The finite-difference method applied to the full 3D wave equation is a rather time-consuming process. However, in the 2.5D case, we can take advantage of the medium symmetry. By taking the Fourier transform with respect to the out-of-plane direction (the symmetry axis) and then, the 3D problem can be reduced to a repeated 2D problem. The third dimension is taken into account by a sum over the corresponding wave-vector component. A criterion for where to end this theoretically infinite sum derives from the stability conditions of the finite-difference schemes employed. In this way, the computation time of the finite-difference calculations can be considerably reduced. The quality of the modelling results obtained with this 2.5D finite-difference scheme is comparable to that obtained using a standard 3D finite-difference scheme. In this work we apply this idea to the anisotropic pseudo-acoustic wave equation.

Fuzzy Transform (F-transform) has been introduced as an approximation method which encompasses both classical transforms as well as approximation methods studied in fuzzy modelling and fuzzy control. It has been proved that, under some conditions, Ftransform can remove a periodical noise and it can significantly reduce random noise. In this work we apply the F-transform methodology on the study of numerical solutions of partial differential equations with noisy initial conditions.

Standard real-valued finite-difference (FD) and Fourier finite-difference (FFD) migrations cannot handle evanescent waves correctly, which can lead to numerical instabilities in the presence of strong velocity variations. A possible solution to these problems is the complex Padé approximation, which avoids problems with evanescent waves by rotating the branch cut of the complex square root. We have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable FD and hybrid FD/FFD migrations for such media. Our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips. Our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60° for FD migration, as confirmed by numerical impulse responses and work with synthetic data.

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

Many migration methods are based on paraxial approximationsto the wave equation. These approximationsare used to describe the wave propagationin a preferred direction. We apply the idea ofa directed propagation to the so-called anisotropicacoustic wave equation. We derive paraxial approximationsfor this equation, the first ones being generalizationsof the 15◦ and 45◦ approximations to theisotropic acoustic wave equation. Moreover, we derivehigher-order approximations using Pad´e approximations.A set of numerical experiments demonstratesthat our paraxial equations provide reasonableapproximations to the solution of the full equation.Moreover, their computational execution ischeaper than using a direct implementation.

Seismic imaging in depth is limited by the accuracy of velocity model estimation. Slope tomography uses the slowness components and traveltimes of picked reflection or diffraction events for velocity model building. The unavoidable data incompleteness requires additional information to assure stability to inversion. One natural constraint for ray-based tomography is a smooth velocity model. We propose a new, reflection-angle-based kind of smoothness constraint as regularization in slope tomography and have compared its effects to three other, more conventional constraints. The effects of these constraints were evaluated through angle-domain common-image gathers, computed with wave-equation migration using the estimated velocity model. We found that the smoothness constraints have a distinct effect on the velocity model but a weaker effect on the migrated data. In numerical tests on synthetic data, the new constraint leads to geologically more consistent models.

Seismic migration by downward continuation using the one-way wave-equation approximations has two shortcomings: imaging steep-dip reflectors and handling evanescent waves. Complex Pade approximations allow a better treatment of evanescent modes, stabilizing finite-difference migration without requiring special treatment for the migration domain boundaries. Imaging of steep-dip reflectors can be improved using several terms in the Pade expansion. We discuss the implementation and evaluation of wide-angle complex Pade approximations for finite-difference and Fourier finite-difference migration methods. The dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the Pade expansion. This ensures stability for a prescribed maximum propagation direction. The implementations are validated on the Marmousi model data set and SEG/EAGE salt model data.

Apresentamos dois algoritmos estáveis para a migração por diferenças finitas e migração híbrida por Fourier finite difference.Os algoritmos utilizam expansão de Padé complexa para aproximar o operador de raiz quadrada associado à equação da onda unidirecional.O algoritmo proposto de migração apresenta resposta ao impulso semelhante à migração PSPI, como indicam simulações da resposta ao impulso e de migração pósempilhamento usando o modelo Marmousi.O algoritmo também foi utilizado para migração pré-empilhamento apresentando resultados satisfatórios e a um custo computacional muito menor que o método Fourier finite difference plus interpolation.