Relationship between wavefield-extrapolation imaging and inverse imaging

At the scattering point, we can define a "distance" or norm as  

$$\textit{E}\left(R \right)=\sum_{\omega_{min}}^{\omega_{max}}\left( \textit{U}_{S}-\textit{U}_{I}R\right) ^{2} d\omega ,$$ (39)

where R is the reflectivity, $\textit{U}_{S}$ is the wavefield downward extrapolated to a reflector, and $\textit{U}_{I}$ is the wavefield downward propagated to the reflector. The scattering wavefield $\textit{U}_{S}$ should be equal to or close to the convolution result between the wavefield $\textit{U}_{I}$ and the reflectivity. Letting 

$$\frac{\partial \textit{E}}{\partial R}=0 ,$$ (40)

we have 

$$-2\sum\limits_{\omega_{min}}^{\omega_{max}} \left( \textit{... ...ht)R \right) \textit{U}_{I}\left(\omega \right) d \omega =0 ,$$ (41)

If the incident wavefield equals zero, equation () is satisfied. However this case has no physical meaning. If the incident wavefield does not equal zero, then,  

$$R=\frac{\sum\limits_{\omega_{min}}^{\omega_{max}}\textit{U}_{S}} {\sum\limits_{\omega_{min}}^{\omega_{max}}\textit{U}_{I}}.$$ (42)

In the complex domain, we can rewrite equation () as  

$$R=\frac{\sum\limits_{\omega_{min}}^{\omega_{max}}\textit{U}_... ...{\omega_{min}}^{\omega_{max}}\textit{U}_{I}\textit{U}_{I}^{*}}.$$ (43)

If the incident wave is quite weak, the following regularization should be introduced:  

$$R=\frac{\sum\limits_{\omega_{min}}^{\omega_{max}}\textit{U}_... ...}\left( \textit{U}_{I}\textit{U}_{I}^{*}+\varepsilon \right) },$$ (44)

where $\varepsilon$ is the regularization coefficient. In fact, the reflectivity is related to the incident angle to a reflector of the plane-wave component of a seismic wave. Therefore, we should modify equation () into the following form to reach the angle gathers:  

$$R\left( p \right) =\frac{\sum\limits_{\omega_{min}}^{\omega_... ...}_{I}^{*}\left(\omega,\textit{p} \right)+\varepsilon \right) },$$ (45)

where $\textit{U}_{S}\left(\omega,\textit{p} \right)$ and $\textit{U}_{I}^{*}\left(\omega,\textit{p} \right)$ are a scattering plane wavefield and an incident plane wavefield, respectively. In fact, the extrapolated wavefield can be defined as  

$$\textit{U}_{S}=\left( \hat{L}^{*}\right)^{T}\hat{\textbf{d}}^{obs}.$$ (46)

Therefore, in the frequency domain, equation () can be rewritten as  

$$R=\frac{\sum\limits_{\omega_{min}}^{\omega_{max}} \left( \h... ...{L}^{D} \left(\hat{L}^{*D}\right)^{T}+\varepsilon \right) }.$$ (47)

We will further discuss this topic later to clarify the relationship.