Jean Virieux

P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method

I present a finite-difference method for modeling P-SV wave propagation in heterogeneous media. This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid. The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poissons ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poissons ratio. Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid-solid interface. Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results. The weathered-layer and corner-edge models show in seismograms the same converted phases obtained by previous authors. This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half-space. The head wave preserves the correct amplitude. Finally, the corner-edge model illustrates a more complex geometry for the liquid-solid interface. As the Poissons ratio v increases from 0.25 to 0.5, the shear converted phases are removed from seismograms and from the time section of the wave field.

An overview of full-waveform inversion in exploration geophysics

Full-waveform inversion FWI 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 strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS 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 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 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 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

SH-wave propagation in heterogeneous media; velocity-stress finite-difference method

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 (quarter-plane 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.

Two-dimensional nonlinear inversion of seismic waveforms; numerical results

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 lp;nonquadratic) function in a space of many (104to106) 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 ...

Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion

Quantitative imaging of the elastic properties of the subsurface at depth is essential for civil engineering applications and oil- and gas-reservoir characterization. A realistic synthetic example provides for an assessment of the potential and limits of 2D elastic full-waveform inversionFWIof wide-aperture seismic data for recovering high-resolution P- and S-wave velocity models of complex onshore structures. FWI of land data is challengingbecauseoftheincreasednonlinearityintroducedbyfreesurface effects such as the propagation of surface waves in the heterogeneous near-surface. Moreover, the short wavelengths of the shear wavefield require an accurate S-wave velocity starting modeliflowfrequenciesareunavailableinthedata.Weevaluated different multiscale strategies with the aim of mitigating the nonlinearities. Massively parallel full-waveform inversion was implemented in the frequency domain. The numerical optimization relies on a limited-memory quasi-Newton algorithm that outperforms the more classic preconditioned conjugate-gradient algorithm. The forward problem is based upon a discontinuous Galerkin DG method on triangular mesh, which allows accuratemodelingoffree-surfaceeffects.Sequentialinversionsofincreasingfrequenciesdefinethemostnaturallevelofhierarchyin multiscale imaging. In the case of land data involving surface waves, the regularization introduced by hierarchical frequency inversions is not enough for adequate convergence of the inversion. A second level of hierarchy implemented with complexvalued frequencies is necessary and provides convergence of the inversion toward acceptable P- and S-wave velocity models. Amongthepossiblestrategiesforsamplingfrequenciesintheinversion, successive inversions of slightly overlapping frequency groups is the most reliable when compared to the more standard sequential inversion of single frequencies. This suggests that simultaneous inversion of multiple frequencies is critical when consideringcomplexwavephenomena.

Probabilistic Earthquake Location in 3D and Layered Models

Probabilistic earthquake location with non-linear, global search methods allows the use of 3D models and produces comprehensive uncertainty and resolution information represented by a probability density function over the unknown hypocentral parameters. We describe a probabilistic earthquake location methodology and introduce an efficient Metropolis-Gibbs, non-linear, global sampling algorithm to obtain such locations. Using synthetic travel times generated in a 3D model, we examine the locations and uncertainties given by an exhaustive grid-search and the Metropolis-Gibbs sampler using 3D and layered velocity models, and by a iterative, linear method in the layered model. We also investigate the relation of average station residuals to known static delays in the travel times, and the quality of the recovery of known focal mechanisms. With the 3D model and exact data, the location probability density functions obtained with the Metropolis-Gibbs method are nearly identical to those of the slower but exhaustive grid-search. The location PDFs can be large and irregular outside of a station network even for the case of exact data. With location in the 3D model and static shifts added to the data, there are systematic biases in the event locations. Locations using the layered model show that both linear and global methods give systematic biases in the event locations and that the error volumes do not include the “true” location — absolute event locations and errors are not recovered. The iterative, linear location method can fail for locations near sharp contrasts in velocity and outside of a network. Metropolis-Gibbs is a practical method to obtain complete, probabilistic locations for large numbers of events and for location in 3D models. It is only about 10 times slower than linearized methods but is stable for cases where linearized methods fail. The exhaustive grid-search method is about 1000 times slower than linearized methods but is useful for location of smaller number of events and to obtain accurate images of location probability density functions that may be highly-irregular.

3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study

We present a finite-difference frequency-domain method for 3D visco-acoustic wave propagation modeling. In the frequency domain, the underlying numerical problem is the resolution of a large sparse system of linear equations whose right-hand side term is the source. This system is solved with a massively parallel direct solver. We first present an optimal 3D finite-difference stencil for frequency-domain modeling. The method is based on a parsimonious staggered-grid method. Differential operators are discretized with second-order accurate staggered-grid stencils on different rotated coordinate systems to mitigate numerical anisotropy. An antilumped mass strategy is implemented to minimize numerical dispersion. The stencil incorporates 27 grid points and spans two grid intervals. Dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3D domain. To assess the feasibility of the method for frequency-domain full-waveform inversion, we computed simulations in the 3D SEG/EAGE overthrust model for frequencies 5, 7, and 10 Hz. Results confirm the huge memory requirement of the factorization (several hundred Figabytes) but also the CPU efficiency of the resolution phase (few seconds per shot). Heuristic scalability analysis suggests that the memory complexity of the factorization is O(35N(4)) for a N-3 grid. Our method may provide a suitable tool to perform frequency-domain full-waveform inversion using a large distributed-memory platform. Further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time- and frequency-domain modeling of wave propagation to perform 3D full-waveform inversion.

Seismic Evidence for a Low-Velocity Zone in the Upper Crust Beneath Mount Vesuvius

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.

Three‐dimensional seismic tomography from P wave and S wave microearthquake travel times and rock physics characterization of the Campi Flegrei Caldera

[1] The Campi Flegrei (CF) Caldera experiences dramatic ground deformations unsurpassed anywhere in the world. The source responsible for this phenomenon is still debated. With the aim of exploring the structure of the caldera as well as the role of hydrothermal fluids on velocity changes, a multidisciplinary approach dealing with three-dimensional delay time tomography and rock physics characterization has been followed. Selected seismic data were modeled by using a tomographic method based on an accurate finite difference travel time computation which simultaneously inverts P wave and S wave first-arrival times for both velocity model parameters and hypocenter locations. The retrieved P wave and S wave velocity images as well as the deduced V p /V s images were interpreted by using experimental measurements of rock physical properties on CF samples to take into account steam/water phase transition mechanisms affecting P wave and S wave velocities. Also, modeling of petrophysical properties for site-relevant rocks constrains the role of overpressured fluids on velocity. A flat and low V p /V s anomaly lies at 4 km depth under the city of Pozzuoli. Earthquakes are located at the top of this anomaly. This anomaly implies the presence of fractured overpressured gas-bearing formations and excludes the presence of melted rocks. At shallow depth, a high V p /V s anomaly located at 1 km suggests the presence of rocks containing fluids in the liquid phase. Finally, maps of the V p *V s product show a high V p * V s horseshoe-shaped anomaly located at 2 km depth. It is consistent with gravity data and well data and might constitute the on-land remainder of the caldera rim, detected below sea level by tomography using active source seismic data.

ITERATIVE ASYMPTOTIC INVERSION IN THE ACOUSTIC APPROXIMATION

We propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation. For this purpose, we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods. The different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image. In the linearized inversion, we use the background velocity model obtained by velocity analysis. We determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model. Because the inverse operator is not exact, and because the source and station distribution is limited, the first iteration of our asymptotic inversion technique is not exact. We improve the images by an iterative procedure. Since the background velocity does not change between iterations. There is no need to retrace rays,...