Appendix A: Perturbed Traveltime Integral

The eikonal equation can be used to derive the traveltime integral associated with a perturbed slowness medium. This integral is the keystone equation by which the slowness model can be efficiently updated in traveltime tomography.

Let the slowness perturbation from a background slowness field $\mbox{ $\space {s}(\vec{x}) $ }$ be given by $\mbox{ $\space \delta {s}(\vec{x}) $ }$, and let the corresponding perturbed traveltime field be given by $\mbox{ $\space {t}(\vec{x}) $ } +\mbox{ $\space \delta {t}(\vec{x}) $ }$. Here, $\mbox{ $\space {t}(\vec{x}) $ }$ is the unperturbed traveltime field and $\mbox{ $\space \delta {t}(\vec{x}) $ }$ is the traveltime perturbation. The perturbed traveltime field honors the eikonal equation

$$\vert \nabla \mbox{ $ {t}(\vec{x}) $ } + \nabla \mbox{ $ \del... ...t}(\vec{x}) $ } +\vert\mbox{ $ \delta {t}(\vec{x}) $ } \vert^2$$

$$= \mbox{ $ {s}(\vec{x}) $ } ^2 + 2 \mbox{ $ \delta {s}(\vec{x... ...box{ $ {s}(\vec{x}) $ } + \mbox{ $ \delta {s}(\vec{x}) $ } ^2.$$ (2.12)

Subtracting equation 2.12 from the unperturbed eikonal equation we get

$$2 ~ \nabla \mbox{ $ {t}(\vec{x}) $ } \cdot \nabla \mbox{ $ ... ...box{ $ {s}(\vec{x}) $ } + \mbox{ $ \delta {s}(\vec{x}) $ } ^2,$$ (2.13)

and neglecting the terms second order in the perturbation parameters this becomes

$$\nabla \mbox{ $ {t}(\vec{x}) $ } \cdot \nabla \mbox{ $ \delta... ...vert ~ d\hat{l} \cdot \nabla \mbox{ $ \delta {t}(\vec{x}) $ }$$

$$= \mbox{ $ \delta {s}(\vec{x}) $ } \mbox{ $ {s}(\vec{x}) $ } ,$$ (2.14)

where $d\hat{l}$ is defined to be the unit vector parallel to the unperturbed ray direction, so that $\nabla \mbox{ $\space {t}(\vec{x}) $ } = \vert\nabla \mbox{ $\space {t}(\vec{x}) $ } \vert ~ d\hat{l}$. Defining the directional derivative along $d\hat{l}$to be $d/dl = d\hat{l} \cdot \nabla$, and dividing equation 2.14 by $\vert\nabla \mbox{ $\space {t}(\vec{x}) $ } \vert=\mbox{ $\space {s}(\vec{x}) $ }$ gives

$$d \mbox{ $ \delta {t}(\vec{x}) $ } / dl = \mbox{ $ \delta {s}(\vec{x}) $ } .$$ (2.15)

Multiplying both sides by dl and integrating along the old raypath finally yields the perturbed traveltime integral

$$\mbox{ $ \delta {t}(\vec{x}) $ } = \int_{raypath} \delta s(\mbox{ $ \vec{x} $ } ') dl',$$ (2.16)

which is correct to first order in the perturbation parameters. Equation 2.16 says that the traveltime perturbation due to a slowness perturbation is given by an integration over the old raypath weighted by the slowness perturbation. This can be quite cost efficient because the traveltime perturbation calculation uses the old raypaths and does not require the retracing of rays through the perturbed slowness model.