Kirchhoff migration using eikonal-based computation of traveltime source-derivatives

Theory and Implementation

We consider the isotropic eikonal equation:

$$\nabla T (\mathbf{x}) \cdot \nabla T (\mathbf{x}) = \frac{1}{v^2 (\mathbf{x})} \equiv W (\mathbf{x})\;,$$ (1)

where $\mathbf{x}$ is a point in space, $T(\mathbf{x})$ is the traveltime and $v(\mathbf{x})$ is the velocity. For 2D models, $\mathbf{x}$ is a vector containing the depth and the inline position. For 3D models, $\mathbf{x}$ also includes the crossline position. For conciseness, we define $W(\mathbf{x})$ as slowness-squared. Equation 1 can be derived by inserting the ray-theory series into the wave-equation and setting the coefficient of the leading-order term to zero (Chapman, 2004). We are interested in particular in point-source solutions of the eikonal equation, i.e. with the initial condition $T(\mathbf{x_s}) = 0$ where $\mathbf{x_s}$ denotes the source location.