Wavelet coherence is a method that is commonly used in hydrology to extract scale-dependent, nonstationary relationships between time series. However, we show that the method cannot always determine why the time-domain correlation between two time series changes in time. We show that, even for stationary coherence, the time-domain correlation between two time series weakens if at least one of the time series has changing skewness. To overcome this drawback, a nonlinear coherence method is proposed to quantify the cross-correlation between nonlinear modes embedded in the time series. It is shown that nonlinear coherence and auto-bicoherence spectra can provide additional insight into changing time-domain correlations. The new method is applied to the El Niño–Southern Oscillation (ENSO) and all-India rainfall (AIR), which is intricately linked to hydrological processes across the Indian subcontinent. The nonlinear coherence analysis showed that the skewness of AIR is weakly correlated with that of two ENSO time series after the 1970s, indicating that increases in ENSO skewness after the 1970s at least partially contributed to the weakening ENSO–AIR relationship in recent decades. The implication of this result is that the intensity of skewed El Niño events is likely to overestimate India's drought severity, which was the case in the 1997 monsoon season, a time point when the nonlinear wavelet coherence between AIR and ENSO reached its lowest value in the 1871–2016 period. We determined that the association between the weakening ENSO–AIR relationship and ENSO nonlinearity could reflect the contribution of different nonlinear ENSO modes to ENSO diversity.

The South Asian monsoon, which is the dominant precipitation source for the Indian subcontinent, has been a target for seasonal prediction for well over a century (Blanford, 1884). Despite this long heritage of research, skillful prediction remains a challenge, driving extensive and ongoing research on statistically and dynamically based prediction methods (e.g., REFS). It is difficult to overstate the importance of the South Asian monsoon to the well-being of citizens in India. Strong monsoon years have caused catastrophic flooding (Kale, 2012; Sanyal and Lu, 2005) and large landslides (Dortch et al., 2009), while weak monsoons have led to water shortages (Mishra et al., 2016) and crop losses (Parthasarathy et al., 1988; Prasanna, 2014) that resulted in significant food shortages in the past (Fagan, 2009). Thus, while the majority of monsoon forecast studies target the prediction of rainfall totals, the hydrological and agricultural impacts of monsoon variability provide the most pressing motivation for our work.

Much of the research on South Asian monsoon prediction has focused on the relationship between the El Niño–Southern Oscillation (ENSO; Walker and Bliss, 1932) and monsoon strength. During El Niño years, droughts are favored, while rainfall surpluses are favored during La Niña years (Shukla and Paolino, 1983; Kripalani and Kulkarni, 1997). However, there is no one-to-one relationship between ENSO and Indian rainfall. As a result, summer rainfall predictions based on ENSO have proven challenging. For example, the 1997–1998 El Niño event was extremely strong, yet climatological Indian monsoon conditions were observed (Shen and Kimoto, 1999; Slingo and Annamalai, 2000). It is therefore important to understand why certain El Niño events are not accompanied by monsoon failures.

There are a few reasons for the challenges faced when predicting Indian rainfall using ENSO. The first reason is that the relationship between ENSO and India's rainfall is nonstationary. As shown by Torrence and Webster (1999), the relationship between ENSO and India's rainfall cycles between periods of high and low coherence. Kumar et al. (1999) found that the relationship between Indian rainfall and ENSO weakened in the 1970s and hypothesized that a southward shift in Walker circulation anomalies associated with ENSO events and increased Eurasian spring and winter surface temperatures were responsible for the weakening relationship. Other works suggest that the changing ENSO–Indian rainfall relationship is the result of the modulating influence of tropical Atlantic sea surface temperatures (SSTs) and the Atlantic Multi-decadal Oscillation (AMO; Lu et al., 2006; Kucharski et al., 2007, 2009; Chen et al., 2010). In contrast, Kumar et al. (2006) and Fan et al. (2017) argued that the occurrence of different ENSO flavors (Johnson, 2013), such as the eastern Pacific and central Pacific types, could explain the ENSO–Indian rainfall relationship changes. Other investigators adopted another perspective to explain changes in the ENSO–Indian rainfall relationship and concluded that temporal undulations in the ENSO–Indian rainfall relationship are related to statistical undersampling and stochastic fluctuations (Gershunov et al., 2001; van Oldenborgh and Burgers, 2005; Delsole and Shukla, Cash et al., 2017). In a recent analysis, Yun and Timmermann (2018) showed that changes in the ENSO–Indian rainfall relationship are consistent with a stochastically perturbed ENSO signal and argued that changes in the ENSO–Indian monsoon relationship may not be related to external climate-forcing mechanisms.

The second reason for the ENSO-related prediction challenges is that ENSO itself is a nonstationary phenomenon. Using wavelet analysis, Kestin et al. (1998) found that the interannual variability of ENSO from 1930 to 1960 was dominated by a 4- to 7-year periodicity, whereas from 1960 to 1990 the interannual variability was also dominated by a 2- to 5-year periodicity. A wavelet power spectral analysis conducted by Torrence and Webster (1999) and Schulte (2016a) showed that ENSO signal energy in the 2- to 7-year band undulates throughout the historical period and increases after the 1960s (Schulte, 2016a). These changes in spectral characteristics are relevant to Indian monsoon prediction because differing spectral characteristics among predictors (e.g., ENSO) and predictands (e.g., Indian rainfall) can negatively impact the predictive skill of statistical models (Jiang et al., 2020). Using a wavelet-based variance transformation method, Jiang et al. (2020) demonstrated that accounting for differences in spectral characteristics can improve prognostic skill. That study suggests that Indian monsoon prediction could be improved by using wavelet-based methods instead of time-domain correlation and regression methods.

The nonlinear characteristics (e.g., skewness) of ENSO are also nonstationary and undergo interdecadal changes (Wu and Hsieh, 2003). Numerous studies have reported an ENSO regime shift in the 1970s in which ENSO began to evolve more nonlinearly than in previous decades (An, 2004, 2009; An and Jin, 2004). It is a curious fact that the ENSO regime shift in the 1970s coincided with the weakening ENSO–Indian rainfall relationship, as documented by Kumar et al. (1999). This observation creates a question that begs to be answered, namely whether nonlinear ENSO regime changes are related to changes in the ENSO–Indian rainfall relationship.

Various mechanisms have been proposed to explain the cause of ENSO skewness. Kang and Kug (2002) suggested that the asymmetry between the magnitude of El Niño and La Niña events is related to the relative westward displacement of zonal wind stress anomalies during La Niña events compared to El Niño events. Jin et al. (2003) and An and Jin (2004) found that ENSO asymmetry is related to nonlinear dynamical heating (NDH), where the magnitude of NDH is related to the propagation characteristics of ENSO. As shown by An and Jin (2004), NDH during strong El Niño events like the 1982–1983 and 1997–1998 events tends to be stronger than during weak El Niño events because SST anomalies tend to propagate eastward. Since the late 1970s there has been a propensity for eastward propagation characteristics of ENSO (Santoso et al., 2013), contrasting with the time period before the 1970s that consisted of the relatively weak El Niño events of 1957–1958 and 1972–1973 (An and Jin, 2004; An, 2009). More recently, Su et al. (2010) showed that vertical temperature advection may have an opposing effect on ENSO asymmetry, and that the asymmetry in the extreme eastern equatorial Pacific is related to meridional ocean temperature advection.

Previous investigators have used different metrics to quantify ENSO asymmetry. To measure the nonlinear character of ENSO, An and Jin (2004) used time-domain metrics, such as skewness and maximum potential intensity (MPI), to quantify the skewness of SST anomalies and the skewness of individual ENSO events, respectively. An (2004) applied a principal component analysis (PCA) to a 21-year moving window of tropical Pacific SST skewness and found that the first PCA mode is characterized by positive skewness across the eastern equatorial Pacific and negative skewness across the central equatorial Pacific. This pattern means that interdecadal changes in the nonlinearity of ENSO are associated with positively skewed SST anomalies across the eastern equatorial Pacific, implying that El Niño events are stronger than La Niña events. While the methods implemented in the aforementioned studies provided important insights, they cannot reveal the frequency modes of ENSO that are contributing to the skewness.

Recognizing the limitations of time-domain approaches, Timmermann (2003)
conducted a bispectral analysis of the Niño 3 anomaly time series,
where a peak (

In this study, the deficiencies associated with the abovementioned techniques are addressed using higher-order wavelet analysis, which allows for the quantification of frequency-dependent and nonstationary nonlinearities in time series (Van Milligen et al., 1995; Elsayed, 2006; Schulte, 2016b). More specifically, the objectives of the paper are the following: (1) to quantify the nonlinearity of ENSO using higher-order wavelet analysis together with recently developed statistical tests, (2) to determine if different nonlinear modes of ENSO are associated with distinct SST patterns, and (3) to develop nonlinear wavelet coherence methods to test the hypothesis that the breakdown of the ENSO–Indian rainfall relationship in recent decades is related to the shift of ENSO from a linear regime to a nonlinear one. The paper is organized as follows: in Sect. 2, the data used are described. Section 3 includes the description of the implemented methodologies. Results are presented in Sect. 4, and concluding remarks are provided in Sect. 5.

The variability in India rainfall from 1871 to 2016 was analyzed using the all-India rainfall (AIR; Parthasarathy et al., 1994) time series, which was created by averaging representative rain gauges at various locations across India. The full monsoon season (June–September) and the late monsoon (August–September) season were used to identify possible within-season variations in the ENSO–all-India rainfall (ENSO–AIR) relationships. To remove the influence of the annual cycle, AIR time series were converted into anomaly time series by subtracting the 1871–2016 long-term mean for each month from the individual monthly values. The AIR anomaly (hereafter AIR) time series were subsequently standardized by dividing them by their 1871–2016 standard deviations. Because wavelet analysis focuses on specific frequency components that are not impacted by long-term time-domain trends, no detrending of the data was performed.

The monthly data for the Niño 1 and 2, Niño 3, Niño 3.4, and
Niño 4 indices (available at:

The monthly SST data from 1871 to 2016 were based on the Hadley Centre Global Sea Ice and Sea Surface Temperature (HadISST1; Rayner et al., 2003). The data at each grid point were converted to monthly anomalies in the same way as they were computed for the ENSO and AIR time series.

To better diagnose the changes in time series statistics associated with AIR and ENSO, we adopted a wavelet analysis. For a time series,

Linear wavelet coherence (Table 1) was used to quantify the linear
relationship between two time series as a function of frequency and time.
The linear wavelet coherence between two time series

Wavelet quantities and the relationships they measure.

In the context of the Indian monsoon, strong coherence between rainfall and
a climate pattern (e.g., ENSO) at a scale

Although the wavelet power spectrum is useful for quantifying the signal
energy at a scale

In this paper, quadratic nonlinearities giving rise to time series skewness
were quantified using local and global wavelet-based auto-bicoherence
methods (Schulte, 2016b). Global auto-bicoherence (Table 1) was computed
as follows:

To determine if the strength of the quadratic-phase coupling was a function
of time, the local auto-bicoherence spectrum (Schulte, 2016b) given by the following:

To be consistent with the wavelet power and coherence analyses, results for
the higher-order wavelet analysis were casted in terms of the Fourier period
rather than wavelet scale. The Fourier period corresponding to

The statistical significance of all wavelet spectra was evaluated using the
cumulative area-wise test (Schulte, 2016a, 2019a) to account for the
simultaneous testing of multiple hypotheses (Maraun and Kurths, 2004; Maraun
et al., 2007). This test evaluated the statistical significance of points in
the wavelet domain based on the area of contiguous regions of point-wise
significance (i.e., patches) to which they belong, so a larger area implies a greater statistical significance. Given that the patch area can change as the point-wise significance changes, the cumulative area-wise test was used to evaluate significance based on the patch area averaged across a set of
point-wise significance levels (Schulte, 2019a). The test was applied at the
5 % cumulative area-wise significance level, using point-wise significance levels ranging from 0.02 to 0.18, because this choice of point-wise significance levels was shown to result in the cumulative area-wise test outperforming the point-wise test in terms of true positive detection for moderate to high signal-to-noise ratios, even though the cumulative area-wise test is more stringent. The test was performed using the Advanced Biwavelet Wavelet R software package (available at:

To assess the statistical significance of the global auto-bicoherence
estimates, a modified version of the cumulative area-wise test was applied.
In the modified version of the cumulative area-wise test, the normalized
area of patches was computed by dividing the patch area by the product

Although wavelet coherence spectra can provide information regarding how the
relationship between two climate variables changes at a scale

Recognizing the skewness associated with two time series was correlated in frequency space using the following:

Higher-order wavelet analysis can also be interpreted in terms of linear and
nonlinear modes. A linear mode

To better understand nonlinear coherence, we supposed that, in the following:

The relative biphase difference

In this paper, we focused on nonlinear coherence computed along the diagonal
slices (

To demonstrate the concept of nonlinear coherence, we considered a simple example in which the nonlinear climate forcing time series was given by the following:

As shown in Fig. 1a,

An inspection of the local auto-bicoherence spectrum of

The lack of nonlinear coherence at timescales for which

The weakening relationship shown in Fig. 1b could lead a researcher
studying a hydrological process to believe that another direct-forcing
mechanism must be influencing the hydrological process. This belief could
lead to the application of partial wavelet coherence (Ng and Chan, 2012)
and partial correlation analyses to identify another influential forcing
mechanism. However, in this case, there are no other direct-forcing
mechanisms; the weakening time-domain relationship is solely related to how

The contrasting Niño

The time series of the

The 20-year sliding skewness time series of the Niño

The 20-year sliding skewness of

Interestingly, a 20-year sliding skewness analysis of AIR (Fig. 4) reveals
that the skewness of June–September AIR remains close to zero until the
1990s, despite the upward trend in Niño

The 20-year sliding correlation between anomalies for June–September AIR, and the time series for the June–September Niño 1 and 2 and Niño 4 indices.

The differences in skewness shown in Fig. 4 suggests that the correlation
between the ENSO time series and AIR degrades after the 1970s, which is
confirmed by the 20-year sliding correlation between the June–September AIR and ENSO time series (Fig. 5a). The relationship with the Niño

The stronger AIR–Niño 4 index relationship compared to the AIR–Niño

The wavelet power spectra associated with the Niño

Wavelet power spectrum of the

The linear wavelet coherence spectrum, shown in Fig. 7, indicates that the
AIR relationship with the Niño

Wavelet coherence spectrum between AIR and time series for the

Figure 8 shows that the local auto-bicoherence spectra of all ENSO time
series contain statistically significant local auto-bicoherence, but the
spectrum of the Niño 4 index is only associated with a few statistically
significant regions, such as the one around 2015 at a period of 32 months.
For the Niño

Local auto-bicoherence spectra of the

To confirm that the nonlinear-phase coupling identified in Fig. 8 is
associated with skewed waveforms, we inspected the corresponding local
biphase spectra (not shown). It was found that the biphase in the 42 to
64 month band is generally 0

The results shown in Fig. 9 indicate that the nonlinear wavelet coherence
between AIR and the time series for the Niño

Nonlinear wavelet coherence between the full AIR time series and
full times series for the

The 20-year sliding mean of the ENSO auto-bicoherence, coherence, and
nonlinear coherence averaged in the 32 to 64 month band further
highlights the impact of ENSO nonlinearity. As shown in Fig. 10a, the
sliding mean nonlinear coherence between the Niño

To better understand the association between ENSO nonlinearity and the
AIR–ENSO relationship, the global auto-bicoherence spectra associated with
the ENSO time series were first computed (Fig. 11). Then, the
auto-bicoherence of SSTs associated with a few select peaks (

Global auto-bicoherence spectra of the

The spatial structure of global auto-bicoherence corresponding to the peaks in the Niño 3.4 auto-bicoherence spectrum are shown in Fig. 12. The auto-bicoherence associated with the pair (31, 31) is greatest across the central equatorial Pacific, with the overall spatial pattern being reminiscent of a central Pacific El Niño (Lee and McPhaden, 2010). This result suggests that the phase coupling between the 31 month mode and the 15.5 month mode could be related to the occurrence of central Pacific El Niño events (Sect. 5). In contrast, the auto-bicoherence pattern associated with the pair (56, 56) is more uniform, with auto-bicoherence slightly greater across the extreme eastern equatorial Pacific than the central equatorial Pacific. This pattern is reminiscent of an eastern Pacific El Niño. Like the pattern corresponding to the pair (31, 31), the auto-bicoherence for the pair (105, 57) tends to be greater across the central equatorial Pacific. Our findings suggest that different nonlinear modes contribute to different ENSO flavors. Although An and Jin (2004) and Burgers and Stephenson (1999) showed that the skewness is greatest across the eastern equatorial Pacific, we determined that such a time-domain approach is unable to capture frequency-dependent patterns in nonlinearity.

The spatial auto-bicoherence plots associated with the peaks in the Niño

The nonlinear nature of ENSO was examined using higher-order wavelet
methods. The auto-bicoherence spectra of the Niño

The evolution of SSTs across the Niño 4, Niño 3.4, Niño 3, and
Niño

The results from the present and previous studies (Fan et al., 2017) support
the idea that changes in the ENSO–AIR relationship are related to ENSO
flavors because ENSO nonlinearity appears to be related to ENSO flavors
(Figs. 12 and 13), which is in opposition to the findings of other work showing that the changes are related to sampling variability or to noise. According to Yun and Timmermann (2018), the changes in the time-domain correlation between AIR and ENSO are consistent with the assumption that AIR is the sum of the ENSO signal and Gaussian white noise (i.e., AIR

The fact that nonlinear coherence between rainfall and ENSO is determined by linear coherence between ENSO and rainfall at two or three frequencies means that the changing time-domain correlation could be more fully understood by determining why linear coherence changes at the frequencies that contribute to ENSO skewness. Such an analysis could provide a more mechanistic perspective than the theoretical perspective adopted in this study. A preliminary analysis showed that there was enhanced linear coherence between the North Atlantic Oscillation index and AIR after 1995 in the 16 to 64 month band associated with ENSO nonlinearity. This result suggests that conditions across the North Atlantic (Kakade and Dugam, 2000; Bhatla et al., 2016) could influence the nonlinear coherence between ENSO and AIR and, thus, the corresponding time-domain correlation.

The tools used and developed in this study may have important applications
for understanding how forecasting systems replicate Indian rainfall and its
associated teleconnections. These methods, for example, could determine if
forecasting systems can reproduce nonlinear characteristics of climate time
series. As such, an R software package has been developed to implement these
methods (available at:

Data for the Indian rainfall can be accessed through

The supplement related to this article is available online at:

JS conceived the study design and performed the experiments. BZ and FP provided feedback about the experiments and helped with writing the paper.

The authors declare that they have no conflict of interest.

We thankful for the helpful suggestions provided by James Doss-Gollin and the anonymous reviewer.

This research has been supported by the National Aeronautics and Space Administration (grant no. NNX26AN38G.).

This paper was edited by Pierre Gentine and reviewed by James Doss-Gollin and one anonymous referee.