Geophysical Prospecting, 2018 doi: 10.1111/1365-2478.12636 Travel time computations using a compact eikonal equation for vertical transverse isotropic media Peyman Moghaddam 1∗ , Reza Khajavi 1 and Henk Keers 2 1 Earthquake Research Center, Ferdowsi University of Mashhad, Mashhad, Iran, and 2 Department of Geosciences, University of Bergen, Norway Received May 2017, revision accepted March 2018 ABSTRACT Eikonal solvers often have stability problems if the velocity model is mildly hetero- geneous. We derive a stable and compact form of the eikonal equation for P-wave propagation in vertical transverse isotropic media. The obtained formulation is more compact than other formulations and therefore computationally attractive. We im- plemented ray shooting for this new equation through a Hamiltonian formalism. Ray tracing based on this new equation is tested on both simple as well as more realistic mildly heterogeneous velocity models. We show through examples that the new equa- tion gives travel times that coincide with the travel time picks from wave equation modelling for anisotropic wave propagation. Key words: Anisotropy, Numerical modelling, Ray tracing. INTRODUCTION Over the past few decades, the growing need for fast and ac- curate prediction of waveform attributes (especially the travel time) in heterogeneous 2D and 3D media has spawned a pro- lific number of grid- and ray-based solvers. It is important to have an efficient travel time solver as they form the back- bone of pre-stack depth migration methods such as pre-stack Kirchoff depth migration (Alkhalifah 2011) and pre-stack Gaussian beam depth migration (Hill 2001). Traditionally, the method of choice has been ray tracing (Julian and Gubbins 1977; Cerveny 1987; Virieux and Farra 1991; Cerveny 2001), in which the trajectory of paths cor- responding to wavefront normals are computed between two points. This approach is often highly accurate and efficient, and naturally lends itself to the prediction of various seis- mic wave properties. However, it is not always robust, and may fail to converge to a true two-point path even in mildly heterogeneous media. Grid-based schemes, which usually involve the calcula- tion of travel times to all points of a regular grid that spans the velocity medium, have become increasingly popular in ∗ E-mail: ppm1407@hotmail.com recent times. They are often based on finite difference solu- tion of the eikonal equation (Kim and Cook 1999; Popovici and Sethian 2002; Rawlinson and Sambridge 2004; Fomel, Luo and Zhao 2009; Luo and Qian 2012; Waheed, Alkhal- ifah and Wang 2015) or shortest path (network) methods (Nakanishi and Yamaguchi 1986; Moser 1991; Cheng and House 1996), both of which tend to be computationally effi- cient and highly robust, which makes them viable alternatives to ray tracing. Wavefronts and rays can be obtained a poste- riori if required by either contouring the travel time field or following the travel time gradient from receiver to source. In the ray-shooting method, which was popular in the 1990s and early 2000s, in the presence of smooth velocity variations, the kinematic ray tracing equations provide the desirable solution, but as soon as complex velocity hetero- geneity is introduced, ray bending distorts the calculation to become complicated or in some cases virtually impossible to solve (Cerveny 1987; Sambridge and Kennett 1990; Virieux and Farra 1991; Velis and Ulrych 1996, 2001; Rawlinson and Sambridge 2004; Waheed et al. 2013). Given the po- tential pitfalls of using an iterative non-linear solver in two point shooting methods, it would appear that fully non-linear solvers would be at least worthy of investigation. However, there are relatively few examples in the recent literature, 1 C  2018 European Association of Geoscientists & Engineers