Multifractal Modeling and Analyses of Crustal Heterogeneity

Marsan, David; Bean, Christopher J.

doi:10.1007/978-1-4615-0103-9_8

Cited by 15 publications

(14 citation statements)

References 64 publications

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“…filtering procedures, detrending) and to the use of several algorithms for assessing the coefficients (spectral or Hurst Exponents). Goff and Holliger (1999), covariance; (2) Jones and Holliger (1997), robust power spectrum; (3) Leonardi and Kümpel (1998), power spectrum (periodogram); (4) Leonardi and Kümpel (1998), rescaled-range; (5) Marsan and Bean (1999), power spectrum; (6) Marsan and Bean (1999), structure function, (7) Marsan and Bean (2003), power spectrum; (8) Wu et al (1994), power spectrum, (9) Dolan and Bean (1997), power spectrum; (10) Dolan and Bean (1997), rescaled-range; (11) Holliger (1996), autocovariance; (12) Maus and Dimri (1994), power spectrum; (13) Zhou and Thybo (1998), power spectrum; In our work, the obtained H values are close those derived from the previous researches. That establishes the robustness of the proposed approaches.…”

Section: Resultsmentioning

confidence: 99%

Local regularity analysis of strata heterogeneities from sonic logs

Gaci

Zaourar

Hamoudi

et al. 2010

Nonlin. Processes Geophys.

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Abstract. Borehole logs provide geological information about the rocks crossed by the wells. Several properties of rocks can be interpreted in terms of lithology, type and quantity of the fluid filling the pores and fractures.Here, the logs are assumed to be nonhomogeneous Brownian motions (nhBms) which are generalized fractional Brownian motions (fBms) indexed by depth-dependent Hurst parameters H(z). Three techniques, the local wavelet approach (LWA), the average-local wavelet approach (ALWA), and Peltier Algorithm (PA), are suggested to estimate the Hurst functions (or the regularity profiles) from the logs.First, two synthetic sonic logs with different parameters, shaped by the successive random additions (SRA) algorithm, are used to demonstrate the potential of the proposed methods. The obtained Hurst functions are close to the theoretical Hurst functions. Besides, the transitions between the modeled layers are marked by Hurst values discontinuities. It is also shown that PA leads to the best Hurst value estimations.Second, we investigate the multifractional property of sonic logs data recorded at two scientific deep boreholes: the pilot hole VB and the ultra deep main hole HB, drilled for the German Continental Deep Drilling Program (KTB). All the regularity profiles independently obtained for the logs provide a clear correlation with lithology, and from each regularity profile, we derive a similar segmentation in terms of lithological units. The lithological discontinuities (strata' bounds and faults contacts) are located at the local extrema of the Hurst functions. Moreover, the regularity profiles are compared with the KTB estimated porosity logs, showing a significant relation between the local extrema of the Hurst functions and the fluid-filled fractures. The Hurst function may then constitute a tool to characterize underground heterogeneities.

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Section: Resultsmentioning

confidence: 99%

Local regularity analysis of strata heterogeneities from sonic logs

Gaci

Zaourar

Hamoudi

et al. 2010

Nonlin. Processes Geophys.

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“…25a, we show an example of the multiple scaling of various moments of the KTB porosity estimates taken from Marsan and Bean (2003), other examples of more classical semi variograms (q=2) are more common, see e.g. Cheng (2000a) for an application to Cu concentrations.…”

Section: Observables and Nonconservative Multifractalsmentioning

confidence: 91%

“…4.6). Other relevant examples of multifractal analysis are soil moisture (Dubayah et al, 1997), LAND-SAT TM channels (Cheng, 1999), sonic velocities (Marsan and Bean, 1999) and neutron porosity (Marsan and Bean, 2003); Figs. 25a, b (the latter two in the KTB borehole).…”

Section: Evidence For Multiscaling Of Statistical Momentsmentioning

confidence: 99%

“…Structure function analysis of the first three moments of the neutron porosity estimated from the KTB main borehole. Reproduced from Marsan and Bean (2003).…”

Section: Bare/dressed Soc Divergence Of Momentsmentioning

confidence: 99%

“…Also shown is the envelope of the errors (continuous curves, and well as the ξ(q) estimated by the method of Extended Self-Similarity (this plots one of the moments, usually the third against the others, see Benzi et al, 1993). Reproduced from Marsan and Bean (2003).…”

Section: Observables and Nonconservative Multifractalsmentioning

confidence: 99%

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Scaling and multifractal fields in the solid earth and topography

Lovejoy

Schertzer

2007

Nonlin. Processes Geophys.

124

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Abstract. Starting about thirty years ago, new ideas in nonlinear dynamics, particularly fractals and scaling, provoked an explosive growth of research both in modeling and in experimentally characterizing geosystems over wide ranges of scale. In this review we focus on scaling advances in solid earth geophysics including the topography. To reduce the review to manageable proportions, we restrict our attention to scaling fields, i.e. to the discussion of intensive quantities such as ore concentrations, rock densities, susceptibilities, and magnetic and gravitational fields.We discuss the growing body of evidence showing that geofields are scaling (have power law dependencies on spatial scale, resolution), over wide ranges of both horizontal and vertical scale. Focusing on the cases where both horizontal and vertical statistics have both been estimated from proximate data, we argue that the exponents are systematically different, reflecting lithospheric stratification whichwhile very strong at small scales -becomes less and less pronounced at larger and larger scales, but in a scaling manner. We then discuss the necessity for treating the fields as multifractals rather than monofractals, the latter being too restrictive a framework. We discuss the consequences of multifractality for geostatistics, we then discuss cascade processes in which the same dynamical mechanism repeats scale after scale over a range. Using the binomial model first proposed by de Wijs (1951) as an example, we discuss the issues of microcanonical versus canonical conservation, algebraic ("Pareto") versus long tailed (e.g. lognormal) distributions, multifractal universality, conservative and nonconservative multifractal processes, codimension versus dimension formalisms. We compare and contrast different scaling models (fractional Brownian motion, fractional Levy motion, continuous (in scale) cascades), showing that they are all based on fractional integrations of noises built up from singularity Correspondence to: S. Lovejoy (lovejoy@physics.mcgill.ca) basis functions. We show how anisotropic (including stratified) models can be produced simply by replacing the usual distance function by an anisotropic scale function, hence by replacing isotropic singularities by anisotropic ones.

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Fractal density modeling of crustal heterogeneity from the KTB deep hole

Chen

Cheng

2017

JGR Solid Earth

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Fractal or multifractal concepts have significantly enlightened our understanding of crustal heterogeneity. Much attention has focused on 1/f scaling natures of physicochemical heterogeneity of Earth crust from fractal increment perspective. In this study, fractal density model from fractal clustering point of view is used to characterize the scaling behaviors of heterogeneous sources recorded at German Continental Deep Drilling Program (KTB) main hole, and of special contribution is the local and global multifractal analysis revisited by using Haar wavelet transform (HWT). Fractal density modeling of mass accumulation generalizes the unit of rock density from integer (e.g., g/cm3) to real numbers (e.g., g/cmα), so that crustal heterogeneities with respect to source accumulation are quantified by singularity strength of fractal density in α‐dimensional space. From that perspective, we found that the bulk densities of metamorphic rocks exhibit fractal properties but have a weak multifractality, decreasing with the depth. The multiscaling natures of chemical logs also have been evidenced, and the observed distinct fractal laws for mineral contents are related to their different geochemical behaviors within complex lithological context. Accordingly, scaling distributions of mineral contents have been recognized as a main contributor to the multifractal natures of heterogeneous density for low‐porosity crystalline rocks. This finally allows us to use de Wijs cascade process to explain the mechanism of fractal density. In practice, the proposed local singularity analysis based on HWT is suggested as an attractive high‐pass filtering to amplify weak signatures of well logs as well as to delineate microlithological changes.

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Multifractal Modeling and Analyses of Crustal Heterogeneity

Cited by 15 publications

References 64 publications

Local regularity analysis of strata heterogeneities from sonic logs

Local regularity analysis of strata heterogeneities from sonic logs

Scaling and multifractal fields in the solid earth and topography

Fractal density modeling of crustal heterogeneity from the KTB deep hole

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