Statistical Analysis of Fracture-Length Distribution Sampled Under the Truncation and Censoring Effects
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Statistical Analysis of Fracture-Length Distribution Sampled Under the Truncation and Censoring Effects Dmitriy Kolyukhin · Jan Tveranger
Received: 9 August 2013 / Accepted: 19 December 2013 © International Association for Mathematical Geosciences 2014
Abstract The present paper addresses statistical analysis and estimation of fracturelength distributions at scales influenced by the truncation and censoring effects. The computational method employed here uses fracture-length distributions of a given set of measurements and information about observational constraints (i.e., window of observation) to estimate the probability density of the truncated and censored parts of fracture data sets. The results are benchmarked against power-law based maximal likelihood estimations commonly used for the same purpose. The relationship between the accuracy of estimates and size of the window of observation is studied. The utility of employing statistical models with arbitrary probability distributions of fracture lengths in order to provide a valid statistical model approximation is also considered. A verification of the suggested approximation using the Kolmogorov–Smirnov test applied to truncated and censored data is proposed. Numerical computations show that the proposed method can represent an essential improvement compared to other commonly employed techniques. Keywords Truncation effect · Censoring effect · Power-law distribution · Bayesian analysis · Metropolis-Hastings method
D. Kolyukhin · J. Tveranger Uni CIPR, P. O. Box 7810, 5020 Bergen, Norway Present address: D. Kolyukhin (B) Trofimuk Institute of Petroleum Geology and Geophysics SB RAS. 3, Akademika Koptyuga Prosp., Novosibirsk 630090, Russia e-mail: [email protected]
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Math Geosci
1 Introduction Observational data sets of linear geological features, such as faults and fracture lengths, normally only cover a limited range of scales defined by the window of observation of the acquisition method employed. Statistical models are therefore commonly used to estimate fracture properties at and beyond the upper and lower scale boundaries of the window of observation. Although previous studies have assumed power-law distributions for fracture-length probability density, measured survival functions do not produce linear relations in logarithmic scale (Yielding et al. 1995; Fossen and Rørnes 1996; Odling et al. 1999). The most likely cause of this mismatch is sampling bias during data acquisition. This work deals with sampling bias caused by truncation and censoring (Manzocchi 2009; Torabi and Berg 2011). The term truncation effect describes the fact that there is a lower limit of resolution to any sampling method used for collecting fracture data. The term censoring effect here describes the incomplete sampling of large fractures extending outside the sample domain. Statistical methods more complex than linear regression analysis in logarithmic scale could potentially provide better estimates of fracture lengths at all scales (Main et al. 1999; Kolyukhin a
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