\section{Results}
By comparing the migrated image in Figure \ref{fig:mig-stack} with the reflection coefficients shown in Figure \ref{fig:refle-new} we can see that there are events in the migration that do not correspond to the geology. They are the artifacts that correspond to internal multiples  generated at the top and bottom of the salt body. Their amplitudes can overcome that of the weak primaries expected to be boosted by the inversion. 
Because the Sigsbee2B model Hessian (Figure \ref{fig:hmatrix-Sis-new}) was computed using one-way wave-equation Green functions, multiples reflections are not computed in the modeling nor correctly propagated during migration. They should be treated as noise $******* what kind of noise? residuals? Is inversion expected to treat noise? *******$  for the inversion. Later, I will show how the multiples can make the inversion fail to recover the reflectors in the shadow zones.

\plot{migang1}{width=6in}{Sigsbee shot-profile migration (reflection angle) using cross-correlation imaging condition. The front face corresponds to the image at $0^\circ$ reflection angle, and the side face corresponds to the reflection-angle gather at  $x=36800$ ft position.}

Figure \ref{fig:migang1} shows the migration in the reflection angle domain. The front face corresponds to the image at $0^\circ$ reflection angle, and the side face corresponds to the reflection-angle gather at position (below or inside?) the shadow zone ($x=36800$ ft). Notice how multiples and primaries are well separated in the angle gather having the multiples a frowning shape and the primaries being flat.
Figure \ref{fig:migoff1} shows the migration result in the same area but in the subsurface-offset domain. The front face corresponds to the image at zero subsurface offset, and the side face corresponds to the subsurface-offset gather at position inside the shadow zone ($x=36800$ ft). Since the primaries are illuminated at few reflection angles  their signature in the subsurface offset is an slanted line. The internal multiples are also illuminated at a very narrow range of angles. The bouncing at the dipping salt bottom makes the reflection angle of internal multiples to decrease. Consequently, they appear as almost horizontal events in the subsurface offset gathers. However, in the x-z plane they show dips following the base of salt. These different dip patterns will be used to attenuate the multiple amplitudes before inversion by a $k_x-k_h$ filter. As I will show in the field data chapter this behavior of the multiples and primaries in the shadow zones  can be found in field datasets.
 

\plot{migoff1}{width=6in}{Sigsbee shot-profile migration (subsurface-offset) using cross-correlation imaging condition. The front face corresponds to the image at zero subsurface-offset, and the side face corresponds to the subsurface-offset gather at  $x=36800$ ft position. }

I used a 3 stage pre-processing strategy to attenuate the multiple energy. It includes discriminating the multiple energy in the $k_x-k_h$ plane; performing amplitude and phase correction of the discriminated multiples in a least-square sense; and subtract them from the original data. The discrimination of the multiple energy relies on the differences in dip patterns as described previously: multiples show up in the $k_x-k_h$ plane at low $k_h$ and high $k_x$, and primaries at high $k_h$ and low $k_x$. Therefore, the bandpass version of the filter, to be applied at every depth steps, assumes the shape as shown in Figure ***. The multiples are obtained after subtraction of the filtered result from the original data. 

The multiple attenuation process can be ameliorated if the amplitude and phase of the obtained multiples are adjusted to minimize its difference from the original data in a least-squares sense. The scheme we used is the simultaneous adaptive matching of primaries and multiples with non-stationary filters computed in small patches introduced by \longcite{Gabriel1.sepphd.125}. In Equation{e:eq2} bla bla bla ...
\begin{eqnarray}
\label{e:eq1}
        \begin{bmatrix} \mathbf{M} & \mu \mathbf{P} \end{bmatrix} \begin{bmatrix} \mathbf{f_m} \\
                \mathbf{f_p} \end{bmatrix}&\approx &\mathbf{d}\\
\label{e:eq2}
        \epsilon \mathbf{A} \begin{bmatrix} \mathbf{f_m}\\ \mathbf{f_p} \end{bmatrix}&\approx &\mathbf{0}
\end{eqnarray}
where $\mathbf{f_m}$ and $\mathbf{f_p}$ are the matching filters for the multiples and primaries respectively, $\mu$ is a parameter to balance the relative importance of the two components of the fitting goal, $\mathbf{d}$ is the data (primaries and multiples), $\mathbf{A}$ is a regularization operator, and $\epsilon$ controls how strong is the regularization. I used a mask to limit the areal extent of the attenuation process in such a way that the dipping fault are not affected.


\plot{migoff1-filt}{width=6in}{Sigsbee shot-profile migration (subsurface-offset) using cross-correlation imaging condition. The front face corresponds to the image at zero subsurface-offset, and the side face corresponds to the subsurface-offset gather at  $x=36800$ ft position. }

After filtering, the multiples had been largely attenuated, making the migration image more suitable for the inversion. Figure \ref{fig:migoff1-filt} shows the filtered migration in the subsurface-offset domain. As in Figure \ref{fig:migoff1} the front face corresponds to the image at zero subsurface offset, and the side face corresponds to the subsurface-offset gather at position inside the shadow zone ($x=36800$ ft). 
This result is corroborated in the angle domain (Figure \ref{fig:migang1-filt}). The front face corresponds to the image at $0^\circ$ reflection angle, and the side face corresponds to the reflection-angle gather at position below inside the shadow zone ($x=36800$ ft).  
