\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}
An accurate depth-velocity model is the key for obtaining good quality and reliable depth images in areas of complex geology. In such 
areas, velocity-model definition should use methods that describe the complexity of wavefield propagation, such as focusing and defocus
ing, multiple arrivals, and frequency-dependent velocity sensitivity. Wave-equation tomography in the image space has the ability to ha
ndle these issues because it uses wavefields as carriers of information. However, its high cost and low flexibility for parametrizing t
he model space has prevented its routine industrial use.
This thesis aims at overcoming those limitations by using new wavefields as carriers of information: the image-space generalized wavef
ields. These wavefields are synthesized by using a pre-stack generalization of the exploding-reflector model. Cost of wave-equation tom
ography in the image space is decreased because only a small number of image-space generalized wavefields are necessary to accurately d
escribe the kinematics of velocity errors and because these wavefields can be easily used in a target-oriented way. Flexibility is natu
rally incorporated into wave-equation tomography in the image space by using these wavefields because their modeling have as the initia
l conditions some key  selected reflectors, allowing a layer-based parametrization of the model space.
To use the image-space generalized wavefields in wave-equation tomography in the image space, the method is extended from the shot-pro
file domain to the image-space generalized-sources domain. In this new domain, the velocity updates are very fast. Migration with the o
ptimized velocity model provides good quality and reliable depth images, as can be seen in a 3D-field data example.
\end{abstract}

\section{introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{pre-stack-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, the entire prestack image is spatially convolved with the source wavefield, generating the receiver wavefield, whose back-propagation and collection at the surface originate data. In order to guarantee illumination similar to the original, several shot records need to be modeled. Notice that both 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 SODCIG of a reflector. Since we do not know beforehand which shots contribute to forming the image at a point in the subsurface, again we would have to perform several modeling experiments to synthesize these shot records. 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[]{biondi***}, 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 downward and the upward 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 downward and upward 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[]{} when applying the multi-offset image condition \cite[]{}. In 3D, the rotation is given by 
*
*
where ***. A complete description of the rotation is given by \cite{guerra_thesis}.
\par
PERM as described by equations \ref{eq:perm_bU} 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 SODCIGs equally spaced by certain $x$ and $y$ decorrelation distances 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 downward and upward 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. If the periods of the 2D--comb function are chosen to be the decorrelation distances, no crosstalk between wavefields from different SODCIGs occurs during migration. The decorrelation distances correspond to twice the $x$ and $y$ subsurface-offset range. If more than one reflector is to be used, each reflector is 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. Two different strategies can be used to mitigate both kinds of crosstalk. 
\par
One strategy mitigates the reflector crosstalk, ****taking advantage**** that the image of a reflector is formed at the zero time of wavefield propagation when the migration velocity is accurate. When the migration velocity is inaccurate, the focusing of the image departs from the zero time, and this can be potentially used to update the velocity model \cite[]{sava:S209,yang:3715}. Therefore, 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 data 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}. Phase-encoding is usually performed in the data space. However, wavefields can also 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 reflector crosstalk but also crosstalk 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. \cite{guerra2008b} also use Gold codes \cite[]{gold}, sequences widely used in third- and fourth-generation cellphones, to phase-encode the modeling. The pseudo-random phase-encoding function causes the frequency components of the initial conditions to be randomly injected into the modeling. Downward and upward image-space phase-encoded wavefields (ISPEW) initiated at the same SODCIG and from the same reflector are equally encoded, whereas downward and upward 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 an inaccurate velocity model (Figure \ref{fig:velmarmousi}) by shot-profile migration of 375 shot records from Marmousi is used as the initial conditions after rotation according to the apparent geological dip. 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 SODCIGs of Figure \ref{fig:***} reflects this velocity inaccuracy. We use the four reflectors indicated in Figure \ref{fig:***} to model 35 pairs of PERM data (Figure \ref{fig:***}) and 11 pairs of ISPEW (Figure \ref{fig:***}). Migration of PERM data (Figure \ref{fig:***}) used the time-windowed-imaging condition. Migration of ISPEW is shown in Figure \ref{fig:***}. Reflectors are satisfactorily imaged and crosstalk is avoided. For MVA it is important that the prestack kinematics is correct and the residual moveout is maintained.  
\par
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 we selected four reflectors 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 Figures \ref{fig:ismarm08}c.  
\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. 
\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, and c) areal-shot migration of 11 ISPEWs corresponding to a single random realization. The moveout information is basically the same.} 

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