\chapter{Introduction}
\label{chap:chap01}
The last decade has seen the development of migration-velocity analysis (MVA) by wavefield extrapolation \cite[]{biondi:1723,shen:2132,sava2004, shen2004}. Provided the finite frequency nature of wavefield extrapolation, this MVA technique does not suffer from the limitations caused by the high-frequency approximation, which are present in the ray-based methods, namely the need of smooth velocity contrasts However, despite its theoretical superiority, MVA by wavefield extrapolation has rarely been used in 3D projects \cite[]{fei}. This is because of its higher cost and because it is less flexible than its ray-based counterpart in parameterizing the velocity model. Therefore, decreasing its cost and improving its flexibility is crucial to implement MVA by wavefield extrapolation as a routine process. 
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Because of the linearity of wavefield propagation 


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We start by describing the prestack exploding-reflector model and how using this new concept can drastically decrease data size especially for 3D applications. Then, using phase-encoding techniques, we introduce the image-space phase-encoded wavefields, further decreasing data size. After introducing the theory of MVA by wavefield extrapolation using image-space generalized wavefields, we illustrate their use to optimize the migration velocity for the Marmousi model. Finally, we show the application on a 3D field data from the North Sea, which presents different challenges to migration-velocity definition. 

Applying the properties of linear, time-invariant systems enables us to consider source functions other than punctual. This characterizes generalized source functions defined in the generalized source domain. For instance, according to the superposition principle, a hypothetical experiment in which all the point sources are initiated in unison generates a horizontal plane wave. Another thought experiment would be to initiate all the sources of the seismic survey at random times, using the superposition and the time-shift properties. This concept is used in simultaneous Vibroseis acquisition, where different arrays of vibrators are initiated independently and with no synchronization between them, allowing great improvement in productivity \cite[]{howe:109}.
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Those experiments can be easily synthesized in the seismic processing environment, by combining the  recorded wavefields with the same scheme as that used for point sources. In this way, the migrated image computed using the combined source functions and the combined recorded wavefields is similar to the one we would obtain by migrating the original data, provided that certain conditions particular to each combination method are fulfilled. However, the imaging cost can be smaller by orders of magnitude with generalized wavefields than with conventional wavefields. 
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The idea of synthesizing generalized sources during processing is not new in seismic exploration. Plane-wave synthesis \cite[]{whitmore}, controlled illumination \cite[]{rietveld01}, and random-phase encoding \cite[]{romero} are methods that synthesize generalized sources. In the plane-wave synthesis method, linear time shifts are applied to the shot records to simulate slanted plane waves. In the controlled illumination method, a wavefield with a pre-defined shape is upward propagated and collected at the surface, defining a synthesis operator to be convolved with the original data. The generalized wavefields synthesized by the controlled illumination method tend to assume the shape of the pre-defined wavefield during the downward propagation. In the random-phase encoding method, source functions and the corresponding receiver gathers are encoded with the same random-code function, so that during migration cross-correlation of unrelated wavefields is attenuated, whereas the cross-correlation of related wavefields is minimally affected.
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The methods described above for generating generalized sources are illustrated in Figures \ref{fig:intro02}-\ref{fig:intro04}. In these figures, on the top left is the generalized source function, on the top right is the generalized receiver gather, and on the bottom is the areal-shot migrated image. In Figure \ref{fig:intro02} the plane-wave synthesis method creates horizontal plane waves at the surface. In Figure \ref{fig:intro03} a horizontal plane wave at depth 2300 m is synthesized by the method of controlled illumination. In Figure \ref{fig:intro04}, the random-phase encoding method  is used to combine every 20 shots into one generalized receiver gather. Migrating data from only one generalized source is not sufficient to recover a similar image quality as the migration of the original 375 shot-profiles (Figure \ref{fig:intro01}). Therefore, it is usual to synthesize more generalized sources to achieve a reasonable quality. Even so, the cost of migrating generalized sources is much smaller than that of migrating the original shot-profiles.

\sideplot{intro01}{width=3.in, height=2.in}{ Migration of the original 375 shot-profiles.}
  
\plot{intro02}{width=6.in, height=4.in}{Horizontal plane-wave synthesis. a) generalized source function, b) generalized receiver gather, and c) areal-shot migration.}

\plot{intro03}{width=6.in, height=4.in}{Horizontal plane-wave at depth 2300 m by the controlled illumination method. a) generalized source function, b) generalized receiver gather, and c) areal-shot migration.}

\plot{intro04}{width=6.in, height=4.in}{Random-phase encoding combining every 20 shots. a) generalized source function, b) generalized receiver gather, and c) areal-shot migration.}
 
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The methods for computing generalized sources discussed above operate in the data space, characterizing the data-space generalized sources. This thesis introduces a new category of generalized sources that are initiated from selected reflectors, using the pre-stack exploding-reflector model (PERM) \cite[]{biondi2006}. 
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PERM is an extension of the exploding-reflector model \cite[]{loewenthal01}. Because PERM wavefields are initiated in the image space, they are called the image-space generalized sources. The image-space generalized sources are suitable for migration-velocity analysis. After optimizing for the velocity model, any migration scheme can be used to generate the final image using the original data. Figure \ref{fig:intro05}a shows the Marmousi image computed with shot-profile migration using an initial velocity model, which is inaccurate below the top of the anticline at a depth of 1700 m. The inaccuracy of the velocity model is indicated by reflectors being pulled up in the center of the image below 2000 m. Figure \ref{fig:intro05}b shows the Marmousi image computed with shot-profile migration using a velocity model optimized with ISWET. ISWET was performed with 11 pairs of image-space generalized sources synthesized from 12 selected reflectors and collected at a depth of 1500 m. The pull up effect has been corrected, and the reflectors are better focused. Compare Figure \ref{fig:intro05}b with Figure \ref{fig:intro01}, which was computed with the true velocity model. For this example, each iteration of ISWET using the image-space generalized wavefields is approximately 60 times faster than that with the conventional 375 shot-profiles.
\plot{intro05}{width=6.in, height=2.in}{Shot-profile migrated images of Marmousi obtained with: a) An initial velocity model, which is inaccurate below the top of the anticline at depth 1700 m, b) a velocity model optimized with ISWET, using 11 pairs of image-space generalized sources synthesized from 12 selected reflectors and collected at depth 1500 m. }
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As will be shown, in 3D a dramatic data reduction is possible with these generalized sources. Since they are initiated at selected reflectors, a horizon-based strategy to parameterize the velocity model can be naturally incorporated into ISWET. Moreover, image-space generalized sources can be collected at any depth during the upward propagation, making a target-oriented approach also easily integrated. This thesis is one step forward in making 3D-ISWET a standard for depth-imaging projects in the presence of complex geology. 

