Produção Científica

**Artigo em Revista**

A robust interactive estimation of the regularization parameterWe have developed a new and robust method (in the sense of it being applicable to a wide range of situations) to estimate the regularization parameter μ in a regularized inverse problem. For each tentative value of μ, we perturb the observations with J sequences of pseudorandom noise and we track down the instability effect on the solutions. Then, we define a quantitative measure ρ(μ) of the solution instability consisting of the largest value among the Chebyshev norms of the vectors obtained by the differences between all pairs of the perturbed solutions. Despite being quantitative, ρ(μ) cannot be used directly to estimate the best value of μ (the smallest value that stabilizes the solution) because, in practice, instability may depend on the particular and specific interests of the interpreter. Then, we determine that the interpreter, at each iteration of a bisection method, visually compares, in the (x, y, z) space, the pair p^i and p^j of the solutions most distant from each other and associated with the current ρ(μ). From this comparison, the interpreter decides if the current μ produces stable solutions. Because the bisection method can be applied only to monotonic functions (or segments of monotonic functions) and because ρ(μ) has a theoretical monotonic behavior that can be corrupted, in practice by a poor experiment design, the set of values of ρ(μ) can be used as a quality control of the experiments in the proposed bisection method to estimate the best value of μ. Because the premises necessary to apply the proposed method are very weak, the method is robust in the sense of having broad applicability. We have determined part of this potential by applying the proposed method to gravity, seismic, and magnetotelluric synthetic data, using two different interpretation models and different types of pseudorandom noise. |

**Artigo em Revista**

Modeling the wave propagation in viscoacoustic media: An efficient spectral approach in time and space domainWe present an efficient and accurate modeling approach for wave propagation in anelastic media, based on a fractional spatial differential operator. The problem is solved with the Fourier pseudo-spectral method in the spatial domain and the REM (rapid expansion method) in the time domain, which, unlike the finite-difference and pseudo-spectral methods, offers spectral accuracy. To show the accuracy of the scheme, an analytical solution in a homogeneous anelastic medium is computed and compared with the numerical solution. We present an example of wave propagation at a reservoir scale and show the efficiency of the algorithm against the conventional finite-difference scheme. The new method, being spectral in the time and space simultaneously, offers a highly accurate and efficient solution for wave propagation in attenuating media. |

**Artigo em Revista**

Perfectly matched layer boundary conditions for the second-order acoustic wave equation solved by the rapid expansion methodWe derive a governing second-order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite-difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme,numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections. from the model boundaries. |

**Artigo em Revista**

NEW ITERATIVE AND MULTIFREQUENCY APPROACHES IN GEOPHYSICAL DIFFRACTION TOMOGRAPHYSeismic tomography is used in reservoir geophysics as an important method for high-resolution imaging. The classical Born approach, which is used in single-frequency diffraction tomography under the condition of weak scattering, is limited by the requirement to know the background velocity in advance. We propose tomographic inversion approaches within matrix formalism and the Born approximation conditions. These approaches are iterative (in the sense that the background velocity field is updated at each iteration) and do not require knowledge of the true background velocity. In the first approach, a single-frequency that is kept constant is used. In the second approach, several frequencies are also kept constant and are used simultaneously. In the third approach, in addition to the background velocity, the working frequency is also updated. Finally, in the last approach, the multiple frequencies used simultaneously are updated throughout the iteration. The proposed approaches were tested on a synthetic model containing a dipping layer and a paleochannel, with cross-well acquisition geometry, and the data were contaminated with Gaussian noise. When compared to the standard, single-frequency non-iterative approach, the iterative process with the use of multiple frequencies generated results with smaller RMS errors for model parameter, velocity and data.Keywords: seismic inversion, seismic tomography, wave numerical modeling, reservoir characterization. |

**Artigo em Revista**

Inversion of Bottom Hole Temperatures for Gradient Determination by the Damped Least Squares Method for Noise AttenuationThis study consists in obtain the 1-D distribution of the geothermal gradient from the inversion of Bottom Hole Temperature (BHT) data. Before the inversion procedure, Horner correction method was used to determine the correct formation temperature. The inversion was performed in a synthetic model based on real data from Pineview Field (Utah, USA), in this case, to obtain geothermal gradients from nine formations using BHT data from 32 wells. The Z matrix of the geothermal problem contains the elements zi j, i.e., the thickness of the i-th layer logged in the j-th well. The least squares method was used, and, because of the occurrence of noise, damping was required. The numerical implementation of the inversion, i.e., the determination of the inverse operator (ZtZ)+ or (ZtZ+ε1)+ was performed by singular value decomposition. Initial inversions did not produce satisfactory results, but they significantly improved with the introduction of damping. The improvement of the results is quantitatively explained by the fact that the condition number of the matrix to be inverted greatly reduced with the use of the damping. In turn, damping requires the choice of an optimal parameter, and the L-curve was used for this purpose. |

**Artigo em Revista**

Signal decomposition and time–frequency representation using iterative singular spectrum analysisThe application of the singular value decomposition method (SVD) for filtering of seismic data has become common in recent decades, as it promotes significant improvements of the signal-to-noise ratio, highlighting reflections in seismograms. One particular way to apply SVD in a single (or multivariate) time-series is the singular spectrum analysis (SSA) method, normally applied on constant-frequency slices in one or many spatial dimensions. We demonstrate that SSA method applied in the time domain corresponds to filtering the time-series with a symmetric zero-phase filters, which are the autocorrelations of the eigenvectors of the data covariance matrix, preserving the phase of the original data. In this paper, we explore the SSA method in the time domain, and we propose a new recursive-iterative SSA (RI-SSA) algorithm, which uses only the first eigenvector of the data covariance matrix to decompose a discrete time-series into signal components. From the analytic signal of each component we compute a time–frequency representation. By interpretation of the time signals and their time–frequency representations, groups with similar features are summed to produce a smaller number of signal components. The resulting RI-SSA signal decomposition is exact and phase-preserving, but non-unique. Applications to land seismic data for ground-roll removal and to two synthetic signals for time–frequency analysis give good results. |

**Artigo em Revista**

Deep structures seismic enhancement using singular spectral analysis in time and frequency domain: Application in the regional transect of Parnaíba basin - BrazilThe Parnaíba basin is located in the Northeast of Brazil and it started in the Archaean. In a project involving Global Geophysical Services Incorporated and BP Energy do Brasil, a 2D seismic data, 1400 km long and 20 s of two-way travel time was acquired. Because of the acquisition characteristics and large volume of data it was necessary to develop a powerful filtering flow, in order to enhance the signal-to-noise ratio, particularly for deep structures, such as the Moho Discontinuity. For that matter, we have used a two-step recursive-adaptive singular spectral analysis (RA-SSA) to enhance the signal-to-noise ratio. First, we applied the RA-SSA in the t-x domain, along the time variable, for every seismic trace, to attenuate uncorrelated noise, and to enhance the low frequency content of the data. Second, the data was moved to the f-x domain, by means of the Fourier Transform of every single trace, and the RI-SSA method was applied for every frequency, along the x variable, to enhance the correlation of the reflectors between neighboring seismic traces. The filtered results, shown on common offset and CMP gather and on stacked data, show how successful the method was in enhancing the reflectors. We introduce a processing flow capable of enhancing the final stacked image quality, in order to map the Moho Discontinuity and interpret the transect to obtain a better understanding of the Parnaíba basin formation. |

**Artigo em Revista**

Seismic processing applied to shale-gas reservoir characterization in Reconcavo basinThis paper is intended to present all the steps used in the seismic processing to characterize and interpret the 2D seismic line 0026-RL-1624, located between Dom João and Candeias fields, from the perspective of nonconventional reservoirs. It includes a new SVD (Singular Value Decomposition) method used to attenuate the ground-roll and direct wave, which had a very dispersive form in the shot domain, causing several problems in the visualization of reflections and, consequently, in the raw stacked data as well. It was later applied seismic attributes in order to have a better precision in mapping geological structures of interest and also interpret the main horizons present in the log well data. During the development of this work, the processing step were entirely performed by both Seismic Unix and SeisSpace/Promax, software developed by Landmark/Halliburton. |

**Artigo em Revista**

Up/down acoustic wavefield decomposition using a single propagation and its application in reverse time migrationThe separation of up- and downgoing wavefields is an important technique in the processing of multicomponent recorded data, propagating wavefields, and reverse time migration (RTM). Most of the previous methods for separating up/down propagating wavefields can be grouped according to their implementation strategy: a requirement to save time steps to perform Fourier transform over time or construction of the analytical wavefield through a solution of the wave equation twice (one for the source and another for the Hilbert-transformed source), in which both strategies have a high computational cost. For computing the analytical wavefield, we are proposing an alternative method based on the first-order partial equation in time and by just solving the wave equation once. Our strategy improves the computation of wavefield separation, and it can bring the causal imaging condition into practice. For time extrapolation, we are using the rapid expansion method to compute the wavefield and its first-order time derivative and then we can compute the analytical wavefield. By computing the analytical wavefield, we can, therefore, separate the wavefield into up- and downgoing components for each time step in an explicit way. Applications to synthetic models indicate that our method allows performing the wavefield decomposition similarly to the conventional method, as well as a potential application for the 3D case. For RTM applications, we can now use the causal imaging condition for several synthetic examples. Acoustic RTM up/down decomposition demonstrates that it can successfully remove the low-frequency noise, which is common in the typical crosscorrelation imaging condition, and it is usually removed by applying a Laplacian filter. Moreover, our method is efficient in terms of computational time when compared to RTM using an analytical wavefield computed by two propagations, and it is a little more costly than conventional RTM using the crosscorrelation imaging condition. |

**Artigo em Revista**

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansionWe have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method. |