Rytov approximation

The first-order Rytov approximation creates a linear relation between the velocity and wavefield (denoted by $\psi$) phase perturbations: $\Delta\phi\left(\omega\right)=\ln\left[\psi \left(\omega\right)\right]-\ln\left[\psi_0\left(\omega\right)\right]$. For a point source at $\eta$ and a receiver at $\bf\xi$, we have (Woodward, 1992)

$$\Delta\phi\left(\omega,{\mbox{\boldmath$\bf\xi$ }},{\mbox{\bo... ...mbox{\boldmath$\bf\eta$ }}\right)}d{\mbox{\boldmath$\bf x$ }},$$ (13.1)

where $O\left({\mbox{\boldmath$\bf x$ }}\right)$ represents the perturbed velocity field.

Let $k=\frac{\omega}{C_{ref}}$, denoting $f\left({\mbox{\boldmath$\bf x$ }}\right)=n^2\left({\mbox{\boldmath$\bf x$ }}\right)-n_{0}^2\left({\mbox{\boldmath$\bf x$ }}\right)$ as the model function, and replace the phase perturbation with the first arrival traveltime residual $\omega\Delta\tau$ (Schuster and Quintus-Bosz, 1993), we have

$$\Delta\tau\left({\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$... ...box{\boldmath$\bf\eta$ }}\right )}d{\mbox{\boldmath$\bf x$ }},$$ (13.2)

where $n\left({\mbox{\boldmath$\bf x$ }}\right)$= $\frac{C_{ref}}{C\left({\mbox{\boldmath$\bf x$ }}\right)}$, $n_0\left({\mbox{\boldmath$\bf x$ }}\right)$= $\frac{C_{ref}}{ C_0\left({\mbox{\boldmath$\bf x$ }}\right)}$, and C_ref is a reference velocity. Here we assume first arrivals only so that $\Delta\tau$ approximates the first arrival traveltime residual. Using the first term of the geometrical optics approximation, the Green's function can be written as

$$G_0\left(k,{\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$... ...{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$ }}\prime\right)},$$ (13.3)

where n represents the dimension and $\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$ }}\prime\right)$ satisfies the eikonal equation

$$\left(\nabla_{{\mbox{\boldmath$\bf x$ }}}\phi\right)^2=n_{0}^2\left({\mbox{\boldmath$\bf x$ }}\right).$$ (13.4)

Here $n_0\left({\mbox{\boldmath$\bf x$ }}\right)$ represents the slowness and $a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$ }}\prime\right)$ satisfies the transport equation

$$a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$ }}\... ...\boldmath$\bf x$ }},{\mbox{\boldmath$\bf x$ }}\prime\right)=0.$$ (13.5)

Equation () can be written as

$$\Delta\tau\left({\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$... ...box{\boldmath$\bf\eta$ }}\right )}d{\mbox{\boldmath$\bf x$ }},$$ (13.6)

where $a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$= $a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }}\right)a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$/ $\left(C_{ref}a\left( {\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$\bf\eta$ }}\right)\right)$ and $\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$= $\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }}\right) +\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$- $\phi\left({\mbox{\boldmath$\bf\xi$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$.

Equation () is related to the causal GRT (Beylkin, 1985), where analogous integral equations can be derived for fluids with variable density and for elastic solids. For different waves, the phase functions $\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }}\right)$ and $\phi\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$ satisfy different eikonal equations corresponding to the indices of refraction for P and S waves; and the amplitudes $a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\xi$ }}\right)$ and $a\left({\mbox{\boldmath$\bf x$ }},{\mbox{\boldmath$\bf\eta$ }}\right)$ satisfy the corresponding transport equations along the rays connecting points ${\mbox{\boldmath$\bf x$ }}$ with ${\mbox{\boldmath$\bf\xi$ }}$ and ${\mbox{\boldmath$\bf x$ }}$ with ${\mbox{\boldmath$\bf\eta$ }}$, respectively.