\lefthead{Valenciano and Guerra}
\righthead{Preprocessing prior to Inversion}
\email{guerrac@stanford.edu}
\footer{SEP--133}
\title{Preprocessing prior to Inversion}
\author{Alejandro Valenciano and Claudio Guerra}

due to an incomplete model parameterization, 

\ABS{Seismic inversion is very sensitive to the presence of noise. Wave equation inversion using the one-way approximation suffers with the presence of multiples in the data because they are not predicted by the forward modeling. Therefore, noise lives in the residual space and, if not Gaussian, slows down the convergence to an acceptable solution or, eventually, dominates the whole process inhibiting the efficacy of inversion. The ideal solution is to include the noise in the forward modeling, but this can lead to an prohibitively expensive inversion scheme. Another, more practical, approach is to make the data agree with the physical assumptions of the specific inversion scheme. In this sense, noise attenuation is a pre-processing step before inversion. Here, we illustrate the problem by applying one-way wave-equation inversion to a portion of the well known Sigsbee2b data {\cite ***}. The physics of wave propagation is governed by the one-way wave equation, therefore the inversion problem is subjected to its assumptions. In the present example, noise is represented by multiples, not modeled by the one-way wave-equation, which bounce on the salt edges and on the high reflectivity water bottom. After characterizing the noise present in the migrated data, we use a dip filter to estimate it and a non-stationary adaptive filter technique to subtract it from the migrated data. By doing so, The results attest the importance of having the agreement between the physics of the inversion and the data (or vice-versa!).}
 
\vspace{.5cm}

\section{Introduction} 
Conventionally, migration is a process formulated in such a way that, ideally, the only seismic events expected to be present in the input data are primary reflections. When this is not the case, because the pre-processing was not able to completely eliminate events considered noise (multiple reflections, for instance), the final image presents migrated events that do not find correspondence in geology. In other words, these events are not modeled by the physical theory in which the migration method is based on. These events are frequently called migration noise or migration artifacts. Depending on how strong are the amplitudes and on the relative orientation of the noise compared to the reflector, the image produced by migration for the purposes of the exploration team is acceptable. For reservoir characterization studies, however, the requirements on the data are more strict. One of the products delivered by the processing team to the reservoir geophysicist is the so-called frequency-enhanced cube obtained by means of increasing the bandwidth of the migrated image [(Marcus L. Countiss, 2003, Frequency-enhanced imaging of stratigraphically complex, thin-bed reservoirs: A case study from South Marsh Island Block 128 Field, TLE, Sep 2002), (Gavin Hills-Jones, A new technology for high frequency imaging using seismic data comes to the North Sea, FB, Nov 2002), ...]. Sometimes, because of the small signal-to-noise (generally multiples) ratio at the high frequencies, to obtain a reasonable frequency-enhanced cube can be a difficult task involving characterization and attenuation of the noise prior to the frequency recovering. 
\par
Valenciano () bla bla bla .Since the proposed inversion scheme aims to increase the resolution of the seismic image, besides of solving for illumination problems and correcting for acquisition geometries, it can face similar problems as in the frequency-enhanced cube case because of the presence of migrated noise. 

\section{Inversion}

\section{Noise Characterization and Attenuation}
In contrast with Sigsbee2a, Sigsbee2b data has a high reflectivity water-bottom. This characteristic is responsible to generate

\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.

Figure *** shows that the occurrence of shadow zones below the salt body is strongly conditioned by the concavity of its base. Where the base of salt is concave down, the transmitted energy is focused and high illumination values are produced; where the base of salt is concave up, the transmitted energy is spread according to divergent paths producing low illumination zones. Unfortunately, for reflected energy, and for multiple energy too, this geometry acts in the opposite way, concentrating energy where the base of salt is concave up and dispersing energy where it is concave down. Therefore, in the shadow zones it is very likely that the stronger amplitudes corresponds to multiple reflections and converted waves, while in the well illuminated zones these events produce small or no spurious interference with the primary energy. Figure *** shows ***.

One solution to this problem is to attenuate the multiples (events that do not fit the model) from the migration. In order to do that the multiples need to be separated from the primaries in a way that during removal the primaries are minimally affected. Traditional data space demultiple schemes dealing either with periodicity or differential moveout become ineffective in subsalt settings due to two main reasons: a) complex ray path make periodicity assumption fail, and b) salt focusing effect that makes subsalt reflections concentrate in the near offset range, 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 adaptive subtracted from  the migration in the image domain without affecting much the primaries.

\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).  

\section{Conclusions}

\section{Acknowledgments}

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