Homothetic scaling of travel-times

Audebert 1996 shows that if the slowness field is globally scaled, ray geometry is unchanged. Audebert 1996 points out that starting from slowness field $S(\bar{x})_{orig}$, the travel-time from $\bar{x}_1$ to $\bar{x}_2$ is:

$$T_{orig}(\bar{x}_1,\bar{x}_2) = \int_{L_(\bar{x}_1,\bar{x}_2)} S_{orig} dl$$ (1)

where $L_(\bar{x}_1,\bar{x}_2)$ is the ray-path between $\bar{x}_1$ and $\bar{x}_2$. If we scale S_orig by a positive constant $\gamma$:

$$\begin{eqnarray} T_{\gamma}(\bar{x}_1,\bar{x}_2) &=& \int_{L_(\bar{x}_1,\bar{x}_2)} \gamma S_{orig} dl \\ &=& \gamma T_{orig}(\bar{x}_1,\bar{x}_2)\end{eqnarray}$$ (2) (3)

$L_(\bar{x}_1,\bar{x}_2)$ is the same in both cases because the original Snell's law at each interface can be reduced from the scaled travel-time version:

$$\begin{eqnarray} \gamma v_1 sin \theta_1 &=& \gamma v_2 sin \theta_2 \\ v_1 sin \theta_1 &=& v_2 sin \theta_2\end{eqnarray}$$ (4) (5)

As a result, scaled versions of the travel-time field represent valid focusing operators. By finding the scaled (by $\gamma$) version of the travel-time field that best focuses the data we get an estimate of the error in our interval velocity model.