Quantification of Fracture Roughness by Change Probabilities and Hurst Exponents

The objective of the current study is to utilize an innovative method called “change probabilities” for describing fracture roughness. In order to detect and visualize anisotropy of rock joint surfaces, the roughness of one-dimensional profiles taken in different directions is quantified. The central quantifiers, change probabilities, are based on counting monotonic changes in discretizations of a profile. These probabilities, which usually vary with the scale, can be reinterpreted as scale-dependent Hurst exponents. For a large class of Gaussian stochastic processes, change probabilities are shown to be directly related to the classical Hurst exponent, which generalizes a relationship known for fractional Brownian motion. While related to this classical roughness measure, the proposed method is more generally applicable, therefore increasing the flexibility of modeling and investigating surface profiles. In particular, it allows a quick and efficient visualization and detection of roughness anisotropy and scale dependence of roughness.

EEG Signal Processing for Survey of Dynamic Auditory Verbal Learning and Memory Formation in Brain by Fractal Analysis

Learning and memory involve a complex cognitive process to acquire, retain, and retrieve information in the central nervous system. However, the brain mechanism still needs to be well understood. This study aimed to examine the dynamic auditory verbal learning model of the brain mechanism involved in cognitive learning using the scale-free approach by the fractal analysis of electroencephalogram (EEG) data. This illustrates how the complexity of information processing in the brain changes while auditory and verbal learning occurs. Therefore, a standard verbal-auditory cognitive assessment test was used to create a learning paradigm. Eighteen healthy male volunteers (19–23[Formula: see text]years old) were recruited and their verbal memories were assessed using the Rey auditory verbal learning test. Fifteen unrelated words were sequentially presented to the subjects and they were asked to recall the presented words as many as possible. The experiment was repeated five times with no stop in between. EEG recording was performed before, during and after each stage. Subsequently, the Hurst exponents of EEG were calculated and their associations with the recalled words and the learning rate were estimated. The approximate entropy was intended to confirm the Hurst exponent variations of signals. The statistical analysis of the data showed that the increase in the number of the recalled words was positively correlated with an increase in the Hurst exponents of EEG signals (more significant at the temporal channels) and a decrease in the approximate entropy of EEG signals during the learning of trials. These results denoted a reduced complexity pattern in EEG signals while rehearsing auditory and verbal memories.

Roughness of fracture surfaces in numerical models and laboratory experiments

We investigate the influence of stress conditions during fracture formation on the geometry and roughness of fracture surfaces. Rough fracture surfaces have been generated in numerical simulations of triaxial deformation experiments using the discrete element method and in a small number of laboratory experiments on limestone and sandstone samples. Digital surface models of the rock samples fractured in the laboratory experiments were produced using high-resolution photogrammetry. The roughness of the surfaces was analyzed in terms of absolute roughness measures such as an estimated joint roughness coefficient (JRC) and in terms of its scaling properties. The results show that all analyzed surfaces are self-affine but with different Hurst exponents between the numerical models and the real rock samples. Results from numerical simulations using a wide range of stress conditions to generate the fracture surfaces show a weak decrease of the Hurst exponents with increasing confining stress and a larger absolute roughness for transversely isotropic stress conditions compared to true triaxial conditions. Other than that, our results suggest that stress conditions have little influence on the surface roughness of newly formed fractures.

A Quadratic–Exponential Model of Variogram Based on Knowing the Maximal Variability: Application to a Rainfall Time Series

Variogram models are a valuable tool used to analyze the variability of a time series; such variability usually entails a spherical or exponential behavior, and so, models based on such functions are commonly used to fit and explain a time series. Variograms have a quasi-periodic structure for rainfall cases, and some extra steps are required to analyze their entire behavior. In this work, we detailed a procedure for a complete analysis of rainfall time series, from the construction of the experimental variogram to curve fitting with well-known spherical and exponential models, and finally proposed a novel model: quadratic–exponential. Our model was developed based on the analysis of 6 out of 30 rainfall stations from our case study: the Río Bravo–San Juan basin, and was constructed from the exponential model while introducing a quadratic behavior near to the origin and taking into account the fact that the maximal variability of the process is known. Considering a sample with diverse Hurst exponents, the stations were selected. The results obtained show robustness in our proposed model, reaching a good fit with and without the nugget effect for different Hurst exponents. This contrasts to previous models, which show good outcomes only without the nugget effect.

Testing reliability of the spatial Hurst exponent method for detecting a change point

The reliability of using abrupt changes in the spatial Hurst exponent for identifying temporal points of abrupt change in climate dynamics is explored. If a spatio-temporal dynamical system undergoes an abrupt change at a particular time, the time series of spatial Hurst exponent obtained from the data of any variable of the system should also show an abrupt change at that time. As expected, spatial Hurst exponents for each of the two variables of a model spatio-temporal system – a globally coupled map lattice based on the Burgers' chaotic map – showed abrupt change at the same time that a parameter of the system was changed. This method was applied for the identification of change points in climate dynamics using the NCEP/NCAR data on air temperature, pressure and relative humidity variables. Different abrupt change points in spatial Hurst exponents were detected for the data of these different variables. That suggests, for a dynamical system, change point detected using the two-dimensional detrended fluctuation analysis method on a single variable alone is insufficient to comment about the abrupt change in the system dynamics and should be based on multiple variables of the dynamical system.

Scaling Features of Diurnal Variation of Galactic Cosmic Rays

We analyze the scaling properties of the diurnal variation of galactic cosmic rays (GCRs) in Solar Cycle 24 and the solar minima between Solar Cycles 23/24 and 24/25 for 2007 – 2019 based on the count rates of the Oulu, Newark, Hermanus, and Potchefstroom neutron monitors. The scaling features of the GCR diurnal variation are studied by evaluating the Hurst exponent, a quantitative parameter used as an indicator of the state of the randomness of a time series. We estimate the Hurst exponents for GCR diurnal-variation parameters amplitude and phase using structure-function and detrended-fluctuation-analysis methods. Results show that the Hurst exponents for the GCR diurnal variation vary in the range from $\approx0.3$ ≈ 0.3 to $\approx0.9$ ≈ 0.9 , with a general tendency of being systematically above 0.5. It suggests that the GCR diurnal variation reveals a more persistent structure than Brownian motion. However, the time series of GCR diurnal-variation amplitude and phase evolve from a more persistent structure in the solar minimum between Solar Cycles 23/24 in 2007 – 2009 to a more random character in and near the solar maximum 2012 – 2014. This observation seems to be in agreement with the general configuration of the heliosphere through the 11-year solar-activity cycle. Moreover, the temporal profile of the Hurst exponent for GCR diurnal amplitude and phase around the beginning of the solar minimum between Solar Cycles 24/25 (2018 – 2019) differs from the solar minimum between Solar Cycles 23/24 in 2007 – 2009, suggesting a dependence on solar-magnetic polarity. These findings could shed more light on GCR particle transport in the turbulent heliosphere over the solar cycle.

Roughness of Fracture Surfaces in Numerical Models and Laboratory Experiments

We investigate the influence of stress conditions during fracture formation on the geometry and roughness of fracture surfaces. Rough fracture surfaces have been generated in numerical simulations of triaxial deformation experiments using the Discrete Element Method and in laboratory experiments on limestone and sandstone samples. Digital surface models of the rock samples fractured in the laboratory experiments were produced using high resolution photogrammetry. The roughness of the surfaces was analyzed in terms of absolute roughness measures such as an estimated joint roughness coefficient (JRC) and in terms of its scaling properties. The results show that all analyzed surfaces are self-affine, but with different Hurst exponents between the numerical models and the real rock samples. Results from numerical simulations using a wide range of stress conditions to generate the fracture surfaces show a weak decrease of the Hurst exponents with increasing confining stress and a larger absolute roughness for transversely isotropic stress conditions compared to true triaxial conditions. Other than that, our results suggest that stress conditions have little influence on the surface roughness of newly formed fractures.

A Simplified Multifractal Model for Self-Similar Traffic Flows in High-Speed Computer Networks

<div>This paper proposes a multifractal model, with the aim of providing a possible explanation for the locality phenomenon that appears in the estimation of the Hurst exponent in stationary second order temporal series representing self-similar traffic flows in current high-speed computer networks. It is shown analytically that this phenomenon occurs if the network flow consists of several components with different Hurst exponents.</div>