to be aware of the drawbacks associated with the (M)EMD

The code for the MEMD is freely available on GitHub (

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.

As mentioned in Sect.

First, we remove the smooth trend/seasonal cycle

We divide data by their standard deviations (

Since we use more than one variable (i.e. SST, surface wind stress, thermocline depth) in the analysis, we concatenate (denoted with

Then, we reduce the dimensionality by computing spatial patterns (empirical orthogonal functions, EOFs) and their time series (principal components, PCs) via singular value decomposition (SVD).

Finally, we only retain the first 20 PCs that explain the majority of the variance in the field

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. PC

To compute an index, such as the eastern Pacific SST (Niño3), we can average over a latitude–longitude region (Table

Once we have computed the IMFs, we need to test if they are statistically significant. The importance of each IMF can be assessed by computing the explained variance of each IMF relative to the input field (e.g. retaining those IMFs that explain more than 0.1 % variance) or through other significance tests

The red noise test can be performed in different ways (see also Sect.

The red noise significance test for spatio-temporal IMFs (from MEMD) via SST averaged over the Niño3 region (for details see text and Appendices A and B). Each blue dot represents the average squared amplitude (

Here, we first test for potential quasi-periodic variability using SST time series that are relevant for (eastern Pacific) ENSO variability, i.e. the eastern Pacific SST (Niño3) from input data and the corresponding eastern Pacific SST (Niño3) from spatio-temporal IMFs (as described in Sect.

The shape of the red noise fit (red solid line in Fig.

As MEMD in conjunction with a red noise test has not been applied in climate science before, we perform an extensive analysis of the method itself (in addition to an analysis of ENSO dynamics – see below). Then, we compare it to the basic band-pass filtering (fifth-order Butterworth filter) and to Fourier transform analysis to ensure consistency with other methods. Please recall that while other spectral methods often require prior knowledge about the spatial/temporal patterns of interest in order to construct appropriate indices, the MEMD method allows for the objective extraction of significant patterns and modes of variability from data without pre-existing knowledge (see also below and Appendices A and B).

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.

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.

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.

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

Above we have established that the Niño3 index in the tropical Pacific exhibits two quasi-periodic modes of variability with average periods

Time series (1871–2010) of the Niño3 index from IMF 11 (red dotted line) and IMF 12 (red dashed line) obtained via MEMD. The sum of the two IMF indices (black dashed line) and the band-pass-filtered (16–53 months) Niño3 index (black solid line) are also shown. All data are standardised.

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

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.

Additionally, time series of the Niño3 index extracted from IMF 11 and 12 (Fig.

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.

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.

Similarly, IMFs can capture different propagation directions of SST anomalies (Fig.

ENSO events have also been characterised as east Pacific (EP) or central Pacific (CP) depending on the longitude where SST anomalies maximise

Time–longitude Hovmöller diagram of tropical Pacific SST anomalies (in K) averaged between 5

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.

Figures

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.

The dynamics of ENSO typically involves positive (e.g. Bjerknes) and negative feedbacks between the atmosphere and ocean. The Bjerknes feedback

As ENSO dynamics is primarily related to the evolution of the ocean surface zonal wind stress (

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

Latitude–longitude phase composite (phases 0 to 11 as labelled) of IMF 12: shading for SST, contours for thermocline depth (contour interval is the same as in the colour bar with solid contours representing positive values, and dashed contours represent negative values), and arrows for

Figure

Phase composites of the eastern Pacific SST (Niño3) (black solid line), off-equatorial western Pacific isotherm/thermocline depth (Niño6) (black dashed line), Pacific mean isotherm/thermocline depth (blue dotted line), central Pacific

Together the two figures (Figs.

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);

as La Niña weakens (phases 8–10), the westerly wind stress in the Niño5 region and thermocline depth averaged across the tropical Pacific peak, starting the El Niño cycle;

as the SST warms, the eastern Pacific thermocline (Niño3) becomes deeper, the central Pacific wind stress (Niño4) becomes westerly, and the thermocline in the western Pacific (including the Niño6 region) becomes shallower (phases 11,0,1);

El Niño weakens (phases 2–4), the western Pacific

the cycle repeats.

The evolution described above is also seen in the band-pass-filtered (29–53 months; 2.5–4.5 years) data (Fig.

While Fig.

IMF 12's standardised time series of the eastern Pacific SST (Niño3) (black solid line), off-equatorial western Pacific thermocline depth (Niño6) (black dashed line), Pacific mean thermocline depth (blue dotted line), central Pacific

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.

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

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

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

Figure

As in Fig.

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.

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.

In this study we have used a recently developed nonlinear and nonstationary method for identifying intrinsic variability of multivariate systems, the multivariate empirical mode decomposition (MEMD;

Additionally, a red noise significance test has been developed to robustly identify quasi-periodic modes of variability in the given data, which had not been used before in the framework of MEMD. This means that MEMD can now be used as an alternative method for objective identification of the timescale of quasi-periodic motions in the climate system. Since the red noise test can be applied on every grid point separately, MEMD together with the red noise test can also be used for identifying potential new regions of quasi-periodic variability (similar to Fig.

We demonstrate that MEMD can identify physical quasi-periodic modes of variability within the climate system by analysing the tropical Pacific SST variability. We have identified a clear quasi-periodic behaviour on a timescale of about 2–3 years (16–53 months) in the tropical Pacific. This timescale falls within the typical timescale range of ENSO, i.e. 2–8 years, and the dynamics of this quasi-periodic variability is consistent with ENSO dynamics. While ENSO quasi-periodic variability is well known, an identification (via MEMD) of a frequency range linked to the two dominant quasi-periodic modes of variability (i.e. 16–53 months) has still led to a few interesting results.

By analysing composites (e.g. Figs.

On shorter and longer timescales (12–19 months and 42–135 months; Fig.

MEMD analysis could be extended in several ways, for example (i) to assess ENSO dynamics in models as they typically struggle with the representation of the western Pacific processes

Overall, this study has analysed the variability in the tropical Pacific (using MEMD with a red noise test), identified two quasi-periodic modes of variability (on

To find the intrinsic variability of our 3-D field, i.e.

Now we can use the 20 PCs (PC

Each PC is associated with a spatial pattern (EOF

Note that from here on (and in the main text) IMFAs are referred to as IMFs for simplicity.

Typically we can test if the modes (IMFs) are different from white or red noise, depending on the distribution of our data. In the climate system, variables often exhibit behaviour that resembles white or red noise. The IMFs that are significant (i.e. different from both red and white noise) likely represent quasi-oscillations, indicating higher potential for predictability of processes that correspond to the timescale of the significant IMF. Thus, this distinction is very important in climate science. Therefore, we discuss the white and red noise tests (for 1-D data, i.e. time series) below, whereas the robustness of IMFs from the MEMD analysis and the relevant significance tests are discussed in the main text (Sects.

Note that the white and red noise tests are performed on 1-D time series; hence, EMD (univariate decomposition; see main text) is first used to test the performance of IMFs that arise from the EMD analysis. The multivariate data (via MEMD) in the main text are analysed with the simplest and most relevant test (i.e. theoretical red noise test described below).

The white noise significance test has been derived by

Note that the frequency (and thus also period) of each IMF is computed using Hilbert transform by first generating an analytical signal

The relationship between the logarithms of energy density and average period of the IMFs (Eq.

Significance tests for EMD modes:

Alternatively, we can test whether the input data are different from white noise by constructing multiple (

A comparison with the IMFs from the input data (Niño3 index; blue dots in Fig.

To test if our data (e.g. Niño3 index) is purely red noise or it has inherent oscillations that we can identify through the (M)EMD analysis, we generate

Once we obtain the red noise time series

Note that

Alternatively, one can compute a theoretical power spectrum of the red noise (see

This red noise test is typically used in climate science to determine the significance of power spectra peaks in our data (using

Red noise power spectrum (

The significance of the IMFs from the input data is tested using a

Figure

Similarly, we can use the theoretical red noise significance test on IMFs obtained via MEMD, which yields two significant modes of variability in the eastern Pacific SST (Niño3) index (see the main text). Note, however, that we do not necessarily expect exactly the same results from the EMD (Fig.

Alternatively, one can also compute significant modes by computing the red noise test at each grid point and then average the results over all grid points, but we have not used this here. Instead, we use an additional test on map plots in Sect.

Despite some differences between the MEMD and EMD modes of variability (primarily due to different input time series), the two methods agree on the quasi-periodic timescale of 2–3 years in the Niño3 region. This is also consistent with the significant periods inferred from the usual 1-D wavelet transform (not shown) and the power spectrum analysis of the Niño3 index (1-D) obtained via Fourier transform (Fig.

Power spectrum of the Niño3 index. The power spectra are first computed for 500-month-long chunks (overlapped by 250 months) and then averaged over all cases (grey solid line). The black solid line represents a 10-point running mean of the black dotted line (to increase the number of degrees of freedom, which is

EMD and MEMD Python codes are available on GitHub (

SODA2 data can be downloaded from

The supplement related to this article is available online at:

LB performed the analysis, prepared the figures, and wrote the first draft of the manuscript. NEO and NSK provided additional insight and helped improve the manuscript for the final version.

The contact author has declared that none of the authors has any competing interests.

This work reflects the analysis and views of the author Lina Boljka. No reader should interpret this work to present the views of any third party. Assumptions, opinions, views, and estimates constitute the author's judgement as of the date given and are subject to change without notice and without duty to update.Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work was supported by the Trond Mohn Foundation (project BCPU, grant no. BFS2018TMT01) and was performed on NIRD/Sigma2 (project NS9039K). Noel S. Keenlyside and Nour-Eddine Omrani acknowledge support from the Research Council of Norway (grant no. 312017, RACE; grant no. 316618, ROADMAP). We thank three anonymous reviewers and William Roberts for their constructive comments that helped improve the original manuscript. We also thank Lander Crespo for helpful discussions and Ingo Bethke for help with the data.

This research has been supported by the Trond Mohn stiftelse (grant no. BFS2018TMT01) and the Research Council of Norway (grant no. 312017, RACE; grant no. 316618, ROADMAP).

This paper was edited by William Roberts and reviewed by three anonymous referees.