The Madagascar software package is designed for analysis of large-scale multidimensional data, such as those occurring in exploration geophysics. Madagascar provides a framework for reproducible research. By "reproducible research" we refer to the discipline of attaching software codes and data to computational results reported in publications. The package contains a collection of (a) computational modules, (b) data-processing scripts, and (c) research papers. Madagascar is distributed on SourceForge under a GPL v2 license https://sourceforge.net/projects/rsf/. By October 2013, more than 70 people from different organizations around the world have contributed to the project, with increasing year-to-year activity. The Madagascar website is http://www.ahay.org/.

Wave-equation migration velocity analysis (MVA) is a technique similar to wave-equation tomography because it is designed to update velocity models using information derived from full seismic wavefields. On the other hand, wave-equation MVA is similar to conventional, traveltime-based MVA because it derives the information used for model updates from properties of migrated images, e.g., focusing and moveout. The main motivation for using wave-equation MVA is derived from its consistency with the corresponding wave-equation migration, which makes this technique robust and capable of handling multipathing characterizing media with large and sharp velocity contrasts. The wave-equation MVA operators are constructed using linearizations of conventional wavefield extrapolation operators, assuming small perturbations relative to the background velocity model. Similar to typical wavefield extrapolation operators, the wave-equation MVA operators can be implemented in the mixed space-wavenumber domain using approximations of differentorders of accuracy. As for wave-equation migration, wave-equation MVA can be formulated in different imaging frameworks, depending on the type of data used and image optimization criteria. Examples of imaging frameworks correspond to zero-offset migration (designed for imaging based on focusing properties of the image), survey-sinking migration (designed for imaging based on moveout analysis using narrow-azimuth data), and shot-record migration (also designed for imaging based on moveout analysis, but using wide-azimuth data). The wave-equation MVA operators formulated for the various imaging frameworks are similar because they share elements derived from linearizations of the single square-root equation. Such operators represent the core of iterative velocity estimation based on diffraction focusing or semblance analysis, and their applicability in practice requires efficient and accurate implementation. This tutorial concentrates strictly on the numeric implementation of those operators and not on their use for iterative migration velocity analysis.

Extended common-image-point-gathers (CIP) contain all the necessary information for decomposition of reflectivity as a function of the reflection and azimuth angles at selected locations in the subsurface. This decomposition operates after the imaging condition applied to wavefields reconstructed by any type of wideazimuth migration method, e.g. using downward continuation or time reversal. The reflection and azimuth angles are derived from the extended images using analytic relations between the space-lag and time-lag extensions. The transformation amounts to a linear Radon transform applied to the CIPs obtained after the application of the extended imaging condition. If information about the reflector dip is available at the CIP locations, then only two components of the space-lag vectors are required, thus reducing computational cost and increasing the affordability of the method. Applications of this method include the study of subsurface illumination in areas of complex geology where ray-based methods are not usable, and the study of amplitude variation with reflection and azimuth angles if the subsurface subsurface illumination is sufficiently dense. Migration velocity analysis could also be implemented in the angle domain, although an equivalent implementation in the extended domain is cheaper and more effective.

We introduce a computationally efficient and robust method to regularize acquisition geometries of 3-D prestack seismic data before prestack migration. The proposed method is based on a formulation of the geometry regularization problem as a regularized leastsquares problem. The model space of this least-squares problem is composed of uniformly sampled common offset-azimuth cubes. The regularization term fills the acquisition gaps by minimizing inconsistencies between cubes with similar offset and azimuth. To preserve the resolution of dipping events in the final image, the regularization term includes a transformation by Azimuth Moveout (AMO) of the common offset-azimuth cubes. The method is computationally efficient because we applied the AMO operator in the Fourierdomain, and we precondition the least-squares problem. Therefore, no iterative solution is needed and excellent results are obtained by applying the adjoint operator followed by a diagonal weighting in the model domain. We tested the method on a 3-D land data set from South America. Subtle reflectivity features are better preserved after migration when the proposed method is employed as compared to more standard geometry regularization methods. Furthermore, a dipping event at the reservoir depth (more than 3 km) is better imaged using the AMO regularization as compared to a regularization operator that simply smoothes the data over offsets.

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Transmission anomalies sometimes create AVO effects by focusing the reflected seismic wavefields, which impedes AVO analysis. The AVO anomalies caused by focusing are distinguishable by surface consistent patterns. We analyze the previous efforts to define, describe and eliminate spurious AVO anomalies. We also propose using wave equation migration velocity analysis to build an accurate velocity model. The transmission-related AVO can then be eliminated by downward continuation through this velocity model.