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Fractal Fract
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Fractals in Earthquake and Atmospheric Science
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Fractals in Earthquake and Atmospheric Science

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 30 June 2026 | Viewed by 5206

Special Issue Editor

Prof. Dr. Dimitrios Nikolopoulos

E-Mail Website
Guest Editor
Department of Industrial Design and Production Engineering, University of West Attica, Petrou Ralli & Thivon 250, GR 122 44 Aigaleo, Greece
Interests: fractal analysis; fractal dimension; long-memory; Hurst exponent; DFA; symbolic dynamics; R/S analysis; entropy; Tsallis entropy; earthquakes; pre-seismic precursors; radon; radon progeny; radon in soil; kHz-MHz electromagnetic radiation and ionising radiation physics; radiation dosimetry; radiation exposure; radiation protection and X-rays
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Several types of systems can be described by fractals. This is because nature exhibits scaling behaviour in various processes. This behaviour is revealed when the corresponding systems are dilated, translated, or rotated in space. Many times the related physical procedures are characterised as self-affine or self-similar, in a way that any part of the system is a small- or large-scale representation of it. Due to this, fractal systems can be described by analysing their parts. Moreover, the scaling and fractal properties are linked with the properties of long memory and complexity, in contrast to simple systems which are characterised by linear mechanisms and order. Complex fractal systems with long memory show complex non-Markovian associations of present, past, and future in a way that there are solutions which yield the unavoidable evolution–collapse of the system in terms of space and time. Fractal approaches are robust and significant for the scientific analysis of such systems.

Fractals have been used extensively in Earthquake and Atmospheric Science. In Earthquake Science fractals employed in electromagnetic disturbances from ultra-low frequencies (ULF) between 0.001 Hz and 1 Hz, low frequencies (LF) between 1 and 10 kHz, high frequencies (HF) between 40 and 60 MHz up to very high frequencies (VHF) of the order of 300 MHz are a subject of analysis. Remote sensing techniques and satellite data are also used nowadays since they provide a multi-process framework. For many years radon has been acknowledged as an undoubted earthquake precursor. The related research includes radon in soil, atmosphere, and groundwater; other gases; ions in the atmosphere; and most importantly in active faults. Fractals in Earthquake Science extend within the general framework of the Lithosphere–Ionosphere coupling. In Atmospheric Science fractals have been employed in PM10 and PM2.5 urban air pollution time series; in ozone data; and in NO2, SO2, and CO variations.

Fractal methods in Earthquake and Atmospheric Science include, among others, DFA and MFDFA, R/S and Power-law spectral analysis with wavelets and Fourier Transform, fractal dimensions, Fourier analysis, Hurst and Lyapunov exponents, entropy analysis, symbolic dynamics, and several signal processing methods. Since the related procedures are multi-faceted, the related analysis focuses on the dynamic exchange between the fractal and stochastic behaviour of the investigated systems.

Due to the aforementioned reasons, I would like to invite you to submit papers on your most recent work, experimental study, and case studies related to the subjects mentioned above. Papers that discuss how the aforementioned subjects are related are highly welcomed.

In order to determine early on if the contribution you want to submit fits with the goals of this Special Issue, I kindly ask that you email me a brief synopsis stating the purpose of the research and the main findings.

Prof. Dr. Dimitrios Nikolopoulos
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractals
  • self-organised systems
  • non-linear dynamics and chaos
  • radon
  • electromagnetism
  • geodesy
  • urban air pollution
  • earthquakes
  • modelling and simulation
  • data analysis: algorithms and implementation

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Published Papers (6 papers)

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Research

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25 pages, 7570 KB  
Article
Relationship of Multifractal and Entropic Properties of Global Seismic Noise with Major Earthquakes, 1997–2025
by Alexey Lyubushin and Eugeny Rodionov
Fractal Fract. 2026, 10(4), 267; https://doi.org/10.3390/fractalfract10040267 - 17 Apr 2026
Viewed by 433
Abstract
A method for analyzing long-term (1997–2025) continuous records of low-frequency global seismic noise measured at a network of 229 broadband seismic stations distributed across the Earth’s surface is proposed in this study. The method is based on the use of nonlinear multifractal and [...] Read more.
A method for analyzing long-term (1997–2025) continuous records of low-frequency global seismic noise measured at a network of 229 broadband seismic stations distributed across the Earth’s surface is proposed in this study. The method is based on the use of nonlinear multifractal and entropy statistics, evaluated daily in successive time intervals, of first-principal component analysis, correlation analysis, and parametric models of point process intensity. The relationships between changes in seismic noise properties and the response of noise properties to the irregularity of the Earth’s rotation with the sequence of strong earthquakes, including those of a predictive nature, are investigated. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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33 pages, 2206 KB  
Article
Preliminary Multifractal Rainfall Analysis in the Tunis Region
by Hanen Ghanmi and Cécile Mallet
Fractal Fract. 2026, 10(3), 137; https://doi.org/10.3390/fractalfract10030137 - 24 Feb 2026
Viewed by 410
Abstract
This study investigates the scaling properties of rainfall in Tunis over temporal scales ranging from 5 min to 2.5 years using high-resolution rain gauge data from three recording stations. We employ the Universal Multifractal (UM) framework to characterize scaling properties across multiple temporal [...] Read more.
This study investigates the scaling properties of rainfall in Tunis over temporal scales ranging from 5 min to 2.5 years using high-resolution rain gauge data from three recording stations. We employ the Universal Multifractal (UM) framework to characterize scaling properties across multiple temporal regimes. The UM model was selected over alternative multifractal approaches because of its parsimonious three-parameter formulation (C1, α, H). It explicitly accounts for non-conservative processes through the Fractionally Integrated Flux (FIF) extension and includes established bias correction methods for highly intermittent signals. This framework has demonstrated universality across diverse climatic conditions and enables direct comparison with existing rainfall studies in Mediterranean environments. Spectral analysis reveals three distinct scaling regimes: micro-scale (5 min–2 h 40 min), meso-scale (2 h 40 min–7 days), and synoptic scale (>7 days). The non-conservative nature of the micro-scale regime is addressed through a multifractal fractionally integrated flux model. A key challenge in applying UM analysis to rainfall data is the prevalence of low and zero rain rates (>98% zeros in our dataset). This extreme intermittency introduces significant bias in parameter estimation. Existing correction methods require either continuous rain sequences—scarce in semi-arid climates—or are limited to moderate intermittency levels. We propose an empirical correction method that extends the existing semi-empirical approach by explicitly linking the percentage of zero values to biased UM parameters through empirical relationships applicable to sequences with as few as 50% rainy observations. This advancement enables reliable parameter estimation from highly intermittent datasets. In such conditions, traditional event-by-event analysis yields insufficient samples (only five continuous events longer than 2 h 40 min over 2.5 years in Tunis). The corrected estimates (α = 1.63, C1 = 0.16 for micro-scales) demonstrate strong consistency with continuous rainfall events and align well with high-resolution studies, validating our approach for extreme intermittency conditions characteristic of Mediterranean semi-arid climates. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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18 pages, 4397 KB  
Article
Multifractal and Entropic Properties of Seismic Noise in the Japanese Islands
by Alexey Lyubushin
Fractal Fract. 2026, 10(2), 122; https://doi.org/10.3390/fractalfract10020122 - 12 Feb 2026
Cited by 1 | Viewed by 636
Abstract
This article examines the behavior of seismic noise fields over the Japanese islands recorded by the F-net seismic network for 1997–2025. This paper uses nonlinear noise statistics: the entropy of the wavelet coefficient distribution, the Donoho–Johnston (DJ) wavelet index, and the multifractal singularity [...] Read more.
This article examines the behavior of seismic noise fields over the Japanese islands recorded by the F-net seismic network for 1997–2025. This paper uses nonlinear noise statistics: the entropy of the wavelet coefficient distribution, the Donoho–Johnston (DJ) wavelet index, and the multifractal singularity spectrum support width. These parameters were chosen because their changes reflect the complication or simplification of the noise structure. Changes in the structure of seismic noise properties are analyzed in comparison with a sequence of strong earthquakes. Using a model of the intensity of interacting point processes, the effect of the leading of local noise property extrema relative to the seismic event times is estimated. Using the Hilbert–Huang decomposition, the synchronization of the amplitudes of the envelopes of noise property time series for different IMF levels is estimated. A sequence of weighted probability density maps of extreme values of noise properties is analyzed in comparison with the mega-earthquake of 11 March 2011 and the preparation of another possible strong seismic event. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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18 pages, 3437 KB  
Article
The Hidden Order of Cosmic Rays: Fractal Scaling and Temporal Complexity
by Costas Varotsos
Fractal Fract. 2025, 9(11), 748; https://doi.org/10.3390/fractalfract9110748 - 18 Nov 2025
Cited by 2 | Viewed by 1079
Abstract
This article explores the temporal dynamics and fractal characteristics of cosmic ray (CR) intensity by conducting a comprehensive analysis of their intrinsic scaling properties. The study utilizes sophisticated methodologies, including standard Detrended Fluctuation Analysis (DFA) and the Multifractal DFA (MF-DFA) approach to robustly [...] Read more.
This article explores the temporal dynamics and fractal characteristics of cosmic ray (CR) intensity by conducting a comprehensive analysis of their intrinsic scaling properties. The study utilizes sophisticated methodologies, including standard Detrended Fluctuation Analysis (DFA) and the Multifractal DFA (MF-DFA) approach to robustly evaluate long-memory, self-similarity, and singularity spectra within extensive CR time series. By systematically investigating measurements from two neutron monitor stations with long data archives, the analysis demonstrates the prevalence of multifractal behavior with persistent long-range correlations. Building on the fractal regime revealed in CR time series, this work utilizes the Natural Time Analysis (NTA) tool that is based in the order of occurrence of the extreme cosmic ray events (ECREs). The operational utility of this tool is demonstrated through a case analysis of CR fluctuations during the severe geomagnetic disturbances observed from 9 to 15 May 2024, capturing early-warning signatures and complex temporal responses. Furthermore, the Modified NTA (M-NTA) is used to estimate the occurrence rate of future ECREs. Our findings contribute to a deeper understanding of the scaling laws governing CR intensity and their potential for improving the ECRE modeling, with direct implications for space weather risk mitigation and solar–terrestrial interactions. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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25 pages, 7449 KB  
Article
Influence of Volumetric Geometry on Meteorological Time Series Measurements: Fractality and Thermal Flows
by Patricio Pacheco Hernández, Gustavo Navarro Ahumada, Eduardo Mera Garrido and Diego Zemelman de la Cerda
Fractal Fract. 2025, 9(10), 639; https://doi.org/10.3390/fractalfract9100639 - 30 Sep 2025
Cited by 1 | Viewed by 1014
Abstract
This work analyzes the behavior of the boundary layer subjected to stresses by obstacles using hourly measurements, in the form of time series, of meteorological variables (temperature (T), relative humidity (RH), and magnitude of the wind speed (WS)) in a given period. The [...] Read more.
This work analyzes the behavior of the boundary layer subjected to stresses by obstacles using hourly measurements, in the form of time series, of meteorological variables (temperature (T), relative humidity (RH), and magnitude of the wind speed (WS)) in a given period. The study region is Santiago, the capital of Chile. The measurement location is in a rugged basin geography with a nearly pristine atmospheric environment. The time series are analyzed through chaos theory, demonstrating that they are chaotic through the calculation of the parameters Lyapunov exponent (λ > 0), correlation dimension (DC < 5), Kolmogorov entropy (SK > 0), Hurst exponent (0.5 < H < 1), and Lempel–Ziv complexity (LZ > 0). These series are simultaneous measurements of the variables of interest, before and after, of three different volumetric geometries arranged as obstacles: a parallelepiped, a cylinder, and a miniature mountain. The three geometries are subject to the influence of the wind and present the same cross-sectional area facing the measuring instruments oriented in the same way. The entropies calculated for each variable in each geometry are compared. It is demonstrated, in a first approximation, that volumetric geometry impacts the magnitude of the entropic fluxes associated with the measured variables, which can affect micrometeorology and, by extension, the climate in general. Furthermore, the study examines which geometry favors greater information loss or greater fractality in the measured variables. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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Review

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27 pages, 1994 KB  
Review
Bridging the Scaling Gap: A Review of Nonlinear Paradigms for the Estimation and Understanding of Extreme Rainfall from Heavy Storms
by Kevin K. W. Cheung
Fractal Fract. 2025, 9(12), 827; https://doi.org/10.3390/fractalfract9120827 - 18 Dec 2025
Cited by 1 | Viewed by 643
Abstract
Short-duration extreme rainfall is a major trigger of flash floods and urban inundation, yet its quantification remains a profound challenge due to the scarcity of high-resolution observations. This review synthesizes how three central paradigms of nonlinear science, multifractal cascade theory, self-organized criticality (SOC) [...] Read more.
Short-duration extreme rainfall is a major trigger of flash floods and urban inundation, yet its quantification remains a profound challenge due to the scarcity of high-resolution observations. This review synthesizes how three central paradigms of nonlinear science, multifractal cascade theory, self-organized criticality (SOC) and chaos theory, provide critical insights and practical methodologies for bridging this observational gap. We examine how multifractal temporal downscaling leverages scale-invariance to derive sub-hourly rainfall statistics from coarser data. The SOC paradigm is discussed for its ability to explain the power-law statistics of rainfall extremes and cluster properties, offering a physical basis for estimating rare events. The role of chaos theory and its modern evolution into complex network analysis is explored for diagnosing predictability and spatiotemporal organization. By comparing and integrating these perspectives plus recent developments in stochastic hydrology, this review highlights their collective potential to advance the estimation, understanding, and prediction of short-duration extreme rainfall, ultimately informing improved risk assessment and climate resilience strategies. Full article
(This article belongs to the Special Issue Fractals in Earthquake and Atmospheric Science)
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