Rytov traveltime sensitivity

The first Rytov approximation (or the phase-field linearization method, as it is also known) provides a linear relationship between the slowness and complex phase perturbations.

$${\bf \delta \Psi} = {\bf R \; \delta S},$$ (8)

where ${\bf \Psi}=\exp({\bf U})$, and the Rytov operator, ${\bf R}$,is a discrete implementation of equation (19), which is also described in Appendix A.

Traveltime is related to the complex phase by the equation, $\Im (\delta \psi)=\omega \, \delta t$.For a band-limited arrival with amplitude spectrum, $F(\omega)$,traveltime perturbation can be calculated simply by summing over frequency Woodward (1992),

$${\bf \delta T} = \sum_\omega \frac{F(\omega)}{\omega} \; \Im... ...ac{F(\omega)}{\omega} \;\Im \left( {\bf R \; \delta S}\right).$$ (9)

Of the two approximations, several authors Beydoun and Tarantola (1988); Woodward (1989) note that the Born approximation is the better choice for modeling reflected waves, while the Rytov approximation is better for transmitted waves. Differences tend to zero, however, as the scattering becomes small.