Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.01047.x Geostatistical traveltime tomography in elliptically anisotropic media Bernard Giroux ∗ and Erwan Gloaguen Institut national de la recherche scientifique, Centre Eau Terre Environnement, 490, de la Couronne, Qu´ ebec, Qc, G1K 9A9, CANADA Received May 2011, revision accepted November 2011 ABSTRACT In geological materials, anisotropy may arise due to different mechanisms and can be found at different scales. Neglecting anisotropy in traveltime tomographic reconstruc- tion leads to artefacts that can obscure important subsurface features. In this paper, a geostatistical tomography algorithm to invert cross-hole traveltime data in elliptically anisotropic media is presented. The advantages of geostatistical tomography are that the solution is regularized by the covariance of the model parameters, that known model parameters can be used as constraints and fitted exactly or within a prescribed variance and that stochastic simulations can be performed to appraise the variabil- ity of the solution space. The benefits of the algorithm to image anisotropic media are illustrated by two examples using synthetic georadar data and real seismic data. The first example confirms suspected electromagnetic anisotropy in the vadose zone caused by relatively rapid water content variations with respect to wavelength at geo- radar frequencies. The second presents how sonic log data can be used to constrain the inversion of cross-well seismic data and how geostatistical simulations can be used to infer parameter uncertainty. Results of both examples show that considering anisotropy yields a better fit to the data at high ray angles and reduces reconstruction artefacts. Key words: Anisotropy, Tomography, Inverse problem, Electromagnetics, Seismics. INTRODUCTION In geological materials, anisotropy may arise due to different mechanisms and can be found at different scales: preferred mineral alignments in rocks and intrinsically anisotropic crys- tals, lithologic depositional processes leading to elongated or flattened grains, stress-induced micro-cracks aligned by de- viatoric stresses, fluid-filled cracks and fractures can all give rise to anisotropy in electric, magnetic and seismic properties (Thomsen 1986; Negi and Saraf 1989). Further, it is known that if a layered sequence of different media (isotropic or not) is probed with a wave of wavelength much longer than the typical layer thickness, the wave propagates as though it were in a homogeneous, but anisotropic, medium (Backus 1962). ∗ E-mail: bernard.giroux@ete.inrs.ca In sedimentary rocks, clays and fine layering have been men- tioned as the main causes of seismic anisotropy (Wang 2002). Studies in clays have also shown that velocity anisotropy is influenced by the content of organic matter and the inter- action with pore fluids (Vernik and Liu 1997). Macroscopi- cally, anisotropy occurs as a result of the natural fractures of rocks (Schoenberg and Sayers 1995). Anisotropy has reper- cussion for seismic imaging (Tsvankin et al. 2010) but also for ground-penetrating radar since the dielectric constant of water is close to 80 and that for most dry rocks it is 4-8 and a set of aligned, water-filled fractures may present a high degree of dielectric anisotropy. Sch ¨ on (2004) listed the dif- ference in the dielectric permittivity anisotropy ratio between dry and wet samples for a few types of rock, where decreases of about 10-15% can be observed for wet limestones. Besides, anisotropy in the electrical conductivity also affects the pat- tern of the electromagnetic (EM) velocity wave front, albeit C  2012 European Association of Geoscientists & Engineers 1