## 2 Description of the Method
### Identification of the Model Characteristic Component  
In order to identify the model characteristic component \(Q^{char}(t)\), we shall apply the multiple-scale analysis (MSA) [29; 30; 42]. We consider \(L^{2}(R)\), the function space with finite energy (Lebesgue space [29; 30]), as the signal space. Assume that the initial signal \(f\) belongs to the subspace \(V_{j}\subset L^{2}(R)\) of resolution \(2^{-j}\):  
\[V_{j}=clos_{L^{2}(R)}\big{(}\phi_{j,n}:n\in Z\big{)},\ \phi_{j,n}=2^{\frac{j}{ 2}}\phi\Big{(}2^{j}t-n\Big{)},\ j\in Z.\]  
panel (Figure 1e-h) presents the results for the period of 1-30 September 2021. Diurnal values of K-index are indicated in the upper part of the panels. Smoothed components (Figure 1b,f) approximates signal trend and the detailing components (Figure 1c,d) contain short-period variations of different structures and duration. Analysis of the results shows that during calm periods (K-index has low values), MSA smoothed components allow us to obtain an approximation to the field calm variation. During disturbed periods, a significant increase of detailing component coefficients characterizes the field disturbance degree. Figure 1: Application of MSA to Paratunka observatory data applying the Daubechies 3 wavelet. (**a**,**e**) H-component; (**b**,**f**) \(f_{j-6}\) component; (**c**,**g**) \(g_{j-3}\) component; (**d**,**h**) \(g_{j-6}\) component. The smoothed component \(f_{j-m}\) describes geomagnetic signal variations during quiet periods. The obtained approximation error depends on the decomposition level \(m\); the problem arises in determining the decomposition level \(m^{*}\), which minimizes the error. Sq-curve can be used as a reference function, reflecting quiet geomagnetic field diurnal variation at an observation site [41]. As described above, Sq-curve is a mean smoothed curve of the five <<quietest>> intervals of the geomagnetic field over a current month. In this case, the approximation error in wavelet space can be determined as  
\[U_{m}=\sqrt{\sum_{n=1}^{L}\left|c_{j-m,n}-c_{j-m,n}^{Sq}\right|^{2}}, \tag{3}\]  
where \(c_{j-m,n}\) are the signal smoothed component coefficients at the decomposition level \(m\); \(c_{j-m,n}^{Sq}\) are the Sq-curve smoothed component coefficients at the decomposition level \(m\); \(n\) are readings; \(L\) is the signal component length. When choosing the decomposition level \(m^{*}\), one should also take into account the fact that according to (2), during the change from the decomposition level \(m\) to the decomposition level \(m+1\), a part of the information on the signal turns to the detailing component \(g_{j-l}\) (see (2)) that increases the losses  
\[P_{m}=\sqrt{\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j,n}^{sig}\right|^{2}}, \tag{4}\]  
where \(c_{j,n}^{m}\) are the reconstructed signal coefficients obtained on the basis of wavelet recovery \(f_{j-m}\) and using only approximation coefficients; \(c_{j,n}^{sig}\) are the initial signal coefficients, \(N\) is the signal length. Thus, according to (3) and (4), we have a problem of multicriteria optimization in which the objective functions  
\[\Phi^{U_{m}}=\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j-m,n}^{m,Sq}\right|^{2}\to min, \tag{5}\] \[\Phi^{P_{m}}=\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j,n}^{sig}\right|^ {2}\to min.\]  
are mutually conflicting. In this case, the optimum level \(m^{*}\) can be obtained by the method of constraint change (\(\varepsilon\)-constraints):  
\[\Phi^{U_{m}}=\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j-m,n}^{m,Sq}\right|^{2}\to min, \tag{6}\] \[\Phi^{P_{m}^{\prime\prime}}=\frac{1}{N}\sum_{n=1}^{N}\left|c_{j, n}^{m}-c_{j,n}^{sig}\right|^{2}\leq\varepsilon.\]  
The value \(\varepsilon\) is considered as an admissible level for \(\Phi^{P_{m}^{\prime\prime}}\), \(\varepsilon_{i}\) were estimated for each station \(i\) as the dispersion of geomagnetic field variations during <<quiet>> periods (absence of magnetic storms and substorms). From the above said we obtain the algorithm to choose the decomposition level \(m^{*}\):  
1.