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Partial differential equations for the geometrical spreading of image rays

In this section, we derive the partial differential equations for $ Q$ in 2-D and 3-D. From now on, we will denote the square of the Dix velocity by $ f$ in 2-D and the corresponding matrix by $ \mathbf{F}$ in 3-D, to avoid the subscript:

$\displaystyle \tensor{F}\equiv\frac{\partial}{\partial t_0} \left(\tensor{K}(\mathbf{x}_0,t_0)\right).$ (12)

Furthermore, we imply that $ t_0$ denotes the one-way traveltime along the image rays. Finally, we assume that our domain does not contain caustics, i.e., the image rays do not cross on the interval of time we consider.