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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, , that
act to scatter the wavefield.
The total slowness is given by the sum,
| |
(10) |

Similarly, the total wavefield, *U*, can be considered
as the sum of a background wavefield, *U*_{0}, and a
scattered field, , so that

| |
(11) |

where *U*_{0} satisfies the Helmholtz equation in the background
medium,
| |
(12) |

and the scattered wavefield is given by the (exact) non-linear
integral equation Morse and Feshbach (1953),
| |
(13) |

In equation (13), *G*_{0} is the Green's function for
the Helmholtz equation in the background medium:
i.e. it is a solution of the equation
| |
(14) |

Since the background medium is smooth, in this paper I use Green's
functions of the form,
| |
(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 ,results in the infinite Born series, which is a Neumann series
solution Arfken (1985) to equation (13).
The first term in the expansion is given below: it corresponds
to the component of wavefield that interacts with scatters only once.

| |
(16) |

The approximation implied by equation (16) is known as the
first-order Born approximation. It provides a linear relationship
between and , and it can be computed more easily
than the full solution to equation (13).
The Rytov formalism starts by assuming the heterogeneity perturbs the
phase of the scattered wavefield. The total field, ,is therefore given by

| |
(17) |

The linearization based on small leads to the
infinite Rytov series, on which the first term is given by
| |
(18) |

| (19) |

The approximation implied by equation (19) is known as
the first-order Rytov approximation. It provides a linear relationship
between and .

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Stanford Exploration Project

4/27/2000