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, $\delta U$,

$${\bf \delta U} = {\bf B \; \delta S}.$$ (5)

The Born operator, ${\bf B}$, is a discrete implementation of equation (16), 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

$${\bf \delta T} = {\bf C \; \delta U} = {\bf C B \; \delta S}$$ (6)

where ${\bf C}$ is a (linearized) picking operator, and a function of the background wavefield, U₀.

Cross-correlating the total wavefield, U(t), with U₀(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 $\delta U(t)$, 

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

where dots denote differentiation with respect to t, and t₁ and t₂ define a temporal window around the event of interest. Equation (7) is only valid for small time-shifts, $\delta T \ll \lambda s_0$.