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From Scale Dependence to Scale Invariance: A Multi-metric REV Framework for Representative Surface Roughness Characterization.

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Τίτλος: From Scale Dependence to Scale Invariance: A Multi-metric REV Framework for Representative Surface Roughness Characterization.
Συγγραφείς: Singh, Kuldeep1 (AUTHOR) ckuldeep@kent.edu, Paliwal, Nitin1 (AUTHOR), Gromofsky, Gabrielle1 (AUTHOR)
Πηγή: Mathematical Geosciences. Jul2026, p1-52. 52p.
Θεματικοί όροι: *Surface roughness, *Scale invariance (Statistical physics), *Power spectra, *Measurement uncertainty (Statistics), Fractal dimensions
Περίληψη: Surface roughness characterization lacks standardized protocols for representativity and uncertainty, limiting reproducibility and functional property prediction. We frame this as a convergence problem: determining the minimum measurement area, termed representative elementary volume (REV), such that descriptor estimates converge within tolerance and uncertainty falls below prescribed bounds. This study introduces a multi-metric REV framework with explicit convergence criteria and uncertainty thresholds for geometric (Sa,Sq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${S}_{a},{S}_{q}$$\end{document}), correlational (Lc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L}_{c}$$\end{document}), fractal (D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D$$\end{document}), Hurst scaling (H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}), and spectral (power spectral density, PSD) descriptors using corner-based two-dimensional expansion algorithms applied to high-resolution confocal microscopy data. All metrics exhibit pronounced scale dependence requiring distinct REVs, although not all provide equivalent diagnostic utility. The arithmetic roughness (Sa\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${S}_{a}$$\end{document}) and Hurst exponent (H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}) achieve correlated convergence (REVSa≈REVH\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${REV}_{Sa}\approx {REV}_{H}$$\end{document}), identifying transitions from multifractal to scale-invariant behavior. Correlation length (Lc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L}_{c}$$\end{document}) displays strong scale dependence with material-specific convergence patterns—challenging its traditional treatment as intrinsic—while coarse surfaces often preclude reliable REVC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${REV}_{C}$$\end{document} determination. The global fractal dimension (D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D$$\end{document}) converges reliably but provides no material differentiation, demonstrating that statistical convergence does not guarantee interpretive value. Corner-based uncertainty metrics (σR,σC,σ1/e\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\sigma}_{R},{\sigma}_{C},{\sigma}_{1/e}$$\end{document}) quantify spatial-method variability, enabling assessment of measurement reliability and undersampling costs at sub-REV scales. PSD\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$PSD$$\end{document} analysis shows spectral–spatial disconnect, where roll‑off wavelengths occur substantially below physical REV scales and exhibit monofractal behavior, contrasting with multifractal signatures detected through spatial methods. Collectively, these findings support a transition from fixed-value roughness characterization toward scale-aware, uncertainty-aware, and descriptor-specific frameworks in which representativity itself becomes a measurable surface property. [ABSTRACT FROM AUTHOR]
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  Data: From Scale Dependence to Scale Invariance: A Multi-metric REV Framework for Representative Surface Roughness Characterization.
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Geosciences%22">Mathematical Geosciences</searchLink>. Jul2026, p1-52. 52p.
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  Data: Surface roughness characterization lacks standardized protocols for representativity and uncertainty, limiting reproducibility and functional property prediction. We frame this as a convergence problem: determining the minimum measurement area, termed representative elementary volume (REV), such that descriptor estimates converge within tolerance and uncertainty falls below prescribed bounds. This study introduces a multi-metric REV framework with explicit convergence criteria and uncertainty thresholds for geometric (Sa,Sq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${S}_{a},{S}_{q}$$\end{document}), correlational (Lc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L}_{c}$$\end{document}), fractal (D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D$$\end{document}), Hurst scaling (H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}), and spectral (power spectral density, PSD) descriptors using corner-based two-dimensional expansion algorithms applied to high-resolution confocal microscopy data. All metrics exhibit pronounced scale dependence requiring distinct REVs, although not all provide equivalent diagnostic utility. The arithmetic roughness (Sa\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${S}_{a}$$\end{document}) and Hurst exponent (H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}) achieve correlated convergence (REVSa≈REVH\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${REV}_{Sa}\approx {REV}_{H}$$\end{document}), identifying transitions from multifractal to scale-invariant behavior. Correlation length (Lc\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L}_{c}$$\end{document}) displays strong scale dependence with material-specific convergence patterns—challenging its traditional treatment as intrinsic—while coarse surfaces often preclude reliable REVC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${REV}_{C}$$\end{document} determination. The global fractal dimension (D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D$$\end{document}) converges reliably but provides no material differentiation, demonstrating that statistical convergence does not guarantee interpretive value. Corner-based uncertainty metrics (σR,σC,σ1/e\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\sigma}_{R},{\sigma}_{C},{\sigma}_{1/e}$$\end{document}) quantify spatial-method variability, enabling assessment of measurement reliability and undersampling costs at sub-REV scales. PSD\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$PSD$$\end{document} analysis shows spectral–spatial disconnect, where roll‑off wavelengths occur substantially below physical REV scales and exhibit monofractal behavior, contrasting with multifractal signatures detected through spatial methods. Collectively, these findings support a transition from fixed-value roughness characterization toward scale-aware, uncertainty-aware, and descriptor-specific frameworks in which representativity itself becomes a measurable surface property. [ABSTRACT FROM AUTHOR]
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s11004-026-10314-9
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 52
        StartPage: 1
    Subjects:
      – SubjectFull: Surface roughness
        Type: general
      – SubjectFull: Scale invariance (Statistical physics)
        Type: general
      – SubjectFull: Power spectra
        Type: general
      – SubjectFull: Measurement uncertainty (Statistics)
        Type: general
      – SubjectFull: Fractal dimensions
        Type: general
    Titles:
      – TitleFull: From Scale Dependence to Scale Invariance: A Multi-metric REV Framework for Representative Surface Roughness Characterization.
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            NameFull: Singh, Kuldeep
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            NameFull: Paliwal, Nitin
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            NameFull: Gromofsky, Gabrielle
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            – D: 08
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 18748961
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            – TitleFull: Mathematical Geosciences
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