## 2 Description of the Method
### Identification of the Model Characteristic Component
We apply the MSA for the initial data estimated for each month of Sq-curves and obtain the representations \(f_{j-m}(t)=\sum_{n=1}^{N}c_{j-m,n}\phi_{j-m,n}(t)\), \(f_{j-m}^{Sq}(t)=\sum_{n=1}^{N}c_{j-m,n}^{Sq}\phi_{j-m,n}(t)\), \(m=1,2,\ldots,J\), where \(J\leq\log_{2}N\) (\(N\) is the signal length);For each decomposition level \(m\) we carry out the reconstruction \(f_{j-m}\) and \(f_{j-m}^{Sq}:f_{j}^{m}(t)=\sum_{n=1}^{N}c_{j,n}^{m}\phi_{j,n}(t),f_{j}^{m,Sq}(t) =\sum_{n=1}^{N}c_{j,n}^{m,Sq}\phi_{j,n}(t)\) and estimate the error \(U_{m}=\sqrt{\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j,n}^{m,Sq}\right|^{2}}\) and losses \(P_{m}=\sqrt{\sum_{n=1}^{N}\left|c_{j,n}^{m}-c_{j,n}^{sig}\right|^{2}}\);
3. We determine the decomposition level \(m^{*}\) providing the least error \(U_{m}\) under admissible losses \(P_{m}\) (conditions (5)). The obtained estimates of \(U_{m}\) and \(P_{m}\) depend on the applied basic wavelet, the choice of which can be based on minimizing the loss function \(P_{m}\) on the set of wavelet basis dictionaries. Figure 2 illustrates the calculation results of the losses for different basic wavelets. Daubechies wavelets of the 1-st-10-th orders, Db1-Db10 were used [30]. The estimates were calculated according to months (for January-June 2021) and for the whole period of the first half of 2021. To analyze geomagnetic activity, Figure 2 presents the K-index summary values for the periods under analysis. The results show close values of \(P_{m}\) (see ratio (4)), including the decomposition 10-th level. A comparison with geomagnetic activity data shows loss increase when the geomagnetic field disturbance degree grows. The result indicates the complicated structure of geomagnetic data, containing features of different forms. Close loss values indicate the possibility of applying different wavelets for geomagnetic data approximation. Figure 2: Results of estimation of losses \(P_{m}\) for different basic wavelets, Paratunka observatory data were used. (**a**) January–June 2021; (**b**) first half of 2021. To assess the method efficiency, the results were compared with the method for construction of nonlinear approximations in wavelet basis described in the paper [42]. Decompositions into wavelet packages were applied. They allow one to obtain the best signal approximations of finite length by minimizing the concave cost function (entropy) [42]. When considering the dictionary \(D\) of orthonormal bases  
\[D=\cup_{\lambda\in\Lambda}\mathfrak{B}^{\lambda},\]  
the approximation cost in the basis \(\mathfrak{B}^{\lambda}=\left\{g_{m}^{\lambda}\right\}_{1\leq m\leq N}\) can be estimated by Shur concave sum [43]  
Figure 1:0  
Then the optimal basis \(\mathfrak{B}^{\alpha}\) minimizing the losses  
Figure 1:1  
where \(\hat{f}\) is the estimate of the function \(f\) is determined as  
Figure 1:2  
In the package space [42]  
Figure 1:3  
the optimal basis is [43]  
\[W_{j}^{p}=\left\{ \begin{array}{cc}W_{j+1}^{2p}\cup W_{j+1}^{2p+1},&if\ C \Big{(}f,W_{j+1}^{2p}\Big{)}+C\Big{(}f,W_{j+1}^{2p+1}\Big{)}<C\Big{(}f, \mathfrak{B}_{j}^{p}\Big{)},\\ W_{j}^{p},&if\ C\Big{(}f,W_{j+1}^{2p}\Big{)}+C\Big{(}f,W_{j+1}^{2p+1}\Big{)} \geq C\Big{(}f,\mathfrak{B}_{j}^{p}\Big{)}.\end{array} \right. \tag{7}\]  
Recursive computation of bases (7), when moving upwards along a wavelet package tree, allows us to obtain an optimal basis, minimizing the cost (6). Construction of an optimal basis is described more in detail in this paper [43]. Estimate results, obtained according to the data of the Paratunka observatory for January 2021 (K-index total value = 196) and for March 2021 (K-index total value = 371), are illustrated in Tables 1 and 2, respectively. They show that the losses \(P\) of the optimal basis are significantly less than the losses \(P_{m}\) of the suggested method.