## 2 Materials and Methods
### Empirical Mode Decomposition (EMD)  
EMD is an adaptive time-series analysis tool specifically for the examination of nonstationary and nonlinear time series via the decomposition of a singular nonstationary time series into a series of IMFs and a residue (trend) function through sifting processes [33; 37]. The IMFs are components of a time series that involve only one mode of oscillation at any given time and can give rise to a meaningful instantaneous frequency everywhere.  
In the EMD algorithm, two factors are very important: defining an IMF and establishing a suitable stoppage criterion in the sifting iterations [33; 38]. First, an IMF should be determined via the following two requirements for defining a meaningful instantaneous frequency from the original signal: (1) an IMF should have zero or one difference between the numbers of zero-crossings and the numbers of local extrema, and (2) an IMF must have symmetric envelops defined by local maxima and minima, respectively, which yield a zero mean. In that case, an IMF can be transformed into an analytic function via the Hilbert transform and give rise to a single assigned instantaneous frequency at a time [33]. Second, a stoppage criterion is needed for obtaining physically meaningful IMFs because there is an inevitable conflict between the IMF definition and the EMD algorithm based on spline fitting. That is, if we adhere to the strict definition of the IMF [33], we ultimately obtain the IMFs, which are physically meaningless [38]. Thus, an appropriate stoppage criterion is needed for the sifting process. There are various stoppage criteria, and the corresponding IMFs have distinct characteristics depending on the adopted stoppage criterion. In this study, we adopted the Cauchy-type criterion among all available stoppage criteria [33; 38]. In the following, we present a brief procedure for EMD implementation.  
A basic sifting process for EMD is summarized as follows [38]:  
1. All the local extrema are identified for a given time series, \(x(t)\).
2. By interpolating all the local maxima and minima with cubic spline curves, respectively, we obtain the upper, \(u(t)\), and lower, \(l(t)\), envelops.
3. An average envelop is obtained from the upper and lower envelops via \(m(t)=[u(t)+l(t)]/2\).
4. A proto-IMF is obtained via the following differences, \(h(t)=x(t)-m(t)\).
5. By checking if the proto-IMF satisfies two requirements, the definition of IMF and the stoppage criterion, the proto-IMF is determined to be an IMF.
6. If the proto-IMF does not satisfy these requirements, steps 1 through 5 on \(h(t)\) are repeated until they are satisfied.
7. When the requirements are satisfied, the proto-IMF, \(h(t)\), is assigned as an IMF component, \(c(t)\).
8. For the next IMF, we repeat steps 1 through 7 on the residue, \(r(t)=x(t)-c(t)\).
9. All the iterations end when the final residue contains no more than one extremum.  
Here, the adopted Cauchy-type stoppage criterion is given as follows [38]:  
\[SD=\frac{[h_{k-1}(t)-h_{k}(t)]^{2}}{h_{k-1}^{2}(t)} \tag{1}\]  
where \(SD\) denotes a local standard deviation of the two consecutive sifting results and \(h_{k}\) denotes the proto-IMF at \(k\)-th iteration of the sifting process. Finally, we obtain a set of IMFs, \(\{c_{i}(t)\}\), and a residue, \(r_{n}(t)\), satisfying the following relation:  
\[r_{n}(t)=x(t)-\sum_{i=1}^{n}c_{i}(t) \tag{2}\]  
where the \(i\) denotes a mode number and the residue, \(r_{n}(t)\), represents the trend of the time series, \(x(t)\). Due to the nature of the sifting process, an IMF has a gradually decreasing frequency as its mode number increases. In this study, by adopting the \(SD=0.001\) as a stoppage criterion, we obtained 10 IMFs from each SSTA time series.