Produção Científica

**Artigo em Revista**

SoluĂ§Ăµes de Problemas envolvendo EquaĂ§Ăµes Diferenciais Sujeitas a IncertezasEste trabalho objetiva analisar, atravĂ©s de alguns exemplos, a influĂŞncia de se considerar aleatoriedades na soluĂ§ĂŁo de equaĂ§Ăµes diferenciais com dados e/ou parĂ˘metros aleatĂłrios. Um comparativo das mĂ©dias das soluĂ§Ăµes das equaĂ§Ăµes estocĂˇsticas com as soluĂ§Ăµes das equaĂ§Ăµes determinĂsticas simplificadas, nas quais substituĂmos os parĂ˘metros aleatĂłrios por suas mĂ©dias, Ă© apresentado. Estes exemplos mostram que a mĂ©dia da soluĂ§ĂŁo, que normalmente Ă© uma informaĂ§ĂŁo relevante em aplicaĂ§Ăµes, pode ser qualitativamente diferente da aproximaĂ§ĂŁo obtida pela soluĂ§ĂŁo de uma equaĂ§ĂŁo diferencial determinĂstica na qual substituĂmos os parĂ˘metros aleatĂłrios por suas mĂ©dias. |

**Artigo em Revista**

A spaceâ€“time multiscale method for computing statistical moments in strongly heterogeneous poroelastic media of evolving scalesA new multiscale procedure is proposed to compute flow in compressible heterogeneous porous media with geology characterized by power-law covariance structure. At the fine scale, the deformable medium is modeled by the partially coupled formulation of poroelasticity with Youngâ€™s modulus and permeability treated as stationary random fields represented by their Karhunenâ€“LoĂ¨ve decompositions. The framework underlying the multiscale procedure is based on mapping these random parameters to an auxiliary domain and constructing a family of equivalent stochastic processes at different length scales characterized by the same ensemble mean and covariance function. The poromechanical variables inherit a spaceâ€“time version of the scaling relations of the random input parameters which allows for constructing a set of multiscale solutions of the same governing equations posed at different space and time scales. A notable feature of the multiscale method proposed herein is the feasibility of solving both the poroelastic model and the Fredholm integral equation for the eigenpairs of the Karhunenâ€“LoĂ¨ve expansion in an auxiliary domain with much lower computational effort and then derive the long term behavior at a coarser scale from a straightforward rescaling of the auxiliary solution. Within the framework of the finite element approximation, in conjunction with the Monte Carlo algorithm, numerical simulations of fluid withdrawal and injection problems in a heterogeneous poroelastic reservoir are performed to illustrate the potential of the method in drastically reducing the computational burden in the computation of the statistical moments of the poromechanical unknowns in large-scale simulations. |

**Artigo em Revista**

A Numerical Comparison Between Quasi-MonteCarlo and Sparse Grid Stochastic Collocation MethodsQuasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations.These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse gridmethods in the context of groundwater flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance. |

**Artigo em Revista**

Effect of Element Distortion on the Numerical Dispersion of Spectral Element MethodsSpectral element methods are well established in the field of wave propagation,in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries. |

**Artigo em Revista**

Hypocentral relocation using clustering-along-planes constraints: implications for fault geometryHypocentre location is an ill-posed inverse problem even assuming that the velocity model is known, because different sets of hypocentre locations may satisfy the fitting criterion. We present a regularized hypocentre inversion in which the constraints of spatial proximity of the hypocentres to target planes are used. This constraint introduces the geological bias that earthquakes might occur along fault planes. Here, the target planes may be either (1) planes specified by the interpreter or (2) planes fitting groups of events. We assume also that initial estimates of hypocentres and origin times are available. Then, the initial hypocentre estimates, origin times and target planes are used as input to an inversion problem to relocate the hypocentres so that the maximum-possible clustering of events along the given planes is attained, matching the observed traveltimes. We use L1 norm for data fitting, L2 norm for the plane proximity criterion and a polytope algorithm to minimize the functional. Results from synthetic and real data indicate that the plane proximity constraint allows for hypocentre relocation presenting a high degree of clustering along planes. The real-data example is an intraplate earthquake sequence in NE Brazil. Our methodology defined the geometry and strike of fault segments close to known geology and focal mechanism data. In addition, the new method indicates that the fault is characterized by a splay geometry in its southern end and that more than three fault segments are necessary to explain the hypocentre distribution. |

**Artigo em Revista**

Evidences of buried loads in the base of the crust of Borborema Plateau (NE Brazil) from Bouguer admittance estimatesIn the Borborema Province (BP) e northeastern Brazil e two important Cenozoic events occurred at the surface: the Macau magmatism and the Borborema Plateau epeirogenesis. To obtain appropriated-scale geophysical data to explain the deep origins of these two events, different gravimetric/elevation databases were integrated with new surveys. Bouguer admittance estimates reveal that isostatic condition of the BP, especially in the Borborema Plateau, can be explained using elastic models to the lithosphere only if surface and buried loadings are combined. If the buried load is applied in the base of the crust, the ratio between buried and surface weights is circa 15 for a lithosphere with effective elastic thickness around 15 km and crust thickness around 33 km. From an nterpretative viewpoint of the buried load, it is assumed that the lower crust under the Borborema Plateau might have an anomalous high value of density. Magmatic underplating might explain this fact as well as the observed surface magmatism and epeirogenesis. Crustal thickening of about 4 km under the Borborema Plateau and intracrustal seismic velocity discontinuity with high Vp/Vs ratio are geophysical facts consistent with magmatic underplating. However, the surface magmatism presents low volume and mainly alkaline composition e facts that are not entirely consistent with the hypothesis of magmatic underplating. Regardless the validity of this hypothesis, Cenozoic-to-present events in BP might be somewhat associated with imbalances in lithosphere-asthenospheric mantle and/or crust-lithospheric mantle systems. The existence of free-air anomalies showing no null integral over area and of an expressive positive geoid anomaly are geophysical evidences of these imbalances. Possibly, the Borborema Plateau is still suffering epeirogenesis. Post-depositional deformation found in Barreiras Formation strata, Late Quaternay fault reactivations, and AFT thermochronology analysis suggesting the existence of a cooling stage between 20 and 0 Ma might be geologic evidences of the continued action of peirogenesis until the present. In addition, the relatively high level of the present intraplate seismicity recorded in several regions of the BP is another unequivocal geophysical evidence that the crust of the province is still submitted to accommodation processes. |

**Artigo em Revista**

A fast modified parabolic radon transform.We propose a fast and efficient frequency-domain implementation of a modified parabolic Radon transform (modified PRT) based on a singular value decomposition (SVD) with applications to multiple removal. The problem is transformed into a complex linear system involving a single operator after merging the curvature-frequency parameters into a new variable. A complex SVD is applied to this operator and the forward transform is computed by means of a complex back-substitution that is frequency independent. The new transform offers a wider curvature range at signal frequencies than the other PRT implementations, allowing the mapping in the transform domain of low-frequency events with important residual moveouts (long period multiples). The method is capable of resolving multiple energy from primaries when they interfere in a small time interval, a situation where most frequency-domain methods fail to discriminate the different wave types. Additionally, the method resists better to amplitude variations with offset (AVO) effects in the data than does the iteratively reweighted least-squares (IRLS) method.The proposed method was successfully applied to a deep-water seismic line in the Gulf of Mexico to attenuate water-bottom multiples and subsequent peg-legs originating from multiple paths in the water column. Combining the suggested method with the surface-related multiple elimination (SRME) has led to the best attenuation results in removing residual multiple energy in the stack. Â©2011 Society of Exploration Geophysicists |

**Artigo em Revista**

Total variation regularization for depth-to-basement estimate: Part 1 â€” Mathematical details and applicationsWe have developed an inversion approach that estimates the basement relief of a fault-bounded sedimentary basin. The sedimentary pack is approximated by a grid of 3D or 2D vertical prisms juxtaposed in the horizontal directions of a right-handed coordinate system. The prisms' thicknesses represent the depths to the basement and are the parameters to be estimated from the gravity data. To obtain depth-to-basement estimates, we introduce the total variation (TV) regularization as a stabilizing function. This approach lets us estimate a nonsmooth basement relief because it does not penalize sharp features of the solution. We have deduced a compact matrix form of the gradient vector and the Hessian matrix of the approximation to the TV function that allows a regularized Gauss-Newton minimization approach. Because the Hessian matrix of the approximation to the TV function is ill conditioned, we have modified this Hessian matrix to improve its condition and to accelerate the convergence of the Gauss-Newton algorithm. Tests conducted with synthetic data show that the inversion method can delineate discontinuous basements presenting large slips or sequences of small-slip step faults. Tests on field data from the Almada Basin, Brazil, and from the San Jacinto Graben, California, U.S.A., confirm the potential of the method in detecting and locating in-depth normal faults in the basement relief of a sedimentary basin. Â©2011 Society of Exploration Geophysicists |

**Artigo em Revista**

Total variation regularization for depth-to-basement estimate: Part 2 â€” Physicogeologic meaning and comparisons with previous inversion methodsWe applied the mathematical basis of the total variation (TV) regularization to analyze the physicogeologic meaning of the TV method and compared it with previous gravity inversion methods (weighted smoothness and entropic Regularization) to estimate discontinuous basements. In the second part, we analyze the physicogeologic meaning of the TV method and compare it with previous gravity inversion methods (weighted smoothness and entropic regularization) to estimate discontinuous basements. Presenting a mathematical review of these methods, we show that minimizing the TV stabilizing function favors discontinuous solutions because a smooth solution, to honor the data, must oscillate, and the presence of these oscillations increases the value of the TV stabilizing function. These three methods are applied to synthetic data produced by a simulated 2D graben bordered by step faults. TV regularization and weighted smoothness are also applied to the real anomaly of Steptoe Valley, Nevada, U.S.A. In all applications, the three methods perform similarly. TV regularization, however, has the advantage, compared with weighted smoothness, of requiring no a priori information about the maximum depth of the basin. As compared with entropic regularization, TV regularization is much simpler to use because it requires, in general, the tuning of just one regularization parameter. Â©2011 Society of Exploration Geophysicists |

**Artigo em Revista**

Partitioned least-squares operator for large-scale geophysical inversion Geophysics 75, R121, 2010Least-squares (LS) problems are encountered in many geophysical estimation and data analysis problems where a large number of observations (data) are combined to determine a model (some aspect of the earth structure). Examples of least squares in seismic exploration include several data processing algorithms, theoretically accurate LS migration, inversion for reservoir parameters, and background velocity estimation. A frequently encountered problem is that the volume of data in 3D is so large that the matrices required for the LS solution cannot be stored within the memory of a single computer. A new technique is described for parallel computation of the LS operator that is based on a partitioned-matrix algorithm. The classical LS method for solution of block-Toeplitz systems of normal equation (NE) to the general case of block-Hermitian and non-Toeplitz systems of NE. is generalized. Specifically, a solution of a block-Hermitian system of NE is shown that may be obtained recursively by linearly combining the solutions of lesser order that are related to the forward and backward subsystems of equations. This results in an efficient parallel algorithm in which each partitioned system can be evaluated independently. The application of the algorithm to the problem of 3D plane wave transformation is demonstrated. Â©2010 Society of Exploration Geophysicists |