Representative-volume sizing in finite cylindrical computed tomography by low-wavenumber spectral convergence

Abdullah Alqubalee; Christian Tantardini; Fernando Alonso-Marroquin

arxiv: 2601.09283 · v3 · submitted 2026-01-14 · ❄️ cond-mat.soft · physics.geo-ph

Representative-volume sizing in finite cylindrical computed tomography by low-wavenumber spectral convergence

Fernando Alonso-Marroquin , Abdullah Alqubalee , Christian Tantardini This is my paper

Pith reviewed 2026-05-16 14:38 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.geo-ph

keywords representative volume elementCT imagingspectral convergenceaxial detrendingThalassinoidesburrowsitynonstationary fieldstwo-point covariance

0 comments

The pith

Suppressing axial drift and checking low-wavenumber spectral stabilization determines reproducible REV sizes in cylindrical CT scans of trending rock data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a workflow to size representative element volumes from finite cylindrical CT scans when a field variable shows slow axial trends from geology or acquisition artifacts. It first removes the axial drift to produce a residual field, then identifies the smallest inscribed cylinder diameter at which the low-wavenumber part of the isotropic spectrum stabilizes within a chosen tolerance. This matters for samples like Thalassinoides burrow networks, where heterogeneity occurs at scales comparable to the core diameter so that ordinary averaging produces position-dependent statistics. Applied to segmented data, the method produces concrete dimensions D_REV approximately 93 mm and H_REV approximately 83 mm that support consistent correlation-scale reporting and connectivity-sensitive property calculations.

Core claim

Representativeness under nonstationary conditions is established by axial detrending of the scalar burrowsity field followed by convergence of the two-point covariance and its isotropic spectrum on nested cylinders. The smallest diameter at which the low-wavenumber plateau of the spectrum becomes stable is adopted as D_REV, with height chosen analogously; the procedure applied to the Thalassinoides core yields D_REV ≃ 93 mm and H_REV ≃ 83 mm.

What carries the argument

Axial detrending to obtain a residual field, followed by identification of the smallest inscribed-cylinder diameter at which the low-wavenumber plateau of the estimated isotropic spectrum stabilizes.

If this is right

Subvolume statistics for covariance and spectrum become independent of position once the identified REV is reached.
Correlation-scale quantities can be reported reproducibly even when the original data contain slow axial trends.
Digital-rock property estimates that depend on connectivity become insensitive to exact subvolume choice at the converged size.
The same detrending-plus-plateau criterion can be applied to other cylindrical CT datasets that exhibit acquisition or geological drift.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Testing the workflow on synthetic stationary fields with superimposed linear trends would quantify how well the tolerance choice recovers the known true statistics.
The approach could be extended to anisotropic spectra when burrow orientations produce directional dependence in the covariance.
Similar low-wavenumber convergence checks might adapt to REV sizing in 2D slices or other 3D imaging modalities of porous media.

Load-bearing premise

Removing the axial trend leaves a residual field whose low-wavenumber spectral plateau stabilization reliably marks the point at which further volume increase would not alter the statistics.

What would settle it

Showing that the low-wavenumber spectral value at the reported D_REV and H_REV changes by more than the tolerance when the subvolume is translated along the axis or when the diameter is increased by 20 percent.

Figures

Figures reproduced from arXiv: 2601.09283 by Abdullah Alqubalee, Christian Tantardini, Fernando Alonso-Marroquin.

**Figure 2.** Figure 2: FIG. 2. Axial detrending of the slice-wise phase-fraction [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Slice-wise statistics for a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Choosing a representative element volume (REV) from finite cylindrical Computed Tomography (CT) scans becomes ambiguous when a key field variable exhibits a slow axial trend, reflecting either geological variability or CT acquisition/reconstruction artifacts. In such cases, estimated statistics may vary systematically with subvolume size and position rather than converging by simple averaging. We present a practical workflow for sizing an REV under nonstationary conditions by first suppressing axial drift/trend to obtain a residual field suitable for second-order analysis, and then selecting the smallest analysis diameter for which the low-wavenumber spectral content stabilizes within a prescribed tolerance. The method is demonstrated on \textit{Thalassinoides}-bearing rocks, where branching burrow networks introduce heterogeneity at length scales comparable to laboratory core diameters, making imaging-based microstructural statistics and digital-rock estimates sensitive to subvolume choice. From segmented data, we define a scalar ``burrowsity'' field capturing burrow-related pore spaces and infills. Axial detrending, with optional normalization, mitigates acquisition drift and nonstationary trends, while covariance/spectral convergence is evaluated on nested cylinders consistent with the core geometry. Representativeness is posed as diameter convergence on nested inscribed cylinders: the two-point covariance and isotropic spectrum $\widehat{C}$ are estimated, and the smallest diameter at which the low-wavenumber plateau becomes stable is selected. Applied to a segmented \textit{Thalassinoides} core, the method gives $D_{\mathrm{REV}}\simeq 93~\mathrm{mm}$ and $H_{\mathrm{REV}}\simeq 83~\mathrm{mm}$, enabling reproducible correlation-scale reporting and connectivity-sensitive property estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable procedure for picking REV sizes in cylindrical CT with axial trends by detrending then checking low-wavenumber spectral stability, but the jump to connectivity estimates rests on second-order stats that may not capture it.

read the letter

The core contribution is a straightforward workflow: remove slow axial drift from the segmented burrowsity field in a finite cylinder, then find the smallest inscribed diameter where the low-wavenumber plateau of the isotropic spectrum stabilizes inside a user tolerance. They show this on one Thalassinoides core and report D_REV around 93 mm and H_REV around 83 mm. That combination of detrending plus nested-cylinder spectral check is not standard in the REV literature they cite, so the specific recipe is new enough to be worth noting for people who already work with cylindrical CT of heterogeneous rocks. The geometry match and the explicit tolerance on the spectrum make the choice reproducible, which is the practical win here. The demonstration is internally consistent with the steps described. The soft spot is the claim that the resulting REV supports connectivity-sensitive property estimates. Two-point covariance and its spectrum are second-order measures; they can plateau while branching topology or percolation paths in the burrow network still shift with diameter. The paper does not appear to test whether connectivity metrics themselves converge at the same size, so the link between the chosen REV and those estimates is not yet tight. No sensitivity checks on the tolerance or on multiple cores are mentioned, which leaves the robustness open. This is the kind of targeted methods paper that digital-rock groups will read for the workflow even if they later adapt the tolerance or add their own connectivity checks. It is coherent on its own terms and addresses a real pain point in nonstationary CT data, so it deserves a serious referee rather than a desk reject. I would bring it to a reading group for the discussion on what second-order convergence actually buys for topological properties.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a workflow for sizing a representative volume element (REV) in finite cylindrical CT scans of heterogeneous materials exhibiting axial nonstationarity. The procedure involves axial detrending of a scalar 'burrowsity' field derived from segmented Thalassinoides burrow networks to produce a residual suitable for second-order statistics, followed by selection of the smallest inscribed cylinder diameter at which the low-wavenumber plateau of the isotropic spectrum of the two-point covariance stabilizes within a prescribed tolerance. Demonstration on one core yields D_REV ≈ 93 mm and H_REV ≈ 83 mm, purportedly enabling reproducible correlation-scale reporting and connectivity-sensitive property estimation.

Significance. Should the central claim hold, the work addresses a practical challenge in digital rock physics and CT-based microstructural analysis: determining REV sizes in cylindrical samples with trends or artifacts. By leveraging spectral convergence rather than simple variance stabilization, it offers a potentially more robust criterion for nonstationary fields. The specific REV dimensions reported for the Thalassinoides example could serve as a benchmark if the method is shown to generalize and if the link to connectivity properties is substantiated.

major comments (2)

[Abstract] Abstract: The assertion that the determined REV enables 'connectivity-sensitive property estimation' is not supported by the method. The convergence criterion relies on stabilization of the low-wavenumber plateau of the two-point covariance spectrum Ĉ, a second-order statistic. Topological features such as burrow branching, spanning clusters, or percolation paths are not captured by two-point correlations and may vary independently with diameter. A direct check that connectivity metrics converge at the reported D_REV ≃ 93 mm is needed to support the claim.
[Abstract] Abstract: The stabilization tolerance is described only as 'user-prescribed' with no quantitative definition, error estimation procedure, or sensitivity analysis provided. This leaves the specific values D_REV ≃ 93 mm and H_REV ≃ 83 mm without a clear reproducibility path and weakens the central claim that the workflow yields a well-defined REV under nonstationary conditions.

minor comments (2)

The exact computation of the scalar 'burrowsity' field from the segmented burrow networks should be specified (e.g., binary indicator, volume fraction, or intensity map) to allow independent implementation.
[Abstract] Notation: Define Ĉ explicitly as the Fourier transform of the covariance or the power spectral density, and clarify how the isotropic spectrum is obtained from the cylindrical geometry.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments point by point below and outline the revisions we will make to strengthen the work.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that the determined REV enables 'connectivity-sensitive property estimation' is not supported by the method. The convergence criterion relies on stabilization of the low-wavenumber plateau of the two-point covariance spectrum Ĉ, a second-order statistic. Topological features such as burrow branching, spanning clusters, or percolation paths are not captured by two-point correlations and may vary independently with diameter. A direct check that connectivity metrics converge at the reported D_REV ≃ 93 mm is needed to support the claim.

Authors: We agree that two-point covariance statistics primarily capture second-order information and do not directly address topological connectivity such as branching or percolation paths. Our workflow is designed to determine an REV based on spectral convergence of the two-point function under nonstationarity, which is a necessary step for reliable property estimation. To address this, we will revise the abstract to remove the unsubstantiated claim about 'connectivity-sensitive property estimation' and instead emphasize the method's utility for correlation-scale reporting. If space permits, we will add a note that higher-order metrics could be checked separately in future work. revision: yes
Referee: [Abstract] Abstract: The stabilization tolerance is described only as 'user-prescribed' with no quantitative definition, error estimation procedure, or sensitivity analysis provided. This leaves the specific values D_REV ≃ 93 mm and H_REV ≃ 83 mm without a clear reproducibility path and weakens the central claim that the workflow yields a well-defined REV under nonstationary conditions.

Authors: The tolerance is intentionally user-prescribed to accommodate varying data quality and application requirements. However, we recognize the need for greater specificity to ensure reproducibility. In the revised manuscript, we will define the tolerance quantitatively as the point where the relative variation in the low-wavenumber plateau falls below 5% (or another specified threshold), include an error estimation based on the variance of spectral estimates across subregions, and add a sensitivity analysis demonstrating the stability of D_REV and H_REV with respect to tolerance choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity in REV sizing workflow

full rationale

The paper defines a practical workflow: construct a burrowsity scalar field from segmented data, apply axial detrending to obtain a residual, then select the smallest inscribed-cylinder diameter at which the low-wavenumber plateau of the isotropic spectrum Ĉ stabilizes inside a user-prescribed tolerance. This convergence criterion is stated externally and does not reduce the reported D_REV ≃ 93 mm or H_REV ≃ 83 mm to any fitted parameter drawn from the same data by construction. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the derivation chain. The method therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on one domain assumption about the suitability of the detrended field and one user-chosen tolerance parameter; no new physical entities are introduced.

free parameters (1)

stabilization tolerance
User-prescribed tolerance within which the low-wavenumber plateau must stabilize to declare convergence; value not specified in abstract.

axioms (1)

domain assumption After axial detrending the residual field is suitable for second-order statistical analysis
Invoked when the workflow proceeds directly to covariance and spectral estimation on the residual.

pith-pipeline@v0.9.0 · 5599 in / 1300 out tokens · 28572 ms · 2026-05-16T14:38:59.630554+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

[1]

D. Knaust,The trace fossil thalassinoides paradoxicus kennedy, 1967 revisited from its type locality (albian– cenomanian chalk, se england), Palaeogeography, Palaeo- climatology, Palaeoecology634, 111913 (2024)

work page 1967
[2]

H. A. Eltom and A. M. Alqubalee,Quantitative variabil- ity of burrow percentage estimated from 2d views: Exam- ple from thalassinoides-bearing strata, central saudi ara- bia, PALAIOS37, 35–43 (2022)

work page 2022
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H. A. Eltom, A. Alqubalee, and M. A. Yassin,Potential overlooked bioturbated reservoir zones in the shallow ma- rine strata of the hanifa formation in central saudi arabia, Marine and Petroleum Geology124, 104798 (2021)

work page 2021
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A. A. Ekdale and R. G. Bromley,Paleoethologic interpre- tation of complex thalassinoides in shallow-marine lime- stones, lower ordovician, southern sweden, Palaeogeog- raphy, Palaeoclimatology, Palaeoecology192, 221–227 (2003)

work page 2003
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R. G. Bromley and R. W. Frey,Redescription of the trace fossil gyrolithes and taxonomic evaluation of tha- lassinoides, ophiomorpha and spongeliomorpha, Bulletin of the Geological Society of Denmark23, 311–335 (1974), legacy article; DOI not assigned

work page 1974
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H. A. Eltom, A. M. Alqubalee, A. S. Sultan, A. A. Barri, and K. Abdelbasit,Understanding the permeability of 9 burrow-related gas reservoirs through integrated labora- tory techniques, Journal of Natural Gas Science and En- gineering90, 103917 (2021)

work page 2021
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M. Ostoja-Starzewski,Material spatial randomness: From statistical to representative volume element, Prob- abilistic Engineering Mechanics21, 112–132 (2006)

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T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin,Determination of the size of the representa- tive volume element for random composites: Statistical and numerical approach, International Journal of Solids and Structures40, 3647–3679 (2003)

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S. Schl¨ uter, A. P. Sheppard, K. Brown, and D. Wilden- schild,Image processing of multiphase images obtained via X-ray microtomography: A review, Water Resources Research50, 3615–3639 (2014)

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Capillarity in Stationary Random Granular Media: Distribution-Aware Screening and Quantitative Supercell Sizing

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work page internal anchor Pith review Pith/arXiv arXiv 2025
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R. A. Brooks and G. Di Chiro,Beam hardening in x- ray reconstructive tomography, Radiology118, 373–377 (1976)

work page 1976
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Sijbers and A

J. Sijbers and A. Postnov,Reduction of ring artefacts in high resolution micro-CT reconstructions, Physics in Medicine and Biology49, N247–N253 (2004)

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N. Otsu,A threshold selection method from gray-level his- tograms, IEEE Transactions on Systems, Man, and Cy- bernetics9, 62–66 (1979)

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Iassonov, T

P. Iassonov, T. Gebrenegus, and M. Tuller,Segmentation of x-ray computed tomography images of porous materi- als: A crucial step for characterization and quantitative analysis of pore structures, Water Resources Research45, W09415 (2009)

work page 2009
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R. A. Ketcham and R. D. Hanna,Beam hardening cor- rection for x-ray computed tomography of heterogeneous natural materials, Computers & Geosciences67, 49–61 (2014)

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[23]

Wildenschild and A

D. Wildenschild and A. P. Sheppard,X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems, Ad- vances in Water Resources51, 217–246 (2013)

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Singh, K

A. Singh, K. Regenauer-Lieb, S. D. C. Walsh, R. T. Arm- strong, J. J. M. van Griethuysen, and P. Mostaghimi,On representative elementary volumes of grayscale micro-ct images of porous media, Geophysical Research Letters 47, e2020GL088594 (2020)

work page 2020
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Arias-Calluari, M

K. Arias-Calluari, M. N. Najafi, M. S. Harr´ e, Y. Tang, and F. Alonso-Marroquin,Testing stationarity of the de- trended price return in stock markets, Physica A: Statis- tical Mechanics and its Applications587, 126487 (2022)

work page 2022

[1] [1]

D. Knaust,The trace fossil thalassinoides paradoxicus kennedy, 1967 revisited from its type locality (albian– cenomanian chalk, se england), Palaeogeography, Palaeo- climatology, Palaeoecology634, 111913 (2024)

work page 1967

[2] [2]

H. A. Eltom and A. M. Alqubalee,Quantitative variabil- ity of burrow percentage estimated from 2d views: Exam- ple from thalassinoides-bearing strata, central saudi ara- bia, PALAIOS37, 35–43 (2022)

work page 2022

[3] [3]

H. A. Eltom, A. Alqubalee, and M. A. Yassin,Potential overlooked bioturbated reservoir zones in the shallow ma- rine strata of the hanifa formation in central saudi arabia, Marine and Petroleum Geology124, 104798 (2021)

work page 2021

[4] [4]

A. A. Ekdale and R. G. Bromley,Paleoethologic interpre- tation of complex thalassinoides in shallow-marine lime- stones, lower ordovician, southern sweden, Palaeogeog- raphy, Palaeoclimatology, Palaeoecology192, 221–227 (2003)

work page 2003

[5] [5]

R. G. Bromley and R. W. Frey,Redescription of the trace fossil gyrolithes and taxonomic evaluation of tha- lassinoides, ophiomorpha and spongeliomorpha, Bulletin of the Geological Society of Denmark23, 311–335 (1974), legacy article; DOI not assigned

work page 1974

[6] [6]

H. A. Eltom, A. M. Alqubalee, A. S. Sultan, A. A. Barri, and K. Abdelbasit,Understanding the permeability of 9 burrow-related gas reservoirs through integrated labora- tory techniques, Journal of Natural Gas Science and En- gineering90, 103917 (2021)

work page 2021

[7] [7]

H. A. Eltom and R. H. Goldstein,Use of geostatistical modeling to improve understanding of permeability up- scaling in isotropic and anisotropic burrowed reservoirs, Marine and Petroleum Geology129, 105067 (2021)

work page 2021

[8] [8]

Andr¨ a, N

H. Andr¨ a, N. Combaret, J. Dvorkin, E. Glatt, J. Han, M. Kabel, Y. Keehm, F. Krzikalla, M. Lee, C. Madonna, M. Marsh, T. Mukerji, E. H. Saenger, R. Sain, N. Sax- ena, S. Ricker, A. Wiegmann, and X. Zhan,Digital rock physics benchmarks—part II: Computing effective prop- erties, Computers & Geosciences50, 33–43 (2013)

work page 2013

[9] [9]

Bultreys, W

T. Bultreys, W. De Boever, and V. Cnudde,Imaging and image-based fluid transport modeling at the pore scale: A review, Earth-Science Reviews155, 93–128 (2016)

work page 2016

[10] [10]

Bear,Dynamics of Fluids in Porous Media(American Elsevier Publishing Company, New York, 1972)

J. Bear,Dynamics of Fluids in Porous Media(American Elsevier Publishing Company, New York, 1972)

work page 1972

[11] [11]

Ostoja-Starzewski,Material spatial randomness: From statistical to representative volume element, Prob- abilistic Engineering Mechanics21, 112–132 (2006)

M. Ostoja-Starzewski,Material spatial randomness: From statistical to representative volume element, Prob- abilistic Engineering Mechanics21, 112–132 (2006)

work page 2006

[12] [12]

Kanit, S

T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin,Determination of the size of the representa- tive volume element for random composites: Statistical and numerical approach, International Journal of Solids and Structures40, 3647–3679 (2003)

work page 2003

[13] [13]

Torquato,Random Heterogeneous Materials: Mi- crostructure and Macroscopic Properties(Springer, New York, 2002)

S. Torquato,Random Heterogeneous Materials: Mi- crostructure and Macroscopic Properties(Springer, New York, 2002)

work page 2002

[14] [14]

Al-Raoush and A

R. Al-Raoush and A. Papadopoulos,Representative el- ementary volume analysis of porous media using X-ray computed tomography, Powder Technology200, 69–77 (2010)

work page 2010

[15] [15]

Schl¨ uter, A

S. Schl¨ uter, A. P. Sheppard, K. Brown, and D. Wilden- schild,Image processing of multiphase images obtained via X-ray microtomography: A review, Water Resources Research50, 3615–3639 (2014)

work page 2014

[16] [16]

Capillarity in Stationary Random Granular Media: Distribution-Aware Screening and Quantitative Supercell Sizing

C. Tantardini and F. Alonso-Marroquin, Capillarity in stationary random granular media: Distribution- aware screening and quantitative supercell sizing (2025), arXiv:2509.21350 (v2, revised 8 Dec 2025), arXiv:2509.21350 [cond-mat.soft]

work page internal anchor Pith review Pith/arXiv arXiv 2025

[17] [17]

R. A. Brooks and G. Di Chiro,Beam hardening in x- ray reconstructive tomography, Radiology118, 373–377 (1976)

work page 1976

[18] [18]

J. F. Barrett and N. Keat,Artifacts in CT: Recognition and avoidance, Radiographics24, 1679–1691 (2004)

work page 2004

[19] [19]

Sijbers and A

J. Sijbers and A. Postnov,Reduction of ring artefacts in high resolution micro-CT reconstructions, Physics in Medicine and Biology49, N247–N253 (2004)

work page 2004

[20] [20]

Otsu,A threshold selection method from gray-level his- tograms, IEEE Transactions on Systems, Man, and Cy- bernetics9, 62–66 (1979)

N. Otsu,A threshold selection method from gray-level his- tograms, IEEE Transactions on Systems, Man, and Cy- bernetics9, 62–66 (1979)

work page 1979

[21] [21]

Iassonov, T

P. Iassonov, T. Gebrenegus, and M. Tuller,Segmentation of x-ray computed tomography images of porous materi- als: A crucial step for characterization and quantitative analysis of pore structures, Water Resources Research45, W09415 (2009)

work page 2009

[22] [22]

R. A. Ketcham and R. D. Hanna,Beam hardening cor- rection for x-ray computed tomography of heterogeneous natural materials, Computers & Geosciences67, 49–61 (2014)

work page 2014

[23] [23]

Wildenschild and A

D. Wildenschild and A. P. Sheppard,X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems, Ad- vances in Water Resources51, 217–246 (2013)

work page 2013

[24] [24]

Singh, K

A. Singh, K. Regenauer-Lieb, S. D. C. Walsh, R. T. Arm- strong, J. J. M. van Griethuysen, and P. Mostaghimi,On representative elementary volumes of grayscale micro-ct images of porous media, Geophysical Research Letters 47, e2020GL088594 (2020)

work page 2020

[25] [25]

Arias-Calluari, M

K. Arias-Calluari, M. N. Najafi, M. S. Harr´ e, Y. Tang, and F. Alonso-Marroquin,Testing stationarity of the de- trended price return in stock markets, Physica A: Statis- tical Mechanics and its Applications587, 126487 (2022)

work page 2022