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, s₀, 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, U₀, and a scattered field, $\delta U$, so that

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

where U₀ 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 (13), G₀ 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 A₀ and T₀ are ray-traced traveltimes and amplitudes respectively.

A Taylor series expansion of U about U₀ for small $\delta s$,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.  

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

The approximation implied by equation (16) 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 (13).

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

$$\begin{eqnarray} \delta \psi_{\rm Rytov} ({\bf x},\omega) & = & \frac{\delta U_{... ...')\,U_0({\bf x},\omega; {\bf x}') \delta s({\bf x}') dV({\bf x}').\end{eqnarray}$$ (18) (19)

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