Geophysical Prospecting, 2010, 58, 577–597 doi: 10.1111/j.1365-2478.2009.00856.x Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part I: 1D ray propagation ‡ Igor Ravve ∗ and Zvi Koren Paradigm Geophysical, Research & Development, Gav-Yam Center, No. 9 Shenkar St., PO Box 2061, Herzliya B 46120, Israel Received May 2009, revision accepted November 2009 ABSTRACT Anisotropy in subsurface geological models is primarily caused by two factors: sed- imentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hy- perbolic and non-hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In gen- eral, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the move- out velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high-order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing. Key words: Anisotropy, Modelling, Inversion, Parameter estimation, Velocity analysis. INTRODUCTION Transversely isotropic models with both vertical and hori- zontal symmetry axes have been extensively studied (e.g., Thomsen 1986). Within a given horizontal transversely isotropic (HTI) layer, the fractures are considered aligned with a specific azimuth (e.g., Bakulin, Grechka and Tsvankin 2000a,b). In this study we extend the existing work on move- ‡ This paper is based on extended abstracts P0126 and P0128 pre- sented at the 71 th EAGE Conference & Exhibition Incorporating SPE EUROPEC 2009, 8–11 June 2009 in Amsterdam, the Netherlands. ∗ E-mail: igor.ravve@pdgm.com out approximation in HTI layered medium (e.g., Al-Dajani and Tsvankin 1998; Grechka and Tsvankin 1999), account- ing for the deviation of the ray velocity from the phase velocity plane at each HTI layer. Consequently, the resulting relation- ships include the direction of the phase velocity, described by its zenith angle, θ phs (angle between the phase velocity and the vertical axis) and azimuth angle, ϕ phs . For a flat reflector, the zenith angle of the phase velocity is also the reflection an- gle. This study consists of two parts. In Part I we study the hyperbolic and non-hyperbolic approximations of the move- out for a package of horizontal transversely isotropic, vertical transversely isotropic and isotropic layers, including the trav- eltime, the magnitude and the direction of the offset on the C  2010 European Association of Geoscientists & Engineers 577