# Unveiling the Core Patterns of High-Latitude Electron Density Distribution at Swarm Altitude
## 3 Method: EMD and its Multivariate Extension  
Empirical Mode Decomposition (EMD) is a data analysis technique first introduced by [PERSON] et al. (2013) with the aim of adaptively representing non-stationary signals. The fundamental concept is that a signal can be decomposed in a finite, often small number of intrinsic oscillatory modes, each one of them reproducing the repeating behaviour of the signal at a specific time scale ([PERSON] et al., 2013). The EMD method allows signals that are not compatible with the Fourier Transform restrictions to be analyzed. In fact, Fourier analysis is a powerful method of extracting the energy-frequency distribution of a signal, and is  
Figure 1: Polar view of the average spatial distribution of electron density (\(N_{\lambda}\)) for low (**a**,**c**,**e**) and high (**b**,**d**,**f**) geomagnetic activity conditions in the Northern Hemisphere as represented in the QD–magnetic latitude (MLat) and MLT reference system. Each column shows the average spatial distribution of electron density during three different periods of the year: equinoxes, summer, and winter. The maps are based on data collected by the Swarm A satellite from 1 January 2016 to 31 December 2021, and reflect an average altitude of approximately 460 km. The concentric circles are plotted in 10\({}^{\circ}\) intervals, with the outermost circle corresponding to 50\({}^{\circ}\). widely used for its simplicity and its general applicability; however, it cannot produce physically meaningful results when dealing with nonstationary data [23]. In such cases, EMD can resolve the role of reducing the signal \(s(t)\) into a set of basis signals  
\[s(t)=\sum_{n=1}^{N}c_{n}(t)+res(t) \tag{1}\]  
which are derived directly from the data itself [24]. The empirical modes \(c_{n}(t)\) must fulfill two criteria: first, when considering the entire dataset, each of these function must have the same number of extrema (local maxima and/or minima) and zero crossings or differ by one at most; second, when considering any point, the mean value between the envelopes defined by the local maximum and local minimum must be zero [23]. The resulting functions are simple oscillatory modes, which differ from simple harmonic components in that their frequency and amplitude can change along the time frame. The procedure for extracting the empirical modes, called sifting, involves the following steps:  
1. The local extrema of the considered dataset are identified. 2. A cubic spline is used to connect all the local maxima in order to obtain the upper envelope. 3. Another cubic spline is used to connect all the local minima in order to obtain the lower envelope. 4. The mean between the two envelopes (\(m_{1}\)) is computed. 5. The first component \(h_{1}\) is obtained by subtracting \(m_{1}\) from the data \(h_{1}=x(t)-m_{1}\). 6. If \(h_{1}\) is a function with zero mean, it is accepted as the first empirical mode, \(c_{1}=h_{1}\); otherwise, \(h_{1}\) undergoes steps 1-5 itself and the procedure is repeated until \(h_{1}\) fulfills the zero mean criterion. In addition, it is necessary to apply a stopping criteria to the number of sifting iterations; in this way, the obtained empirical modes preserve the amplitude and frequency modulation as well as their physical sense [23]. One potential stopping criterion is to determine a threshold limit for the size of the standard deviation between two consecutive sifting steps \(k\) and \(k+1\):  
\[\sigma_{k}=\sum_{j}\frac{\left[\delta_{k}(t_{j})-\delta_{k+1}(t_{j})\right]^{ 2}}{\delta_{n}(t_{j})^{2}}<\epsilon \tag{2}\]  
in which \(\epsilon\) is usually fixed at 0.3 [23]. To obtain the following empirical modes, the process is successively applied to the residual \(r_{1}=s(t)-c_{1}\). The procedure described above is then repeated until the residual becomes a constant, a monotonic function, or a function with only one maximum and minimum from which it is not possible to extract other modes [24].