CONNECTION OF THE LINEARIZED EIKONAL EQUATION AND TRAVELTIME TOMOGRAPHY

The eikonal equation (1) can be rewritten in the form  

$$\bold{n} \cdot \nabla \tau = n \;,$$ (12)

where $\bold{n}$ is the unit vector, pointing in the traveltime gradient direction. The integral solution of equation (12) takes the form  

$$\tau = \int_{\Gamma (\bold{n})} n dl\;,$$ (13)

which states that the traveltime $\tau$ can be computed by integrating the slowness n along the ray $\Gamma (\bold{n})$,tangent at every point to the gradient direction $\bold{n}$ .

Similarly, we can rewrite the linearized eikonal equation (5) in the form  

$$\bold{n_0} \cdot \nabla \tau_1 = n_1 \;,$$ (14)

where $\bold{n_0}$ is the unit vector, pointing in gradient direction for the initial traveltime $\tau_0$. The integral solution of equation (14) takes the form  

$$\tau_1 = \int_{\Gamma (\bold{n_0})} n_1 dl\;,$$ (15)

which states that the traveltime perturbation $\tau_1$ can be computed by integrating the slowness perturbation n₁ along the ray $\Gamma (\bold{n_0})$, defined by the initial slowness model n₀ . This is exactly the basic principle of traveltime tomography.

I have borrowed this proof from Lavrentiev et al. (1970), who used linearization of the eikonal equation as the theoretical basis for traveltime inversion.