ORIGINAL PAPER Regularity analysis applied to sonic logs data: a case study from KTB borehole site Said Gaci & Naima Zaourar & Louis Briqueu & Matthias Holschneider Received: 16 November 2009 / Accepted: 6 February 2010 / Published online: 19 March 2010 # Saudi Society for Geosciences 2010 AbstractBorehole logs provide in situ information about the fluctuations of petrophysical properties with depth and thus allow the characterization of the crustal heterogeneities. A detailed investigation of these measurements may lead to extract features of the geological media. In this study, we suggesta regularity analysis based on thecontinuous wavelet transform to examine sonic logs data. The descrip- tion of the local behavior of the logs at each depth is carried outusing the local Hurstexponent estimated by two (02) approaches: the local wavelet approach and the average– localwavelet approach. Firstly,a synthetic log, generated using therandom midpoints displacement algorithm, is processed by the regularity analysis. The obtained Hurst curvesallowed thediscernment of thedifferentlayers composing the simulated geological model.Next, this analysis is extended to real sonic logs data recorded at the Kontinentales Tiefbohrprogramm (KTB) pilot borehole (Continental Deep Drilling Program, Germany). The results show a significant correlation between the estimated Hurst exponents and the lithological discontinuities crossed by t well.Hence,the Hurst exponent can be used as a tool to characterize underground heterogeneities. Keywords Regularity analysis . Wavelet transform . Well log . Fractal . Lithology Introduction Kontinentales Tiefbohrprogramm (KTB) pilot and main boreholes provide an opportunity to study heterogeneities of the upper crust,since they dispose a large amount of high quality information, related to earth's properties, recorded at significant depths. The KTB velocity logs have been statistically examined in several researches (Wu et a 1994;Kneib 1995; Holliger 1996; Dolan and Bean 1997; Jones and Holliger 1997;Leonardiand Kümpel 1998; Marsan and Bean 1999). However, the obtained results widely differ, with characteristic scales varying from about 1 m to some kilometers, or without characteristic scale in the case of fractal representation. For the first time, Li and Wu (1993) and Wu et al. (1994) have shown that the measurements of the global power spectrum behave as power-law spectrum and have inferre thatthe geological medium in the KTB site is self-similar. This impliesthatthe resultsof the correlation length estimations are hard to interpret. Kneib (1995) proposed that the autocovariance function of the KTB sonic log data could be approximated by a superposition of two exponen tial functions with correlation lengths of 1 and 20 m. While Holliger (1996) used a von Karman function and estimated a correlation length of 150 m for the KTB pilot borehole S. Gaci (*) : N. Zaourar Département Géophysique—FSTAT, Université des Sciences et de la Technologie Houari Boumediene (USTHB), BP 32 El Alia, 16111 Algiers, Algeria e-mail: said_gaci@yahoo.com N. Zaourar e-mail: naimazaourar@gmail.fr L. Briqueu Laboratoire Géosciences, Université Montpellier 2, Place E.Bataillon, 34095 Montpellier Cedex 05, France e-mail: louis.briqueu@gm.univ-montp2.fr M. Holschneider Institute for Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany e-mail: hols@math.uni-potsdam.de Arab J Geosci (2011) 4:221–227 DOI 10.1007/s12517-010-0129-y