\title{Fast 3D migration-velocity analysis by wavefield extrapolation using the prestack exploding-reflector model}

\lefthead{Guerra and Biondi}
\righthead{MVA with the prestack exploding-reflector}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%\ms{GEO-2008-0443}

\address{
\footnotemark[1]Formerly Stanford Exploration Project, Geophysics Department, 
Stanford University, Stanford, CA 94305. Presently Petrobras, Rio de Janeiro, Brazil.\\
E-mail: claudio.guerra@petrobras.com.br \\
\footnotemark[2]Stanford Exploration Project, Geophysics Department, 
Stanford University, Stanford, CA 94305. E-mail: biondo@sep.stanford.edu \\
%Manuscript submitted on 12/24/2008. Revised manuscript submitted on 4/29/2009.
}

\author{Claudio Guerra\footnotemark[1] and Biondo Biondi\footnotemark[2]}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
In areas of complex geology, migration-velocity estimation should use methods that describe the complexity of wavefield propagation, such as focusing and defocusing, multipathing, and frequency-dependent velocity sensitivity. Migration-velocity analysis by wavefield extrapolation has the ability to address these issues because it uses wavefields as carriers of information, contrasting with ray-based methods. However, its high cost and lack of flexibility with respect to model parametrization and target-oriented analysis have  prevented its routine industrial use. We overcome those limitations by using new wavefields as carriers of information: the image-space generalized wavefields. These wavefields are synthesized from a prestack image computed with wavefield-extrapolation methods, using the pre-stack exploding-reflector model. Cost of migration-velocity analysis by wavefield extrapolation is decreased because only a small number of image-space generalized wavefields are necessary to accurately describe the kinematics of velocity errors and because these wavefields can be easily used in a target-oriented way. Flexibility is naturally incorporated because modeling these wavefields have as the initial conditions  selected reflectors, which allows using a horizon-based parametrization of the model space. In a 3D example of the North Sea, we show that using wavefields synthesized by the prestack exploding-reflector model greatly improves efficiency of migration-velocity analysis by wavefield-extrapolation, while yielding a final accurate migration-velocity model evidenced by good quality and reliable depth images.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{introduction}
Migration-velocity analysis (MVA) solves for earth models that explain, under some norm, a residual-moveout parameter measured in the post-migrated domain. Ray-based MVA is the most widely used method due its low cost and flexibility to parametrize the model space, which can improve convergence to a geologically reasonable velocity model. However, in the presence of high lateral velocity variation and irregular interfaces, ray-based MVA methods are prone to fail because they do not describe the entire complexity of wavefield propagation. Hence, in those situations it is desirable to use wavefields to define the velocity model.
\par
The last decade has seen the development of MVA by wavefield extrapolation \cite[]{biondi:1723,sava2004,shen2004,shen:VE49}, which promises to overcome the limitations imposed by the high-frequency approximation present in ray methods. MVA by wavefield extrapolation is solved in the image space, using wavefields as carriers of information. Despite its theoretical superiority to ray methods, this relatively new technology has been rarely used in 3D projects \cite[]{fei:2283} because of its higher cost and because it is less flexible than its ray-based counterpart in parameterizing the velocity model and in solving velocity problems in a target-oriented fashion. We overcome these limitations by using image-space generalized wavefields \cite[]{yax2008, guerra2009a, guerra_thesis}. These wavefields are initiated in the image space, having as the initial condition key reflectors selected from a prestack image obtained with a wavefield-extrapolation migration. 
\par
The image-space generalized wavefields are computed with the pre-stack exploding-reflector model (PERM) \cite[]{biondi2006}, which is an extension of the exploding-reflector model \cite[]{loewenthal01} in a sense that migration of PERM data can produce a prestack image. The fundamental idea of PERM is to model downgoing and upgoing wavefields necessary to provide a prestack image of a single point on a reflector with kinematic information consistent to that of the original prestack image. Synthesizing data sets from a subset of the subsurface has been previously explored by \cite{Rietveld02} and further developed into the concept of Common-Focus Point (CFP) \cite[]{berkhout:1895, thorbecke:S273}. CFP technology aims at synthesizing data gathers that contain all the reflections that would have been recorded at the surface if we had placed a point source in the subsurface.
A CFP gather is composed of traces resulting from the time convolution between the shot gather and a focusing operator plus stacking of the stationary phases. The focusing operator is computed by modeling the seismic response of one point at the subsurface.
In contrast with CFP, PERM data result from the wave propagation of the prestack image at one point at the subsurface, without operating on the original shot gathers. In this sense, PERM data is more closely related to the CFP operator itself than to the CFP data.
\par
To image the entire reflector or group of reflectors, several independent PERM experiments would have to be performed, which could potentially generate data even bigger than the original. However, because of the linearity of wavefield propagation, the modeling experiments can be combined, causing a dramatic data reduction, especially in 3D. When the wavefields are linearly combined after encoding with stochastic phase functions, the image-space phase-encoded wavefields (ISPEW) are generated. Since image-space generalized wavefields are initiated at selected reflectors, a horizon-based strategy to parametrize the velocity model can be naturally incorporated into MVA by wavefield extrapolation. Moreover, these wavefields can be generated for a small subset of the image cube and collected at any depth during their propagation, making a target-oriented approach also easily integrated. 
\par
This paper is organized as follows. First, we describe PERM and how to synthesize ISPEW. Next, we introduce MVA by wavefield extrapolation in the image-space generalized-sources domain, illustrating with the Marmousi model. Finally, we run MVA by wavefield extrapolation, using only 30 pairs of ISPEW, to estimate the 3D-migration-velocity model in the North Sea with the occurrence of a salt body. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{the prestack exploding-reflector model} 
A prestack image computed by wavefield-extrapolation methods with an inaccurate velocity model presents energy departing from the zero subsurface offset. In conventional Born modeling, this entire prestack image is spatially convolved with the source wavefield, generating the receiver wavefield, whose back-propagation and collection at the surface create data. Source and receiver wavefields must be  propagated with the same inaccurate velocity model used in the original migration to preserve the correct kinematics. Now, let us consider the modeling of data necessary to correctly image a reflector in a single subsurface-offset common-image gather (SODCIG). Since we do not know beforehand which shots contribute to forming the image at a point in the subsurface, we would have to perform several modeling experiments to guarantee illumination similar to that of the original SODCIG. However, ideally, we would like to synthesize a small amount of data with the condition that migration produces kinematics consistent to that of the initial prestack reflector. This is the fundamental idea of the prestack exploding-reflector model (PERM) \cite[]{biondi2006}, whose primary objective is to accelerate migration-velocity analysis.
\par
Areal data is generated by PERM, propagating energy of a prestack image.  PERM operates in a way similar to the exploding-reflector model. However, since migration of PERM data can produce a prestack image, PERM can be considered a generalization of the exploding-reflector model. The modeling of PERM  wavefields can be carried out by any wavefield-extrapolation scheme. Here, we use the following one-way wave equations:
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z} - i\sqrt{\omega ^2 s_0^2({\bf x})-|{\bf k}| ^2}\right) D({\bf x},\omega;{\bf x_m}) = I_{\scriptscriptstyle D}({\bf x_m},{\bf h}) \\
D(x,y,z=z_{\rm max},\omega;{\bf x_m}) = 0 \end{array} \right.,
\label{eq:perm_bD}
\end{eqnarray}
and
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z} + i\sqrt{\omega ^2 s_0^2({\bf x})-|{\bf k}| ^2}\right) U({\bf x},\omega;{\bf x_m}) = I_{\scriptscriptstyle U}({\bf x_m},{\bf h}) \\
U(x,y,z=z_{\rm max},\omega;{\bf x_m}) = 0 \end{array} \right.,
\label{eq:perm_bU}
\end{eqnarray}
where $\omega$ is the radial frequency, $s_0$ is the same background velocity used to produce the initial prestack image, $D$ and $U$ represent the downgoing and the upgoing PERM wavefields, respectively, $I_{\scriptscriptstyle D}$ and $I_{\scriptscriptstyle U}$ are subsets of the prestack image centered at ${\bf x_m}$ for a single reflector, suitable for the initial conditions for modeling downgoing and upwgoing PERM wavefields, respectively. The information along the subsurface-offset $\bf h$ dimension of the initial conditions is projected onto the $\bf x$ dimension for the modeling. The subsurface offsets is parameterized as ${\bf h}=(h_x,h_y)$, where $h_x$ and $h_y$ are the inline- and the crossline-subsurface offsets, respectively. Notice that if the prestack image has energy focused at zero subsurface offset, the initial conditions can be parameterized only by the spatial coordinates, and PERM is equivalent to ERM.  
\par
The initial conditions are obtained by rotating the original unfocused SODCIGs according to the apparent geological dip of the reflector.  This operation corrects for the image-point dispersal due to velocity inaccuracy and reflector dip, causing events with different reflection angles from the same reflection point to be imaged at different locations. This is a consequence of not shifting the wavefields along the apparent geological-dip direction \cite[]{biondi:1283} when applying the multi-offset image condition \cite[]{james2002}. The 2D geometry that explains the rotation is depicted in Figure \ref{fig:pdisp} for the cases of migration velocity which is slower (Figure \ref{fig:pdisp}a) and faster (Figure \ref{fig:pdisp}b) than the correct one.
\par
\plot{pdisp}{width=0.8\textwidth}{Geometry for the computation of SODCIGs. Source, receiver and image points are labeled with $\mathsf S$, $\mathsf R$ and $\mathsf I$, respectively. The subscript $\mathsf {hx}$ corresponds to subsurface offsets computed with horizontal shift. The subscript $\mathsf {hg}$ corresponds to subsurface offsets computed by shifting along the apparent geological dip $\alpha$. a) Underestimated velocity, and b) overestimated velocity. Modified from \cite{biondi:1283}.}
The angles $\gamma + \alpha$ and $\gamma - \alpha$ are the 
source and receiver ray angles, respectively. They are the propagation directions of the wavefields locally at the image point. In 2D, $\alpha$ and $\gamma$ are
 related to slopes in the pre-stack image $\mathsf I_{hg}$ according to
 \begin{eqnarray}
 \tan \alpha = -\frac{dz_m}{dx_m}
 \label{eq:talpha}
 \end{eqnarray}
 \begin{eqnarray}
 \tan \gamma = -\frac{dz}{dh_x},
 \label{eq:tgamma}
 \end{eqnarray}
 where the subscript $m$ in equation \ref{eq:talpha} refers to the local nature of the relationship. The solutions of the differential equations \ref{eq:talpha}
 and \ref{eq:tgamma} define slant-stack paths, which allow us to transform the 2D pre-stack image $I(x,z,h_x)$ into $I(x,z,\alpha,\gamma)$.
 To align the initial conditions with the geological dip, the dip along the subsurface-offset axis is changed, 
 yielding the new subsurface offset $\widetilde{h_{x_d}}$ and $\widetilde{h_{x_u}}$ for the initial conditions of the modeling of downgoing and upgoing PERM wavefields,
  respectively. This is accomplished by solving the following differential equations:
 \begin{eqnarray}
 \tan (\gamma + \alpha) = -\frac{dz}{d\widetilde {h_{x_d}}}, 
 \label{eq:theta1}
 \end{eqnarray}
 \begin{eqnarray}
 \tan (\gamma - \alpha) = \frac{dz}{d\widetilde {h_{x_u}}}.
 \label{eq:theta2}
 \end{eqnarray}
 The solutions of equations \ref{eq:theta1} and \ref{eq:theta2} define new slant-stack operations, which 
  reduce the dimensionality of the decomposed pre-stack image by transforming $I(x,z,\alpha,\gamma)$ into 
 $I_D(x,z,\widetilde{h_{x_d}})$ and $I_U(x,z,\widetilde{h_{x_u}})$.
In 3D, the cross-line subsurface offsets also must be rotated.  
A complete description of the 3D rotation is given by \cite{guerra_thesis}.
\par
PERM as described by equations \ref{eq:perm_bD} and \ref{eq:perm_bU} generates sufficient data to image a single SODCIG. However, to fully image a reflector, we would have to model data that could potentially be even greater than the original data. Using the linearity of wavefield propagation, the modeling experiments can be combined in a way that PERM data size is decreased. Hence, a group of equally spaced SODCIGs is simultaneously injected into the modeling according to
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z}-i\sqrt{\omega ^2 s_0^2({\bf x})-|{\bf k}| ^2}\right) \widehat {D}({\bf x},\omega) = \widehat {I_D}({\bf x-h}) \\
\widehat{D}(x,y,z=z_{\rm max},\omega) = 0 \end{array} \right.,
\label{eq:comb_bD}
\end{eqnarray}
and
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z}+i\sqrt{\omega ^2 s_0^2({\bf x})-|{\bf k}| ^2}\right) \widehat{U}({\bf x},\omega) = \widehat {I_U}({\bf x+h}) \\
\widehat{U}(x,y,z=z_{\rm max},\omega) = 0 \end{array} \right.,
\label{eq:comb_bU}
\end{eqnarray}
where $\widehat{I_D}$ and $\widehat{I_U}$ are the combination of SODCIGs for a single reflector to be used as the initial conditions for the modeling of combined downgoing  and upgoing wavefields, $\widehat{D}$ and $\widehat{U}$, respectively. 
\par
The selection of SODCIGs can be thought of as the multiplication of the pre-stack image by a 2D--comb function, which is shifted laterally to select new set of SODCIGs to initiate the modeling of another pair of combined wavefields. After shifting along one period of the sampling function in the $x$ and $y$ directions, all the points on the reflector are used in the modeling. Consequently, the number of modeling experiments equals the number of lateral shifts of the sampling function.
\par
Depending on the spatial periods of the sampling function, crosstalk between wavefields modeled from different SODCIGs  occurs during migration. If the periods of the 2D--comb function are chosen to be the decorrelation distances of twice the $x$ and $y$ subsurface-offset range,  no crosstalk is generated. If more than one reflector is used, each reflector must be separately injected into the modeling, so that wavefields from different reflectors do not cross-correlate during imaging, avoiding reflector crosstalk. These procedures to avoid crosstalk limit data size reduction with PERM. To overcome this limitation, two different strategies can be used.
\par
One strategy mitigates the reflector crosstalk, benefiting from the fact that the image of a reflector is formed at the zero lag of the cross-correlation between source and receiver wavefields, which, when using PERM data, corresponds to the zero time of wavefield propagation when the migration velocity is accurate. When migration velocity is inaccurate, focusing of the image departs from the zero time. This can be potentially used to update the velocity model \cite[]{sava:S209,yang:3715}. Depending on the magnitude of the velocity errors and the distance between reflectors, selecting events with propagation time close to zero can avoid cross-correlating unrelated events. This is the principle of the time-windowed imaging condition, which for a single pair of PERM wavefields reads
\begin{eqnarray}
I_{\scriptscriptstyle {W}}({\bf x},{\bf h}) = \sum_{\frac{-t_{\scriptscriptstyle w}}{2} \leq t \leq \frac{t_{\scriptscriptstyle w}}{2}} {\mathcal F}^{-1} \left [{\widehat D}^{*}({\bf x}-{\bf h}) \right ] {\mathcal F}^{-1}\left [{\widehat U}({\bf x}+{\bf h}) \right ],
\label{eq:wic_bI}
\end{eqnarray}
where $t_{\scriptscriptstyle w}$ is the length of the time window. When using one-way propagators, the wavefields are inverse Fourier transformed to time by ${\mathcal F}^{-1}$.  
\par
Another strategy is to phase-encode the modeling experiments. Phase-encoding is a well stablished technique for decreasing the cost of seismic imaging by linearly combining the shot records \cite[]{Schultz,whitmore,romero,Sun01,liu01,duquet2006}. In general, phase-encoding is performed in the data space. PERM wavefields can be phase-encoded in the image space, being called the image-space phase-encoded wavefields (ISPEW). We achieve this by phase-encoding the initial conditions  ${\widetilde I}$, according to:
\begin{eqnarray}
{\widetilde I}(\widehat{\bf x}{\bf ,h},{\bf q},\omega) = \sum_m \sum_j \delta({\widehat{\bf x}-m\Delta{\bf x}}) \beta(\widehat{\bf x},j,{\bf q},\omega) W_j(\widehat{\bf x}{\bf ,h}) I(\widehat{\bf x}{\bf ,h}),
\label{eq:ispew_Itil}
\end{eqnarray}
where $W_j$ selects the reflector $j$ by identifying and windowing it in the pre-stack image $I$, and $\beta(\widehat{\bf x},j,{\bf q},\omega)$ is a pseudo-random phase-encoding function defined as
\begin{eqnarray}
\beta(\widehat{\bf x},j,{\bf q},\omega) = e^{i\epsilon(\widehat{\bf x},j,{\bf q},\omega)},
\label{eq:ispew_rnd}
\end{eqnarray}
with $\epsilon(\widehat{\bf x},j,{\bf q},\omega)$ usually being a uniformly distributed pseudo-random sequence with zero mean. Another sequences, Gold codes for instance, used in third- and fourth-generation cellphones can also be used to phase-encode the modeling  \cite[]{guerra2008b}. Phase-encoding the modeling experiments mitigates not only crosstalk between wavefields modeled from different reflectors as well as crosstalk between wavefields modeled from different SODCIGs. Downgoing and upgoing ISPEW initiated at the same SODCIG and from the same reflector are equally encoded, whereas downgoing and upgoing ISPEW initiated at different SODCIGs and from the different reflectors have different codes assigned to them. Computing more realizations of ISPEW makes crosstalk attenuation more effective, according to the law of large numbers. 
\par
Image-space generalized wavefields can be collected at any depth, which can be the top of a region of inaccurate velocity, having key reflectors affected by the velocity inaccuracy as the initial conditions. By doing so, velocity update will be restricted to the inaccurate velocity region. The use of  key reflectors characterizes a target-oriented strategy to perform MVA by wavefield extrapolation.  These two features along with the decrease of data size drastically decreases the cost of MVA by wavefield extrapolation. 
\par
To illustrate the generation of PERM data and ISPEW, a prestack image computed with the inaccurate velocity model of Figure \ref{fig:ismarm01}b is used as the initial conditions after rotation according to the apparent geological dip. Data comprise 375 split-spread shots, with 6000 m maximum offset, computed by one-way Born modeling using the smooth Marmousi velocity model of Figure \ref{fig:ismarm01}a along with the reflectivity derived from the true Marmousi model. The inaccurate velocity model is exactly the same as the correct velocity model up to the black horizon. From this horizon down, velocity is strongly smoothed and multiplied by 0.9. The curvature of the reflectors in the SODCIG and the pull-up of reflectors in the center of Figure \ref{fig:ismarm02}a reflect the velocity inaccuracy. We use the four reflectors shown in Figure \ref{fig:ismarm02}b to model 35 pairs of PERM data (Figure \ref{fig:ismarm03}) and 11 pairs of ISPEW (Figure \ref{fig:ismarm06a}). Since 17 subsurface offsets are computed in migration, which is less than half of the spatial sampling period, no SODCIG crosstalk is expected to occur when using 35 pairs of PERM data. Migration of the 35 pairs of PERM data using the conventional imaging condition is shown in Figure \ref{fig:ismarm04}a and using the time-windowed-imaging condition is shown in Figure \ref{fig:ismarm04}b. Notice that this modified imaging condition almost completely avoids reflector crosstalk. Migration of the 11 pairs of ISPEW is shown in Figure \ref{fig:ismarm07}. Reflectors are satisfactorily imaged and crosstalk is dispersed throughout the image.  
\par
\plot{ismarm01}{width=0.8\textwidth}{Edited velocity models for the Marmousi example: a) Smooth velocity model used to model the one-way Born data. b) Background velocity model used to migrate the Born data, and to model and migrate PERM data.}
\plot{ismarm02}{width=0.8\textwidth}{a) Pre-stack image computed with the background velocity model. b) Selected reflectors from the background image to perform modeling of wavefields.}  
\plot{ismarm03}{width=0.8\textwidth}{PERM wavefields for the Marmousi example: a) Downgoing wavefield. b) upgoing wavefield.}  
\plot{ismarm06a}{width=0.8\textwidth}{ISPEW for the Marmousi example: a) Downgoing wavefield. b) upgoing wavefield.}  
\plot{ismarm04}{width=0.8\textwidth}{Pre-stack image computed with 35 pairs of PERM wavefields and background velocity model using: a) the conventional imaging condition, and b) the time-windowed imaging condition. Reflector crosstalk is avoided when reflectors are sufficiently separated. However, some residual crosstalk is still present, labeled as RC in the panel on the left.}
\plot{ismarm07}{width=0.8\textwidth}{Pre-stack  images computed with 11 ISPEW.}
Residual-moveout panels from images computed with the original shot records (Figure \ref{fig:ismarm08}a), with 35 pairs of PERM data (Figure \ref{fig:ismarm08}b), and with 11 ISPEW (Figure \ref{fig:ismarm08}c) are used to analyze the quality of the moveout information. Notice that in the angle-domain common-image gathers (ADCIGs) corresponding to images computed with 35 pairs of PERM data and 11 pairs of ISPEW, less events are present since only four reflectors were selected to model these wavefields. The corresponding reflectors in the original shot-profile image are highlighted by green boxes in Figure \ref{fig:ismarm08}a. The residual moveout information in the four panels is very similar. However, some crosstalk not entirely rejected by the time-windowed imaging condition causes the residual-moveout information lose vertical resolution in Figure \ref{fig:ismarm08}b when compared to Figure \ref{fig:ismarm08}c.  
\plot{ismarm08}{width=0.8\textwidth}{ADCIGs (top) and $\rho$-panels (bottom) corresponding to images computed by: a) Shot-profile migration of 375 shot gathers, b) areal-shot migration of 35 PERM wavefields using the time-windowed imaging condition, c) areal-shot migration of 11 ISPEW. The moveout information is basically the same.} 
\par
The cost for obtaining the images in Figure \ref{fig:ismarm08} widely varies. For instance, migrating 11 ISPEW is approximately 30 times faster than migrating all the 375 original shots. This difference in performance is even more dramatic in MVA by wavefield extrapolation when we consider several iterations of migrations and gradient computations. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{MVA by wavefield-extrapolation}
Optimization of the migration velocity is a non-linear inverse problem. When using wavefield-extrapolation methods, we search for an optimal background velocity which minimizes an objective function defined in the image space. The residual that is used to compute the gradient of the objective function with respect to velocity is represented by the perturbed image $\Delta {\widehat I}$, which, in turn, is derived from the background image ${\widehat I}_0$ computed with the background slowness $s_0$. 
\par
The perturbed image can be computed by the linearized-residual prestack-depth migration \cite[]{sava2003}, and it is used in the 
wave-equation migration-velocity analysis \cite[]{biondi:1723, sava2004}. Also, the perturbed image can be computed by the differential-semblance-optimization (DSO) operator \cite[]{symes:654}, and it is used in the differential-semblance velocity optimization (DSVA) \cite[]{shen2004, shen:VE49}. 
\par
Under the $\ell_2$ norm, the DSVA objective function $J_{\rm DSO}$ is
\begin{eqnarray}
J_{\rm DSO} = \frac{1}{2}||\Delta {\widehat I}||_2 = \frac{1}{2}||{\bf H}[{\widehat I}_0]||_2,
\label{jdso}
\end{eqnarray}
where $\bf H$ is the  DSO operator either in the subsurface-offset domain or in the angle domain. In the subsurface-offset domain,  DSO penalizes energy departing from the zero offset, by weighting the background image with the absolute value of the subsurface offset. In the angle domain, DSO penalizes the lack of flattening of reflectors, by taking the derivative of the background image along the aperture angle.
\par
DSVA is solved by gradient-based optimization techniques, such as the nonlinear conjugate-gradient method, to which the gradient must be explicitly computed. The gradient ${\bf \nabla}J_{\rm DSO}$ of the objective function of equation \ref{jdso} with respect to the slowness ${\bf s}$ is
\begin{eqnarray}
{\bf \nabla} J_{\rm DSO} = {\bf T}^{\prime}{\bf H}^{\prime}{\bf H} I_0, \label{eq_grad_dso}
\end{eqnarray}
where ${\bf T} = \left.\frac{\partial I}{\partial s}\right|_{s=s_0}$ is the wave-equation tomographic operator (WETOM), evaluated at $s=s_0$, which is composed of several operators and linearly maps the slowness perturbation ${\Delta s}$ into the perturbed image.
\par
 WETOM has been evaluated in the source-receiver domain \cite[]{sava2004}, in the shot-profile domain \cite[]{shen2004}, and in the generalized-sources domain \cite[]{tang2008}. We apply WETOM in the image-space generalized-sources domain. To illustrate the process we use 35 ISPEW modeled from 12 reflectors  selected from the background image computed with the velocity model of Figure \ref{fig:ismarm101}b. This velocity model is obtained by smoothing the Marmousi velocity model (Figure \ref{fig:ismarm101}a) and multiplying it by 0.9 only below the black horizon. In Figure \ref{fig:ismarm201}, is represented the initial conditions for the modeling of upgoing PERM wavefields. 
\plot{ismarm201}{width=0.8\textwidth}{Selected reflectors used to model 35 pairs of ISPEW.}
\plot{ismarm101}{width=0.8\textwidth}{Velocity models used in the explanation of MVA by wavefield extrapolation with the image-space generalized wavefields: a) True Marmousi velocity model. b) Background velocity model used in the initial migration and in the modeling of ISPEW.}
\par
In the areal-shot migration of image-space generalized wavefields, they are downward continued with the following one-way wave equations:
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z}+i\sqrt{\omega ^2 s^2({\bf x})-|{\bf k}| ^2}\right){\widehat D_0}({\bf x},{\bf p},\omega) = 0 \\
{\widehat D_0}(x,y,z=z_{min},{\bf p},\omega) =  {\widetilde  D}(x,y,z=z_{min},{\bf p},\omega)
\end{array} \right., \label{aeq_wveq_bD}
\end{eqnarray}
and
\begin{eqnarray}
\left\{ \begin{array}{l}
\left( \frac{\partial}{\partial z}-i\sqrt{\omega ^2 s^2({\bf x})-|{\bf k}| ^2}\right){\widehat U_0}({\bf x},{\bf p},\omega) = 0 \\
{\widehat U_0}(x,y,z=z_{min},{\bf p},\omega) = {\widetilde  U}(x,y,z=z_{min},{\bf p},\omega)
 \end{array} \right., \label{aeq_wveq_bU}
 \end{eqnarray}
 where ${\widehat D_0}({\bf x},{\bf p},\omega)$ is the image-space generalized background downgoing wavefield for a single frequency, ${\widehat U_0}({\bf x},{\bf p},\omega)$ is the image-space generalized background upgoing wavefield for a single frequency, ${\bf p}$ is the index of the areal shot, and ${\widetilde  D}(x,y,z=z_{min},{\bf p},\omega)$ and ${\widetilde  U}(x,y,z=z_{min},{\bf p},\omega)$ are the data synthesized with PERM and collected at $z=z_{min}$, which denotes the top of a target zone. These data serve as the boundary conditions of equations \ref{aeq_wveq_bD} and \ref{aeq_wveq_bU}, respectively. Snapshots of the background ISPEW are shown in Figure \ref{fig:ismarm202}.
 \plot{ismarm202}{width=0.8\textwidth}{Snapshots of background ISPEW: a) downgoing, and b) upgoing wavefields.}
 \par
 The crosscorrelation imaging condition produces the background image ${\widehat I}({\bf x},{\bf h})$ (Figure \ref{fig:ismarm203})
 \begin{eqnarray}
 {\widehat I}({\bf x},{\bf h}) = \sum_{\bf p}\sum_{\omega} {\widehat D}^*({\bf x}-{\bf h},{\bf p},\omega) {\widehat U}({\bf x}+{\bf h},{\bf p},\omega).
 \label{aeq_ic_bI}
 \end{eqnarray}
 \plot{ismarm203}{width=0.8\textwidth}{Background image computed with the image-space generalized background wavefields of Figure \ref{fig:ismarm202}.}
\par
 The perturbed image is derived by applying the differentiation-product rule to equation \ref{aeq_ic_bI}, which gives
 \begin{eqnarray}
 \Delta {\widehat I}({\bf x},{\bf h})  &=& \sum_{\bf p}\sum_{\omega} \Delta {\widehat D}^*({\bf x}-{\bf h},{\bf p},\omega) {\widehat U}_0({\bf x}+{\bf h},{\bf p},\omega) + \nonumber \\
 & & {\widehat D}_0^*({\bf x}-{\bf h},{\bf p},\omega) \Delta {\widehat U} ({\bf x}+{\bf h},{\bf p},\omega), \label{aeq_ic_dI}
 \end{eqnarray}
 where $\Delta {\widehat D}({\bf x}-{\bf h},{\bf p},\omega)$ and $\Delta {\widehat U}({\bf x}+{\bf h},{\bf p},\omega)$ are the image-space generalized perturbed downgoing wavefield and the image-space generalized perturbed upgoing wavefield, respectively. These image-space generalized perturbed wavefields are the response to a slowness perturbation. They satisfy the following one-way wave equations linearized with respect to the slowness:
 \begin{eqnarray}
 \left\{ \begin{array}{l}
 \left( \frac{\partial}{\partial z}+i\sqrt{\omega ^2 {s}_0^2 ({\bf x})-|{\bf k}|^2}\right) \Delta {\widehat D}({\bf x},{\bf p},\omega)= {\widehat D}_{\scriptscriptstyle SC}({\bf x},{\bf p},\omega)
  \\
  \Delta {\widehat D}(x,y,z=z_{min},{\bf p},\omega) = 0 \end{array} \right. , \label{aeq_wveq_dD}
  \end{eqnarray}
  and
  \begin{eqnarray}
  \left\{ \begin{array}{l}
  \left( \frac{\partial}{\partial z}-i\sqrt{\omega ^2 s_0^2 ({\bf x})-|{\bf k}|^2}\right) \Delta {\widehat U}({\bf x},{\bf p},\omega)={\widehat U}_{\scriptscriptstyle SC}({\bf x},{\bf p},\omega)\\
  \Delta {\widehat U}(x,y,z=z_{min},{\bf p},\omega) = 0 \end{array} \right.. \label{aeq_wveq_dU}
  \end{eqnarray}
  Snapshots of the image-space generalized perturbed wavefields are shown in Figure \ref{fig:ismarm205}.
  \plot{ismarm205}{width=0.8\textwidth}{Snapshots of image-space generalized perturbed wavefields: a) downgoing, and b) upgoing.}
 No scattering occurs above the black horizon of Figure \ref{fig:ismarm101} because the initial conditions above it are null. The wavefields in the right-hand side of equations \ref{aeq_wveq_dD} and \ref{aeq_wveq_dU} are the image-space generalized scattered downgoing and upgoing wavefields, respectively, which result from the interaction of the image-space generalized background wavefields with a slowness perturbation according to
  \begin{eqnarray}
  {\widehat D}_{\scriptscriptstyle SC}({\bf x},{\bf p},\omega)=\frac{i\omega \Delta s({\bf x})}{\sqrt{1-\frac{|{\bf k}|^2}{\omega^2 {s}_0^2({\bf x})}}}{\widehat D}_0({\bf x},{\bf p},\omega)
  \label{aeq_Dsc}
  \end{eqnarray}
  and 
  \begin{eqnarray}
  {\widehat U}_{\scriptscriptstyle SC}({\bf x},{\bf p},\omega)=\frac{- i\omega \Delta s({\bf x})}{\sqrt{1-\frac{|{\bf k}|^2}{\omega^2 s_0^2({\bf x})}}}{\widehat U}_0({\bf x},{\bf p},\omega).
  \label{aeq_Usc}
  \end{eqnarray}
  These wavefields are injected at every depth level during the recursive downward propagation of the perturbed  wavefields. The image-space generalized perturbed downgoing and upgoing  wavefields are used along with the precomputed image-space generalized background downgoing and upgoing wavefields in equation \ref{aeq_ic_dI} to generate the perturbed image (Figure \ref{fig:ismarm204}). 
  \plot{ismarm204}{width=0.8\textwidth}{Perturbed image computed with equation \ref{aeq_ic_dI}.}
  \par
  The adjoint WETOM ${\bf T}'$ is obtained by the folowing operations. First, the adjoint-imaging condition is applied to compute the image-space generalized perturbed downgoing and upgoing wavefields according to the following convolutions:
  \begin{eqnarray}
  \Delta {\widehat D}({\bf x},{\bf p},\omega) &=& \sum_{\bf h} \Delta {\widehat I}({\bf x},{\bf h}) {\widehat U}_0({\bf x}+{\bf h},{\bf p},\omega) \nonumber \\
  \Delta {\widehat U}({\bf x},{\bf p},\omega) &=& \sum_{\bf h} \Delta {\widehat I}({\bf x},{\bf h}) {\widehat D}_0({\bf x}-{\bf h},{\bf p},\omega). \label{aeq_aic_dI}
  \end{eqnarray}
  The image-space generalized perturbed wavefields are upward propagated using the adjoint counterparts of equations \ref{aeq_wveq_dD} and \ref{aeq_wveq_dU}. At every depth of their upward propagation, the image-space generalized perturbed downgoing wavefield is cross-correlated with the image-space generalized scattered downgoing wavefield, and the image-space generalized perturbed upgoing wavefield is cross-correlated with the image-space generalized scattered upgoing wavefield to generate the slowness perturbation according to
  \begin{eqnarray}
  \Delta {\widehat s} ({\bf x}) &=& \sum_{{\bf p}}\sum_{\omega} {\widehat D}_{\scriptscriptstyle SC}^*({\bf x},{\bf p},\omega) \Delta {\widehat D}({\bf x},{\bf p},\omega) + \nonumber \\
  & & {\widehat U}_{\scriptscriptstyle SC}^*({\bf x},{\bf p},\omega) \Delta {\widehat U}({\bf x},{\bf p},\omega).
  \label{aeq_asc}
  \end{eqnarray}
  The slowness perturbation for the Marmousi example computed in the image-space generalized sources domain using ISPEWs is shown in Figure \ref{fig:ismarm206}.
  \plot{ismarm206}{width=0.8\textwidth}{Slowness perturbation from the back-projected image perturbations computed with 11 ISPEWs.}
\par
When performing velocity optimization using DSVA, for instance, the perturbed image in equation \ref{aeq_aic_dI} is computed with DSO at the begining of every nonlinear iteration.
DSO easily automates migration-velocity optimization. However, neither the phase nor the amplitudes of the DSO perturbed image are consistent with those of the perturbed image computed by the forward one-way WETOM \cite[]{vyas:4077,fei:4071}. These differences prevent the use of linear conjugate-gradient methods, and therefore the objective function computed with DSO is typically minimized by nonlinear optimization methods.
  \par
  The gradient of the MVA by wavefield extrapolation objective function is sensitive to amplitude variations of the prestack image caused by uneven illumination. Since these amplitude variations are not related to velocity inaccuracy, we should ideally attenuate them using some sort of illumination compensation scheme \cite[]{valenciano:WCA109,tang:WCA95}. Instead, to prevent these amplitude variations we apply a B-spline smoothing to the gradient, which consists of representing the gradient as B-spline basis functions, using the adjoint operator $\bf B'$, and transforming it back to the Cartesian space, using the forward operator $\bf B$. 
  \par
 When using image-space generalized wavefields, a target-oriented strategy can be adopted if the velocity model is sufficiently accurate for shallower layers. A mask operator $\bf M$ is applied to the gradient, zeroing out amplitudes in the accurate velocity region, preventing the velocity model from being updated. 
  \par
  Since the gradient is not properly scaled, we normalize it by the inverse of its maximum absolute value multiplied by the minimum slowness, which characterizes the diagonal operator $\bf F$. To improve convergence, we would like to limit the velocity update from one iteration with respect to its previous values. In other words, we would like the new velocity to vary within a range defined by a percentage of the velocity from the previous iteration. This can be implemented by applying to the gradient either a nonlinear (since it depends on the velocity) diagonal operator, or a diagonal operator $\bf W$ linearized around the initial velocity. Therefore, in contrast with the gradient of equation \ref{eq_grad_dso}, the one  we use to update the velocity model is
  \begin{eqnarray}
  {\bf \nabla} J_{\scriptscriptstyle DSO} = {\bf WFMBB^{\prime}} \left . {\bf T}^{\prime}\right |_{s=s_0} {\bf H}^{\prime}{\bf H} \left .{\widetilde I}\right |_{s=s_0} \label{eq_grad_is}.
  \end{eqnarray}
  \par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{3D example }
We estimate the migration-velocity model with 3D ISPEW synthesized from prestack images of the North Sea. The challenges for defining the velocity model for this dataset are due to a possibly irregular salt body, the intense faulting, the amplitude variations caused by irregular acquisition, the short source-receiver offsets, and the limited source-receiver azimuths. Because of the narrow azimuthal configuration, the 3D dataset was transformed to common-azimuth data by azimuth-moveout (AMO), and common-azimuth migration (CAM)  images are used as the initial conditions for the modeling of very few 3D ISPEW. 
Thirty nodes of Dual Nehalem 5520 with 24Gb RAM were used, accounting to 240 CPUs. To generate 30 pairs of 3D ISPEWs, using 196 frequencies, it took 10 minutes. On average, each iteration of the velocity optimization, consisting of one function evaluation, the gradient computation, and two additional function evaluations for the line search, took approximately two hours.   The velocity estimation presented here could have been easily performed interactively if a larger, but commonly available in industrial setting, number of computer nodes have been used to perform the computations. 
\par
The 3D North Sea dataset spans over an area of approximately 55 km$^2$, with 13.5 km in-lines and 4 km cross-lines. It was acquired using dual sources at intervals of 25 m in the in-line direction and 50 m in the cross-line direction, with three cables 100 m apart and a maximum offset 3600 m. The limited cross-line offsets resulting from this acquisition configuration impose limitations in the azimuthal distribution. The offset distribution is spatially quite irregular, as can be seen in the fold of coverage map (Figure \ref{fig:3dex1025}a). The acquisition footprint is evident, with a wide low-fold region occurring for in-lines around 4000 m in the y direction. In spite of the regularization provided by AMO, amplitude variations caused by the offset irregularity still persists as can be seen in Figure \ref{fig:3dex105}b. It shows a  time slice at  2.8 s through the trace envelope for the stacked volume of the regularized data.
\plot{3dex1025}{width=0.8\textwidth}{a) Fold of coverage map, and b) time slice through the trace-envelope cube of stacked AMO data.} 
\par
Although the fold irregularity and amplitude imbalance  are in the data-space, we will see that the acquisition footprint is evident in the amplitudes of the gradient of the objective function at approximately the same spatial position. 
\par
The velocity model provided along with the data (Figure \ref{fig:3dex104}), herein called the original velocity model, presents a general layered structure with an overhanging salt dome connected to a deeper layer with the same velocity as that of the salt dome. The shallower layer with a domed structure is a chalk layer, herein called chalk layer. The final velocity model we derive shows remarkable differences when compared to the original velocity model.
\plot{3dex104}{width=0.8\textwidth}{Slices through the original velocity model.} 
\par
To derive the initial velocity model, the sediment velocity above the chalk layer was refined using residual prestack depth-migration scans. Below the top of chalk, velocity was heavily smoothed using a 5000 m wide 2D median smoother. In addition, to increase the inaccuracy of the initial velocity model, velocity was scaled down by a factor of 0.9 (Figure \ref{fig:3dex105}). 
\par
A layer-stripping approach was used to define the velocity for the chalk layer, considering a sufficiently accurate velocity for the sediments above. Then, the top salt was interpreted, and a salt flooding procedure enabled the interpretation of the base of salt. The interpretation of the base of salt can be considered the main source of uncertainty of this example, since no previous geological  information was available. Finally, a group of reflectors below the salt is used to define the velocity structure for the deeper part. 
\par
\plot{3dex105}{width=0.8\textwidth}{Slices through the  initial velocity model used in the velocity optimization.} 
The initial velocity model produces the CAM image of Figure \ref{fig:3dex1066}, which shows the volume for the zero subsurface offset, and Figure \ref{fig:3dex405}, which shows the zero subsurface offset on the top, and ADCIGs at the bottom for a central in-line. The effects of migrating with a too low velocity are evidenced by poorly collapsed diffractions close to the salt flank, poorly imaged faults, and reflectors curving up in the ADCIGs.
\plot{3dex1066}{width=0.8\textwidth}{Slices through the CAM image computed with the initial velocity model of Figure \ref{fig:3dex105}, showing poorly collapsed diffractions close to the salt flank and poorly imaged faults.} 
\plot{3dex405}{width=0.8\textwidth}{Central in-line of the CAM image computed with the initial velocity model. On the top is the zero-subsurface  offset section, and ADCIGs at the bottom.} 
\par
 The base of chalk was interpreted in the 3D pre-stack volume and used as input for the modeling of 30 ISPEW. In-line and cross-line intervals of the CAM image were interpolated from 20 m to 30 m, which are the in-line and cross-line intervals used for optimizing the migration velocity. 
\par
Since no cross-line offset is computed, the CAM initial conditions can be continuously sampled in the cross-line direction, reducing by at least one order of magnitude the number of image-space generalized wavefields to be synthesized, following \cite{guerra_thesis}. For the base of chalk, only 30 3D ISPEWs were modeled and  collected at 600 m depth. A pair of 3D-ISPEW is shown in Figure \ref{fig:3dex109}. Since the number of subsurface offsets that will be used during velocity optimization is 25 and the spatial sampling in the x direction of 30 SODCIGs is used to model the 3D ISPEWs, crosstalk is expected to be sufficiently attenuated. 
\plot{3dex109}{width=0.8\textwidth}{A pair of 3D downgoing  (a) and upgoing (b) ISPEWs computed for the base of chalk.} 
\par
In the velocity optimization for the chalk layer, velocity update is constrained to a maximum of 10$\%$ variation between iterations. All the wavefield propagation is performed between the depth at which the 30 3D ISPEWs were collected (600 m) and a maximum depth of 3300 m. Velocity is updated up to the top of the chalk layer.
\par
 To illustrate the problem of amplitude variations in the raw gradient, we compute the slowness perturbation without applying smoothing (Figure \ref{fig:3dex2102}a). The acquisition footprint is clear, specially around the y coordinate 4000 m, which is a region with low-fold of coverage (Figure \ref{fig:3dex1025}). The gradient of the objective function must be smooth to yield velocity updates consistent with the Born approximation. We apply a B-spline smoothing with node intervals of 420 X 420 X 160 m in the in-line, cross-line and depth directions, respectively. Applying B-spline smoothing on the DSO slowness perturbation of Figure \ref{fig:3dex2102}a mitigates the amplitude-variation problems and yields consistent slowness perturbations (Figure \ref{fig:3dex2102}b). 
\plot{3dex2102}{width=0.8\textwidth}{a) Slowness perturbation without smoothing, and b) Slowness perturbation after B-spline smoothing.} 
\par
Two runs of velocity optimization were necessary to define the velocity model for the chalk layer. The first used wavefields modeled from all the extension of base of chalk. Figure \ref{fig:3dex213} shows the evolution of the objective function for the first run. Since CAM with the resulting velocity revealed velocity inaccuracies close to the salt flanks, new wavefields were modeled having the initial conditions limited to an area approximately 2 km wide around the salt edge. By doing this, we explore the localized nature of these wavefields, so that this second run is targeted for updating velocity close to salt flanks. In both runs, two function evaluations were used in the line search. 
\plot{3dex213}{width=0.8\textwidth}{Evolution of the DSVA objective function for the first run of velocity optimization for the base of chalk.}
\par
The optimized velocity models for the first and second runs of velocity optimization is shown in Figure \ref{fig:3dex216a}. Detailed views of CAM images computed with these velocity models are shown in Figure \ref{fig:3dexzoom}. Only slices through the zero subsurface offset are shown. Overall, the image computed with the velocity of the second run of velocity optimization is better focused. 
\par
\plot{3dex216a}{width=0.8\textwidth}{Slices through the optimized velocity for the chalk layer from: a) the first run of velocity optimization, and b) the second run of velocity optimization. The dashed white line approximately represents the base of chalk.} 
\plot{3dexzoom}{width=0.8\textwidth}{Detailed view of slices through the zero subsurface offset of the CAM migrated images computed with velocity from the a) first run of velocity optimization, and b) second run of velocity optimization. On the left is an in-line and on the right, a cross-line.}
Once a sufficiently accurate velocity for the chalk layer had been defined, salt flooding was used to  delineate the salt body. The top salt was interpreted, and below it velocity was replaced by a constant value of 4500 m/s. CAM image computed with the salt-flooded velocity supported the interpretation of the base of the salt. Insufficient illumination due to the limited azimuthal coverage and irregular shape of the salt body caused the base of salt to be discontinuous. In this situation, prior  geological information would be extremely helpful to constrain the interpretation. The lack thereof is a source of uncertainty for defining the velocity model below the salt.
\par
After the base of salt was interpreted, salt velocity was delineated and the initial velocity was inserted below the salt and the chalk layer (Figure \ref{fig:3dex313}). This new velocity model produced the CAM  image in Figure \ref{fig:3dex400} from which seven deeper reflectors are used to model new ISPEWs to optimize sub-salt and sub-chalk velocity. These wavefields were collected at a depth of 1650 m, which is the minimum depth used in the velocity optimization. The maximum depth is 4800 m. 

\plot{3dex313}{width=0.8\textwidth}{Slices through the velocity volume after interpretation of the base of salt.} 
\plot{3dex400}{width=0.8\textwidth}{Slices through the CAM image computed with the migration velocity of Figure \ref{fig:3dex313}, showing in yellow the selected reflectors for the modeling of 30 3D ISPEWs to be used in the sub-salt velocity optimization: a) the zero subsurface offset, and b) a central in-line.}
\par
In the sub-salt velocity optimization, the intervals between B-spline nodes is 1050 m in the $x$ and $y$ directions and 150 m in the $z$ direction. A maximum of 5$\%$ local velocity variation is allowed between iterations, and two function evaluations are performed in the line search. The final objective function dropped only 10$\%$ with respect to its initial value. This decrease is small compared to the 40$\%$ decrease for the velocity optimization of the chalk layer. Apart from the poorer illumination in the sub-salt and sub-chalk case, which prevents the complete focusing at the zero subsurface offset, this difference can be explained by the different levels of crosstalk. For the chalk layer, only crosstalk from different SODCIGs occurs, whereas for the sub-salt case, crosstalk from different SODCIGs and reflector crosstalk were generated. To decrease the effect crosstalk has on the objective function for the sub-salt-velocity optimization, more random realizations could have beeen used as well as increasing the SODCIG sampling interval.  
%\sideplot{3dex402}{width=\textwidth}{Evolution of the DVSA objective function of the sub-salt velocity  optimization.}
\par
 The evolution of the velocity model through iterations is shown in Figure \ref{fig:3dex403}. CAM with the optimized velocity model pf Figure \ref{fig:3dex403}d can be seen in Figure \ref{fig:3dex404}. For comparison, CAM with the  initial velocity model and CAM with the original velocity model are shown in Figures \ref{fig:3dex405} and \ref{fig:3dex406}. When compared with the results using the initial velocity model, the improvements obtained with the optimized velocity model are clear: flatter angle gathers, better focusing of the reflectors, and imaging of the faults. The improvements compared to the original (unmodified) velocity model are also clear: better focusing of reflectors and slightly flatter angle gathers below the salt as well as close to its flanks.
\plot{3dex403}{width=0.8\textwidth}{Slices through velocity models for: a) first iteration, b) third iteration, c) fifth iteration, and d) seventh iteration.} 
\plot{3dex404}{width=0.8\textwidth}{Central in-line of the CAM image with the final velocity model after optimization for the chalk layer, salt flooding, and sub-salt velocity optimization. On the top is the zero-subsurface offset section, and at the bottom ADCIGs.} 
\plot{3dex406}{width=0.8\textwidth}{Central in-line of the CAM image with the original velocity model. On the top is the zero-subsurface offset section, and at the bottom ADCIGs.} 
\par
Figure \ref{fig:3dex502} shows additional comparisons between the CAM images obtained with the initial, with the original, and with the final velocity models, for different in-lines and cross-lines. The images are displayed as vertical slices through the migrated cube and show the zero subsurface offset. The initial image is on the top, the original image in the middle, and the final image at the bottom.  The left panel of each figure shows the in-line, and the right panel shows the cross-line. Ovals highlight the main differences. Overall, the final images present better quality than do the initial and original images, expressed by better focusing and continuity of reflectors in the chalk layer as well as in the sub-salt region.  The optimized velocity model allowed imaging of a complex fault system on the  right of the salt body, collapsing diffractions from the salt flank.
\plot{3dex502}{width=0.8\textwidth}{CAM images obtained with: a) the initial velocity model, b)  the original velocity model, and  c) the final velocity model. The final image shows better focusing and continuity for the sub-salt reflectors, better definition of subtle faults in the chalk layer.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{conclusions}
We introduced the image-space generalized wavefields, which are synthesized with the prestack exploding-reflector model. When phase encoded, these wavefields originate the image-space phase-encoded wavefields. 
Using 3D ISPEW has proven to greatly accelerate 3D-MVA by wavefield extrapolation, thanks to the small number of wavefields needed to satisfactorily describe the kinematics of the prestack image and the fact that they can be computed in a target-oriented manner. Because of these computational efficiencies, we could solve a 3D migration-velocity problem
employing only limited computational resources. Considering the computational resources available in the industry, using 3D ISPEW can turn MVA by wavefield extrapolation an interactive process, which can yield more accurate and geologically reasonable solutions.
\par
In a 3D example from the North Sea, with only 30 ISPEW we were able to determine an accurate migration-velocity model, using different strategies like layer stripping and based on horizons. We also used a target-oriented strategy, which  allowed updating the velocity model not only within a limited depth range, but also within a limited lateral extent. This is achieved by selecting for synthesizing the wavefields a specific portion of a reflector or group of reflectors that still present residual moveout.
\par
Besides the computational gain, 3D ISPEWs were able to provide reliable velocity updates. Using the final velocity model produces a CAM image with quality superior to that obtained with the initial velocity model, as expected. Moreover, the image computed with the final velocity model is more accurate than that computed with the original velocity model, with better focusing and continuity  of the sub-salt reflectors and better fault imaging. The computational efficiency, flexibility, and velocity accuracy obtained with data computed by the prestack exploding-reflector model enables the use of MVA by wavefield extrapolation as a routine procedure to define migration velocity in areas of complex geology. 

\section{Acknowledgements}
We would like to thank the sponsors of the Stanford Exploration Project for the financial support. The authors would like to thank TotalFinaElf for the field data and the Stanford Center for Computational Earth $\&$ Environmental Science in whose computational infrastrutcure all the computations were carried out. Claudio Guerra is indebted to Petrobras for all kinds of support during his studies at Stanford University. 

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