Perturbation field: Forward operator

If we perturb the velocity model, we also introduce a perturbation in the wavefield. In other words, the perturbation in slowness generates a secondary scattered wavefield.

1.

Scattering and downward continuation

If we consider the perturbation in the wavefield at the surface, we can recursively downward continue it, adding at every depth step the scattered wavefield:  

$$\Delta u^{z+1} = T_0^{z} \Delta u^{z} + \Delta v^{z+1}$$ (4)

where

• $\Delta u^{z}(\omega_{\!})$ is the perturbation in the wavefield generated by the perturbation in velocity and downward continued from the surface, and
• $\Delta v^{z+1}(\omega_{\!})$ represents the scattered wavefield caused at depth level z+1 by the perturbation in velocity from depth level z.

In the first-order Born approximation, the scattered wavefield can be written as  

$$\Delta v^{z+1} = T_0^{z} G_0^{z} u_0^{z} \Delta s^{z}$$ (5)

where

• $G_0^{z}(\omega_{\!},s_0)$ is the scattering operator at depth z,
• $\Delta s^{z}(\omega_{\!})$ is the perturbation in slowness at depth z, and
• $u_0^{z}(\omega_{\!})$ is the background wavefield at depth z.

If we introduce equation (5) into  (4) we find that  

$$\Delta u^{z+1} = T_0^{z} \left[ \Delta u^{z} + G_0^{z} u_0^{z} \Delta s^{z} \right]$$ (6)

2.

Imaging

As for the background image, the perturbation in image ($\Delta \i^{z}$), caused by the perturbation in slowness, is obtained by a summation over all the frequencies $\omega_{\!}$: 

$$\Delta \i^{z} = \sum_1^{N_{\omega_{\!}}} \Delta u^{z}(\omega_{\!})$$ (7)

Equations (6) and (7) establish a linear relation between the perturbation in slowness ($\Delta s^{z}$) and the perturbation in image ($\Delta \i^{z}$). We can use this linear relation in an iterative algorithm to invert for the perturbation in slowness based on the perturbation in the image.