Revealing Sea-Level Dynamics Driven by El Niño–Southern Oscillation: A Hybrid Local Mean Decomposition–Wavelet Framework for Multi-Scale Analysis

Abstract

Analysis of global mean sea-level (GMSL) variations provides insights into their spatial and temporal characteristics. To analyze the sea-level cycle and its correlation with the El Niño–Southern Oscillation (ENSO, represented by the Oceanic Niño Index), this study proposes an enhanced analytical framework integrating Local Mean Decomposition with an improved wavelet thresholding technique and wavelet transform. The GMSL time series (January 1993 to July 2020) underwent multi-scale decomposition and noise reduction using Local Mean Decomposition combined with improved wavelet thresholding. Subsequently, the Morlet continuous wavelet transform was applied to analyze the signal characteristics of both GMSL and the Oceanic Niño Index. Finally, cross-wavelet transform and wavelet coherence analyses were employed to investigate their correlation and phase relationships. Key findings include the following: (1) Persistent intra-annual variability (8–16-month cycles) dominates the GMSL signal, superimposed by interannual fluctuations (4–8-month cycles) related to climatic and seasonal forcing. (2) Phase analysis reveals that GMSL generally leads the Oceanic Niño Index during El Niño events but lags during La Niña events. (3) Strong El Niño episodes (May 1997 to May 1998 and October 2014 to April 2016) resulted in substantial net GMSL increases (+7 mm and +6 mm) and significant peak anomalies (+8 mm and +10 mm). (4) Pronounced negative peak anomalies occur during La Niña events, though prolonged events are often masked by the long-term sea-level rise trend, whereas shorter events exhibit clearly discernible and rapid GMSL decline. The results demonstrate that the proposed framework effectively elucidates the multi-scale coupling between ENSO and sea-level variations, underscoring its value for refining the understanding and prediction of climate-driven sea-level changes.

1. Introduction

Global mean sea-level (GMSL) change serves as a critical indicator of climate change, with significant implications for marine science, ecosystems, and socio-economic systems. Recent observations indicate an accelerated rate of GMSL rise. From 2005 to 2015, the acceleration reached 0.27 ± 0.17 mm/yr², tripling the rate observed between 1993 and 2014 [1]. Approximately two-thirds of this rise is attributed to seawater mass increase, while the remaining third stems from thermosteric expansion [2]. Furthermore, extensive research has established the influence of the El Niño–Southern Oscillation (ENSO) on interannual sea-level variability. ENSO alters sea temperatures in the equatorial Pacific and modulates global precipitation–evaporation patterns and ocean heat content [3]. Consequently, developing accurate methods to analyze GMSL sequences and their relationship with ENSO holds substantial scientific and practical value.

Traditional analytical approaches for global mean sea-level (GMSL) variation time series can be classified into three categories: 1. spectral analysis methods, 2. empirical orthogonal function-based methods, and 3. other techniques. Spectral analysis methods, such as Fast Fourier Transform (FFT) [4], wavelet analysis [5], Empirical Mode Decomposition (EMD) [6], and Variational Mode Decomposition (VMD) [7], are widely used. Although FFT efficiently processes large-scale datasets, it struggles with non-stationary and long-term signals [8]. Wavelet analysis effectively separates multi-frequency components and suppresses noise [9]; however, its performance heavily depends on the selection of basic functions and parameter tuning. EMD excels in extracting multi-scale information and denoising, but is susceptible to mode mixing [10]. In contrast, VMD mitigates mode mixing and end effects through predefined parameters, yet its accuracy is highly sensitive to initial parameterization [11]. In summary, spectral methods can track the evolution of frequency components over time, which is crucial for studying non-stationary processes like climate change. However, selecting an appropriate method remains essential. Empirical orthogonal function-based methods include the empirical orthogonal function (EOF) [12], Principal Component Analysis (PCA) [13], Singular Spectrum Analysis or Multivariate Singular Spectrum Analysis (SSA or MSSA) [14], and Independent Component Analysis (ICA) [15]. Among these, EOF can effectively reconstruct variability but may underestimate trends [16]. PCA efficiently identifies and extracts major modes of variation in data [17]; however, it faces challenges such as computational burden and reduced statistical accuracy when handling high-dimensional data [18]. SSA or MSSA decomposes the original time series into several independent components for denoising and prediction, but a major limitation is its sensitivity to outliers, which can lead to biased results and erroneous conclusions [19]. ICA is used to separate independent source signals from mixed signals, yet the quality of separation is highly affected by noise. Overall, empirical orthogonal methods are powerful tools for analyzing the spatial heterogeneity of GMSL and identifying dominant spatial patterns of sea-level change. However, they struggle to distinguish signals from different physical processes that share similar spatiotemporal scales and are less efficient in extracting globally uniform signals, such as global thermosteric expansion. Other methods primarily include machine learning-based techniques, such as Hybrid Deep Learning Models [20] and Autoregressive Machine Learning methods [21]. These methods offer strong nonlinear fitting capabilities and noise resistance but often lack interpretability regarding the underlying physical mechanisms. Additionally, they require large amounts of data for training, and their performance depends heavily on the distribution of the training data.

Considering the trade-offs among the aforementioned three categories of methods, this study employs Local Mean Decomposition (LMD) to perform multi-scale decomposition of the sea-level time series. As an adaptive time–frequency analysis method developed from EMD, LMD effectively resolves amplitude-modulated multicomponent signals. LMD adaptively smooths local extrema to reduce mode mixing while resolving amplitude–frequency-modulated signals [22]. However, its susceptibility to noise contamination may compromise decomposition accuracy. To mitigate this, we propose an improved wavelet thresholding technique that hybridizes soft and hard threshold functions. This novel approach selectively eliminates high-frequency noise components within LMD-derived modes. Subsequently, both detrended GMSL and the Oceanic Niño Index (ONI) undergo continuous wavelet transform (CWT) using the Morlet basis [23]. The resultant CWT scalograms characterize time–frequency properties of GMSL variability and ENSO dynamics. Critical phase-coherent interactions are then quantified via cross-wavelet Transform (XWT) [24], which generates two diagnostic metrics: (1) cross-wavelet power spectra identifying shared high-energy periods and (2) wavelet coherence spectra evaluating localized correlation strength. This framework reveals asymmetric GMSL-ENSO coupling: El Niño phases induce sea-level rises, with strong events generating pronounced peaks (>3 mm) amplified by larger shared periodicities. Conversely, short-duration La Niña events trigger rapid sea-level declines yet generate only moderate negative anomalies. During prolonged La Niña conditions, however, these declines are substantially attenuated against a background of persistent secular acceleration (3.204 ± 0.326 mm/yr), resulting in negligible net sea-level reduction or even localized rises. Extreme weather anomalies (e.g., droughts, cold surges) further contribute to episodic reductions.

2. Adopted Datasets and Research Methodology

2.1. Adopted Datasets

2.1.1. GMSL Data

The global mean sea-level (GMSL) data used in this study were obtained from the reconstructed dataset developed by Church and White [25] (available at https://www.cmar.csiro.au/sealevel/sl_data_cmar.html; accessed on 20 April 2024). This product synergistically combines monthly averaged tide gauge records from the Permanent Service for Mean Sea Level (PSMSL) database with empirical orthogonal functions (EOFs) derived from a 12-year TOPEX/Poseidon and Jason-1 satellite altimetry record. This methodology yields a globally reconstructed sea-level time series of high credibility. It is widely regarded as an essential climate variable for tracking long-term sea-level trends and assessing the impacts of climate change on oceanic systems [26].

The data used in this study cover the period from January 1993 to July 2020 and span the latitudinal range of 65° S to 65° N, with a monthly temporal resolution. As the data retain non-inverse barometer-corrected signals and seasonal components, some noise may be introduced. To address this issue, a dynamic thresholding algorithm was applied for outlier detection, using a 12-month sliding window and a threshold multiplier of 2 standard deviations. Outliers were identified through point-by-point computation of the local mean and standard deviation within each window. Representative results of the outlier detection are illustrated in Figure 1.

As shown in Figure 1, noise is present in the GMSL time series signal. Therefore, it is necessary to remove the noise.

2.1.2. ONI Data

The Oceanic Niño Index (ONI) data were obtained from the Climate Prediction Center (CPC) of the National Centers for Environmental Prediction (NCEP)/National Oceanic and Atmospheric Administration (NOAA) (available at https://www.cpc.ncep.noaa.gov, accessed on 20 April 2024). As a primary indicator for monitoring the El Niño–Southern Oscillation (ENSO) phenomenon [26], the ONI is defined as the three-month running mean of sea surface temperature (SST) anomalies in the Niño 3.4 region (5° N–5° S, 170° W–120° W). An El Niño or La Niña event is identified when the ONI meets or exceeds +0.5 °C or falls below −0.5 °C, respectively, for at least five consecutive overlapping seasons [27]. The intensity of these events is classified as weak (ONI between ±0.5 °C and ±0.9 °C), moderate (ONI between ±1.0 °C and ±1.4 °C), or strong (ONI ≥ ±1.5 °C). Periods with ONI values between −0.5 °C and +0.5 °C are considered ENSO-neutral. By quantifying SST anomalies in this key region, the ONI provides a robust measure of the oceanic component of the coupled ocean–atmosphere ENSO phenomenon [28].

2.2. Methodology

Figure 2 outlines the core workflow of this study. Red annotations indicate the principal methodological procedures.

Global mean sea-level data are inherently nonlinear and non-stationary, incorporating components spanning multiple time scales. Local Mean Decomposition (LMD) leverages its adaptive decomposition capability to isolate these multi-scale components. This isolation facilitates the analysis of key drivers of sea-level change, including climatic forcing, ocean circulation dynamics, and anthropogenic influences. Fundamentally, LMD decomposes a complex signal into a finite set of Product Functions (PFs). Each PF is a mono-component amplitude-modulated and frequency-modulated (AM-FM) signal, allowing direct extraction of physically significant instantaneous amplitude and frequency characteristics. The LMD procedure comprises the following steps:

For the time series, all local extreme points $n$ of the original signal are identified first. The mean points and envelope estimates of adjacent local extreme points are calculated as follows:

Step 1. Identify all local extrema $n_i$ of the signal. Compute the local means ${m_i}_{\mathrm{i}}$ between consecutive extrema using Equation (1), and envelope estimates $a_i$ using Equation (2):

$$\begin{array}{c}{m}_i={\displaystyle \frac{{|n}_{i+}{n}_{i+1}|}{2}}\end{array}$$

(1)

$$\begin{array}{c}{a}_i={\displaystyle \frac{n_i-n_{i+1 }}{2}}\end{array}$$

(2)

Step 2. Adjacent $m_i$ and $a_i$ are connected with broken lines, respectively. A moving average is then applied as per Equation (3) to obtain the local mean function $m_{11}$ and the envelope estimate function $a_{11}$.

$$\begin{array}{c}{Y}_s\left(i\right)={\displaystyle \frac{1}{2R+1}}(Y\left(i+R\right)+Y\left(i+R-1\right)+\cdots Y\left(i-R\right))\end{array}$$

(3)

Step 3. Utilizing $m_{11}$ and $a_{11}$ obtained from Step (3), separate the local mean function from the original signal $x\left(t\right)$ to yield $h_{11}$ (Equation (4)), and then demodulate it to obtain the frequency-modulated (FM) signal $s_{11}$ (Equation (5)):

$$\begin{array}{c}{h}_{11}\left(t\right)=x\left(t\right)-m_{11}\left(t\right)\end{array}$$

(4)

$$\begin{array}{c}{s}_{11}\left(t\right)={\displaystyle \frac{h_{11}\left(t\right)}{a_{11}\left(t\right)}}\end{array}$$

(5)

Step 4. Let the envelope estimation function of $s_{11}\left(t\right)$ be denoted as $a_{12}\left(t\right)$. If $a_{11}\left(t\right)$ is not identically equal to 1, then $s_{11}\left(t\right)$ is not a pure FM signal. In this case, repeat step 1 to step 3 above until the condition $a_{1\mathrm{n}}\left(t\right)$ ≈ 1 is satisfied. The envelope signal (instantaneous amplitude function) $a_{1\mathrm{q}}\left(t\right)$ is then calculated using Equation (6). The first Product Function component ${PF}_1$ of the original signal is obtained using Equation (7):

$$\begin{array}{c}{\displaystyle \prod _{q=1}^n}{a}_{1q}\left(t\right)\end{array}$$

(6)

$$\begin{array}{c}{PF}_1=a_1\left(t\right)S_{1n}\left(t\right)\end{array}$$

(7)

Step 5. Separate ${PF}_1$ from $x\left(t\right)$ to yield a new signal $u_1\left(\mathrm{t}\right)$. Treat $u_1\left(\mathrm{t}\right)$ as the new original signal and repeat step 1 to step 4. This process is iterated $k$ times until $u_k\left(\mathrm{t}\right)$ becomes a monotonic function.

$$\begin{array}{c}x\left(t\right)={\displaystyle \sum _{p=1}^k}{PF}_p+u(t)\end{array}$$

(8)

In the LMD results, ${PF}_1$ to ${PF}_k$ represent the components ordered from the highest to lowest frequency. The term $u\left(t\right)$ represents the residual component signifying the GMSL change trend. The envelope signal $a\left(t\right)$ corresponds to the instantaneous amplitude of its corresponding $PF$ component, while the pure FM signal $s\left(t\right)$ represents the instantaneous frequency of the $PF$ component. The algorithm terminates when the residual component $u\left(t\right)$ becomes monotonic. Ultimately, the original signal is decomposed into a set of $PF$ components, ${PF}_1$ to ${PF}_k$, and a residual component $u\left(t\right)$.

After decomposing the GMSL time series using the LMD method, the noise typically resides in the first few $PF$ components. To prevent signal loss caused by filtering artifacts, a secondary signal–noise separation is applied to noise-dominant components. Wavelet thresholding is a widely adopted denoising technique [29]. By setting appropriate thresholds, it effectively suppresses noise while preserving essential signal features. Conventional soft and hard thresholding functions suffer from limitations such as discontinuity and pseudo-oscillations. To address these issues, we propose a hybrid thresholding function that integrates both schemes, which is applied to the high-frequency $PF$ components from LMD for secondary denoising. The improved thresholding function is illustrated in Figure 3.

Prior to implementing the improved wavelet threshold denoising, a continuous mean square error (CMSE) criterion is employed to determine the demarcation point between high-frequency and low-frequency components [30]. This enables targeted selection of high-frequency components for secondary denoising using the proposed wavelet threshold method. The CMSE criterion, mathematically defined in Equation (9), quantifies the optimal separation threshold between high- and low-frequency components.

$$\begin{array}{c}{CMSE(x}_p,x_{p+1})={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^n}{\left[x_p\right(t_i)-x_{p+1}(t_i\left)\right]}^2={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^n}{\left[{PF}_p\right(t_i\left)\right]}^2\end{array}$$

(9)

where $x_p$ and $x_{p+1}$ denote two adjacent components, ${PF}_p\left(t_i\right)$ represents the difference between consecutive components, $n$ is the total number of $PF$ components, and $N$ indicates the length of the signal.

The high-frequency components selected by Equation (9) undergo wavelet threshold denoising. The conventional hard- and soft-thresholding functions are, respectively, defined in Equations (10) and (11).

$$\begin{array}{c}{\hat{w}}_{j,k}=\left\{\begin{array}{c}{w}_{j,k} \left|w_{j,k}\right|\ge \sigma \\ 0 \left|w_{j,k}\right|<\sigma \end{array}\right.\end{array}$$

(10)

$$\begin{array}{c}{\hat{w}}_{j,k}=\left\{\begin{array}{c}{sgn(w}_{j,k})(w_{j,k}-\sigma ) \left|w_{j,k}\right|\ge \sigma \\ 0 \left|w_{j,k}\right|<\sigma \end{array}\right. \end{array}$$

(11)

To overcome limitations of these methods, we propose an improved thresholding function (Equation (12)) by hybridizing both approaches.

$$\begin{array}{c}{\hat{w}}_{j,k}=\left\{\begin{array}{c}{\displaystyle \frac{w_{j,k}+sign(w_{j,k})\sqrt{{w_{j,k}}^2-{{\sigma }_j}^2}}{2}} \left|w_{j,k}\right|\ge {\sigma }_j\\ \begin{array}{c}sign(w_{j,k}){\displaystyle \frac{{w_{j,k}}^2}{2{\sigma }_j}} \left|w_{j,k}\right|<{\sigma }_j\\ \end{array}\end{array}\right. \end{array}$$

(12)

The high-frequency $PF$ components ${PF}_1$ to ${PF}_n$ identified in Equation (9) are denoised using the improved thresholding function (Equation (11)). Subsequently; the denoised components ${PF}_1^{\prime }$ to ${PF}_n^{\prime }$; the retained low-frequency $PF$ components ${PF}_{n+1}$ to ${PF}_k$ selected by the CMSE criterion; and the residual term $u_k\left(\mathrm{t}\right)$ are reconstructed through Equation (8) to obtain the denoised signal $x\left(t\right)$. To evaluate the denoising performance of the proposed improved threshold and various established thresholding methods (e.g., SURE threshold; Minimax threshold); we utilized the Signal-to-Noise Ratio (SNR) and root mean square error (RMSE) [31] as quantitative performance metrics; with their mathematical expressions provided in Equations (13) and (14), respectively

$$\begin{array}{c}SNR=10\times ln({\displaystyle \frac{\sum _{i=1}^N{x\left(i\right)}^2}{\sum _{i=1}^N{\left(x\right(i)-\hat{x}\left(i\right))}^2}})\end{array}$$

(13)

$$\begin{array}{c}RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}\left(\right|\left(x\right(i)-\hat{x}\left(i\right)|)}\end{array}$$

(14)

where $x\left(i\right)$ represents the original signal, and $\hat{x}\left(i\right)$ represents the denoised signal.

Following denoising with each thresholding function, the SNR and RMSE of the processed signals were computed. The optimal denoising result is indicated by the simultaneous occurrence of the highest SNR value and the lowest RMSE value.

To conduct multi-scale analysis between the LMD-derived global sea-level change series and ENSO signals, cross-wavelet transform is employed to examine their synchronous phase relationships and correlation properties. Cross-wavelet transform requires continuous wavelet coefficients of two input signals. First, the continuous wavelet transform (CWT) is applied to signals $m\left(t\right)$ and $n\left(t\right)$. The CWT of $m\left(t\right)$ is defined as

$$\begin{array}{c} w_m(a,b)={\displaystyle \frac{1}{\sqrt{\left|a\right|}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}m(t){\phi }^{\mathrm{*}}({\displaystyle \frac{a-b}{a}})dt \end{array}$$

(15)

where $a$ is the scale parameter ($a\ne 0$) controlling wavelet dilation, $b$ is the translation parameter governing wavelet shifting, $\phi$ denotes the mother wavelet, and ${\phi }^{\mathrm{*}}$ represents its complex conjugate.

The selection of a mother wavelet is critical for continuous wavelet transform (CWT) analysis. While the Bump wavelet offers high spectral concentration and excellent frequency resolution, it suffers from poor temporal resolution, making it difficult to accurately identify the onset and termination of ENSO events. The Morse wavelet, as a parameterized family, provides considerable flexibility but requires careful tuning of its symmetry and time-bandwidth parameters for optimal performance. In contrast, the Morlet wavelet achieves a balance between time and frequency resolution. Its phase-supporting properties and similarity to a modulated sinusoid ensure interpretable and comparable results. Therefore, the Morlet wavelet was chosen as the mother wavelet [32] for CWT, with its explicit form given by

$$\begin{array}{c} w_m(a,b)={\displaystyle \frac{1}{\sqrt{\left|a\right|}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}m(t)({\pi }^{-\frac{1}{4}}{e}^{-i{w}_0{\displaystyle \frac{t-b}{2}}}{e}^{{\displaystyle \frac{{-\left(t-b\right)}^2}{2a^2}}})\mathrm{*}dt \end{array}$$

(16)

where $w_0$ is the central frequency, typically set to 5.0 or 6.0 to satisfy the admissibility condition.

After computing CWTs for both signals, the cross-wavelet power spectrum is analyzed. The relative phase angle between signals, indicated by arrows in the spectrum, is calculated as

$$\begin{array}{c}\varphi =arctan{\displaystyle \frac{\xi \left\{W_{mn}\right(\alpha ,\tau \left)\right\}}{R\left\{W_{mn}\right(\alpha ,\tau \left)\right\}}} \end{array}$$

(17)

where $\xi \left\{W_{mn}\right(\alpha ,\tau \left)\right\}$ and $R\left\{W_{mn}\right(\alpha ,\tau \left)\right\}$ denote the imaginary and real parts of the cross-wavelet transform $W_{mn}$.

The wavelet coherence spectrum quantifying energy correlation is defined as

$$\begin{array}{c}{\gamma }_{m,n}^2={\displaystyle \frac{{\left|S\right({\alpha }^{-1}{C}_{m,n}\left)\right|}^2}{S\left({\alpha }^{-1}{P}_{mk}\right)S\left({\alpha }^{-1}{P}_{nk}\right)}} \end{array}$$

(18)

where ${\gamma }_{m,n}^2$ measures cross-wavelet energy correlation, and $S\left({\alpha }^{-1}{P}_{nk}\right)$ denotes the smoothing operator, with its formulation being

$$\begin{array}{c}S\left(W\right)=S_{scale}\left(S_{time}\right(W\left)\right) \end{array}$$

(19)

where $S_{scale}\left(W\right)$ and $S_{time}\left(W\right)$ represent smoothing along the scale and translation axes, respectively.

Statistical significance of correlations is assessed using the power spectrum of a first-order autoregressive (AR1) model:

$$\begin{array}{c}{P}_k={\displaystyle \frac{1-{\alpha }^2}{{|1-{\alpha e}^{-2j\pi k}|}^2}}\end{array}$$

(20)

where $P_k$ is the Fourier power at frequency $k$, and $\alpha$ is the AR1 parameter.

The probability that wavelet power exceeds significance level $P$ is

$$\begin{array}{c}D\int \left({\displaystyle \frac{w_x(\alpha ,\tau )}{{\alpha }_X^2}}<p\int \right)={\displaystyle \frac{1}{2}}{P}_k{X}_v^2(p)\end{array}$$

(21)

where $v$ denotes degrees of freedom; $D$, scaling factor; ${\alpha }_X^2$, signal variance; and $X_v^2\left(p\right)$, chi-square distribution.

Finally, the cross-wavelet power distribution is

$$\begin{array}{c}{w}_x(\alpha ,\tau )w_y(\alpha ,\tau ){\displaystyle \frac{{\sigma }_x{\sigma }_y}{Z_v\left(p\right)}}\end{array}$$

(22)

where ${\sigma }_x$ and ${\sigma }_y$ are standard deviations.

3. Results and Analysis

3.1. LMD Results

The monthly GMSL anomalies from January 1993 to July 2020 were decomposed using LMD. The decomposition process terminated when the residual component became monotonic. The decomposition results are presented in Figure 4.

As shown in Figure 4, the residual component $u\left(t\right)$ exhibits a monotonically increasing trend after extracting two $PF$ components, indicating robust decomposition performance. The residual component represents the linear trend of GMSL rise, which is 3.204 ± 0.326 mm/yr. The high-frequency PF components comprise noise from the tidal signals, variations in atmospheric pressure, oceanic turbulence, instrumental noises, and regional variations in the signal. These components were therefore selected for secondary denoising to suppress noise contamination. The CMSE algorithm determined the optimal demarcation point separating high-frequency noise from low-frequency signals, enabling extraction of noise-contaminated high-frequency components. Equation (9) was applied to compute the mean square error between consecutive PF components and the residual component generated by LMD. The resulting CMSE values are quantitatively presented in

Table 1. Continuous mean square errors between consecutive PF components and residual components in LMD.

	Adjacent Components	CMSE
1	(${PF}_1,{PF}_2$)	14.1850
2	(${PF}_2,u\left(t\right)$)	658.3767

.

As shown in Table 1, the minimum CMSE among the decomposed components of the sea-level signal is 14.1850. The transition point separating high-frequency from low-frequency components occurs between ${PF}_1$ and ${PF}_2$, Therefore, the ${PF}_1$ component was selected as the high-frequency component and introduced into the improved wavelet threshold method for denoising.

3.2. Denoising the High-Frequency Component Using the Improved Wavelet Threshold Method

3.2.1. Results of Improved Wavelet Threshold Method

The high-frequency component ${PF}_1$ was processed using Equation (12) for denoising. Figure 5 compares the original noisy ${PF}_1$ component, the denoised ${PF}_1$ result, and the removed noise component.

A comparison of the denoised and original PF₁ components in Figure 5 reveals that the denoised PF₁ (using the improved wavelet threshold method) is smoother, exhibits more pronounced periodic components, and has significantly reduced sharp features compared to the original signal. Analysis of the extracted noise component shows that it contains a relatively stable, low-amplitude signal with a periodicity of approximately 12 months. Intermittently, the noise component exhibits significant transient anomalies with large fluctuations and high amplitudes. Additionally, it includes irregular fluctuations of varying amplitudes.

3.2.2. Comparison of Denoising Results

The ${PF}_1$ component was denoised using various soft and hard thresholds. Based on quantitative performance metrics (SNR and RMSE), the two most effective thresholds were selected for comparison with the proposed improved threshold; the results are shown in Figure 6.

The denoising performance of five thresholding methods—SURE hard threshold, SURE threshold, Minimax hard threshold, Minimax soft threshold, and the improved threshold—was evaluated using the SNR and RMSE as quantitative performance metrics. The results of this evaluation are summarized in

Table 2. SNR and RMSE values for denoising the PF₁ component using five thresholding methods.

	SNR	RMSE
SURE hard threshold	7.3622	1.6188
SURE soft threshold	7.0152	1.6848
Minimax hard threshold	8.1278	1.4822
Minimax soft threshold	7.0912	1.6701
Improved threshold	8.9945	1.3415

.

As shown in Table 2, the proposed improved wavelet threshold achieves a significantly higher SNR (8.9945) and lower RMSE (1.3415) compared to the best-performing conventional thresholding methods (SURE and Minimax, soft and hard variants), demonstrating its superior effectiveness.

3.3. Wavelet-Based Analysis of GMSL-ONI Relationships

3.3.1. ONI-Based ENSO Event Categorization

In this study, ENSO is characterized using the ONI. The ONI time series is presented in Figure 7.

Figure 7 depicts the classification of ENSO phases based on the ONI. ONI values above the solid red horizontal line indicate strong El Niño conditions; values above the dashed red horizontal line indicate El Niño conditions. Conversely, ONI values below the dashed blue horizontal line indicate La Niña conditions, while values below the solid blue horizontal line indicate strong La Niña conditions. ENSO-neutral conditions correspond to ONI values between the dashed red and dashed blue horizontal lines.

3.3.2. CWT and Periodicity Analysis of GMSL and ONI

The residual trend component was removed from the GMSL data. The denoised high-frequency and low-frequency components were then summed to reconstruct the detrended GMSL signal. The CWT was applied separately to this detrended GMSL signal and to the ONI to obtain their respective wavelet power spectra. In these spectra, the color bar represents the power of the oscillations, with warmer colors indicating higher power values. As the wavelet transform is implemented through convolution between the wavelet and the signal, edge artifacts can arise at the temporal boundaries of the data. To mitigate this effect, the cone of influence (COI) is used to delineate statistically valid regions; areas within the cone are considered reliable, while regions outside are excluded from interpretation. Furthermore, contours enclose areas where oscillatory features are statistically significant (p < 0.05). The resulting Morlet continuous wavelet power spectra for the GMSL and the ONI are presented in Figure 8a,b, respectively.

Figure 8a reveals two dominant, persistent oscillations with periods of 4–16 months during May 1994 to October 2007 and November 2008 to January 2020, which are marked by highly significant spectral power density. Additionally, four episodic shorter-term cycles (4–8 months) exhibiting strong spectral signatures are identified from February 1998 to October 2000, February 2002 to September 2004, July 2003 to December 2006, and May 2014 to November 2015. Pronounced sea-level fluctuations occurred during all these intervals. Figure 8b demonstrates statistically significant spectral signatures across the entire analysis domain for the ONI from January 1993 to July 2020. Dominant oscillations with 8–64-month periodicities persist from June 1994 to November 2018. Additionally, three distinct shorter-term cycles (2–8 months) emerge during August 2006 to February 2007, August 2014 to January 2015, and January 2018 to March 2019, exhibiting characteristic phase-locked behavior associated with ENSO evolution.

3.3.3. Cross-Wavelet Coherence Analysis of GMSL and ONI

Figure 9 presents a comparative analysis of the time–frequency co-variability between GMSL and ONI using two complementary methods: the cross-wavelet transform (XWT) and wavelet coherence (WTC). Figure 9a displays the XWT spectrum, which identifies regions of high common power derived from the continuous wavelet transforms of both signals. Warm-hued zones indicate significant co-oscillations between January 1993 and July 2020. Phase relationships are represented by arrow vectors: “→” denotes in-phase coupling; “←”, anti-phase behavior; “↓”, GMSL leading ONI by a quarter cycle; and “↑”, GMSL lagging by a quarter cycle. Color gradation corresponds to the magnitude of wavelet coherence (0–1), with warmer colors reflecting stronger interdependence. Figure 9b shows the WTC spectrum, which specifically highlights regions of statistically significant common variance, even under low spectral power conditions. The phase relationships within these coherent areas are interpreted based on the XWT analysis presented in Figure 9a.

Figure 9a reveals several significant common power regions in the XWT spectrum between GMSL and ONI. Among these, two sustained oscillatory regions with notably high spectral power density can be observed: one within the 28–40 month period band occurring from January 2007 to February 2016, where GMSL and ENSO are predominantly positively correlated, and another within the 8–24 month period band spanning from June 1995 to March 2019, which exhibits an ambiguous phase relationship, with no consistent lead–lag pattern between GMSL and ENSO. Additionally, several shorter intermittent regions are identified. One such region, characterized by relatively high-power spectral density, occurs within the 28–32 month period band from June 1997 to July 2000, during which the GMSL signal lags ENSO by approximately a quarter period. The remaining intermittent regions show lower-power spectral density and are associated with uncertain duration, wherein the phase relationship between GMSL and ENSO is also inconsistent.

Figure 9b reveals one sustained oscillatory region within the 28–40-month period band, occurring from October 2009 to February 2017, which is predominantly positively correlated—consistent with that shown in Figure 9a. Several shorter intermittent regions are observed within the 2–16-month period band, occurring throughout the entire study period with uncertain duration, wherein the phase relationship between GMSL and ENSO is also inconsistent.

3.3.4. Correlation Analysis of GMSL and ONI

Integrated analysis of the GMSL and ONI records using the XWT-revealed regions of coherent phase coupling between sea-level oscillations and ENSO variability. As demonstrated in

Table 3. Sea-level anomalies during El Niño events.

El Niño Period	Period (Month)	GMSL Increase (mm)	Peak Anomaly (mm)	Strong El Niño (ONI ≥ 1.5 °C)	Phase Lag (Month)
October 1994–March 1995	6–8	−1	4		−1
May 1997–May 1998	10–16/28–32	5	8	Yes	−2/3
June 2002–February 2003	8–16	7	14		2
July 2004–February 2005	8–12	7	4		2
October 2006–January 2007	6–8/10–12	−3	7		−1/1
July 2009–March 2010	16–24/28–40	5	11		0/1
October 2014–April 2016	8–16/30–40	6	10	Yes	0/2
September 2018–May 2019	4–8/18–12	3	6		0/0

, positive sea-level anomalies coincided with El Niño events.

Table 3 shows that El Niño events are typically accompanied by sea-level rise and exhibit a phase-locked relationship with GMSL. Stronger El Niño events tend to produce greater net increases and higher instantaneous peaks in global mean sea level. The two most intense events caused the largest net GMSL increases (+7 and +6 mm) and substantial peak anomalies (+8 and +10 mm). Notably, the events in June 2002 to February 2003 and July 2009 to March 2010 produced even higher peak anomalies (+14 and +11 mm), resulting in net sea-level rises of 7 and 5 mm, respectively. Conversely, La Niña phenomena are associated with negative sea-level anomalies, as quantified in

Table 4. Sea-level anomalies during La Niña events.

La Niña Period	Period (Month)	GMSL Decrease (mm)	Peak Negative Anomaly (mm)	Strong La Niña (ONI ≤ −1.5 °C)	Phase Lag (Month)
August 1995–March 1996	10–20	−3	−5		2
July 1998–February 2001	10–20/28–32	−12	−8	Yes	−2/3
November 2005–March 2006	10–12	10	−1		1
June 2007–June 2008	4–8/10–12	−4	−8	Yes	−2/0
November 2008–March 2009	10–16/32–40	6	−2		−2/0
June 2010–April 2012	10–24/32–40	−4	−11	Yes	0/1
August 2016–December 2016	8–24/32–40	−2	−2		−2/1
October 2017–March 2018	4–8/18–12	2	−4		0/0

.

Table 4 indicates a phase relationship between GMSL and ENSO, with strong La Niña events often corresponding to pronounced negative sea-level anomalies. The three most intense La Niña events produced marked negative peak anomalies (−8 mm, −8 mm, and −11 mm, respectively). The sea-level depressing effect of prolonged La Niña events is frequently counteracted by the overarching trend of sea-level rise; however, shorter-duration events—specifically those occurring in November 2005 to March 2006, November 2008 to March 2009, and October 2017 to March 2018—exhibit clear and rapid GMSL declines of 10 mm, 6 mm, and 2 mm, respectively.

4. Conclusions and Discussion

4.1. Conclusions

This study aims to elucidate the multi-scale coupling mechanisms and asymmetries between ENSO and GMSL variability. The GMSL signal was first decomposed and denoised using a framework incorporating LMD and an improved wavelet thresholding technique. This processed signal preserves the advantages of LMD in retaining long-term trends, interannual variations, and seasonal cycles while also enabling precise noise removal from high-frequency components. The denoised GMSL data, along with the ONI, were then analyzed using CWT to systematically characterize their time–frequency features. Finally, XWT and WTC were applied to quantify the phase-coherent interactions between GMSL and ONI.

The main findings are as follows: (1) The linear trend component indicates a GMSL rise rate of 3.204 ± 0.326 mm/yr during January 1993 to July 2020, consistent with recent satellite altimetry studies [33,34]. (2) Persistent intra-annual variability (8–16-month cycles) dominates the GMSL signal, superimposed with interannual fluctuations (4–8-month cycles) associated with climatic and seasonal forcing. (3) Phase analysis reveals that GMSL generally leads ONI during El Niño episodes, whereas it tends to lag during La Niña events. (4) El Niño events, particularly strong episodes (May 1997 to May 1998 and October 2014 to April 2016), resulted in larger net GMSL increases (+7 mm and +6 mm) and significant peak anomalies (+8 mm and +10 mm). (5) Pronounced negative peak anomalies in sea level are observed during La Niña events. However, the overall declining trend in GMSL is often masked by the long-term background rise in sea level [3,35]. As a result, prolonged La Niña events tend to be offset by this secular trend, whereas shorter events exhibit clearly discernible and rapid declines in GMSL. In summary, these results provide robust evidence for the complex and asymmetric nature of the ENSO–GMSL relationship across multiple time scales, offering valuable insights for understanding and predicting climate-driven sea-level variability.

4.2. Discussion

This study investigates the multi-scale variability of GMSL and its teleconnections with ENSO using a hybrid signal-processing approach. The key findings and their implications are discussed below.

The denoised GMSL signal reveals oscillations across multiple timescales that are strongly indicative of teleconnection with ENSO. The evolving phase relationships identified through XWT analysis—particularly notable shifts from anti-phase to in-phase coupling—suggest that the interaction between GMSL and ENSO is not static but is likely modulated by low-frequency climate models such as Pacific Decadal Oscillation (PDO) [36]. The consistent anti-phase behavior, accompanied by significant phase lags (e.g., GMSL lagging ONI in the 25–44-month band), indicates that sea level acts as an integrator of the oceanic response to ENSO forcing. This reflects the prolonged processes of heat and mass redistribution following major ENSO events. The intermittent yet statistically significant coherence across various periods, as captured by both XWT and WTC, further underscores the role of ENSO as a primary driver of interannual sea-level variability.

The detected noise component also offers physical insights. The stable annual periodic signal (Figure 5c) likely stems from seasonal steric changes driven by variations in seawater temperature, salinity, and wind patterns. In contrast, transient high-amplitude anomalies may be linked to extreme weather events (e.g., storms, typhoons) or episodic geological events such as earthquakes or submarine landslides. Irregular fluctuations may also arise from instrumental errors or anthropogenic disturbances, such as ship traffic [37]. The detected noise component also offers physical insights. The stable annual periodic signal likely stems from seasonal steric changes driven by variations in seawater temperature, salinity, and wind patterns. In contrast, transient high-amplitude anomalies may be linked to extreme weather events (e.g., storms, typhoons) or episodic geological events such as earthquakes or submarine landslides. Irregular fluctuations may also arise from instrumental errors or anthropogenic disturbances, such as ship traffic.

In addition to the strong El Niño events, Table 3 also highlights two shorter-duration El Niño events (June 2002 to February 2003 and July 2009 to March 2010) that were associated with unusually rapid GMSL increases (+5 mm and +5 mm) and the highest peak anomalies (+14 mm and +11 mm) observed. This suggests that factors beyond ENSO itself, such as marine heat content anomalies and other regional oceanic processes, may have synergistically contributed to these pronounced sea-level rise events.

Future studies should aim to attribute these specific noise sources more precisely. Several important limitations and future research directions emerge from this work. First, the phase relationships identified here are influenced by a complex interplay of ocean temperature, steric height, and mass redistribution. To elucidate the precise physical mechanisms behind these phase lags and leads, future work should incorporate ocean reanalysis data into a multivariate framework that includes climate indices such as the PDO [29] and the Indian Ocean Dipole (IOD). This would help disentangle their relative contributions to GMSL variability from those of ENSO. Additionally, the proposed LMD-based denoising method should be quantitatively benchmarked against other advanced techniques (e.g., CEEMDAN, optimized VMD) to validate its performance in geophysical signal processing. The instantaneous amplitude and frequency information derived from LMD offers a unique opportunity to quantify the nonlinearity and non-stationarity of the GMSL signal and its modulation by ENSO phases. Finally, applying this hybrid methodology to regional sea-level records could reveal spatially heterogeneous patterns of ENSO forcing across ocean basins. On a broader scope, future efforts should aim to close the sea-level budget (steric + mass) during extreme El Niño and La Niña events using detrended and denoised data. Such an approach is crucial for identifying the dominant physical drivers—whether thermosteric expansion, halosteric effects, or land ice melt—underlying the observed asymmetric sea-level responses during ENSO events.