\title{Fast 3D wave-equation migration-velocity analysis using the prestack exploding-reflector model}

\lefthead{Guerra and Biondi}
\righthead{WEMVA 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, velocity-model definition should use methods that describe the complexity of wavefield propagation, such as focusing and defocusing
, multiple paths, and frequency-dependent velocity sensitivity. Migration-velocity analysis (MVA) by wavefield extrapolation has the ability to
address these issues because it uses wavefields as carriers of information. However, its high cost and low flexibility for parametrizing 
the model space has 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 by using a pre-stack generalization of the exploding-reflector model. Cost of MVA 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 layer-based parametrization of the model space.
 Migration with the 
optimized velocity model provides good quality and reliable depth images, as we show in a 3D-field data example.
\end{abstract}

\section{introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{prestack exploding-reflector model} 
A prestack image computed with an inaccurate velocity model by wavefield-extrapolation methods 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 originate 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 single subsurface-offset common-image gather (SODCIG) of a reflector. 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 the same kinematics 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 (ERM). However, since migration of PERM data can produce a prestack image, PERM can be considered a generalization of ERM. The modeling of PERM  wavefields can be carried out by any wavefield-continuation 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 offset 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 its 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, 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=0.9\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 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. 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. 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 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 usually performed in the data space. PERM wavefields can be phase-encoded in the image space, using stochastic phase functions parametrized according to the model-space coordinates and selected reflector. 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. This is achieved 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. Sequences used in third- and fourth-generation cellphones can also be used to phase-encode the modeling \cite[]{guerra2008b}. Downgoing and upgoing image-space phase-encoded wavefields (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. According to the law of large numbers, crosstalk attenuation is more effective for more random realizations. 
\par
PERM wavefields can be collected at any depth, which can be the top of a region of inaccurate velocity, having key reflectors within this region 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 yielded by PERM drastically decreases the cost of MVA by wavefield extrapolation. 
\par
To illustrate the computation 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}). 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}{Velocity models for the Marmousi example: a) Smooth velocity model used to model the 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) Upward wavefield. b) Downward wavefield.}  
\plot{ismarm06a}{width=0.8\textwidth}{ISPEW for the Marmousi example: a) Upward wavefield. b) Downward wavefield.}  
\plot{ismarm04}{width=0.8\textwidth}{Pre-stack image computed with PERM wavefields and background velocity model using: a) the conventional imaging condition (equation \ref{eq:I_perm}), and b) the time-windowed imaging condition (equation \ref{eq:wic_bI}). Reflector crosstalk is avoided when reflectors are sufficiently separated. However, some residual crosstalk is still present (RC). Notice the phase difference of the PERM image due to the squaring of the wavelet when compared to the windowed reflectors of Figure \ref{fig:ismarm02}b.}
\plot{ismarm07}{width=0.8\textwidth}{Pre-stack  images computed with: a) Four random realizations of ISPEWs, and b) a single random realization.}
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 ADCIGs corresponding to images computed with PERM wavefields and ISPEWs, 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 the 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 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 360 shot gathers, b) areal-shot migration of 35 PERM wavefields using the time-windowed imaging condition, c) areal-shot migration of 44 ISPEWs corresponding to four random realizations, and d) areal-shot migration of 11 ISPEWs corresponding to a single random realization. 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. Using the same wavefields, computing the gradient of the objective function of MVA by wavefield-extrapolation can be 60 times faster.  This difference in performance is even more dramatic when we consider that several iterations of migrations and gradient computations are performed during migration-velocity  optimization by wavefield-extrapolation.

\section{MVA by wavefield-extrapolation}
Optimization of the migration velocity is a non-linear inverse problem. When using wavefield-extrapolation methods, it searches for an optimal background velocity which minimizes an objective function defined in the image space. The residual that characterizes the objective function 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 at zero offset, by weighting the background image with the absolute value of the subsurface offset. In the angle domain, DSO penalizes flattening of events, by taking the derivative of the background image along the aperture angle.
\par
Gradient-based optimization techniques, such as the nonlinear conjugate-gradient method, can be used in the optimization, 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 and receiver domain \cite[]{sava2004}, in the shot-profile domain \cite[]{shen2004}, and in the generalized-sources domain both in the data and in the image space \cite[]{tang2008}. We discuss WETOM in the image-space generalized-sources domain, using 35 ISPEW modeled from 12 reflectors (Figure \ref{fig:ismarm201}) 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.
\plot{ismarm201}{width=0.6\textwidth}{Selected reflectors used to model 35 pairs of ISPEW.}
\par
In the areal-shot migration of image-space generalized wavefields, wavefields 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}({\bf x},{\bf p},\omega) = 0 \\
{\widehat D}(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}({\bf x},{\bf p},\omega) = 0 \\
{\widehat U}(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 ${\widehat D}({\bf x},{\bf p},\omega)$ is the image-space generalized downgoing wavefield for a single frequency, ${\widehat U}({\bf x},{\bf p},\omega)$ is the image-space generalized 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 using phase encoding or not, and collected at $z=z_{min}$, which denotes the top of a target zone. PERM 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=0.6\textwidth}{Snapshots of kground ISPEW: a) downgoing, and b) upgoing wavefields.}
 \par
 The cross-correlation imaging condition produces the 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=0.6\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 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 ${\widehat D}_0({\bf x}-{\bf h},{\bf p},\omega)$ and ${\widehat U}_0({\bf x}+{\bf h},{\bf p},\omega)$ are the image-space generalized background downgoing and upgoing wavefields computed with the background slowness; and $\Delta {\widehat D}({\bf x}-{\bf h},{\bf p},\omega)$ 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. The image-space generalized perturbed wavefields 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=0.6\textwidth}{Snapshots of image-space generalized perturbed wavefields: a) source, and b) receiver.}
  Since the boundary conditions at $z_{min}$ are null, no scattering occurs at depth levels shallower than $z_{min}$. 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 source and receiver wavefields are used along with the precomputed image-space generalized background source and receiver wavefields in equation \ref{aeq_ic_dI} to generate the perturbed image (Figure \ref{fig:ismarm204}). 
  \plot{ismarm204}{width=0.6\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 back-projected image perturbations computed with 35 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. 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 wave-equation tomography 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 and sometimes guarantee 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 field-data example }
We estimate the migration-velocity model for a 3D field-data 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 submitted to azimuth-moveout (AMO), and common-azimuth migration (CAM)  images are used as the initial conditions for the modeling of very few 3D ISPEW. 
All the computations were carried out on computer nodes of the Stanford Center for Computational Earth $\&$ Environmental Science (CEES). 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.    
\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 maps for different offset ranges (Figure \ref{fig:3dex102}). 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:3dex103}. It shows time slices at  2.8 s through the trace envelope for different offset cubes taken from the regularized data.
\plot{3dex102}{width=\textwidth}{Fold of coverage plots: a) full offset, b) 0 -- 1200 m offset, c) 1200 -- 2400 m offset, and d) 2400 -- 3600 m offset.} 
\plot{3dex103}{width=\textwidth}{Time slices through the trace envelope for different offset cubes from the regularized data: a) offset 200m, b) offset 1200 m, c) offset 2400 m, and d) offset 3000 m.} 
\par
Even though the fold irregularity (Figure \ref{fig:3dex102}) and amplitude imbalance (Figure \ref{fig:3dex103}) are in the data-space, we will see that they will be 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 final velocity model we derive shows remarkable differences when compared to the original velocity model,  especially in the salt body shape. 
\plot{3dex104}{width=\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}). 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 the 3D-field data 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=\textwidth}{Slices through the  initial velocity model used in the velocity optimization.} 
The initial velocity model produces the CAM image of Figure \ref{fig:3dex106}, which shows the volume for the zero subsurface offset, and Figure \ref{fig:3dex1073180}, which shows the zero subsurface offset on the left, and ADCIGs on the right for in-line 3180. 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{3dex106}{width=1.2\textwidth, angle=90}{Slices through the CAM image with the initial velocity model of Figure \ref{fig:3dex105}. Notice poorly collapsed diffractions close to the salt flank (A and D), and poorly imaged faults (B and C) caused by migrating with an inaccurate velocity.} 
\plot{3dex1073180}{width=1.3\textwidth, angle=90}{In-line 3180 of the CAM image with the initial velocity model of Figure \ref{fig:3dex105}. On the left is the zero-subsurface offset section, and on the right ADCIGs. Notice the strong residual moveout on the ADCIGs. The continuous blue line is the top of chalk, and the dashed blue line it the base of chalk.} 
\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. 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 ISWET 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 strongly attenuated during imaging in ISWET.
\plot{3dex109}{width=\textwidth}{A pair of 3D source (a) and receiver (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 the maximum depth (3300 m). Velocity is updated up to the top of the chalk layer.
\par
 To illustrate the amplitude variation problem, we compute the slowness perturbation without applying smoothing (Figure \ref{fig:3dex210}). The acquisition footprint is clear, specially around the y coordinate 4000 m, which is a region with low-fold of coverage (Figure \ref{fig:3dex102}). 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:3dex211} mitigates the amplitude-variation problems and yields consistent slowness perturbations (Figure \ref{fig:3dex212}). 
\plot{3dex210}{width=\textwidth}{Slowness perturbation without smoothing.} 
\plot{3dex212}{width=\textwidth}{Slowness perturbation after B-spline smoothing the slowness perturbation of Figure \ref{fig:3dex211}.} 
\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. Since CAM with the resulting velocity revealed velocity inaccuracy close to the salt flanks, new ISPEWs were modeled having the initial conditions limited to approximately 2 km around the salt body. 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. Figure \ref{fig:3dex213} shows the evolution of the objective function for the first run. 
\plot{3dex213}{width=\textwidth}{Evolution of the DVSA objective function for the first run of ISWET for the base of chalk.}
\par
The optimized velocity for the first run of ISWET is shown in Figure \ref{fig:3dex216}. The initial background image and the background image of the seventh iteration can be seen in Figures \ref{fig:3dex214} and \ref{fig:3dex215}, respectively. On the left is the zero-subsurface-offset section, and on the right are the SODCIGs. Overall, the focusing around the zero subsurface offset improved. However, close to the salt flanks, SODCIGs still show events curving down.
\plot{3dex214}{width=1.3\textwidth, angle=90}{In-line 3520 of the initial background image of the first run of ISWET for the base of chalk.} 
\plot{3dex215}{width=1.3\textwidth, angle=90}{In-line 3520 of the background image computed with the optimized velocity model of the first run of ISWET for the base of chalk.} 
\plot{3dex216}{width=\textwidth}{Slices through the optimized velocity from the first run of ISWET for the base of chalk. The dashed white line approximately represents the base of chalk. } 
\par
The optimized velocity for the second run of ISWET is shown in Figure \ref{fig:3dex217}. CAM using this optimized velocity is shown in Figure \ref{fig:3dex218}, which shows the volume for the zero subsurface offset, and Figure \ref{fig:3dex2193180}, which shows the zero subsurface offset on the top, and ADCIGs at the bottom for in-line 3180. Compare these figures with Figures \ref{fig:3dex106} and \ref{fig:3dex1073180}, respectively. The optimized velocity model allowed imaging of a complex fault system on the  right of the salt body, collapsing diffractions from the salt flank, and flattening reflectors in the ADCIGs.
\plot{3dex217}{width=\textwidth}{Slices through the optimized velocity from the second run of ISWET for the base of chalk. The dashed white line approximately represents the base of chalk.} 
\plot{3dex218}{width=1.2\textwidth, angle=90}{Slices through the CAM image with the optimized velocity model of Figure \ref{fig:3dex217}. Notice the imaging of a big fault close to the salt flank (A), and the focusing of a complex fault system (B, C, and D).} 
\plot{3dex2193180}{width=1.3\textwidth, angle=90}{In-line 3180 of the CAM image with the optimized velocity model of Figure \ref{fig:3dex217}. On the left is the zero-subsurface offset section, and on the right ADCIGs. Notice flatter reflectors  in the ADCIGs compared to those in Figure \ref{fig:3dex1073180}. The continuous blue line is the top of chalk, and the dashed blue line it the base of chalk.} 
\par
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 the velocity was replaced by a constant value of 4500 m/s.
The 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.
%\plot{3dex312}{width=\textwidth, angle=90}{Slices through the CAM migrated image computed with the salt flooding velocity model of Figure \ref{fig:3dex311}.} 
\par
After the base of salt was interpreted, the velocity was edited again (Figure \ref{fig:3dex313}), so that the salt velocity was confined within the salt body and the initial velocity was inserted below the salt and the chalk layer. With this new velocity model, another run of CAM generated the image in Figure \ref{fig:3dex314}  from which deeper reflectors are used to model new ISPEWs to be used in ISWET for deeper layers.
\plot{3dex313}{width=\textwidth}{Slices through the velocity volume after interpretation of the base of salt.} 
\plot{3dex314}{width=\textwidth, angle=90}{Slices through the CAM migrated image computed with the velocity model of Figure \ref{fig:3dex313}.} 

\section{Solving for sub-salt velocities}
On the prestack image of Figure \ref{fig:3dex400}, resulting from CAM with the sufficiently accurate velocity model for the chalk layer and the salt body, seven reflectors were interpreted (Figure \ref{fig:3dex401}) to be used as the initial conditions for the modeling of 3D ISPEWs for defining sub-salt velocities. Again, 30 pairs of 3D-source and -receiver ISPEWs were generated. These wavefields were collected at a depth of 1650 m, which is the minimum depth used in the velocity optimization. The prestack image used as the initial conditions for the modeling has 33 in-line subsurface offsets with 30 m intervals and a maximum offset 480 m. The CMP interval in the x and y directions is 30 m.
\plot{3dex400}{width=\textwidth}{Slices through the prestack image computed with the migration velocity of Figure \ref{fig:3dex313}: a) the zero subsurface offset, and b) the in-line 3520.} 
\plot{3dex401}{width=\textwidth}{Slices through the windowed prestack image, showing 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) the in-line 3520.} 
\par
The initial velocity used in the sub-salt velocity optimization is shown in Figure \ref{fig:3dex313}. The shallower limit for velocity updating is represented by the base of chalk and base of salt, and the deeper limit is 4800 m. The interval 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. 
\par
The evolution of the objective function normalized by its initial value is shown in Figure \ref{fig:3dex402}. Velocity optimization stopped  after 7 iterations because the update of the objective function was smaller than 0.1$\%$. The final objective function dropped 10$\%$. This decrease is small compared to the 40$\%$ decrease of the objective function for the velocity optimization of the base of chalk. For the chalk case, only crosstalk from different SODCIGs was generated because only one reflector was used to generate the 3D ISPEWs. For the sub-salt case, in addition to the crosstalk from different SODCIGs, reflector crosstalk was generated since seven reflectors were used as the initial conditions. Hence, the amount of background dispersed crosstalk  is greater in the sub-salt case than in the chalk case, explaining the smaller decrease of the objective function in the sub-salt velocity optimization.
\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}. 
\plot{3dex403}{width=\textwidth}{Slices through velocity models for: a) first iteration, b) third iteration, c) fifth iteration, and d) seventh iteration.} 
\par
CAM with the optimized velocity model can be seen in Figure \ref{fig:3dex404a4060}. For comparison, CAM with the  initial velocity model and CAM with the original velocity model are shown in Figures \ref{fig:3dex405a4060} and \ref{fig:3dex406a4060}. 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{3dex404a4060}{width=1.3\textwidth, angle=90}{In-line 4060 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{3dex405a4060}{width=1.3\textwidth, angle=90}{In-line 4060 of the CAM image with the initial velocity model. On the top is the zero-subsurface  offset section, and at the bottom ADCIGs.} 
\plot{3dex406a4060}{width=1.3\textwidth, angle=90}{In-line 4060 of the CAM image with the original velocity model. On the top is the zero-subsurface offset section, and at the bottom ADCIGs.} 
\par
Next, I show more systematic comparisons of the results computed with the initial, original and the final velocity models.
\section{Comparisons}
Here, I show three 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 left panel of each figure shows the in-line, and the right panel shows the cross-line.
\par
Figure captions explain the differences highlighted by ovals. 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. Certainly, the final image would ease interpretation and understanding of geological and structural history, adding more value to seismic information in an exploratory study.
\plot{3dex500}{width=0.99\textwidth}{On the left panel is the in-line 3320 and on the right panel is the cross-line 5220 of the CAM images obtained with: a) the initial velocity model, b)  the original velocity model, and  c)the final velocity model after optimization for the chalk layer, salt flooding, and sub-salt velocity optimization. The oval in the in-line of the final image show better focusing and continuity for the sub-salt reflectors as well as for the salt flank. Besides focusing and continuity, the oval in the cross-line of the final image show unstructured sub-salt reflectors not pulled up by velocity errors.} 
\plot{3dex501}{width=0.99\textwidth}{On the left panel is the in-line 3680 and on the right panel is the cross-line 6880 of the CAM images obtained with: a) the initial velocity model, b)  the original velocity model, and  c)the final velocity model after optimization for the chalk layer, salt flooding, and sub-salt velocity optimization. The oval in the in-line of the final image show better focusing and continuity for the sub-salt reflectors. Besides focusing and continuity, the oval in the cross-line of the final image show unstructured sub-salt reflectors not pulled up by velocity errors.} 
\plot{3dex502}{width=0.99\textwidth}{On the left panel is the in-line 4060 and on the right panel is the cross-line 9400 of the CAM images obtained with: a) the initial velocity model, b)  the original velocity model, and  c)the final velocity model after optimization for the chalk layer, salt flooding, and sub-salt velocity optimization. The oval in the in-line of the final image show better focusing and continuity for the sub-salt reflectors. In the cross-line, the upper oval of the final image show better definition of subtle faults in the chalk layer, the lower oval highlights better continuity of the reflectors, and the arrow labeled "F" indicates better fault imaging not clear in the initial and original images.} 

\section{conclusions}
The 3D-field data North Sea turned out to be a very challenging dataset. The challenges were represented by the narrow azimuth configuration and irregular fold of coverage, causing incomplete and irregular illumination of the reflectors. Also, the irregular salt body and intense faulting were difficult targets to delineate and image. The problems with the illumination, evidenced by amplitude irregularity of the gradient of the objective function, were mitigated by properly scaling the prestack image used to model the 3D ISPEWs and the gradient of the objective function, and by smoothing the gradient of the objective function.
\par
Using 3D ISPEWs has proved to greatly accelerate 3D ISWET, due to the small number of wavefields needed to satisfactorily describe the kinematics of the prestack image and the fact that they are computed in a target-oriented manner. These synthesized wavefields made it possible to solve 3D ISWET  in an academic environment with limited computational resources. Considering the computational resources available in the industry, using 3D ISPEWs can turn 3D ISWET an interactive process, which can yield more reliable solutions.
\par
Solving for migration velocity in a target-oriented manner with wave-equation methods allows updating the velocity model not only within a limited depth range, but also within a limited lateral extent. This is achieved by selecting a specific portion of a reflector or group of reflectors that still present residual moveout. This feature was demonstrated when updating the velocity model in the vicinity of the salt body, using only a small portion of the reflector corresponding to the base of chalk close to the salt.
\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. 

\section{acknowledgments}
I would like to thank Bob Clapp for making available the common-azimuth migration code and for continuously improving SEP's 3D-interpretation and visualization capability. I am indebted to Dennis Michael, manager of the High Productivity Technical Computing (HTPC) initiative hosted by CEES, for his continuous support.

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