Born traveltime sensitivity

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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,

Cross-correlating the total wavefield, , with , 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

, and

and

define a temporal window around the event of interest. Equation (7) is only valid for small time-shifts, $\delta T \ll \lambda s_0$ .

Next: Rytov traveltime sensitivity Up: Theory Previous: Theory

2013-03-03