Starting from the spatially variable acoustic wave equation we show how to produce the future tim... more Starting from the spatially variable acoustic wave equation we show how to produce the future time response of the seismic wave field in "one step". This method is called the Rapid Expansion Method, REM, and it uses Chebyshev polynomials to compute the time response. We use the pseudo spectral method for the spatial derivatives and the REM method to get highly accurate wave field snap shots in space and time. We then develop a recursive solution for arbitrary time steps for seismic modelling and RTM. We develop each method and show their relation to previous work and illustrate them with numerical examples of subsalt modeling and imaging.
Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a ... more Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados. É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
texto foi preparado para a apresentação no V Simpósio Brasileiro de Geofísica, Salvador, 27 a 29 ... more texto foi preparado para a apresentação no V Simpósio Brasileiro de Geofísica, Salvador, 27 a 29 de novembro de 2012. Seu conteúdo foi revisado pelo Comitê Técnico do V SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
Conventional tools for seismic imaging normally ignore the anisotropy of the media to produce ima... more Conventional tools for seismic imaging normally ignore the anisotropy of the media to produce images of the subsurface, and such omission reduces the quality of the images. Therefore, it is necessary to develop methods which account for subsurface anisotropic properties in order to produce more accurate images. In this paper, we present a least‐squares reverse time migration based on the coupled pseudo‐acoustic equations for tilted transverse isotropic media. Thus, from the stable pseudo‐acoustic wave equation for such media, we derive the linearized modelling (Born modelling) and adjoint migration operators to implement a least‐squares reverse time migration. Then, to save time and effort on development, while ensuring optimal performance of the inversion, we based our implementation on the domain‐specific language Devito, which, in turn, allowed us to easily verify the correctness of the developed operators. Synthetic examples demonstrate the validity of our approach in dealing with tilted transverse isotropic media.
Este texto foi preparado para a apresentação no VI Simpósio Brasileiro de Geofísica, Porto Alegre... more Este texto foi preparado para a apresentação no VI Simpósio Brasileiro de Geofísica, Porto Alegre, 14 a 16 de outubro de 2014. Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
In this work we compare migrated sections reconstructed by the reverse time migration (RTM) metho... more In this work we compare migrated sections reconstructed by the reverse time migration (RTM) method, which uses the full 3-D acoustic wave equation, with the split-step and phase-shift plus interpolation (SS-PSPI) method, which is a one-way 3-D acoustic wave equation solution. The RTM method is implemented using the rapid expansion method (REM) and it shows better results compared to the migrated section obtained with the SS-PSPI method. However, in terms of computational cost the SS-PSPI method is much faster than the RTM method as expected. For this numerical experiment we used the 3-D SEGEAGE dataset and, for comparison, the same computer resources parameters were used in both 3-D migration methods. Introduc¸ ˜ ao
An acoustic approach to description of P-wave travel times in anisotropic media is needed in orde... more An acoustic approach to description of P-wave travel times in anisotropic media is needed in order to avoid the solution of elastic wave equations. Type of stone, lateral and vertical facies variations or preferential direction of fracturing, are some of the causes to produce wave velocity dependence with the angle of propagation in the medium. In this work, we present an equation to the approximation of the relation dispersion for a TTI media. The procedure used in the approximation reduces the fourth order differential equation used to model the P wave to a second order equation. Numerical examples of seismic modeling and migration of TTI media are used to validate the new equation. RTM migration, using the REM expansion in time and pseudoespectral method in space, shows that the P-wave equation improves the migrated image in a VTI media and produces also a stable numerical solutions.
Full-waveform inversion (FWI) is a efficient method and it has been used successfully to invert s... more Full-waveform inversion (FWI) is a efficient method and it has been used successfully to invert subsurface parameters. It consists basically on the minimization of the difference between the predicted and observed data. However, its application using finite-difference schemes is limited to low frequency content and the increase of the range of higher frequency will demand a high computational cost of the wavefield propagation procedure and also the whole inversion scheme. To overcome this problem, we apply the rapid expansion method (REM) for numerical wavefield extrapolation inside the FWI workflow thus increasing the frequency content of the inversion process. Besides that, an efficient preconditioning method using source-receiver illumination (PSRI) for computing the gradient vector in order to increase resolution of the models and better convergence of the objective function has also been proposed. Beyond that, we compared the performance of the preconditioning method with relation to conventional gradient method for two types of source signature. To test the efficiency of our proposed FWI approach, we apply it using a frequency multiscale scheme for a synthetic data set with a complex velocity model. The inversion results show satisfactory inverted velocity models which can be used to produce depth imaging of high quality. Thus we demonstrate the effectiveness and applicability of our FWI scheme using REM combined with a multiscale approach.
One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in ... more One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called "true-amplitude one-way wave equations," provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium, the extrapolation of the wavefields with the true-amplitude oneway wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain trueamplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.
Seu conteúdo foi revisado pela Comissão Tecno-científica do II SR-SBGf mas não necessariamente re... more Seu conteúdo foi revisado pela Comissão Tecno-científica do II SR-SBGf mas não necessariamente representa a opinião da SGBf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
The 3-D acoustic wave equation is generally solved using finite difference schemes on the mesh wh... more The 3-D 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 Split-Step-Plus-Interpolation (SS-PSPI). This method has been proven to be quite adequate for migration problems in 3-D media, providing satisfactory results at low computational cost. In this work, 3-D 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 pre-stack migration of 3-D 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 3-D 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.
Obtaining an accurate image of the subsurface still remains a great challenge for the seismic met... more Obtaining an accurate image of the subsurface still remains a great challenge for the seismic method. Migration algorithms aim mainly on positioning seismic events in complex geological contexts. Multiple reflections are typically not accounted for in this process, which can lead to the emergence of artifacts. In Marchenko imaging, we retrieve the complete up-and downgoing wavefields in the subsurface to construct an image without such artifacts. The quality of this image depends on the type of imaging condition that is applied. In this paper, we propose an imaging condition that is based on stabilized unidimensional deconvolution. This condition is computationally much cheaper than multidimensional deconvolution, which has been proposed for Marchenko imaging earlier. Two specific approaches are considered. In the first approach, we use the full up-and downgoing wavefields for deconvolution. Although this leads to balanced and relatively accurate amplitudes, the crosstalk is not completely removed. The second approach is to incorporate the initial focusing function in the deconvolution process, in such a way that the retrieval of crosstalk is avoided. We compare images with the results of the classical cross-correlation imaging condition, which we apply to reverse-time migrated wavefields and to the up-and downgoing wavefields that are retrieved by the Marchenko method.
Starting from the spatially variable acoustic wave equation we show how to produce the future tim... more Starting from the spatially variable acoustic wave equation we show how to produce the future time response of the seismic wave field in "one step". This method is called the Rapid Expansion Method, REM, and it uses Chebyshev polynomials to compute the time response. We use the pseudo spectral method for the spatial derivatives and the REM method to get highly accurate wave field snap shots in space and time. We then develop a recursive solution for arbitrary time steps for seismic modelling and RTM. We develop each method and show their relation to previous work and illustrate them with numerical examples of subsalt modeling and imaging.
Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a ... more Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados. É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
texto foi preparado para a apresentação no V Simpósio Brasileiro de Geofísica, Salvador, 27 a 29 ... more texto foi preparado para a apresentação no V Simpósio Brasileiro de Geofísica, Salvador, 27 a 29 de novembro de 2012. Seu conteúdo foi revisado pelo Comitê Técnico do V SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
Conventional tools for seismic imaging normally ignore the anisotropy of the media to produce ima... more Conventional tools for seismic imaging normally ignore the anisotropy of the media to produce images of the subsurface, and such omission reduces the quality of the images. Therefore, it is necessary to develop methods which account for subsurface anisotropic properties in order to produce more accurate images. In this paper, we present a least‐squares reverse time migration based on the coupled pseudo‐acoustic equations for tilted transverse isotropic media. Thus, from the stable pseudo‐acoustic wave equation for such media, we derive the linearized modelling (Born modelling) and adjoint migration operators to implement a least‐squares reverse time migration. Then, to save time and effort on development, while ensuring optimal performance of the inversion, we based our implementation on the domain‐specific language Devito, which, in turn, allowed us to easily verify the correctness of the developed operators. Synthetic examples demonstrate the validity of our approach in dealing with tilted transverse isotropic media.
Este texto foi preparado para a apresentação no VI Simpósio Brasileiro de Geofísica, Porto Alegre... more Este texto foi preparado para a apresentação no VI Simpósio Brasileiro de Geofísica, Porto Alegre, 14 a 16 de outubro de 2014. Seu conteúdo foi revisado pelo Comitê Técnico do VI SimBGf, mas não necessariamente representa a opinião da SBGf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
In this work we compare migrated sections reconstructed by the reverse time migration (RTM) metho... more In this work we compare migrated sections reconstructed by the reverse time migration (RTM) method, which uses the full 3-D acoustic wave equation, with the split-step and phase-shift plus interpolation (SS-PSPI) method, which is a one-way 3-D acoustic wave equation solution. The RTM method is implemented using the rapid expansion method (REM) and it shows better results compared to the migrated section obtained with the SS-PSPI method. However, in terms of computational cost the SS-PSPI method is much faster than the RTM method as expected. For this numerical experiment we used the 3-D SEGEAGE dataset and, for comparison, the same computer resources parameters were used in both 3-D migration methods. Introduc¸ ˜ ao
An acoustic approach to description of P-wave travel times in anisotropic media is needed in orde... more An acoustic approach to description of P-wave travel times in anisotropic media is needed in order to avoid the solution of elastic wave equations. Type of stone, lateral and vertical facies variations or preferential direction of fracturing, are some of the causes to produce wave velocity dependence with the angle of propagation in the medium. In this work, we present an equation to the approximation of the relation dispersion for a TTI media. The procedure used in the approximation reduces the fourth order differential equation used to model the P wave to a second order equation. Numerical examples of seismic modeling and migration of TTI media are used to validate the new equation. RTM migration, using the REM expansion in time and pseudoespectral method in space, shows that the P-wave equation improves the migrated image in a VTI media and produces also a stable numerical solutions.
Full-waveform inversion (FWI) is a efficient method and it has been used successfully to invert s... more Full-waveform inversion (FWI) is a efficient method and it has been used successfully to invert subsurface parameters. It consists basically on the minimization of the difference between the predicted and observed data. However, its application using finite-difference schemes is limited to low frequency content and the increase of the range of higher frequency will demand a high computational cost of the wavefield propagation procedure and also the whole inversion scheme. To overcome this problem, we apply the rapid expansion method (REM) for numerical wavefield extrapolation inside the FWI workflow thus increasing the frequency content of the inversion process. Besides that, an efficient preconditioning method using source-receiver illumination (PSRI) for computing the gradient vector in order to increase resolution of the models and better convergence of the objective function has also been proposed. Beyond that, we compared the performance of the preconditioning method with relation to conventional gradient method for two types of source signature. To test the efficiency of our proposed FWI approach, we apply it using a frequency multiscale scheme for a synthetic data set with a complex velocity model. The inversion results show satisfactory inverted velocity models which can be used to produce depth imaging of high quality. Thus we demonstrate the effectiveness and applicability of our FWI scheme using REM combined with a multiscale approach.
One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in ... more One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called "true-amplitude one-way wave equations," provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium, the extrapolation of the wavefields with the true-amplitude oneway wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain trueamplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.
Seu conteúdo foi revisado pela Comissão Tecno-científica do II SR-SBGf mas não necessariamente re... more Seu conteúdo foi revisado pela Comissão Tecno-científica do II SR-SBGf mas não necessariamente representa a opinião da SGBf ou de seus associados.É proibida a reprodução total ou parcial deste material para propósitos comerciais sem prévia autorização da SBGf.
The 3-D acoustic wave equation is generally solved using finite difference schemes on the mesh wh... more The 3-D 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 Split-Step-Plus-Interpolation (SS-PSPI). This method has been proven to be quite adequate for migration problems in 3-D media, providing satisfactory results at low computational cost. In this work, 3-D 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 pre-stack migration of 3-D 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 3-D 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.
Obtaining an accurate image of the subsurface still remains a great challenge for the seismic met... more Obtaining an accurate image of the subsurface still remains a great challenge for the seismic method. Migration algorithms aim mainly on positioning seismic events in complex geological contexts. Multiple reflections are typically not accounted for in this process, which can lead to the emergence of artifacts. In Marchenko imaging, we retrieve the complete up-and downgoing wavefields in the subsurface to construct an image without such artifacts. The quality of this image depends on the type of imaging condition that is applied. In this paper, we propose an imaging condition that is based on stabilized unidimensional deconvolution. This condition is computationally much cheaper than multidimensional deconvolution, which has been proposed for Marchenko imaging earlier. Two specific approaches are considered. In the first approach, we use the full up-and downgoing wavefields for deconvolution. Although this leads to balanced and relatively accurate amplitudes, the crosstalk is not completely removed. The second approach is to incorporate the initial focusing function in the deconvolution process, in such a way that the retrieval of crosstalk is avoided. We compare images with the results of the classical cross-correlation imaging condition, which we apply to reverse-time migrated wavefields and to the up-and downgoing wavefields that are retrieved by the Marchenko method.
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