\chapter{Salt-related multiple removal}
\label{app-mult}

Traditional data-space demultiple schemes dealing with either periodicity or differential moveout become ineffective in subsalt settings because the complex ray path makes periodicity harder to predict. Subsalt reflections  usually appear only in the near offsets, making radon-type demultiple schemes ineffective.
\cite{sava:V10} and \cite{alvarez:V97} demonstrated that primaries and multiples have different behavior in image space (subsurface-offset or reflection-angle), and that multiples can be adaptively subtracted from  the migration in the image domain without significantly affecting the primaries.

\plot{migang1}{width=5.0in}{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 a position inside the shadow zone ($x=36800$ ft). Notice how multiples and primaries are well separated in the angle gather, where the multiples have a frowning shape and the primaries are 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 a position inside the shadow zone ($x=36800$ ft). Since the primaries are illuminated at few reflection angles, their signature in the subsurface offset domain 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 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 were used to attenuate the multiple amplitudes before inversion using a $k_x-k_h$ filter. As I show in chapter \ref{field}, this behavior of the multiples and primaries can also be found in field datasets.

\plot{migoff1}{width=5.0in}{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. }

I used a three-stage pre-processing strategy to attenuate the multiple energy. It included discriminating the multiple energy in the $k_x-k_h$ plane (to generate a model of the multiples); performing amplitude and phase correction of the discriminated multiples in a least-square sense; and subtracting 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, to build a model of the multiples I  bandpassed the migration at every depth step. Figure \ref{fig:not-filtered} shows a depth slice at $z=14150$ ft of the migration in the $k_x-k_h$ plane, and Figure \ref{fig:filtered} shows the result of bandpassing Figure \ref{fig:not-filtered}. The model for the multiples (Figure \ref{fig:diff-filtered}) is obtained after subtraction of the filtered result (Figure \ref{fig:filtered}) from the migration (Figure \ref{fig:not-filtered}). 

\plot{not-filtered}{width=4in}{Depth slice  at $z=14150$ ft of the migration in the $k_x-k_h$ plane.}
\plot{filtered}{width=4in}{Primaries after bandpassing the migration (Figure \ref{fig:not-filtered})  in the $k_x-k_h$ plane.}
\plot{diff-filtered}{width=4in}{The model for the multiples in the $k_x-k_h$ plane, obtained after subtraction of the filtered result (Figure \ref{fig:filtered}) from the migration (Figure \ref{fig:not-filtered}).}

The impact on the primaries of the multiple-attenuation process can be ameliorated if the amplitude and phase of the obtained multiples are adjusted to minimize their differences from the original data in a least-squares sense. After  this adjustment, I used an adaptive subtraction scheme that simultaneously adaptively matched the primaries and multiples with non-stationary filters computed in small patches \cite[]{alvarez:V97}.

%\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=5.0in}{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. }

After filtering, the multiples had been largely attenuated, making the migration image more suitable for 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 a 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 a position inside the shadow zone ($x=36800$ ft).  

\plot{migang1-filt}{width=5.0in}{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.}

