A new type of stochastic fractal for application in subsurface hydrology

Molz, Fred J.; Hyden, Paul

doi:10.1016/j.geoderma.2006.03.005

Cited by 9 publications

(7 citation statements)

References 17 publications

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“…Analogously to fGn and fBm or fLn and fLm, fLan consists of the increments of fLam; fLan is then stationary whereas fLam is not. Fractional Laplace processes have been found to account for hydrological data more adequately than fractional Lévy ones (Molz & Hyden, 2006); whether the same holds for behavioral data is open to investigation.…”

Section: Discussionmentioning

confidence: 99%

A Bayesian classifier for fractal characterization of short behavioral series.

Solfo¹,

Leeuwen²

2023

Psychological Methods

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Serial tasks in behavioral research often lead to correlated responses, invalidating the application of generalized linear models and leaving the analysis of serial correlations as the only viable option. We present a Bayesian analysis method suitable for classifying even relatively short behavioral series according to their correlation structure. Our classifier consists of three phases. Phase 1 distinguishes between monoand possible multifractal series by modeling the distribution of the increments of the series. To the series labeled as monofractal in Phase 1, classification proceeds in Phase 2 with a Bayesian version of the evenly spaced averaged detrended fluctuation analysis (Bayesian esaDFA). Finally, Phase 3 refines the estimates from the Bayesian esaDFA. We tested our classifier with very short series (viz., 256 points), both simulated and empirical ones. For the simulated series, our classifier revealed to be maximally efficient in distinguishing between mono-and multifractality and highly efficient in assigning the monofractal class. For the empirical series, our classifier identified monofractal classes specific to experimental designs, tasks, and conditions. Monofractal classes are particularly relevant for skilled, repetitive behavior. Short behavioral series are crucial for avoiding potential confounders such as mind wandering or fatigue. Our classifier thus contributes to broadening the scope of time series analysis for behavioral series and to understanding the impact of fundamental behavioral constructs (e.g., learning, coordination, and attention) on serial performance.

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Section: Discussionmentioning

confidence: 99%

A Bayesian classifier for fractal characterization of short behavioral series.

Solfo¹,

Leeuwen²

2023

Psychological Methods

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“…A positive time-delay in CCM in P → SM represents the causality from past P to future SM, whereas a negative one represents the causality from future P to past SM, which are defined as Equations 4.1 and 4.2) respectively: Note. CD takes a fractal value despite existence of negative Lyapunov Exponents, for example, SM3 and SM4 in most areas (Molz & Hyden, 2006). The data marked in red are analyzed in research.…”

Section: Causality In Sm-p Couplingmentioning

confidence: 99%

Detecting the Non‐Separable Causality in Soil Moisture‐Precipitation Coupling With Convergent Cross‐Mapping

Zhao,

Ma,

Zhou

et al. 2024

Water Resources Research

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Soil moisture‐Precipitation (SM‐P) coupling, especially the causality from SM to P (SM → P), is an important and highly debated topic. The causal inference from observational data provides a statistical approach for this issue, where the experimental research is infeasible. Various causal inference methods exist, each assuming distinct underlying systems for the targeted variables: pure stochastic or deterministic dynamic system (DDS), which means that these methods detect different types of causality: separable or non‐separable. Acknowledging the inherent deterministic dynamic nature of SM‐P coupling, this study employs a DDS‐based method: Convergent Cross‐mapping (CCM), to detect their non‐separable causality. Centering around SM‐P coupling, we also detect the causality between SMs in shallow and deeper layers, and the causality between evapotranspiration (ET) and SMs in all layers. Notably, before applying CCM, a preconditional procedure is required: verifying the DDS nature of targeted variables P, SM, and ET. Key findings of this research include: (a) In SM‐P coupling, only SMs in shallow layers but not deeper layers could render causalities toward P, while P renders causalities toward SMs in all layers. (b) The time‐delay of causality from SM1 to P in spring/summer is around 4–6 days, and that from P to SM1 is around 2–4 days. (c) Inside SMs, shallow SMs have obvious causalities downwards to deeper SMs, but deeper SMs seem harder to render causalities upwards to shallow SMs, explaining why they hardly render causalities to P. In summary, this study furnishes an indispensable DDS‐based complement to stochastic‐based methodologies for SM‐P coupling.

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“…Self-similarity means that fractal objects are made of smaller components, which are similar to the original shape and have some properties similar to it after a change in scale (Hardy and Beier 1994;Souza and Rostirolla 2011;Molz and Hyden 2006). An example of fractal objects is shown in Fig.…”

Section: The Fractal Theorymentioning

confidence: 99%

Characterization and estimation of reservoir properties in a carbonate reservoir in Southern Iran by fractal methods

Rahimi

Masihi

2017

J Petrol Explor Prod Technol

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Reservoir heterogeneity has a major effect on the characterization of reservoir properties and consequently reservoir forecast. In reality, heterogeneity is observed in a wide range of scales from microns to kilometers. A reasonable approach to study this multi-scale variations is through fractals. Fractal statistics provide a simple way of relating variations on larger scales to those on smaller scales and vice versa. Simple statistical fractal models (fBm and fGn) can be useful to understand the model construction and help the reservoir structure characterization. In this paper, the fractal methods (fGn and fBm) have been applied to characterize and to estimate of reservoir properties. Three methods, namely box-counting, variogram, and R/S analysis, were carried out on log and core data for porosity and permeability data to estimate fractal dimension; a high fractal dimension estimated in this study reveals a high heterogeneity in the reservoir. Moreover, sampling from simulated fractal data at nonexisting data depths enables us to generate appropriate realizations of reservoir permeability with suitable accuracy at a proper computational time.

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A new type of stochastic fractal for application in subsurface hydrology

Cited by 9 publications

References 17 publications

A Bayesian classifier for fractal characterization of short behavioral series.

A Bayesian classifier for fractal characterization of short behavioral series.

Detecting the Non‐Separable Causality in Soil Moisture‐Precipitation Coupling With Convergent Cross‐Mapping

Characterization and estimation of reservoir properties in a carbonate reservoir in Southern Iran by fractal methods

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