3dex/paper.tex:To mitigate the offset irregularity, considering the limited azimuthal distribution, \cite{bob05,clapp:2097} applies least-squares 3D-data regularization using offset volumes transformed to a common offset via AMO \cite []{biondi:574}. In spite of the good imaging results, the amplitude variation caused by the offset irregularity, although diminished, 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 provided by Clapp. 
appendix/paper.tex:where $\Delta r = \Delta r({\bf x},{\bf h})$ is the image perturbation that measures the accuracy of the slowness model. To compute $\Delta r$, a differential residual-focusing operator $\bf M$ is applied to the image $r = r({\bf x},{\bf h})$ obtained with the current slowness \cite[]{biondi2008}, using either differential residual prestack migration \cite[]{sava2004a,sava2004b} or differential-semblance optimization (DSO) operators \cite[]{shen:VE49}. In this paper, operators are represented by bold capital letters.
Intro/paper.tex:The last decade has seen the development of migration-velocity analysis (MVA) by wavefield extrapolation \cite[]{biondi:1723,shen:2132,sava2004, shen2004}. Provided the finite frequency nature of wavefield extrapolation, this MVA technique does not suffer from the limitations caused by the high-frequency approximation, which are present in the ray-based methods, namely the need of smooth velocity contrasts However, despite its theoretical superiority, MVA by wavefield extrapolation has rarely been used in 3D projects \cite[]{fei}. This is because of its higher cost and because it is less flexible than its ray-based counterpart in parameterizing the velocity model. Therefore, decreasing its cost and improving its flexibility is crucial to implement MVA by wavefield extrapolation as a routine process. 
Intro/paper.tex:The methods for computing generalized sources discussed above operate in the data space, characterizing the data-space generalized sources. This thesis introduces a new category of generalized sources that are initiated from selected reflectors, using the pre-stack exploding-reflector model (PERM) \cite[]{biondi2006}. 
ispew/paper.tex:%The generalized source domain can be obtained either by data-space phase encoding or image-space phase encoding. For the data-space phase encoding, the synthesized shot gathers are obtained by linear combination of the original shot gathers after some kind of phase encoding; in particular, here we mainly consider plane-wave phase encoding \cite[]{whitmore1995, zhang2005, duquet2006, liu2006} and random phase encoding \cite[]{romero2000}. As the encoding process is done in the data space, we call it data-space phase encoding. For the image-space phase encoding, the synthesized gathers are obtained by prestack exploding-reflector modeling \cite[]{biondi2006,biondi2007,guerra2008a}, where several subsurface-offset-domain common-image gathers (SODCIGs) and several reflectors are simultaneously demigrated to generate areal source and areal receiver gathers. To attenuate the cross-talk, the SODCIGs and the reflectors have to be encoded, e.g., by random phase encoding. Because the encoding process is done in the image space, we call it image-space phase encoding. We show that in these generalized source domains, we can obtain gradients, which are used for updating the velocity model, similar to that obtained in the original shot-profile domain, but with less computational cost.
ispew/paper.tex:To see how crosstalk from unrelated SODCIGs is formed, let us use the same two-reflectors model as in Chapter \ref{chap:chap02}. PERM data were modeled starting from the rotated images of Figure \ref{fig:dip05} using SODCIGs combined into sets with sampling period of 41 and 81 SODCIGs. Equation \ref{eq:I_perm1} shows that no crosstalk is generated if the sampling period is chosen to be the decorrelation distance of twice the subsurface-offset range, which in the present case must be greater than the distance spanned by 162 SODCIGs. Recall that the number of subsurface offsets in the original image is 81. As can be seen in Figures \ref{fig:ispew01}a and \ref{fig:ispew01}b, crosstalk occurs according periods of one-fourth and one-half of the sampling period, respectively. The corresponding ADCIGS at $x = 0$ m and the ADCIG computed from the image with no crosstalk are shown in the top panels of Figure \ref{fig:ispew02}. In the bottom panels we can see the corresponding $\rho$ scans, computed using equation D-7 in \cite{biondi:1283}. Notice that manual picking can identify the correct $\rho = 0.9$ in Figures \ref{fig:ispew02}a-b. Therefore, ray-based methods for velocity update can back-project the correct moveout information. However, when wavefield-extrapolation methods are used for velocity update, perturbed images computed from Figures \ref{fig:ispew01}a-b or \ref{fig:ispew02}a-b will potentially provide incorrect gradients.
ispew/paper.tex:From Figure \ref{fig:comb03}a, notice that for propagation times less than 0.14 s minus the period of the wavelet in time, no crosstalk will occur. This observation can be used to avoid crosstalk by applying a modified imaging condition. As Figure \ref{fig:comb03} shows, crosstalk is formed at times different from zero. Therefore, if the wavefields are cross-correlated within a time window centered at time zero with a length that excludes the times at which crosstalk is formed, reflector crosstalk can be avoided \cite[]{biondi07}. The time-windowed imaging condition for a single pair of areal shot reads
mvags/paper.tex:Two major variants of ISWET are wave-equation migration velocity analysis (WEMVA) \cite[]{sava2004a, sava2004b} and differential semblance velocity analysis (DSVA) \cite[]{shen2004, shen:VE49}. Both variants seek the optimal velocity by driving an image perturbation to a minimum. However, they differ in the way the image perturbation is computed and, consequently, in the numerical optimization scheme. As \cite{biondi2008} points out, WEMVA is not easily automated. The image perturbation is computed by the linearized-residual prestack-depth migration \cite[]{sava2003}, which uses a manually picked residual-moveout parameter. Since the perturbed image computed with the linearized-residual prestack-depth migration is consistent with the application of the forward wave-equation tomographic operator, WEMVA can be solved using a two-step approach. First, in a nonlinear iteration the background image is computed with the current velocity, a residual-moveout parameter is interpreted using enhanced versions of the background image, and  the current perturbed image  for the interpreted residual moveout is computed. Then, linear iterations using conjugate-gradients search for a perturbation in velocity that better explains the current perturbed image. The corresponding velocity solution is used to compute a new background image for the next nonlinear iteration.
mvags/paper.tex:The perturbed image can be computed by the DSO operator \cite[]{symes:654}, in the DVSA variant of ISWET, and by  linearized-residual prestack-depth migration \cite[]{sava2003}, in the WEMVA variant of ISWET. According to \cite{biondi2008}, a general form of the perturbed image can be expressed as
perm/paper.tex:Shot-profile and areal-shot migrations by wavefield extrapolation compute pre-stack images by means of the multi-offset imaging condition \cite[]{james2002}, in which source and receiver wavefields are laterally shifted prior to time correlation. However, the shift between wavefields might not be restricted to the horizontal direction. For instance, vertical shifts of the wavefields produce the vertical-subsurface-offset gathers, which provide reliable velocity information in the presence of steep dips \cite[]{biondi:1284}.
perm/paper.tex:Ideally, wavefields should be shifted along the geological dip direction. According to \cite{biondi:1283}, SODCIGs computed this way do not suffer from image-point dispersal in the presence of dip and inaccuracies in the migration velocity. The image-point dispersal causes events with different reflection angles from the same reflection point in the subsurface to be imaged at different locations.
perm/paper.tex:\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}.}
perm/paper.tex:\cite{biondi:1283} point out that, at least to the first order, the reflection-angle domain is immune to image-point dispersal. This is because the SODCIG to ADCIG transformation shifts events to the line connecting $\mathsf {I}$ and $\mathsf {I_{hg}}$ in Figure \ref{fig:pdisp} at the same image point shared by all the reflection angles.
perm/paper.tex:%The co-planarity assumption is also present in the formulation of the common-azimuth migration \cite[]{biondi:1822}, which turned out to provide sufficiently accurate images in areas of complex geology. 
perm/paper.tex:Although PERM theory was developed in 3D, all the examples I have shown so far have been 2D. Next, I discuss a 3D example under the common-azimuth approximation \cite[]{biondi:1822} and show that in this case the SODCIGs in the $y$ direction can be continuously sampled, and the number of modeling experiments will depend only on the sampling period in the $x$ direction, drastically decreasing data size.
perm/paper.tex:Despite the recent good migration results obtained in geologically complex areas using wide-azimuth data, narrow-azimuth acquisition is still the industry standard. Narrow-azimuth data can be efficiently imaged by common-azimuth wave-equation migration (CAM) \cite[]{biondi:1822}. CAM reduce the dimensionality of the pre-stack wavefields, and therefore the cost of migration, by assuming zero cross-line offset. That does not mean that the cross-line offset wavenumber is zero. Rather, its asymptotic approximation is a function of the the in-line midpoint and in-line offset wavenumbers. Therefore, instead of a five-dimensional hypercube, CAM images are  four-dimensional hypercubes in $\bf x$ and $h_x$.