# Identification of Fractal Properties in Geomagnetic Data of Southeast Asian Region during Various Solar Activity Levels

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Data Procurement

#### 2.2. Methods

#### 2.2.1. Power Spectrum Analysis (PSA)

_{n}is the geomagnetic time series and m = 0, 1, 2, N − 1. The time series may indeed be a self-affine fractal if the correlation of the power spectrum, P(f) and the frequency, (f)

#### 2.2.2. Rescaled Range Analysis (RRA)

_{n}, with size τ. The range of each subinterval is then calculated

_{n}against τ.

#### 2.2.3. Detrended Fluctuation Analysis (DFA)

_{n}(t), of size τ. The subsequent time series is then put through a detrending process in which the local trend is subtracted in each subinterval,

#### 2.2.4. Robust Detrended Fluctuation Analysis (r-DFA)

_{L}for each section according to the division made by the crossovers. Instead, if crossovers are not identified, a robust regression is used to determine the global scaling exponent, α. As for the determination of the Hurst exponent, the method remains the same as the regular DFA method and can be implemented seamlessly with this method.

#### 2.2.5. Fractional Brownian Motion (fBm)

_{H}(t) is fractional Brownian motion at time t, Γ is gamma function, H is Hurst exponent and B(s) is standard Brownian motion at time s [51]. At H = 0.5, it generalizes into a standard Brownian motion.

## 3. Results and Discussion

#### 3.1. Fractal Properties of Quiet Day Geomagnetic Data

#### 3.2. Fractal Methods to Determine the Hurst Exponent of Geomagnetic Data

#### 3.3. Characterization of Geomagnetic Data during Various Cases of Quiet and Disturbed Days

## 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.**An example of the H-component data. These in particular are single day data of 23 June 2015 (

**a**) and 4 June 2015 (

**b**) taken from DAV station. These data represent disturbed and quiet days, respectively, which are part of the short-term events analyzed in this study.

**Figure 2.**Power spectrum periodogram of 60 quiet days during low (2009), intermediate (2013), and high solar activity levels (2015). The circles on the spectra indicate the spectral peaks present in each year. The “M1” and “M2” markings at the bottom indicate areas with different scaling properties.

**Figure 3.**An example of the DFA visualization. These in particular are single day data of 23 June 2015 (disturbed day) (

**a**) and 4 June 2015 (quiet day) (

**b**) taken from DAV station.

Solar Activity Level | Type of Day | Dates Analyzed (DD/MM/YYYY) |
---|---|---|

Intermediate | Disturbed | 17/03/2013 |

01/06/2013 | ||

29/06/2013 | ||

Quiet | 26/03/2013 | |

High | Disturbed | 17/03/2015 |

18/03/2015 | ||

22/06/2015 | ||

23/06/2015 | ||

07/10/2015 | ||

20/12/2015 | ||

21/12/2015 | ||

Quiet | 04/06/2015 |

**Table 2.**Comparison of spectral peaks found in this study with Rabiu et al. [23] study.

This Study (DAV) | Rabiu et al. [23] (KOU and BNG) | ||
---|---|---|---|

Year | Existing Peaks (h) | Year | Existing Peaks (h) |

2009 | 6, 8, 12, 24 | 1996 | 8, 12, 24 |

2013 | 6, 8, 12, 24 | 2000 | 6 *, 8, 12, 24 |

2015 | 6, 8, 12, 24 | 2002 | 8, 12, 24 |

**Table 3.**Result of the accuracy test for each method using various synthetic fractional Brownian motion signals simulating the geomagnetic time series. The color gradient of orange-yellow-green-dark green indicates the degree of accuracy of each method, with orange being the least accurate while dark green being the most accurate.

Hurst (H) | PSA | RRA | DFA | r-DFA |
---|---|---|---|---|

0.1 | 0.04 ± 0.02 | 0.19 ± 0.02 | 0.09 ± 0.01 | 0.01 ± 0.03 |

0.3 | 0.20 ± 0.02 | 0.35 ± 0.02 | 0.31 ± 0.01 | 0.19 ± 0.03 |

0.5 | 0.40 ± 0.01 | 0.53 ± 0.02 | 0.51 ± 0.01 | 0.37 ± 0.03 |

0.7 | 0.44 ± 0.01 | 0.71 ± 0.03 | 0.70 ± 0.01 | 0.55 ± 0.04 |

0.9 | 0.41 ± 0.00 | 0.81 ± 0.07 | 0.91 ± 0.01 | 0.73 ± 0.04 |

**Table 4.**The results of the short-term data analyses utilizing the DFA method. All results are in the parameter of Hurst exponent (H). Orange highlights indicate persistence tendencies, while green highlight indicates anti-persistence tendencies.

Disturbed Day (All Major Events in One Year; Days with Dst < −200 nT) | Disturbed Day | Quiet Day | ||||
---|---|---|---|---|---|---|

Station | 2013 | 2015 | 17/03/2013 | 23/06/2015 | 26/03/2013 | 04/06/2015 |

DAV | 0.56 ± 0.03 (3d) | 0.58 ± 0.01 (7d) | 0.62 ± 0.03 | 0.63 ± 0.05 | 0.36 ± 0.03 | 0.36 ± 0.03 |

LKW | 0.69 ± 0.04 (3d) | 0.74 ± 0.02 (5d) | 0.68 ± 0.03 | 0.69 ± 0.05 | 0.38 ± 0.03 | 0.41 ± 0.04 |

**Table 5.**Results of the long-term data analyses utilizing the DFA method. Orange highlights indicate persistence tendencies.

Period | Quiet Period | Disturbed Period | |||
---|---|---|---|---|---|

Year/Case | 2009 (60d) | 2013 (59d) | 2013 (42d) (A-Index > 25) | 2015 (60d) | 2015 (58d) (A-Index > 25) |

H | 0.51 ± 0.03 | 0.55 ± 0.02 | 0.55 ± 0.02 | 0.55 ± 0.01 | 0.55 ± 0.01 |

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**MDPI and ACS Style**

Rifqi, F.N.; Hamid, N.S.A.; Rabiu, A.B.; Yoshikawa, A.
Identification of Fractal Properties in Geomagnetic Data of Southeast Asian Region during Various Solar Activity Levels. *Universe* **2021**, *7*, 248.
https://doi.org/10.3390/universe7070248

**AMA Style**

Rifqi FN, Hamid NSA, Rabiu AB, Yoshikawa A.
Identification of Fractal Properties in Geomagnetic Data of Southeast Asian Region during Various Solar Activity Levels. *Universe*. 2021; 7(7):248.
https://doi.org/10.3390/universe7070248

**Chicago/Turabian Style**

Rifqi, Farhan Naufal, Nurul Shazana Abdul Hamid, A. Babatunde Rabiu, and Akimasa Yoshikawa.
2021. "Identification of Fractal Properties in Geomagnetic Data of Southeast Asian Region during Various Solar Activity Levels" *Universe* 7, no. 7: 248.
https://doi.org/10.3390/universe7070248