IMPRIMIR VOLTAR A. Ciências Exatas e da Terra - 6. Geociências - 2. Geofísica MODIFIED KIRCHHOFF PRESTACK DEPTH MIGRATION INTEGRAL USING THE GAUSSIAN BEAM OPERATOR AS GREEN FUNCTION Carlos Augusto Sarmento Ferreira 1   (autor)   casf@ufpa.br João Carlos Ribeiro Cruz 2   (orientador)   jcarlos@ufpa.br 1. Curso de Pós-Graduação em Geofísica, Centro de Geociências, Universidade Federal do Pará - UFPA 2. Departamento Nacional de Produção Mineral - DNPM, Belém-PA INTRODUÇÃO: During the past three decades, Kirchhoff-type seismic migration has evolved from an imaging only technique based on the wave equation, to an inversion operator capable of handling petrophysical attributes to hydrocarbons indicators. Today this theory is known as true amplitude theory and has given rise to a sort of other approaches which are part of the now well known unified approach to seismic imaging. Although several other migration algorithms have been developed so far, the Kirchhoff-type method is still widely investigated due to its flexibility and competitiveness. The Kirchhoff-type migration algorithms described above still make extensive use of the (zero order) ray theory (RT). However, RT application is restricted to smooth media. The paraxial theory (PT) has been an attractive way of dealing with the drawbacks of RT. The real PT has been mainly applied in seismic stacking methods, while the complex PT (i.e., Gaussian Beams or GB’s) has been applied to simulate wavefields in 2D and 3D laterally varying media, as well as in migration methods. In the present approach, we investigate the use of paraxial GB’s as a tool for migration. This is done by inserting a superposition integral of paraxial GB’s, which are used to simulate the RT Green function in the asymptotic sense. We have come to the conclusion that using a superposition of GB’s as a process to obtain the Green function in the asymptotic sense is the same as considering a convolution in the time domain. METODOLOGIA: The migration integral in this work is a 3D weighted, Kirchhoff-type, integral operator, composed of a weight-function W(ξ,M) and a time derivative of the analytic particle displacement U(ξ, t). Fourier transforming this integral to the frequency domain, it becomes an oscillatory integral formed by U(ξ,ω) multiplied by an exponential factor containing τD(ξ, M). In this new kernel, U(ξ,ω) is substituted by another integral, composed of a weight function Ф(γ1,γ2) multiplied by the RT Green function. This representation is a superposition of GB’s, equivalent to the Green function in the asymptotic sense, where we consider a complex valued taveltime. Since we can consider target surfaces that intersect the rays along its propagation through the medium as an integration domain, we introduce specific Jacobians and integrate along structural interfaces. Then we transform from ray coordinates to local Cartesian coordinates and asymptotically analyse the four fold integral using the multidimensional stationary phase approach. The next step is to choose W(ξ,M) such that the solution of the asymptotic analysis yields the RT solution. We have noticed that the only factor that was not specified was the determinant of a complex valued matrix that had not been given a clear physical meaning. In our approach this is completely determined when we determine the migration weight-function. RESULTADOS: The final result is a four fold integral that represents a weighted stacking of a diffraction curve, constructed by a superposition of the GB’s that intersected the surface. In the time domain, the inner integral can also be viewed as a convolution of the second derivative in time of the observed data with a weight-function proportional to the projected Fresnel zone of each trace. CONCLUSÕES: One of the main conclusions of this work is the fact that constructing the diffraction (Huygens) curve using GB’s resctricts the size of Huygens surface just to those traces that are inside the projected Fresnel zone of each trace. The main consequence of this is the decrease in the presence of migration artifacts, principally on the borders of the migrated section. For traces far beyond the projected Fresnel zone, the contributions are fairly small due to the bound of the paraxial region. The measure of the radius of the projected Fresnel zone may serve as a cut off function in this case, and no trace beyond this radius can contribute to the construction of the Huygens curve. A second advantage of our approach is the horizontal resolution of the migrated sections. It is well knonw that the size of the Fresnel zone influences the horizontal resolution of the migrated sections and serves as a criteria for the determination of the migration aperture. In our process, this is taken into account when the points of the Huygens surface, constructed by the superposition of GB’s, are weighted by the size of the projeted Fresnel zone. In this way, the outer integral (i.e., the diffraction stack) automatically is taking this influence in consideration and putting this information in depth. Instituição de fomento: CNPq Palavras-chave:  Gaussian Beam; Prestack migration; Fresnel zones. Anais da 56ª Reunião Anual da SBPC - Cuiabá, MT - Julho/2004