  The eikonal equation is instrumental in many applications in several fields
ranging from computer vision to geoscience. This equation can be efficiently
solved using the iterative Fast Sweeping (FS) methods and the direct Fast
Marching (FM) methods. However, when used for a point source, the original
eikonal equation is known to yield inaccurate numerical solutions, because of a
singularity at the source. In this case, the factored eikonal equation is often
preferred, and is known to yield a more accurate numerical solution. One
application that requires the solution of the eikonal equation for point
sources is travel time tomography. This inverse problem may be formulated using
the eikonal equation as a forward problem. While this problem has been solved
using FS in the past, the more recent choice for applying it involves FM
methods because of the efficiency in which sensitivities can be obtained using
them. However, while several FS methods are available for solving the factored
equation, the FM method is available only for the original eikonal equation.
  In this paper we develop a Fast Marching algorithm for the factored eikonal
equation, using both first and second order finite-difference schemes. Our
algorithm follows the same lines as the original FM algorithm and requires the
same computational effort. In addition, we show how to obtain sensitivities
using this FM method and apply travel time tomography, formulated as an inverse
factored eikonal equation. Numerical results in two and three dimensions show
that our algorithm solves the factored eikonal equation efficiently, and
demonstrate the achieved accuracy for computing the travel time. We also
demonstrate a recovery of a 2D and 3D heterogeneous medium by travel time
tomography using the eikonal equation for forward modelling and inversion by
Gauss-Newton.
