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Born traveltime sensitivity

One approach to building a linear finite-frequency traveltime operator is to apply the first-order Born approximation, to obtain a linear relationship between slowness perturbation, $\delta S$, and wavefield perturbation, $\delta U$,
\begin{displaymath}
{\bf\delta U} = {\bf B \; \delta S}.
\end{displaymath} (5)

The Born operator, ${\bf B}$, is a discrete implementation of equation (O-7), which is described in the Appendix.

Traveltime perturbations may then be calculated from the wavefield perturbation through a (linear) picking operator, ${\bf C}$, such that

\begin{displaymath}
{\bf\delta T} = {\bf C \; \delta U} = {\bf C B \; \delta S}
\end{displaymath} (6)

where ${\bf C}$ is a (linearized) picking operator, and a function of the background wavefield, $U_0$.

Cross-correlating the total wavefield, $U(t)$, with $U_0(t)$, provides a way of measuring their relative time-shift, $\delta T$. Marquering et al. (1999) uses this to provide the following explicit linear relationship between $\delta T$ and $\delta U(t)$,

\begin{displaymath}
\delta T =
\frac{\int_{t_1}^{t_2} {\dot U}(t) \; \delta U(t) \; dt}
{\int_{t_1}^{t_2} {\ddot U}(t) \; U(t) \; dt},
\end{displaymath} (7)

where dots denote differentiation with respect to $t$, and $t_1$ and $t_2$ define a temporal window around the event of interest. Equation (7) is only valid for small time-shifts, $\delta T \ll \lambda s_0$.


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Next: Rytov traveltime sensitivity Up: Theory Previous: Theory

2013-03-03