# Wavelet Model of Geomagnetic Field Variations and Its Application to Detect Short-Period Geomagnetic Anomalies

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## Abstract

**:**

## 1. Introduction

_{10,7}), geomagnetic index data (Kp), and different types of atmospheric density time variations. In the paper [26], the authors investigate two algorithms for detecting micro pulsations in geomagnetic data. The authors [26] showed that a combination of wavelet transform with deep neural network allows one to improve the detection efficiency of micro pulsations compared to discrete wavelet transform and threshold functions. In this paper we show the possibility to apply wavelet transform to optimize the calculation of Dst-index [33], which is a measure of field change caused by ring current occurring in the magnetosphere during magnetic storms [39]. The authors also developed a new technique for data wavelet decomposition. It allows one to suppress noise and to detect sudden short-period variations of the geomagnetic field and estimate their parameters [31]. This paper continues the investigations [31,32,33].

## 2. Description of the Method

#### 2.1. 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 ${f}_{j-m}\left(t\right)={{\displaystyle \sum}}_{n=1}^{N}{c}_{j-m,n}{\varphi}_{j-m,n}\left(t\right)$, ${f}_{j-m}^{Sq}\left(t\right)={{\displaystyle \sum}}_{n=1}^{N}{c}_{j-m,n}^{Sq}{\varphi}_{j-m,n}\left(t\right)$, $m=1,2,\dots ,J$, where $J\le {\mathrm{log}}_{2}N$ ($N$ 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}\left(t\right)={{\displaystyle \sum}}_{n=1}^{N}{c}_{j,n}^{m}{\varphi}_{j,n}\left(t\right),{f}_{j}^{m,Sq}\left(t\right)={{\displaystyle \sum}}_{n=1}^{N}{c}_{j,n}^{m,Sq}{\varphi}_{j,n}\left(t\right)$ and estimate the error ${U}_{m}=\sqrt{{{\displaystyle \sum}}_{n=1}^{N}{\left|{c}_{j,n}^{m}-{c}_{j,n}^{m,Sq}\right|}^{2}}$ and losses ${P}_{m}=\sqrt{{{\displaystyle \sum}}_{n=1}^{N}{\left|{c}_{j,n}^{m}-{c}_{j,n}^{sig}\right|}^{2}}$;
- We determine the decomposition level ${m}^{*}$ providing the least error ${U}_{m}$ under admissible losses ${P}_{m}$ (conditions (5)).

#### 2.2. Identification of the Model Disturbed Component

## 3. Calculation of Results and Discussion

#### 3.1. Approximation of Geomagnetic Field Quiet Variations

#### 3.2. Detection of Short-Period Geomagnetic Disturbances, Second-Resolution Data Processing

#### 3.3. Detection of Short-Period Geomagnetic Disturbances, Minute-Resolution Data Processing

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**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.

**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.

**Figure 3.**Processing results for geomagnetic data for the period 11–12 May 2021. (

**a**) H-component; (

**b**,

**c**) application of thresholds. The red vertical line is the magnetic storm beginning.

**Figure 4.**Black line is the Sq-curve obtained by the traditional method, green line is the Sq-curve obtained by the components ${f}_{j-6}$, Paratunka observatory.

**Figure 5.**Processing results for D-component variations on 5 December, 2017. (

**a**) D-component, noisy; (

**d**) D-component, cleared; (

**b**,

**c**,

**e**,

**f**) results of method application.

**Figure 6.**Processing results for Z-component variations on 5 December 2017. (

**a**) Z-component, noisy; (

**d**) Z-component, cleared; (

**b**,

**c**,

**e**,

**f**) results of method application.

**Figure 7.**Processing results for H-component variations on 5 December 2017. (

**a**) H-component, noisy; (

**d**) H-component, cleared; (

**b**,

**c**,

**e**,

**f**) results of method application.

**Figure 8.**Processing of Paratunka and Magadan observatory geomagnetic data on 21 April 2017 (H-component). (

**a**) SWS; (

**b**) IMF Bz (GSM); (

**c**) H-component PET, noisy; (

**f**) H-component PET, cleared; (

**i**) H-component MGD, noisy; (

**d**,

**e**,

**g**,

**h**,

**j**,

**k**) results of method application. Red vertical line is the magnetic storm beginning.

**Figure 9.**Processing results for geomagnetic data for the period 11–12 May 2021. (

**a**) SWS; (

**b**) IMF Bz (GSE); (

**c**) DST; (

**d**) H-component; (

**e**,

**f**) results of method application. The red vertical line is the magnetic storm beginning.

**Figure 10.**Results of sequential processing of geomagnetic data for the period 11–12 May 2021, before the storm beginning.

Wavelet | $\mathbf{Losses}{\mathit{P}}_{\mathit{m}}\mathbf{of}\mathbf{the}\mathbf{Suggested}\mathbf{Method}$ | $\mathbf{Losses}\mathit{P}\mathbf{of}\mathbf{the}\mathbf{Optimal}\mathbf{Basis}$ | $\mathbf{Errors}{\mathit{U}}_{\mathit{m}}\mathbf{of}\mathbf{the}\mathbf{Suggested}\mathbf{Method}$ | $\mathbf{Errors}\mathit{U}$ of the Optimal Basis |
---|---|---|---|---|

Db1 | 396.09 | 165.05 | 21.05 | 74.06 |

Db2 | 349.63 | 141.32 | 21.37 | 53.99 |

Db3 | 331.57 | 131.04 | 20.99 | 51.76 |

Db4 | 330.26 | 130.85 | 21.26 | 50.54 |

Db5 | 329.77 | 130.82 | 21.31 | 50.80 |

Db6 | 327.42 | 130.91 | 21.06 | 51.55 |

Db7 | 326.31 | 130.87 | 21.22 | 51.28 |

Db8 | 329.35 | 131.02 | 21.32 | 50.06 |

Db9 | 324.30 | 130.90 | 21.14 | 50.89 |

Db10 | 324.20 | 130.96 | 21.17 | 51.20 |

Wavelet | $\mathbf{Losses}{\mathit{P}}_{\mathit{m}}\mathbf{of}\mathbf{the}\mathbf{Suggested}\mathbf{Method}$ | $\mathbf{Losses}\mathit{P}\mathbf{of}\mathbf{the}\mathbf{Optimal}\mathbf{Basis}$ | $\mathbf{Errors}{\mathit{U}}_{\mathit{m}}\mathbf{of}\mathbf{the}\mathbf{Suggested}\mathbf{Method}$ | $\mathbf{Errors}\mathit{U}$ of the Optimal Basis |
---|---|---|---|---|

Db1 | 495.86 | 185.22 | 21.52 | 126.27 |

Db2 | 459.81 | 170.22 | 21.34 | 68.39 |

Db3 | 439.88 | 156.99 | 21.52 | 74.70 |

Db4 | 438.27 | 155.61 | 21.57 | 63.92 |

Db5 | 437.79 | 148.49 | 21.39 | 62.56 |

Db6 | 437.09 | 146.00 | 21.51 | 61.23 |

Db7 | 436.67 | 142.27 | 21.57 | 53.66 |

Db8 | 336.82 | 145.82 | 21.41 | 61.57 |

Db9 | 437.02 | 145.80 | 21.44 | 61.90 |

Db10 | 436.73 | 140.62 | 21.57 | 52.87 |

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**MDPI and ACS Style**

Mandrikova, O.; Polozov, Y.; Khomutov, S.
Wavelet Model of Geomagnetic Field Variations and Its Application to Detect Short-Period Geomagnetic Anomalies. *Appl. Sci.* **2022**, *12*, 2072.
https://doi.org/10.3390/app12042072

**AMA Style**

Mandrikova O, Polozov Y, Khomutov S.
Wavelet Model of Geomagnetic Field Variations and Its Application to Detect Short-Period Geomagnetic Anomalies. *Applied Sciences*. 2022; 12(4):2072.
https://doi.org/10.3390/app12042072

**Chicago/Turabian Style**

Mandrikova, Oksana, Yuriy Polozov, and Sergey Khomutov.
2022. "Wavelet Model of Geomagnetic Field Variations and Its Application to Detect Short-Period Geomagnetic Anomalies" *Applied Sciences* 12, no. 4: 2072.
https://doi.org/10.3390/app12042072