Fractal Exponent Research Articles

Dynamical complexity and multifractal analysis of geomagnetic activities at high temporal scales over three solar cycles

Activities in geospace occur at different time scales. Understanding geomagnetic activity at high temporal scales will give insight into fast dynamics in geospace. This study aims to investigate dynamical complexities in geomagnetic activities at a high temporal scale across three solar cycles. Geomagnetic activities, as represented by 5-min SYM-H data, were considered in this study over three solar cycles (22–24) from 1986 to 2019. Chaos analysis using sample entropy, Lyapunov exponent, and correlation dimension indicates that the geomagnetic activities are driven by intrinsic complex and chaotic processes. Positive Lyapunov exponent values between 0.13 and 0.18, 0.15–0.18, and 0.16–0.19 were obtained for solar cycles 22, 23, and 24 respectively. Furthermore, geomagnetic activities were also found to have multifractal structures driven by high fractal exponents with fine structures. A positive relationship was obtained between the annual mean values of SYM-H and the degree of complexity. It is concluded that geomagnetic activities have a short prediction horizon.

Self-similarity and the direct (enstrophy) cascade in forced two-dimensional fluid turbulence

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$ , in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

Fractal contours: Order, chaos, and art

Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.

A method for calculating fall risk parameters from discrete stride time series regardless of sensor placement

BackgroundTo complement traditional clinical fall risk assessments, research is oriented towards adding real-life gait-related fall risk parameters (FRP) using inertial sensors fixed to a specific body position. While fixing the sensor position can facilitate data processing, it can reduce user compliance. A newly proposed step detection method, Smartstep, has been proven to be robust against sensor position and real-life challenges. Moreover, FRP based on step variability calculated from stride times (Standard deviation (SD), Coefficient of Variance (Cov), fractal exponent, and sample entropy of stride duration) proved to be useful to prospectively predict the fall risk. Research questionsTo evaluate whether Smartstep is convenient for calculating FRP from different sensor placements. Methods29 elderly performed a 6-minute walking test with IMU placed on the waist and the wrist. FRP were computed from step-time estimated from Smartstep and compared to those obtained from foot-mounted inertial sensors: precision and recall of the step detection, Root mean square error (RMSE) and Intraclass Correlation Coefficient (ICC) of stride durations, and limits of agreement of FRP. ResultsThe step detection precision and recall were respectively 99.5% and 95.9% for the waist position, and 99.4% and 95.7% for the wrist position. The ICC and RMSE of stride duration were 0.91 and 54 ms respectively for both the waist and the hand position. The limits of agreement of Cov, SD, fractal exponent, and sample entropy of stride duration are respectively 2.15%, 25 ms, 0.3, 0.5 for the waist and 1.6%, 16 ms, 0.23, 0.4 for the hand. SignificanceRobust against the elderly's gait and different body locations, especially the wrist, this method can open doors toward ambulatory measurements of steps, and calculation of different discrete stride-related falling risk indicators.

Combining detrended cross-correlation analysis with Riemannian geometry-based classification for improved brain-computer interface performance.

Riemannian geometry-based classification (RGBC) gained popularity in the field of brain-computer interfaces (BCIs) lately, due to its ability to deal with non-stationarities arising in electroencephalography (EEG) data. Domain adaptation, however, is most often performed on sample covariance matrices (SCMs) obtained from EEG data, and thus might not fully account for components affecting covariance estimation itself, such as regional trends. Detrended cross-correlation analysis (DCCA) can be utilized to estimate the covariance structure of such signals, yet it is computationally expensive in its original form. A recently proposed online implementation of DCCA, however, allows for its fast computation and thus makes it possible to employ DCCA in real-time applications. In this study we propose to replace the SCM with the DCCA matrix as input to RGBC and assess its effect on offline and online BCI performance. First we evaluated the proposed decoding pipeline offline on previously recorded EEG data from 18 individuals performing left and right hand motor imagery (MI), and benchmarked it against vanilla RGBC and popular MI-detection approaches. Subsequently, we recruited eight participants (with previous BCI experience) who operated an MI-based BCI (MI-BCI) online using the DCCA-enhanced Riemannian decoder. Finally, we tested the proposed method on a public, multi-class MI-BCI dataset. During offline evaluations the DCCA-based decoder consistently and significantly outperformed the other approaches. Online evaluation confirmed that the DCCA matrix could be computed in real-time even for 22-channel EEG, as well as subjects could control the MI-BCI with high command delivery (normalized Cohen's κ: 0.7409 ± 0.1515) and sample-wise MI detection (normalized Cohen's κ: 0.5200 ± 0.1610). Post-hoc analysis indicated characteristic connectivity patterns under both MI conditions, with stronger connectivity in the hemisphere contralateral to the MI task. Additionally, fractal scaling exponent of neural activity was found increased in the contralateral compared to the ipsilateral motor cortices (C4 and C3 for left and right MI, respectively) in both classes. Combining DCCA with Riemannian geometry-based decoding yields a robust and effective decoder, that not only improves upon the SCM-based approach but can also provide relevant information on the neurophysiological processes behind MI.

Fractal-Based Pattern Quantification of Mineral Grains: A Case Study of Yichun Rare-Metal Granite

The quantification of the irregular morphology and distribution pattern of mineral grains is an essential but challenging task in ore-related mineralogical research, allowing for tracing the footprints of pattern-forming geological processes that are crucial to understanding mineralization and/or diagenetic systems. In this study, a large model, namely, the Segmenting Anything Model (SAM), was employed to automatically segment and annotate quartz, lepidolite and albite grains derived from Yichun rare-metal granite (YCRMG), based on which a series of fractal and multifractal methods, including box-counting calculation, perimeter–area analysis and multifractal spectra, were implemented. The results indicate that the mineral grains from YCRMG show great scaling invariance within the range of 1.04~52,300 μm. The automatic annotation of mineral grains from photomicrographs yields accurate fractal dimensions with an error of only 0.6% and thus can be utilized for efficient fractal-based grain quantification. The resultant fractal dimensions display a distinct distribution pattern in the diagram of box-counting fractal dimension (Db) versus perimeter–area fractal dimension (DPA), in which lepidolites are sandwiched between greater-valued quartz and lower-valued albites. Snowball-textured albites, i.e., concentrically arranged albite laths in quartz and K-feldspar, exhibit characteristic Db values ranging from 1.6 to 1.7, which coincide with the fractal indices derived from the fractal growth model. The zonal albites exhibit a strictly increasing trend regarding the values of fractal and multifractal exponents from core to rim, forming a featured “fractal-index banding” in the radar diagram. This pattern suggests that the snowball texture gradually evolved from rim to core, thus leading to greater fractal indices of outer zones, which represent higher complexity and maturity of the evolving system, which supports a metasomatic origin of the snowball texture. Our study demonstrates that fractal analyses with the aid of a large model are effective and efficient in characterizing and understanding complex patterns of mineral grains.

Dense random packing of disks with a power-law size distribution in thermodynamic limit.

The correlation properties of a random system of densely packed disks, obeying a power-law size distribution, are analyzed in reciprocal space in the thermodynamic limit. This limit assumes that the total number of disks increases infinitely, while the mean density of the disk centers and the range of the size distribution are kept constant. We investigate the structure factor dependence on momentum transfer across various number of disks and extrapolate these findings to the thermodynamic limit. The fractal power-law decay of the structure factor is recovered in reciprocal space within the fractal range, which corresponds to the range of the size distribution in real space. The fractal exponent coincides with the exponent of the power-law size distribution as was shown previously by the authors of the work of Cherny et al. [J. Chem. Phys. 158(4), 044114 (2023)]. The dependence of the structure factor on density is examined. As is found, the power-law exponent remains unchanged but the fractal range shrinks when the packing fraction decreases. Additionally, the finite-size effects are studied at extremely low momenta of the order of the inverse system size. We show that the structure factor is parabolic in this region and calculate the prefactor analytically. The obtained results reveal fractal-like properties of the packing and can be used to analyze small-angle scattering from such systems.

Multifractal descriptors ergodically characterize non-ergodic multiplicative cascade processes

Biological and psychological processes routinely break ergodicity, meaning they fail to have stable means (Mean) and independent variation over time that we might find in additive white Gaussian noise (awGn). One possible reason for this failure of ergodicity is the failure of biological and psychological processes to exhibit independence across time. Multifractal evidence has long suggested that biological and psychological processes show strong signatures of nonlinear interactions across scales. These cross-scale interactions sooner befit a cascade-dynamical process than awGn. The present work thus compares awGn to simulations of multiplicative binomial cascades, submitting both types of series and shuffled versions of each to the Thirumalai-Mountain method for estimating ergodicity breaking. Estimating ergodicity breaking for original awGn and cascade series allows us to examine the sources of ergodicity breaking across the sequence, e.g., in temporal correlations specifying nonlinear interactions across scales, and examining ergodicity breaking of the shuffled series allows us to assess the raw, sequence-independent contribution of distributional properties (e.g., the heavy tails of a cascade) without the original temporal sequence. Raw cascade fluctuations and the standard deviation (SD) and root mean square (RMS) series describing those raw fluctuations break ergodicity, but nonlinear, cascade-dynamical descriptors: multifractal spectrum width (Δα) and multifractal nonlinearity (tMF), maintain ergodicity. Interestingly, the fundamentally linear descriptor, fractal Hurst exponent (HfGn) shows moderate ergodicity breaking when describing the fundamentally nonlinear cascade processes, but the linear descriptor coefficient of variation (CV) controls for multiplicative relationships between SD and Mean and maintains ergodicity. We conclude that the ergodicity of statistical descriptors depends on how well they can portray nonlinearity (Δα and tMF) or at least multiplicativity (CV) of the underlying cascade processes.

Clouds and Turbulence Theory: Peculiar Self-Similarity, 4/3 Fractal Exponent and Invariants

In 1982 Lovejoy has published an illustration to Mandelbrot proposal how to characterize the area-perimeter ratio of complicated planar forms and it was found that exponent \(\beta \) for the satellite- and radar-determined cloud and rain areas of such a fractal is 1.35 close to 4/3. Later on it was notified that the same exponent was found also for noctilucent clouds. Such a value might be related to classic turbulence theory of 1941. This text demonstrates this relation using two basic papers by Kolmogorov and Obukhov. The role of prefractal multipliers is revealed, they form a couple of the peculiar invariants for cloud fields and a non-dimensional self-similarity numbers for these fields of sizes \(1 - {{10}^{6}}\,\,{\text{k}}{{{\text{m}}}^{2}}.\) The peculiarity is in their dimensional dependence and in the presence of few invariants, not usual invariants in cloud forms. Further research on random walk of a fluid particle in the 6D phase-space may lead to new discoveries.

Densification of Two Forms of Nanostructured TATB under Uniaxial Die Pressures: A USAXS-SAXS Study.

Sequential ultra-small-angle and small-angle and X-ray scattering (USAXS and SAXS) measurements of hierarchical microstructure of a common energetic material, the high explosive 2,4,6-Triamino-1,3,5-trinitrobenzene (TATB), were performed to follow the microstructure evolution upon applied pressure. The pellets were prepared by two different routes-die pressed from a nanoparticle form and a nano-network form of TATB powder. The derived structural parameters, such as void size, porosity, and the interface area, reflected the response of TATB under compaction. Three populations of voids were observed in the probed q range from 0.007 to 7 nm-1. The inter-granular voids with size larger than 50 nm were sensitive to low pressures and had a smooth interface with the TATB matrix. The inter-granular voids with size of ~10 nm exhibited a less volume-filling ratio at high pressures (>15 kN) as indicated by a decrease of the volume fractal exponent. The response of these structural parameters to external pressures implied that the main densification mechanisms under die compaction were the flow, fracture, and plastic deformation of the TATB granules. Compared to the nanoparticle TATB, the applied pressure strongly influenced the nano-network TATB due to its more uniform structure. The findings and research methods of this work provide insights into the structural evolution of TATB during densification.

Numerical Investigation of Fractional Step-Down ELS Option

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under the fractional Black-Scholes model. We establish Crank-Nicolson scheme under one asset and study the impact of Hurst exponent (H) on return of repayment under fixed stock price. We also explore the impact of stock price on return of repayment under different H. Through numerical experiments, it is found that the return of repayment of options is related to H, and the result of difference scheme will increase with the increase of H. In the case of two assets, we establish implicit scheme, and in the case of three assets, we use operator splitting method (OSM) method to establish semi-implicit scheme. We get the result that the H also influences the return of repayment in two and three assets. We also conduct Greeks analysis. Through Greeks analysis, we find that the long-term correlation of stocks has a huge impact on investment gains or losses. Therefore, we take historical volatility (fractal exponents) into account which can significantly reduce risk and increase revenue for investors.

Universal Langmuir and Fractal Analysis of High-Resolution Adsorption Isotherms of Argon and Nitrogen on Macroporous Silica.

High-resolution isotherms of argon and nitrogen adsorption on macroporous silica have been simulated with universal Langmuir and fractal models. A four-parameter, fractal universal Langmuir equation is a good fit to the data at low pressures. Standard Gibbs energy changes calculated from equilibrium adsorption coefficients show a series of broad peaks that indicate adsorbate structural transformations as a function of pressure and coverage. The Freundlich equation or mean fractal model is also a good fit to isotherms at low pressures. Pressure-varying fractals are accurate fits to the data. Fractal exponents provide information on adsorbate coverage and surface access. Broad peaks in pressure-varying exponents are indicators of adsorbate structure. From adsorptive gas amounts, mean and pressure-varying fractal exponents provide details of adsorbate fractal dimensions and surface roughness. Both Ar and N2 adsorption cause increases in mean surface roughness when compared with pure silica. Surface roughness fluctuations from pressure-dependent adsorptive gas fractal dimensions are associated with adsorbate structure. At one trough, the surface is smooth and is linked to close-packed Ar or N2. For Ar adsorption at 87 K, this structure is a complete monolayer (1.00(4)), while for Ar (77 K), 1.15(4) layers and for N2 (87 K), 2.02(10) layers. The universal Langmuir specific area of the silica is 10.1(4) m2 g-1. Pressure- and coverage-dependent adsorbate structures range from filling defects and holes on the surface to cluster formation to adsorbed Ar or N2 evenly distributed or packed across the surface. The Ar (87 K) isotherm is most sensitive to adsorbate structural transformations.

Clouds and Turbulence Theory: Peculiar Self-Similarity, 4/3 Fractal Exponent and Invariants

Clouds and Turbulence Theory: Peculiar Self-Similarity, 4/3 Fractal Exponent and Invariants

Leveraging a virtual alley with continuously varying width modulates step width variability during self-paced treadmill walking

Increased fall risk in older adults and clinical populations is linked with increased amount and altered temporal structure of step width variability. One approach to rehabilitation seeks to reduce fall risk in older adults by reducing the amount of step width variability and restoring the temporal structure characteristic of healthy young adults. The success of such a program depends on our ability to modulate step width variability effectively. To this end, we investigated how manipulation of the visual walking space in a virtual environment could modulate the amount and temporal structure of step width variability. Nine healthy adults performed self-paced treadmill walking in a virtual alley in a fixed-width Control condition (1.91 m) and two conditions in which the alley’s width oscillated sinusoidally at 0.03 Hz: between 0.38 and 1.14 m and 0.38–2.67 m in Narrow and Wide conditions, respectively. The step width time series from each condition was evaluated using: (i) the standard deviation to identify changes in the amount of variability and (ii) the fractal scaling exponent estimated using detrended fluctuation analysis (DFA) to identify changes in the temporal structure of variability in terms of persistence in fluctuations. The Wide condition neither affected the standard deviation nor the fractal scaling exponent of step width time series. The Narrow condition did not affect the standard deviation of step width time series compared to the Control condition but significantly increased its fractal scaling exponent compared to the Control and Wide conditions, suggestive of more persistent fluctuations characteristic of a healthy gait. These results show that virtual reality based rehabilitative intervention can modulate step width variability to potentially reduce fall risk in older adults and clinical populations.

On the scalar turbulent/turbulent interface of axisymmetric jets

The effect of a zero-mean-flow turbulent ambient on the geometry of the turbulent/turbulent interface (TTI) of an axisymmetric jet is investigated and compared with the traditional turbulent/non-turbulent interface (TNTI). The ambient turbulence is generated using a random jet array. Orthogonal cross-sections of the jet subjected to different levels of ambient turbulence intensities and length scales are captured using planar laser-induced fluorescence. The enhanced radial transport of concentration from the jet core towards the interface and the increased scalar fluctuations within the jet caused by the ambient turbulence result in steeper mean and root-mean-square scalar conditional jumps across the TTI layer compared with those of the TNTI, respectively. When compared with the quiescent ambient, the mean effect of background turbulence is to stretch and corrugate the interface surface area as evident from the wider probability density of the interface radial position, the lowered occurrence of zero-curvature surface elements, the greater misalignment between the radial and normal unit vectors of the interface, the increased tortuosity and the increased magnitude of the fractal exponent.

Levodopa-dependent differences in the non-oscillatory activity of the subthalamic nucleus.

The study of local field potentials (LFP) recorded from the basal ganglia of patients with movement disorders led to significant advancement in the understanding the pathophysiology of Parkinson's disease (PD). The possibility of investigating possible changes in the activity of the brain caused by the levodopa administration may provide a useful tool to evaluate the influence or the side-effects of the treatment from patient to patient. The analysis was carried out through a systematic analysis of the fractal component of the subthalamic local field potentials (STN-LFP) that may reveal, with respect to the classical power spectrum analysis, novel important information about the dynamic modulation caused by the drug intake. Indeed, so far, much of what is known about that is related to the presence of a spectral peak in the beta frequency band then attenuated after the levodopa administration. The nonlinear power-law exponent goes beyond this feature, exploring differences that reflect the fractal (scale-free) behavior of the PD brain dynamics. Here, in order to demonstrate that the presence or absence of the peak has no effect on the computation of the power-law exponent, we used simulated LFP recordings. After that, we performed the fractal analysis in shorts epochs of STN LFPs recordings ( N=24 patients, 12 females and 12 males) before and after Levodopa administration. We found no differences in the nonlinear power-law exponent for simulated data, reinforcing the idea that the parameter was not influenced by the attenuation of the hallmark peak for PD patients. As regard real LFP time series, we found that pharmacological treatment for PD differently altered LFP power of non-oscillatory activity, as well as changed the level of fractal exponent in specific frequency bands. Particularly we observed an increase of the fractal exponent in condition of post-levodopa with significant differences related to the response to levodopa in Parkinson's disease. Clinical Relevance- This study points out a potentially novel non-oscillatory biomarker which could reflect intrinsic properties of complex biological systems thus constituting a potential target parameter for novel and alternative neuroprosthetic applications.

Exploiting fractal features to determine fatigue crack growth rates of metallic materials

An alternative approach to the classical fracture mechanics equation to quantify the fatigue crack growth rate is proposed and examined. The method exploits the fractal features of the propagating fatigue crack. The reference crack growth response of AISI 410 martensitic stainless steel is established using compact tension C(T) specimens. Fractal analysis of microscopic images along the edge length of the crack establishes the fractal dimensions, dF of the fatigue crack. The box-counting algorithm is developed for the fractal analysis based on optical images of the crack at 100X magnification and the resolution of 1090 pixels/mm. Results show that the crack initially exhibits a Euclidean nature (dF ≈ 1). The fractal dimension increases steadily with increasing crack length in the Paris crack growth rate region with 1.05 < dF < 1.24. The corresponding stress intensity factor range varies between 18 ≤ ΔKI ≤ 40 MPa√m. The fractal dimension, dF correlates linearly with the normalized stress intensity factor range, ΔKKIC within the Paris crack growth region. This enables the multifractal fatigue crack growth rate equation to be established in terms of the fractal dimension, Paris coefficient and exponent, and fatigue fracture properties of the material. The fatigue crack growth rate of the material could then be determined using the multifractal fatigue crack growth rate model without requiring the geometry factor of the crack.

Scaling behaviour of braided active channels: a Taylor’s power law approach

At a channel (reach) scale, braided channels are fluvial, geomorphological, complex systems that are characterized by a shift of bars during flood events. In such events water flows are channeled in multiple and mobile channels across a gravel floodplain that remain in unmodified conditions. From a geometrical point of view, braided patterns of the active hydraulic channels are characterized by multicursal nature with structures that are spatially developed by either simple- and multi-scaling behavior. Since current studies do not take into account a general procedure concerning scale measurements, the latter behavior is still not well understood. The aim of our investigation is to analyze directly, through a general procedure, the scaling behavior of hydraulically active channels per transect and per reach analyzed. Our generalized stochastic approach is based on Taylor’s law, and the theory of exponential dispersion distributions. In particular, we make use of a power law, based on the variance and mean of the active channel fluctuations. In this way we demonstrate that the number of such fluctuations with respect to the unicursal behavior of the braided rivers, follows a jump-process of Poisson and compound Poisson–Gamma distributions. Furthermore, a correlation is also provided between the scaling fractal exponents obtained by Taylor’s law and the Hurst exponents.

Comparing adaptive fractal and detrended fluctuation analyses of stride time variability: Tests of equivalence

BackgroundFractal analyses quantify self-similarities in stride-to-stride fluctuations over different time scales. Fractal exponents can be measured with adaptive fractal analysis (AFA) or detrended fluctuation analysis (DFA), though measurements obtained with the algorithms have not been directly compared. Research questionAre stride time fractal exponents measured with AFA and DFA algorithms equivalent? MethodsData from 50 participants with Parkinson’s Disease (n = 15), age-similar healthy adults (n = 15) and healthy young adults (n = 20) were analyzed in this cross-sectional, observational study. Participants completed 6-min walks at self-selected speeds overground on a straight walkway and on a treadmill. Stride times were measured with inertial measurement units. Fractal exponents in stride time data were processed using AFA and DFA algorithms and compared with two one-sided tests of equivalence. Mixed ANOVAs were used to compare exponents between groups and conditions. ResultsFractal exponents computed with AFA and DFA were equivalent neither in the overground (0.796 & 0.830, respectively, p = .587) nor treadmill conditions (0.806 & 0.882, respectively, p = .122). Fractal exponents measured with DFA were higher than when measured with AFA. Standard errors were 22% lower when measured with AFA. Additionally, a group × condition interaction was statistically significant when fractal exponents were processed with the AFA algorithm (F(2,47) = 11.696, p < .001), whereas the group × condition interaction was not statistically significant when DFA exponents were compared (F(2, 47) = 2.144, p = .129). SignificanceAFA and DFA do not produce equivalent estimates of the fractal exponent α in stride time dynamics. Estimates of the fractal exponent α obtained with AFA or DFA algorithms therefore should not be used interchangeably. Standard errors were lower when derived with AFA. Fractal exponents calculated with AFA may be more sensitive to conditions that influence stride time fractal dynamics than are measures calculated with DFA.

Identification of Strata from Irregularities in Well Logs Using Chaos Quantifiers

Summary Estimating a transition between rock types from well logs using traditional methods can be challenging and time-consuming. Developing new approaches to improving the quality of the estimation as well as saving time becomes necessary. This paper presents a new methodology that uses elements of chaos theory to evaluate the variability of well logs to identify rock layers. Four different parameters that quantify chaos were used in the present study: fractal correlation dimension, sample entropy, Hurst exponent, and Lyapunov exponent. Each of them describes a different property of a well log. The method presented in this paper uses all of them together for an extensive characterization of well log irregularities. The study was carried out on a set of 68 well logs from six wells in the Pluto gas field (Australia). The logs were divided into segments of 25 m. A computer program was written to calculate the chaos parameter values of each interval. The parameters were then analyzed statistically. Hierarchical methods and k-means clustering were used to create dendrograms and clusters. The statistical analysis of the results has shown that the well log variability can be used to successfully differentiate rock formations by showing which intervals on a log are similar. In addition, the intervals that correspond to Mungaroo sandstones, which are the reservoir rock of the Pluto gas field, were particularly distinguished from other parts of the log. Therefore, the presented methodology could prove useful to estimate zones of interest in terms of hydrocarbon potential. The presented algorithm accounts for the variability of the well log readings, not the log values themselves. It does not point exactly to a depth where rock layers interface, but it rather allows similar (in terms of irregularities), consecutive intervals to be grouped together. Based on that, one can draw a conclusion that a lithology differs between groups of intervals.