Produção Científica
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DIRECT PROBLEMS FOR POROELASTIC WAVES WITH FRACTIONAL DERIVATIVES We prove the uniqueness and continuous dependence on the data of a weak solution to a problem for poroelastic waves with fractional derivatives both in unbounded and bounded time intervals and in all space dimensions. 
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3D SEISMIC MODELING AND DEPTH MIGRATION COMBINING THE EXTRAPOLATION OF UPGOING AND DOWNGOING WAVEFIELDS The 3D acoustic wave equation is generally solved using finite difference schemes on the mesh which defines the velocity model. However, when numerical solution of the wave equation is done by finite difference schemes, attention should be taken with respect to dispersion and numerical stability. To overcome these problems, one alternative is to solve the wave equation in the Fourier domain. This approach is stabler and makes possible to separate the full wave equation in its unidirectional equations. Thus, the full wave equation is decoupled in two first order differential equations, namely two equations related to the vertical component: upgoing (Z) and downgoing (+Z) unidirectional equations. Among the solution methods, we can highlight the SplitStepPlusInterpolation (SSPSPI). This method has been proven to be quite adequate for migration problems in 3D media, providing satisfactory results at low computational cost. In this work, 3D seismic modeling is implemented using Huygens’ principle and an equivalent simulation of the full wave equation solution is obtained by properly applying the solutions of the two uncoupled equations. In this procedure, a point source wavefield located at the surface is extrapolated downward recursively until the last depth level in the velocity field is reached. A second extrapolation is done in order to extrapolate the wavefield upwards, from the last depth level to the surface level, and at each depth level the previously stored wavefield (saved during the downgoing step) is convolved with a reflectivity model in order to simulate secondary sources. To perform depth prestack migration of 3D datasets, the decoupled wave equations were used and the same process described for seismic modeling is applied for the propagation of sources and receivers wavefields. Thus, depth migrated images are obtained using appropriate image conditions: the upgoing and downgoing wavefields of sources and receivers are correlated and the migrated images are formed. The seismic modeling and migration methods using upgoing and downgoing wavefields were tested on simple 3D models. Tests showed that the addition of upgoing wavefield in seismic migration, provide better result and highlight steep deep reflectors which do not appear in the results using only downgoing wavefields. 
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Q factor estimation from the amplitude spectrum of the time–frequency transform of stacked reflection seismic data Attenuation is one factor that degrades the quality of reflection seismic subsurface imaging. It causes a progressive decrease in the seismic pulse energy and is also responsible for limiting seismic resolution. Currently, many methods exist for inverse Q filtering,which can be used to correct these effects to some extent; however, but all of these methods require the value of the Q factor to be known, and this information is rarely available. In this paper we present and evaluate three different strategies to derive the Q factor from the time–frequency amplitude spectrum of the seismic trace. They are based in the analyses of the amplitude decay trend curves that can be measured along time, along frequency or along a compound variable obtained from the time–frequency product. Some difficulties are highlighted, such as the impossibility to use short time window intervals that prevents the method from providing a precise map of the Q factor value of the subsurface layers. However, the Q factor estimation made in thisway can be used to guide the parameterization of attenuation correction by means of inverse Q filtering applied to a stacked seismic section; this is demonstrated in a real data example. 
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Reduction of crosstalk in blendedshot migration When migrating more than one shot at the same time, the nonlinearity of the imaging condition causes the final image to contain socalled crosstalk, i.e., the results of the interference of wavefields associated with different sources. We studied various ideas of using weights in the imaging condition, called encoding, for the reduction of crosstalk. We combined the ideas of random phase and/or amplitude encoding and random alteration of the sign with additional multiplication with powers of the imaginary unit. This procedure moved part of the crosstalk to the imaginary part of the resulting image, leaving the desired crosscorrelation in the real part. In this way, the final image is less impaired. Our results indicated that with a combination of these weights, the crosstalk can be reduced by a factor of four as compared with unencoded shot blending. Moreover, we evaluated the selection procedure of sources contributing to each group of shots. We compared random choice with a deterministic procedure, in which the random numbers were exchanged for numbers similar to those of a Costas array. These numbers preserve certain properties of a random choice, but avoid the occurrence of patterns in the distribution. Our objective was to avoid nearby source being added to the same group of shots, which cannot be guaranteed with a random choice. Finally, we determined that the crosstalk noise can be reduced after migration by image processing. Keywords: migration, crosscorrelation, imaging, noise 

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Symplectic scheme and the Poynting vector in reversetime migration We developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method (REM). This solution can be used for seismic modeling and reversetime migration (RTM). In seismic modeling and RTM, spatial derivatives are usually calculated by finite differences (FDs) or by the Fourier method, and the time evolution is normally obtained by a secondorder FD approach. If the spatial derivatives are computed by higher order FD schemes, then the time step needs to be small enough to avoid numerical dispersion, therefore increasing the computational time. However, by using REM with the Fourier method for the spatial derivatives, we can apply the proposed method to propagate the wavefield for larger time steps. Moreover, if the appropriate number of expansion terms is chosen, thismethod is unconditionally stable and propagates seismic waves free of numerical dispersion. The use of a symplectic numerical scheme provides the solution of the wave equation and its first time derivative at the current time step. Thus, the Poynting vector can also be computed during the time extrapolation process at very low computational cost. Based on the Poynting vector information, we also used a new methodology to separate the wavefield in its upgoing and downgoing components. Additionally, Poynting vector components can be used to compute common gathers in the reflection angle domain, and the stack of some angle gathers can be used to eliminate lowfrequency noise produced by the RTM imaging condition. We numerically evaluated the applicability of the proposed method to extrapolate a wavefield with a time step larger than the ones commonly used by symplectic methods as well as the efficiency of this new symplectic method combined with REM to successfully handle the Poynting vector calculation. 
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Fast Seismic Inversion Methods Using Ant Colony Optimization Algorithm This letter presents ACOBBR  V, a new computationally efficient antcolonyoptimizationbased algorithm, tailored for continuousdomain problems. The ACOBBR  V algorithm is well suited for application in seismic inversion problems, owing to its intrinsic features, such as heuristics in generating the initial solution population and its facility to deal with multiobjective optimization problems. Here, we show how the ACOBBR  V algorithm can be applied in two methodologies to obtain 3D impedance maps from poststack seismic amplitude data. The first methodology pertains to the traditional method of forward convolution of a reflectivity model with the estimated wavelet, where ACOBBR  V is used to guess the appropriate wavelet as the reflectivity model. In the second methodology, we propose an even faster inversion algorithm based on inverse filter optimization, where ACOBBR  V optimizes the inverse filter that is deconvolved with the seismic traces and results in a reflectivity model similar to that found in well logs. This modeled inverse filter is then deconvolved with the entire 3D seismic volume. In experiments, both the methodologies are applied to a synthetic 3D seismic volume. The results validate their feasibility and the suitability of ACOBBR  V as an optimization algorithm. The results also show that the second methodology has the advantages of a much higher convergence speed and effectiveness as a seismic inversion tool. 
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Migration velocity analysis using residual diffraction moveout in the poststack depth domain Diffraction events contain more direct information on the medium velocity than reflection events. We have developed a method for migration velocity improvement and diffraction localization based on a moveout analysis of over or undermigrated diffraction events in the depth domain. The method uses an initial velocity model as input. It provides an update to the velocity model and diffraction locations in the depth domain as a result. The algorithm is based on the focusing of remigration trajectories from incorrectly migrated diffraction curves. These trajectories are constructed by applying a raytracinglike approach to the imagewave equation for velocity continuation. The starting points of the trajectories are obtained from fitting an ellipse or hyperbola to the picked uncollapsed diffraction events in the depthmigrated domain. Focusing of the remigration trajectories points out the approximate location of the associated diffractor, as well as local velocity attributes. Apart from the migration needed at each iteration, the method has a very low computational cost, but relies on the identification and picking of uncollapsed diffractions. We tested the feasibility of the method using synthetic data examples from three simple constantgradient models and the Sigsbee2B data. Although we were able to build a complete velocity model in this example, we think of our technique as one for local velocity updating of a slightly incorrect model. Our tests showed that, within regions where the assumptions are satisfied, the method can be a powerful tool. 
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Estimating quality factor from surface seismic data: A comparison of current approaches The performances of the spectral ratio (SR), frequency centroid shift (FCS), and frequency peak shift (FPS) methods to estimate the effective quality factor Q are compared. These methods do not demand true amplitude data and their implementations were done following an “as simple as possible” approach to highlight their intrinsic potentials and limitations. We use synthetic zerooffset seismic data generated with a simple layercake isotropic model. The methods can be ranked from simple to complex in terms of automation as: FPS, FCS and SR. This is a consequence of: (i) peak identification consists basically of a sorting procedure, (ii) centroid estimation involves basically the evaluation of two wellbehaved integrals, and (iii) implementation of the SR method involves at least choosing a usable frequency bandwidth and fitting a gradient. The methods can be ranked from robust to sensitive in the presence of noise content in the sequence SR, FCS, and FPS. This is consequence of: (i) the gradient estimate associated to the SR method averages out the noise content in the entire usable frequency bandwidth, (ii) in the presence of moderatetohigh noise level, the centroid estimation is biassed towards overestimating Q due to noise contribution in the tail of the amplitude spectrum, and (iii) peak identification is unstable due to local noise fluctuation in the amplitude spectrum around the peak frequency. Regarding the stability of the estimates relative to the attenuation amount, SR and FCS methods show similar behaviours, whereas FPS method presents an inferior performance. This fact is an indirect consequence of the sensitivity of FPS method to the noise content because the higher is the attenuation the lower is the signaltonoise ratio. Finally, regarding the robustness of the methods to the presence of dipping layers, only SR and FCS methods provide good estimates, at least to typical dips in nonfaulted sedimentary layers, with the estimates obtained with SR method being more accurate that those obtained with FCS method. Except in relation to the automation complexity, which is less important than the performances of the methods, SR method was superior or showed similar performance to FCS method in all scenarios we tried. 
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On the elastic wave equation in weakly anisotropic VTI media The knowledge of the wave equation is of fundamental importance for a good and satisfying understanding of the phenomena of wave propagation. However, it is unsatisfactory and inefficient to work with the full anisotropic wave equation in media that exhibit certain symmetries. We derive a specific elastic wave equation for weakly anisotropic VTI media by linearizing the expression of the stiffness tensor in terms of the Thomsen parameters. The resulting wave equation is a system of three coupled differential equations for the three components of the displacement vector. For δ = 0, the third equation becomes an independent equation for the third component of the particle displacement, identical to the isotropic situation, and the first two equations remain coupled. Using zeroorder ray theory, we derive the associated eikonal and transport equations for qP, qSV and qSH waves. These are finally reduced to the pseudoacoustic case where the vertical Swave velocity is zero. This allows for a better understanding of the pseudoSwave artefact in such media. 
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Using SVD filters for velocity analysis and groundroll attenuation This study investigates the adaptive filtering approach based on the Singular Value Decomposition (SVD) method to improve velocity analysis and groundroll attenuation. The SVD filtering is an adaptive multichannel filtering method where each filtered seismic trace keeps a degree of coherence with the immediate neighboring traces. Before applying the adaptive filtering, in order to flatten the primary reflections the seismogram is corrected using the Normal Move Out (NMO) method. The SVD filtering helps to strengthen the spatial coherence of reflectors. It works as multichannel and can be applied by selecting a set of seismic traces taken from around the target trace. Thus traces from different shots can be represented by a fivepoint areal operator, which we call fivepoint cross operator. In this paper we run this operator along the coverage map of the seismic survey. At each operator position, the filtered trace (center of the operator) is obtained by taking the firstor adding the first eigenimages. Thereby we enhance the coherence corresponding to the primary reflections in detriment of the remaining events (groundroll, multiples, and other noncorrelated events) remained in the other eigenimages. The method was tested on a seismic line of the Tacutu, Brazil. The obtained results show the velocity spectra with better definition, as well as better poststacked section exhibiting better continuity of seismic reflections and lower noise, compared with the raw processing results (without SVD filtering). 