Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?

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Appendix A

Born/Rytov review

Modeling with the first-order Born (and Rytov) approximations [e.g. Beydoun and Tarantola (1988)] can be justified by the assumption that slowness heterogeneity in the earth exists on two separate scales: a smoothly-varying background, $s_0$, within which the ray-approximation is valid, and weak higher-frequency perturbations, $\delta s$, that act to scatter the wavefield. The total slowness is given by the sum,

$$s({\bf x})=s_0({\bf x})+\delta s({\bf x}).$$ (10)

Similarly, the total wavefield, $U$, can be considered as the sum of a background wavefield, $U_0$, and a scattered field, $\delta U$, so that

$$U({\bf x},\omega)=U_0({\bf x},\omega)+\delta U({\bf x},\omega),$$ (11)

where $U_0$ satisfies the Helmholtz equation in the background medium,

$$\left[ \nabla^2 + \omega^2 s_0^2({\bf x})\right] U_0({\bf x},\omega) = 0,$$ (12)

and the scattered wavefield is given by the (exact) non-linear integral equation (Morse and Feshbach, 1953),

$$\delta U({\bf x},\omega)=\frac{\omega^2}{4 \pi} \int_V G_0({... ...bf x},\omega; {\bf x}') \delta s({\bf x}') \; dV({\bf x}').$$ (13)

In equation (O-4), $G_0$ is the Green's function for the Helmholtz equation in the background medium: i.e. it is a solution of the equation

$$\left[ \nabla^2 + \omega^2 s_0^2({\bf x}) \right] G_0({\bf x},\omega ; {\bf x}_s) = -4 \pi \delta({\bf x} - {\bf x}_s).$$ (14)

Since the background medium is smooth, in this paper I use Green's functions of the form,

$$G_0({\bf x},\omega ; {\bf x}_s) = A_0({\bf x},{\bf x}_s) e^{i \omega T_0({\bf x},{\bf x}_s)}.$$ (15)

where $A_0$ and $T_0$ are ray-traced traveltimes and amplitudes respectively.

A Taylor series expansion of $U$ about $U_0$ for small $\delta s$, results in the infinite Born series, which is a Neumann series solution (Arfken, 1985) to equation (O-4). The first term in the expansion is given below: it corresponds to the component of wavefield that interacts with scatters only once.

$$\delta U_{\rm Born}({\bf x},\omega)=\frac{\omega^2}{4 \pi} \... ...U_0({\bf x},\omega; {\bf x}') \delta s({\bf x}') dV({\bf x}').$$ (16)

The approximation implied by equation (O-7) is known as the first-order Born approximation. It provides a linear relationship between $\delta U$ and $\delta s$, and it can be computed more easily than the full solution to equation (O-4).

The Rytov formalism starts by assuming the heterogeneity perturbs the phase of the scattered wavefield. The total field, $U=\exp (\psi)$, is therefore given by

$$U({\bf x},\omega)=U_0({\bf x},\omega) \exp(\delta \psi) = \exp(\psi_0+\delta \psi).$$ (17)

The linearization based on small $\delta \psi / \psi$ leads to the infinite Rytov series, on which the first term is given by

$$\delta \psi_{\rm Rytov} ({\bf x},\omega) = \frac{\delta U_{\rm Born}({\bf x},\omega)}{U_0({\bf x},\omega)}$$ (18)

$$= \frac{\omega^2}{4 \pi U_0({\bf x},\omega)} \int_V G_0({\bf x},\omega; {\bf x}') U_0({\bf x},\omega; {\bf x}') \delta s({\bf x}') dV({\bf x}').$$ (19)

The approximation implied by equation (O-10) is known as the first-order Rytov approximation. It provides a linear relationship between $\delta \psi$ and $\delta s$.

Traveltime sensitivity kernels: Banana-doughnuts or just plain bananas?