Articles | Volume 16, issue 4/5
https://doi.org/10.5194/se-16-367-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-16-367-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
15 May 2025
Research article |
| 15 May 2025
| 15 May 2025
Unbiased statistical length analysis of linear features: adapting survival analysis to geological applications
Gabriele Benedetti
CORRESPONDING AUTHOR
Dipartimento di Scienze dell'Ambiente e della Terra, Università degli Studi di Milano-Bicocca, 20126 Milan, Italy
Invited contribution by Gabriele Benedetti, recipient of the EGU Energy, Resources and the Environment Outstanding Student and PhD candidate Presentation Award 2024.
Stefano Casiraghi
Dipartimento di Scienze dell'Ambiente e della Terra, Università degli Studi di Milano-Bicocca, 20126 Milan, Italy
Daniela Bertacchi
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, 20126 Milan, Italy
Andrea Bistacchi
Dipartimento di Scienze dell'Ambiente e della Terra, Università degli Studi di Milano-Bicocca, 20126 Milan, Italy
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Traditional methods for investigating the subsurface cannot properly investigate fractures between 1m and 100–200m. Digital outcrop models (DOMs) provide a framework for the collection of extensive datasets in outcrop analogues. Here we present a workflow, with a solid statistical foundation, to collect a suite of statistical parameters to be used as input in current stochastic 3D Discrete Fracture Network models, including best practices for an optimal outcrop selection and data acquisition.
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Traditional methods for investigating the subsurface cannot properly investigate fractures between 1m and 100–200m. Digital outcrop models (DOMs) provide a framework for the collection of extensive datasets in outcrop analogues. Here we present a workflow, with a solid statistical foundation, to collect a suite of statistical parameters to be used as input in current stochastic 3D Discrete Fracture Network models, including best practices for an optimal outcrop selection and data acquisition.
Andrea Bistacchi, Silvia Mittempergher, Mattia Martinelli, and Fabrizio Storti
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We present an innovative workflow for the statistical analysis of fracture data collected along scanlines. Our methodology is based on performing non-parametric statistical tests, which allow detection of important features of the spatial distribution of fractures, and on the analysis of the cumulative spacing function (CSF) and cumulative spacing derivative (CSD), which allows the boundaries of stationary domains to be defined in an objective way.
Cited articles
Akaike, H.: A New Look at the Statistical Model Identification, IEEE T. Automat. Contr., 19, 716–723, https://doi.org/10.1109/TAC.1974.1100705, 1974. a, b
Anderson, T. W. and Darling, D. A.: A Test of Goodness of Fit, J. Am. Stat. A., 49, 765–769, https://doi.org/10.1080/01621459.1954.10501232, 1954. a, b
Andersson, J., Shapiro, A. M., and Bear, J.: A Stochastic Model of a Fractured Rock Conditioned by Measured Information, Water Resour. Res., 20, 79–88, https://doi.org/10.1029/WR020i001p00079, 1984. a
Baecher, G. B.: Progressively Censored Sampling of Rock Joint Traces, J. Int. Ass. Math. Geol., 12, 33–40, https://doi.org/10.1007/BF01039902, 1980. a, b
Baecher, G. B.: Statistical Analysis of Rock Mass Fracturing, J. Int. Ass. Math. Geol., 15, 329–348, https://doi.org/10.1007/BF01036074, 1983. a
Gabriele Benedetti: gecos-lab/FracAbility: FracAbility: Python toolbox to analyse fracture networks for digitalized rock outcrops (v1.5.3), Zenodo [code and data set], https://doi.org/10.5281/zenodo.14893964, 2025. a
Bistacchi, A. and Massironi, M.: Post-Nappe Brittle Tectonics and Kinematic Evolution of the North-Western Alps: An Integrated Approach, Tectonophysics, 327, 267–292, https://doi.org/10.1016/S0040-1951(00)00206-7, 2000. a, b
Bistacchi, A., Eva, E., Massironi, M., and Solarino, S.: Miocene to Present Kinematics of the NW-Alps: Evidences from Remote Sensing, Structural Analysis, Seismotectonics and Thermochronology, J. Geodynam., 30, 205–228, https://doi.org/10.1016/S0264-3707(99)00034-4, 2000. a
Bistacchi, A., Dal Piaz, G., Massironi, M., Zattin, M., and Balestrieri, M.: The Aosta–Ranzola Extensional Fault System and Oligocene–Present Evolution of the Austroalpine–Penninic Wedge in the Northwestern Alps, Int. J. Earth Sci., 90, 654–667, https://doi.org/10.1007/s005310000178, 2001. a
Bistacchi, A., Griffith, W. A., Smith, S. A. F., Di Toro, G., Jones, R., and Nielsen, S.: Fault Roughness at Seismogenic Depths from LIDAR and Photogrammetric Analysis, Pure Appl. Geophys., 168, 2345–2363, https://doi.org/10.1007/s00024-011-0301-7, 2011. a
Bistacchi, A., Balsamo, F., Storti, F., Mozafari, M., Swennen, R., Solum, J., Tueckmantel, C., and Taberner, C.: Photogrammetric Digital Outcrop Reconstruction, Visualization with Textured Surfaces, and Three-Dimensional Structural Analysis and Modeling: Innovative Methodologies Applied to Fault-Related Dolomitization (Vajont Limestone, Southern Alps, Italy), Geosphere, 11, 2031–2048, https://doi.org/10.1130/GES01005.1, 2015. a
Bistacchi, A., Mittempergher, S., Martinelli, M., and Storti, F.: On a new robust workflow for the statistical and spatial analysis of fracture data collected with scanlines (or the importance of stationarity), Solid Earth, 11, 2535–2547, https://doi.org/10.5194/se-11-2535-2020, 2020. a, b, c, d, e, f
Bistacchi, A., Massironi, M., and Viseur, S.: 3D Digital Geological Models: From Terrestrial Outcrops to Planetary Surfaces, in: 3D Digital Geological Models, Chap. 1, 1–9, John Wiley & Sons, Ltd, ISBN 978-1-119-31392-2, https://doi.org/10.1002/9781119313922.ch1, 2022. a
Bonnet, E., Bour, O., Odling, N. E., Davy, P., Main, I., Cowie, P., and Berkowitz, B.: Scaling of Fracture Systems in Geological Media, Rev. Geophys., 39, 347–383, https://doi.org/10.1029/1999RG000074, 2001. a, b, c
Brown, S. R. and Bruhn, R. L.: Fluid Permeability of Deformable Fracture Networks, J. Geophys. Res.-Sol. Ea., 103, 2489–2500, https://doi.org/10.1029/97JB03113, 1998. a
Burnham, K. P. and Anderson, D. R. (Eds.): Model Selection and Multimodel Inference, Springer, New York, NY, ISBN 978-0-387-95364-9, https://doi.org/10.1007/b97636, 2002. a
Burnham, K. P. and Anderson, D. R.: Multimodel Inference: Understanding AIC and BIC in Model Selection, Sociol. Method. Res., 33, 261–304, https://doi.org/10.1177/0049124104268644, 2004. a, b
Cacas, M. C., Ledoux, E., de Marsily, G., Tillie, B., Barbreau, A., Durand, E., Feuga, B., and Peaudecerf, P.: Modeling Fracture Flow with a Stochastic Discrete Fracture Network: Calibration and Validation: 1. The Flow Model, Water Resour. Res., 26, 479–489, https://doi.org/10.1029/WR026i003p00479, 1990. a, b
Clauset, A., Shalizi, C. R., and Newman, M. E. J.: Power-Law Distributions in Empirical Data, SIAM Review, 51, 661–703, https://doi.org/10.1137/070710111, 2009. a, b
Cox, D. R.: Analysis of Survival Data, Chapman and Hall/CRC, New York, ISBN 978-1-315-13743-8, https://doi.org/10.1201/9781315137438, 2017. a
Dal Piaz, G. V. and Lombardo, B.: Early Alpine Eclogite Metamorphism in the Penninic Monte Rosa-Gran Paradiso Basement Nappes of the Northwestern Alps, in: Geological Society of America Memoirs, 164, 249–266, Geological Society of America, ISBN 978-0-8137-1164-5, https://doi.org/10.1130/MEM164-p249, 1986. a
Dal Piaz, G. V., Bistacchi, A., and Massironi, M.: Geological Outline of the Alps, Episodes, 26, 175–180, https://doi.org/10.18814/epiiugs/2003/v26i3/004, 2003. a
Davy, P., Darcel, C., Le Goc, R., Munier, R., Selroos, J.-O., and Mas Ivars, D.: DFN, Why, How and What For, Concepts, Theories and Issues, in: 2nd International Discrete Fracture Network Engineering Conference, 20–22 June 2018, Seattle, Washington, USA, OnePetro, 2018. a
Deluca, A. and Corral, Á.: Fitting and Goodness-of-Fit Test of Non-Truncated and Truncated Power-Law Distributions, Acta Geophys., 61, 1351–1394, https://doi.org/10.2478/s11600-013-0154-9, 2013. a
Dershowitz, W. and Herda, H.: Interpretation of Fracture Spacing and Intensity, in: The 33rd US Symposium on Rock Mechanics (USRMS), 3–5 June 1992, Santa Fe, New Mexico, OnePetro, 1992. a
Dershowitz, W., Hurley, N., and Been, K.: Stochastic Discrete Fracture Modelling of Heterogeneous and Fractured Reservoirs, in: ECMOR III – 3rd European Conference on the Mathematics of Oil Recovery, cp, European Association of Geoscientists & Engineers, ISBN 978-90-6275-785-5, https://doi.org/10.3997/2214-4609.201411069, 1992. a
De Toffoli, B., Massironi, M., Mazzarini, F., and Bistacchi, A.: Rheological and Mechanical Layering of the Crust Underneath Thumbprint Terrains in Arcadia Planitia, Mars, J. Geophys. Res.-Planets, 126, e2021JE007007, https://doi.org/10.1029/2021JE007007, 2021. a
Enders, C. K.: Direct Maximum Likelihood Estimation, in: Encyclopedia of Statistics in Behavioral Science, edited by: Everitt, B. S. and Howell, D. C., Wiley, 1st edn., ISBN 978-0-470-86080-9 978-0-470-01319-9, https://doi.org/10.1002/0470013192.bsa174, 2005. a
Eppes, M. C., Rinehart, A., Aldred, J., Berberich, S., Dahlquist, M. P., Evans, S. G., Keanini, R., Laubach, S. E., Moser, F., Morovati, M., Porson, S., Rasmussen, M., and Shaanan, U.: Introducing standardized field methods for fracture-focused surface process research, Earth Surf. Dynam., 12, 35–66, https://doi.org/10.5194/esurf-12-35-2024, 2024. a
Fisher, R. A.: Statistical Methods, Experimental Design, and Scientific Inference, Oxford University Press, Oxford, England, New York, ISBN 978-0-19-852229-4, 1990. a
Forstner, S. R. and Laubach, S. E.: Scale-Dependent Fracture Networks, J. Struct. Geol., 165, 104748, https://doi.org/10.1016/j.jsg.2022.104748, 2022. a, b
Gueguen, Y., David, C., and Gavrilenko, P.: Percolation Networks and Fluid Transport in the Crust, Geophys. Res. Lett., 18, 931–934, https://doi.org/10.1029/91GL00951, 1991. a
Hoek, E.: Strength of Jointed Rock Masses, Géotechnique, 33, 187–223, https://doi.org/10.1680/geot.1983.33.3.187, 1983. a
Hyman, J. D., Karra, S., Makedonska, N., Gable, C. W., Painter, S. L., and Viswanathan, H. S.: dfnWorks: A Discrete Fracture Network Framework for Modeling Subsurface Flow and Transport, Comput. Geosci., 84, 10–19, https://doi.org/10.1016/j.cageo.2015.08.001, 2015. a, b
Kaplan, E. L. and Meier, P.: Nonparametric Estimation from Incomplete Observations, J. Am. Stat. A., 53, 457–481, https://doi.org/10.2307/2281868, 1958. a, b, c, d
Karim, M. R. and Islam, M. A.: Reliability and Survival Analysis, Springer, Singapore, ISBN 978-981-13-9775-2 978-981-13-9776-9, https://doi.org/10.1007/978-981-13-9776-9, 2019. a, b, c, d
Kim, N.: Tests Based on EDF Statistics for Randomly Censored Normal Distributions When Parameters Are Unknown, Communications for Statistical Applications and Methods, 26, 431–443, https://doi.org/10.29220/CSAM.2019.26.5.431, 2019. a, b, c, d
Kleinbaum, D. G. and Klein, M.: Survival Analysis: A Self-Learning Text, Statistics for Biology and Health, Springer New York, New York, NY, ISBN 978-1-4419-6645-2 978-1-4419-6646-9, https://doi.org/10.1007/978-1-4419-6646-9, 2012. a
Koziol, J. A. and Green, S. B.: A Cramér-von Mises Statistic for Randomly Censored Data, Biometrika, 63, 465–474, https://doi.org/10.1093/biomet/63.3.465, 1976. a, b, c
Laubach, S. E., Lander, R. H., Criscenti, L. J., Anovitz, L. M., Urai, J. L., Pollyea, R. M., Hooker, J. N., Narr, W., Evans, M. A., Kerisit, S. N., Olson, J. E., Dewers, T., Fisher, D., Bodnar, R., Evans, B., Dove, P., Bonnell, L. M., Marder, M. P., and Pyrak-Nolte, L.: The Role of Chemistry in Fracture Pattern Development and Opportunities to Advance Interpretations of Geological Materials, Rev. Geophys., 57, 1065–1111, https://doi.org/10.1029/2019RG000671, 2019. a
Manzocchi, T.: The Connectivity of Two-Dimensional Networks of Spatially Correlated Fractures, Water Resour. Res., 38, 1–1–1–20, https://doi.org/10.1029/2000WR000180, 2002. a, b
Marrett, R., Gale, J. F. W., Gómez, L. A., and Laubach, S. E.: Correlation Analysis of Fracture Arrangement in Space, J. Struct. Geol., 108, 16–33, https://doi.org/10.1016/j.jsg.2017.06.012, 2018. a, b
Martinelli, M., Bistacchi, A., Mittempergher, S., Bonneau, F., Balsamo, F., Caumon, G., and Meda, M.: Damage Zone Characterization Combining Scan-Line and Scan-Area Analysis on a Km-Scale Digital Outcrop Model: The Qala Fault (Gozo), J. Struct. Geol., 140, 104144, https://doi.org/10.1016/j.jsg.2020.104144, 2020. a, b, c
Mauldon, M.: Estimating Mean Fracture Trace Length and Density from Observations in Convex Windows, Rock Mech. Rock Eng., 31, 201–216, https://doi.org/10.1007/s006030050021, 1998. a
Mauldon, M., Dunne, W., and Rohrbaugh, M.: Circular Scanlines and Circular Windows: New Tools for Characterizing the Geometry of Fracture Traces, J. Struct. Geol., 23, 247–258, https://doi.org/10.1016/S0191-8141(00)00094-8, 2001. a, b, c, d
Mittempergher, S. and Bistacchi, A.: Image Analysis Algorithms for Semiautomatic Lineament Detection in Geological Outcrops, in: 3D Digital Geological Models, Chap. 6, 93–107, John Wiley & Sons, Ltd, ISBN 978-1-119-31392-2, https://doi.org/10.1002/9781119313922.ch6, 2022. a
Nicosia, U., Marinotti, N., and Sacchi, E.: The Late Cretaceous Dinosaur Tracksite near Altamura (Bari, Southern Italy), Geologica Romana, 35, 237–247, 1999. a
Nyberg, B., Nixon, C. W., and Sanderson, D. J.: NetworkGT: A GIS Tool for Geometric and Topological Analysis of Two-Dimensional Fracture Networks, Geosphere, 14, 1618–1634, https://doi.org/10.1130/GES01595.1, 2018. a, b
Oda, M.: A Method for Evaluating the Effect of Crack Geometry on the Mechanical Behavior of Cracked Rock Masses, Mech. Mater., 2, 163–171, https://doi.org/10.1016/0167-6636(83)90035-2, 1983. a
Pahl, P. J.: Estimating the Mean Length of Discontinuity Traces, Int. J. Rock Mech. Min., 18, 221–228, https://doi.org/10.1016/0148-9062(81)90976-1, 1981. a
Patacca, E. and Scandone, P.: Geology of the Southern Apennines, Bollettino della Società Geologica Italiana, 7, 75–119, 2007. a
Philip, Z. G., Jennings, J. W., Olson, J. E., Laubach, S. E., and Holder, J.: Modeling Coupled Fracture-Matrix Fluid Flow in Geomechanically Simulated Fracture Networks, SPE Reservoir Evaluation & Engineering, 8, 300–309, https://doi.org/10.2118/77340-PA, 2005. a
Ricchetti, G. and Luperto Sinni, E.: Osservazioni Stratigrafiche e Paleontologiche Preliminari Sugli Strati Con Raadshoovenia Salentina e Murciella Cuvillieri Delle Murge e Della Penisola Salentina, Studi geologici e morfologici sulla regione pugliese, 6, 26, 1979. a
Rizzo, R. E., Healy, D., and De Siena, L.: Benefits of Maximum Likelihood Estimators for Fracture Attribute Analysis: Implications for Permeability and up-Scaling, J. Struct. Geol., 95, 17–31, https://doi.org/10.1016/j.jsg.2016.12.005, 2017. a, b
Rohrbaugh, M., Dunne, W., and Mauldon, M.: Estimating Fracture Trace Intensity, Density, and Mean Length Using Circular Scan Lines and Windows, AAPG Bulletin, 86, 2089–2104, https://doi.org/10.1306/61EEDE0E-173E-11D7-8645000102C1865D, 2002. a
Sanderson, D. J. and Nixon, C. W.: The Use of Topology in Fracture Network Characterization, J. Struct. Geol., 72, 55–66, https://doi.org/10.1016/j.jsg.2015.01.005, 2015. a, b, c, d
Smirnov, N. V.: On the Estimation of the Discrepancy between Empirical Curves of Distribution for Two Independent Samples, Moscow University Mathematics Bulletin, 2, 3–14, 1939. a
Storti, F., Bistacchi, A., Borsani, A., Balsamo, F., Fetter, M., and Ogata, K.: Spatial and Spacing Distribution of Joints at (over-) Saturation in the Turbidite Sandstones of the Marnoso-Arenacea Fm. (Northern Apennines, Italy), J. Struct. Geol., 156, 104551, https://doi.org/10.1016/j.jsg.2022.104551, 2022. a, b, c, d, e
Suzuki, K., Oda, M., Yamazaki, M., and Kuwahara, T.: Permeability Changes in Granite with Crack Growth during Immersion in Hot Water, Int. J. Rock Mech. Min., 35, 907–921, https://doi.org/10.1016/S0148-9062(98)00016-3, 1998. a
Tavakkoli, M., Shaheabadi, A., Khajoee, S., Malakooti, R., and Beidokhti, M. S.: Deterministic versus Stochastic Discrete Fracture Network (DFN) Modeling, Application in a Heterogeneous Naturally Fractured Reservoir, in: SPE Kuwait International Petroleum Conference and Exhibition, vol. All Days of SPE Kuwait International Petroleum Conference and Exhibition, SPE–127 086–MS, https://doi.org/10.2118/127086-MS, 2009. a
Tavani, S., Corradetti, A., and Billi, A.: High Precision Analysis of an Embryonic Extensional Fault-Related Fold Using 3D Orthorectified Virtual Outcrops: The Viewpoint Importance in Structural Geology, J. Struct. Geol., 86, 200–210, https://doi.org/10.1016/j.jsg.2016.03.009, 2016. a
Terzaghi, R. D.: Sources of Error in Joint Surveys, Géotechnique, 15, 287–304, https://doi.org/10.1680/geot.1965.15.3.287, 1965. a
Thiele, S. T., Grose, L., Samsu, A., Micklethwaite, S., Vollgger, S. A., and Cruden, A. R.: Rapid, semi-automatic fracture and contact mapping for point clouds, images and geophysical data, Solid Earth, 8, 1241–1253, https://doi.org/10.5194/se-8-1241-2017, 2017. a
Twiss, R. J. and Moores, E. M.: Structural Geology, Freeman, New York, 2nd edn., 3. print edn., ISBN 978-0-7167-4951-6, 2007. a
Vai, G. B. and Martini, I. P. (Eds.): Anatomy of an Orogen: The Apennines and Adjacent Mediterranean Basins, Springer Netherlands, Dordrecht, ISBN 978-90-481-4020-6 978-94-015-9829-3, https://doi.org/10.1007/978-94-015-9829-3, 2001. a
Zhang, L. and Einstein, H. H.: Estimating the Mean Trace Length of Rock Discontinuities, Rock Mech. Rock Eng., 31, 217–235, https://doi.org/10.1007/s006030050022, 1998. a
Short summary
At any scale, the limited size of a study area introduces a bias in the interpretation of linear features, defined as right-censoring bias. We show the effects of not considering such bias and apply survival analysis techniques to obtain unbiased estimates of multiple parametrical distributions in three censored length datasets. Finally, we propose a novel approach to select the most representative model from a sensible candidate pool using the probability integral transform technique.
At any scale, the limited size of a study area introduces a bias in the interpretation of linear...
