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Bibliography

Arfken, G., 1985, Mathematical methods for physicists, 3rd ed.: Academic Press Inc.

Beydoun, W. B., and A. Tarantola, 1988, First Born and Rytov approximations: Modeling and inversion conditions in a canonical example: J. Acoust. Soc. Am., 83.

Biondi, B., and P. Sava, 1999, Wave-equation migration velocity analysis: 69th Ann. Internat. Meeting, Expanded Abstracts, Soc. Expl. Geophys., 1723-1726.

Bishop, T. N., K. P. Bube, R. T. Cutler, R. T. Langan, P. L. Love, J. R. Resnick, R. T. Shuey, D. A. Spindler, and H. W. Wyld, 1985, Tomographic determination of velocity and depth in laterally varying media: Geophysics, 50, 903-923.

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----, 1999, Three-dimensional sensitivity kernels for finite-frequency traveltimes: the banana-doughnut paradox: Geophys. J. Int., 137, 805-815.

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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,
\begin{displaymath}
s({\bf x})=s_0({\bf x})+\delta s({\bf x}).
\end{displaymath} (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

\begin{displaymath}
U({\bf x},\omega)=U_0({\bf x},\omega)+\delta U({\bf x},\omega),
\end{displaymath} (11)

where $U_0$ satisfies the Helmholtz equation in the background medium,
\begin{displaymath}
\left[ \nabla^2 + \omega^2   s_0^2({\bf x})\right]
U_0({\bf x},\omega) = 0,
\end{displaymath} (12)

and the scattered wavefield is given by the (exact) non-linear integral equation (Morse and Feshbach, 1953),
\begin{displaymath}
\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}').
\end{displaymath} (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
\begin{displaymath}
\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).
\end{displaymath} (14)

Since the background medium is smooth, in this paper I use Green's functions of the form,
\begin{displaymath}
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)}.
\end{displaymath} (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.

\begin{displaymath}
\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}').
\end{displaymath} (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

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

The linearization based on small $\delta \psi / \psi$ leads to the infinite Rytov series, on which the first term is given by
$\displaystyle \delta \psi_{\rm Rytov} ({\bf x},\omega)$ $\textstyle =$ $\displaystyle \frac{\delta U_{\rm Born}({\bf x},\omega)}{U_0({\bf x},\omega)}$ (18)
  $\textstyle =$ $\displaystyle \frac{\omega^2}{4 \pi   U_0({\bf x},\omega)}
\int_V G_0({\bf x},\omega; {\bf x}') U_0({\bf x},\omega; {\bf x}')
\delta s({\bf x}') dV({\bf x}').$ (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 $\delta s$.


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Next: About this document ... Up: Rickett: Traveltime sensitivity kernels Previous: Conclusions: does it matter?

2013-03-03