Procedures for accurate ray tracing in complex three-dimensional media with interfaces are propos... more Procedures for accurate ray tracing in complex three-dimensional media with interfaces are proposed. The ray tracing equations and the associated paraxial linear equations are solved either by a numerical solver or by an analytical perturbation approach. Interfaces are described with an explicit representation or an implicit representation using B-spline interpolation. For the implicit representation, we exploit two important properties of B-splines, the convex hull and subdivision properties, in order to determine the intersection of the ray with the interface.
Journal of Volcanology and Geothermal Research, 1998
A high-resolution seismic tomography of Mt.Vesuvius was started in May 1994, with the aim of reco... more A high-resolution seismic tomography of Mt.Vesuvius was started in May 1994, with the aim of reconstructing the detailed shallow crustal structure underneath the volcano and define its feeding system. The first phase of the experiment was to perform a 2D profile, using three underground explosions as active sources. Data from controlled sources and microearthquakes were jointly used to determine the shallow structure of the volcano. A high-velocity body (Vp=3.5–4 km/s) was identified at about 2 km beneath the Somma-Caldera. It is likely to represent a sub-volcanic structure, formed by a dense network of solidified dikes. A prominent converted P-to-S phase at about 10 km of depth indicates the occurrence of a sharp transition to a very low-velocity zone. This may represent the top of an extended magmatic reservoir.
Summary. A preliminary study of the aftershocks of three earthquakes that occurred near to Corint... more Summary. A preliminary study of the aftershocks of three earthquakes that occurred near to Corinth (Greece) in 1981 is combined with observations of the morphology and faulting to understand the evolution of the Eastern Gulf of Corinth. The well located aftershocks form a zone 60km long and 20km wide. They do not lie on the main fault planes and are mostly located between the north-dipping faulting on which the first two earthquakes occurred and the south-dipping faulting associated with the third event. A cluster of aftershocks also lies in the footwall of the eastern end of the south-dipping fault of the third event.Morphologically, it is observed that in the evolution of the Eastern Gulf of Corinth, antithetic faulting apparently predates the appearance of the main faulting at the surface. This evolution can be explained by motion on a deep seated, shallow angle, aseismic normal fault. A model based on such a fault also accounts for the aftershock distribution of the 1981 earthquakes.
We have developed a finite difference method that is especially adapted to the study of dynamic s... more We have developed a finite difference method that is especially adapted to the study of dynamic shear cracks. We studied a number of simple earthquake source models in two and three dimensions with special emphasis on the modeling of the stress field. We compared our numerical results for semi-infinite and self-similar shear crar'ks with the few exact solutions that are avadable in the literature. We then studied spontaneous rupture propagation with the help of a maximum stress criterion. From dimensional arguments and a few simple examples, we showed that the maximum stress criterion depended on the physical dimensions of the fault. For a given maximum stress intensity, the finer the numerical mesh, the higher the maximum stress that had to be adopted. A study of in-plane cracks showed that at high rupture velocities, the numerical results did not resolve the stress concentration due to the rupture front from the stress peak associated with the shear wave propagating in front of the crack. We suggest that this is the reason why transonic rupture velocR,es are found in the numerical solutions of in-plane faulting when the rupture resistance Is rather low. Finally, we studied the spontaneous propagation of an initially circular rupture. Two distinct modes of nucleation of the rupture were studied. In the first, a plane circular shear crack was formed instantaneously in a uniformly prestressed medium. After a while, once stress concentrations had developed around the crack edge, the rupture started to grow, In the second type of nucleation, a preexisting circular crack became unstable at time t = 0 and started to grow, The latter model appeared to us as a more realistm simulation of earthquake triggering. In this case, the initial stress was nonuniform and was the static field of the preexisting fault.
A two-dimensional active seismic experiment was performed on Mount Vesuvius: Explosive charges we... more A two-dimensional active seismic experiment was performed on Mount Vesuvius: Explosive charges were set off at three sites, and the seismic signal along a dense line of 82 seismometers was recorded. A high-velocity basement, formed by Mesozoic carbonates, was identified 2 to 3 kilometers beneath the volcano. A slower (P-wave velocity Vp 3.4 to 3.8 kilometers per second) and shallower high-velocity zone underlies the central part of the volcano. Large-amplitude late arrivals with a dominant horizontal wave motion and low-frequency content were identified as a P to S phase converted at a depth of about 10 kilometers at the top of a low-velocity zone (Vp < 3 kilometers per second), which might represent a melting zone.
Full-waveform inversion ͑FWI͒ is a challenging data-fitting procedure based on full-wavefield mod... more Full-waveform inversion ͑FWI͒ is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategies are designed to mitigate the nonlinearity and ill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from V P and V S velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as ͑1͒ building accurate starting models with automatic procedures and/or recording low frequencies, ͑2͒ defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and ͑3͒ improving computational efficiency by data-compression techniques to make 3D elastic FWI feasible.
Three major destructive earthquakes of Ms 6.7, 6.4, 6.4 occurred in the eastern part of the Gulf ... more Three major destructive earthquakes of Ms 6.7, 6.4, 6.4 occurred in the eastern part of the Gulf of Corinth in February' and March 1981. Associated normal faulting was observed on both the north and the south sides of the Gulf. Examination of teleseismic, local seismic, surface faulting and geomorpbological data suggests that the first and second of these shocks activated major north-dipping normal faults. These faults control the topography and bathymetry and are related to the recent uplift and subsidence of the coastline. It is probable that the first earthquake occurred on a faultwhich outcrops underwater and the second on a fault which is visible on land. It appears that the third shock activated an antithetic normal fault dipping southwards on the north side of the Gulf and that the overall structure of the Gulf of Corinth graben is asymmetric. The northern antithetic fault clearly demonstrates the geomorphological changes associated with developing young normal faulting. In addition, we suggest that observations of recent coastline movement provide a powerful tool for identifying major active fault systems and predicting their historical role in the paleogeography of an area.
The nonlinear problem of inversion of seismic waveforms can be set up using least-squares methods... more The nonlinear problem of inversion of seismic waveforms can be set up using least-squares methods. The inverse problem is then reduced to the problem of minimizing a (nonquadratic) function in a space of many (lo4 to 106) variables. Using gradient methods leads to iterative algorithms, each iteration implying a forward propagation generated by the actual sources, a backward propagation generated by the data residuals (acting as if they were sources), and a correlation at each point of the space of the two fields thus obtained, which gives the updated model. The quality of the results of any inverse method depends heavily on the realism of the forward modeling. Finite-difference schemes are a good choice relative to realism because, although they are time-consuming, they give excellent results. Numerical tests performed with multioffset synthetic data from a two-dimensional model prove the feasibility of the approach. If only surface-recorded reflections are used, the high spatial frequency content of the model (but not the low spatial frequencies) is recovered in few (= 5) iterations. By using transmitted data also (e.g., between two boreholes), all the spatial frequencies are recovered. Since the problem is nonlinear. if the initial guess is far enough from the true solution, the iterative algorithm may conveige inti3 a secondary solution. A nonlinear inversion with 8 shots, each shot recorded at 400 receiver locations. with 700 samples in each seismogram, corresponding to a 2-D model described by 40 000 grid points, takes approximately 1 hour in a CRAY 1S supercomputer. INTRODUCTJON Linearized inversion of seismic waveforms has received considerable attention in the iiterature. For instance, Clayton and Stolt (1981) and Tarantola (1984a) propose methods for solving the linearized multidimensional inverse problem with multioffset seismic reflection data. Woodhouse and Dziewonski (1984) and Tanimoto (1984) also use linearized inversion of waveforms to obtain global Earth models, If a starting model is known to be close enough to the actual medium, linearized inversion will probably perform well. Unfortunately, since there is no practical test to check the accuracy of the linearization, nonlinear inverse techniques are more promising. There are two basic options: a full exploration of the parameter space (systematic or Monte Carlo), or a local descent method. The first approach has the advantage of avoiding local minima, but it is too time-consuming for modern computers. The second approach gives the correct solution when the starting model is inside the valley of the global minimum, irrespective of the choice of starting point. Thus the starting model plays a much less crucial role than in linearized inversion.
A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally ... more A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium. This method uses both velocity and stress in a discrete grid. Density and shear modulus are similarly discretized, avoiding any spatial smoothing. Therefore, boundaries will be correctly modeled under an implicit formulation. Standard problems (quarterplane propagation, sedimentary basin propagation) are studied to compare this method with other methods. Finally a more complex example (a salt dome inside a two-layered medium) shows the effect of lateral propagation on seismograms recorded at the surface. A corner wave. always in-phase with the incident wave, and a head wave will appear, which will pose severe problems of interpretation with the usual vertical migration methods.
Procedures for accurate ray tracing in complex three-dimensional media with interfaces are propos... more Procedures for accurate ray tracing in complex three-dimensional media with interfaces are proposed. The ray tracing equations and the associated paraxial linear equations are solved either by a numerical solver or by an analytical perturbation approach. Interfaces are described with an explicit representation or an implicit representation using B-spline interpolation. For the implicit representation, we exploit two important properties of B-splines, the convex hull and subdivision properties, in order to determine the intersection of the ray with the interface.
Journal of Volcanology and Geothermal Research, 1998
A high-resolution seismic tomography of Mt.Vesuvius was started in May 1994, with the aim of reco... more A high-resolution seismic tomography of Mt.Vesuvius was started in May 1994, with the aim of reconstructing the detailed shallow crustal structure underneath the volcano and define its feeding system. The first phase of the experiment was to perform a 2D profile, using three underground explosions as active sources. Data from controlled sources and microearthquakes were jointly used to determine the shallow structure of the volcano. A high-velocity body (Vp=3.5–4 km/s) was identified at about 2 km beneath the Somma-Caldera. It is likely to represent a sub-volcanic structure, formed by a dense network of solidified dikes. A prominent converted P-to-S phase at about 10 km of depth indicates the occurrence of a sharp transition to a very low-velocity zone. This may represent the top of an extended magmatic reservoir.
Summary. A preliminary study of the aftershocks of three earthquakes that occurred near to Corint... more Summary. A preliminary study of the aftershocks of three earthquakes that occurred near to Corinth (Greece) in 1981 is combined with observations of the morphology and faulting to understand the evolution of the Eastern Gulf of Corinth. The well located aftershocks form a zone 60km long and 20km wide. They do not lie on the main fault planes and are mostly located between the north-dipping faulting on which the first two earthquakes occurred and the south-dipping faulting associated with the third event. A cluster of aftershocks also lies in the footwall of the eastern end of the south-dipping fault of the third event.Morphologically, it is observed that in the evolution of the Eastern Gulf of Corinth, antithetic faulting apparently predates the appearance of the main faulting at the surface. This evolution can be explained by motion on a deep seated, shallow angle, aseismic normal fault. A model based on such a fault also accounts for the aftershock distribution of the 1981 earthquakes.
We have developed a finite difference method that is especially adapted to the study of dynamic s... more We have developed a finite difference method that is especially adapted to the study of dynamic shear cracks. We studied a number of simple earthquake source models in two and three dimensions with special emphasis on the modeling of the stress field. We compared our numerical results for semi-infinite and self-similar shear crar'ks with the few exact solutions that are avadable in the literature. We then studied spontaneous rupture propagation with the help of a maximum stress criterion. From dimensional arguments and a few simple examples, we showed that the maximum stress criterion depended on the physical dimensions of the fault. For a given maximum stress intensity, the finer the numerical mesh, the higher the maximum stress that had to be adopted. A study of in-plane cracks showed that at high rupture velocities, the numerical results did not resolve the stress concentration due to the rupture front from the stress peak associated with the shear wave propagating in front of the crack. We suggest that this is the reason why transonic rupture velocR,es are found in the numerical solutions of in-plane faulting when the rupture resistance Is rather low. Finally, we studied the spontaneous propagation of an initially circular rupture. Two distinct modes of nucleation of the rupture were studied. In the first, a plane circular shear crack was formed instantaneously in a uniformly prestressed medium. After a while, once stress concentrations had developed around the crack edge, the rupture started to grow, In the second type of nucleation, a preexisting circular crack became unstable at time t = 0 and started to grow, The latter model appeared to us as a more realistm simulation of earthquake triggering. In this case, the initial stress was nonuniform and was the static field of the preexisting fault.
A two-dimensional active seismic experiment was performed on Mount Vesuvius: Explosive charges we... more A two-dimensional active seismic experiment was performed on Mount Vesuvius: Explosive charges were set off at three sites, and the seismic signal along a dense line of 82 seismometers was recorded. A high-velocity basement, formed by Mesozoic carbonates, was identified 2 to 3 kilometers beneath the volcano. A slower (P-wave velocity Vp 3.4 to 3.8 kilometers per second) and shallower high-velocity zone underlies the central part of the volcano. Large-amplitude late arrivals with a dominant horizontal wave motion and low-frequency content were identified as a P to S phase converted at a depth of about 10 kilometers at the top of a low-velocity zone (Vp < 3 kilometers per second), which might represent a melting zone.
Full-waveform inversion ͑FWI͒ is a challenging data-fitting procedure based on full-wavefield mod... more Full-waveform inversion ͑FWI͒ is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategies are designed to mitigate the nonlinearity and ill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from V P and V S velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as ͑1͒ building accurate starting models with automatic procedures and/or recording low frequencies, ͑2͒ defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and ͑3͒ improving computational efficiency by data-compression techniques to make 3D elastic FWI feasible.
Three major destructive earthquakes of Ms 6.7, 6.4, 6.4 occurred in the eastern part of the Gulf ... more Three major destructive earthquakes of Ms 6.7, 6.4, 6.4 occurred in the eastern part of the Gulf of Corinth in February' and March 1981. Associated normal faulting was observed on both the north and the south sides of the Gulf. Examination of teleseismic, local seismic, surface faulting and geomorpbological data suggests that the first and second of these shocks activated major north-dipping normal faults. These faults control the topography and bathymetry and are related to the recent uplift and subsidence of the coastline. It is probable that the first earthquake occurred on a faultwhich outcrops underwater and the second on a fault which is visible on land. It appears that the third shock activated an antithetic normal fault dipping southwards on the north side of the Gulf and that the overall structure of the Gulf of Corinth graben is asymmetric. The northern antithetic fault clearly demonstrates the geomorphological changes associated with developing young normal faulting. In addition, we suggest that observations of recent coastline movement provide a powerful tool for identifying major active fault systems and predicting their historical role in the paleogeography of an area.
The nonlinear problem of inversion of seismic waveforms can be set up using least-squares methods... more The nonlinear problem of inversion of seismic waveforms can be set up using least-squares methods. The inverse problem is then reduced to the problem of minimizing a (nonquadratic) function in a space of many (lo4 to 106) variables. Using gradient methods leads to iterative algorithms, each iteration implying a forward propagation generated by the actual sources, a backward propagation generated by the data residuals (acting as if they were sources), and a correlation at each point of the space of the two fields thus obtained, which gives the updated model. The quality of the results of any inverse method depends heavily on the realism of the forward modeling. Finite-difference schemes are a good choice relative to realism because, although they are time-consuming, they give excellent results. Numerical tests performed with multioffset synthetic data from a two-dimensional model prove the feasibility of the approach. If only surface-recorded reflections are used, the high spatial frequency content of the model (but not the low spatial frequencies) is recovered in few (= 5) iterations. By using transmitted data also (e.g., between two boreholes), all the spatial frequencies are recovered. Since the problem is nonlinear. if the initial guess is far enough from the true solution, the iterative algorithm may conveige inti3 a secondary solution. A nonlinear inversion with 8 shots, each shot recorded at 400 receiver locations. with 700 samples in each seismogram, corresponding to a 2-D model described by 40 000 grid points, takes approximately 1 hour in a CRAY 1S supercomputer. INTRODUCTJON Linearized inversion of seismic waveforms has received considerable attention in the iiterature. For instance, Clayton and Stolt (1981) and Tarantola (1984a) propose methods for solving the linearized multidimensional inverse problem with multioffset seismic reflection data. Woodhouse and Dziewonski (1984) and Tanimoto (1984) also use linearized inversion of waveforms to obtain global Earth models, If a starting model is known to be close enough to the actual medium, linearized inversion will probably perform well. Unfortunately, since there is no practical test to check the accuracy of the linearization, nonlinear inverse techniques are more promising. There are two basic options: a full exploration of the parameter space (systematic or Monte Carlo), or a local descent method. The first approach has the advantage of avoiding local minima, but it is too time-consuming for modern computers. The second approach gives the correct solution when the starting model is inside the valley of the global minimum, irrespective of the choice of starting point. Thus the starting model plays a much less crucial role than in linearized inversion.
A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally ... more A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium. This method uses both velocity and stress in a discrete grid. Density and shear modulus are similarly discretized, avoiding any spatial smoothing. Therefore, boundaries will be correctly modeled under an implicit formulation. Standard problems (quarterplane propagation, sedimentary basin propagation) are studied to compare this method with other methods. Finally a more complex example (a salt dome inside a two-layered medium) shows the effect of lateral propagation on seismograms recorded at the surface. A corner wave. always in-phase with the incident wave, and a head wave will appear, which will pose severe problems of interpretation with the usual vertical migration methods.
Uploads
Papers by Jean Virieux