René-Edouard Plessix

Three-dimensional frequency-domain full-waveform inversion with an iterative solver

With the acquisition of wide-aperture seismic data sets, full-waveform inversion is an attractive method for deriving velocity models. Three-dimensional implementations require an efficient solver for the wave equation. Computing 3D time-harmonic responses with a frequency-domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved. Time-domain schemes are an alternative. Nevertheless, existing frequency-domain iterative solvers with an efficient preconditioner are a viable option when full-waveform inversion is formulated in the frequency domain. An iterative solver with a multigrid preconditioner is competitive because of a high-order spatial discretization. Numerical examples illustrated the efficiency of the iterative solvers. Three dimensional full-waveform inversion was then studied in the context of deep-water ocean-bottom seismometer acquisition. Three dimensional synthetic data inversion results showed the behavior of full-waveform inversion with respect to the initial model and the minimum frequency available in the data set. Results on a 3D real ocean-bottom seismometer data set demonstrated the relevance of full-waveform inversion, especially to image the shallow part of the model.

An optimal true-amplitude least-squares prestack depth-migration operator

In order to define an optimal true-amplitude prestack depth migration for multishot and multitrace data, we develop a general methodology based on the least-squares data misfit function associated with a forward model. The amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data. The migrated section is then the gradient of the cost function multiplied by a weight matrix. A study of the Hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this Hessian. Thanks to this matrix, the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least-squares migrations. Applied to a forward model based on Born, ray tracing, and diffracting points approximation, this optimal migration outperforms or at least equals the classic Kirchhoff formula, since the latter belongs to the class of least-squares migrations and is only optimal for one shot and an infinite aperture. Numerical results illustrate this construction and confirm the above expectations.

A comparison between one-way and two-way wave-equation migration

Results for wave-equation migration in the frequency domain using the constant-density acoustic two-way wave equation have been compared to images obtained by its one-way approximation. The two-way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two-way wave equation is sensitive to diving waves, leading to low-frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high-pass spatial filtering. The last is the most effective. Iterative migration based on a least-squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constantdensity acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high-pass spatial filtering for the linearized case, a real-data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real-data example shows only marginal improvement over the linearized case. In two dimensions, the computational cost of the twoway approach has the same order of magnitude as that for the one-way method. With our implementation, the two-way method requires about twice the computer time needed for one-way wave-equation migration.

A Helmholtz iterative solver for 3D seismic-imaging problems

A preconditioned iterative solver for the 3D frequency-domain wave equation applied to seismic problems is evaluated. The preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the (undamped) wave equation. The approximate inverse is computed with one multigrid cycle. Numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength. To evaluate the relevance of this iterative solver, the number of floating-point operations required for two imaging problems are roughly evaluated. This rough estimate indicates that the time-domain migration approach is more than one order of magnitude faster. The full-wave-form tomography, based on a least-squares formulation and a scale separation approach, has the same complexity in both domains.

Application of Acoustic Full Waveform Inversion to a Low-frequency Large-offset Land Data Set

By interpreting the full wavefield, full waveform inversion has the potential to become a key tool to interpret seismic data acquired in complex geological settings. However, its application requires low frequencies and large offsets to avoid ending up in a local minimum. This approach has been illustrated with marine data sets and acoustic full waveform inversion. Acoustic full waveform inversion can also be applied with land data sets when the acquisition and the pre-processing are planned correctly. To demonstrate the relevance of this dedicated approach, we invert a land data set acquired with low frequencies down to 1.5 Hz and 20 km offset. We show that the velocity retrieved by full waveform inversion supersedes the one derived by handpicked nmo-gather velocity analysis even when we start with a crude 1D model. With the acquisition of low frequencies and long offsets and a dedicated preprocessing, a high-resolution seismic migrated image could be obtained over this land area.

Detecting hydrocarbon reservoirs from CSEM data in complex settings : Application to deepwater Sabah, Malaysia

Controlled-source electromagnetic (CSEM) field surveys offer a geophysical method to discriminate between high and low hydrocarbon saturations in a potential reservoir. However, the same geological processes that create the possible hydrocarbon reservoir may also create topography and near-surface variations of resistivity (e.g., shallow gas or hydrates) that can complicate the interpretation of CSEM data. In this paper, we discuss the interpretation of such data over a thrust belt prospect in deepwater Sabah, Malaysia. We show that detailed modeling of the key scenarios can help us understand the contributions of topography, near-surface hydrates, and possible hydrocarbons at reservoir depth. Complexity at the surface and at depth requires a 3D electromagnetic modeling code that can handle realistic ten-million-cell models. This has been achieved by using an iterative solver based on a multigrid preconditioner, finite-difference approach with frequency-dependent grid adaptation.

The use of low frequencies in a full-waveform inversion and impedance inversion land seismic case study

Velocity model building and impedance inversion generally suffer from a lack of intermediate wavenumber content in seismic data. Intermediate wavenumbers may be retrieved directly from seismic data sets if enough low frequencies are recorded. Over the past years, improvements in acquisition have allowed us to obtain seismic data with a broader frequency spectrum. To illustrate the benefits of broadband acquisition, notably the recording of low frequencies, we discuss the inversion of land seismic data acquired in Inner Mongolia, China. This data set contains frequencies from 1.5–80 Hz. We show that the velocity estimate based on an acoustic fullwaveform inversion approach is superior to one obtained from reflection traveltime inversion because after full-waveform inversion the background velocity conforms to geology. We also illustrate the added value of low frequencies in an impedance estimate.

Waveform Inversion of Reflection Seismic Data for Kinematic Parameters by Local Optimization

A reformulation of the usual least-squares waveform inversion problem is proposed to retrieve, from seismic data, the kinematic parameters (the two-dimensional (2D) background velocity and the source and cable depth) by local optimization. These last parameters are of paramount importance for a successful inversion of very high resolution (VHR) seismic data which we are interested in. In our inversion the source and cable depth parameters are treated in the same way as the background velocity. To avoid the problem of local minima, a change of unknowns is performed: the depth reflectivity is replaced by its dual variable, called the time reflectivity. In this way, the current value of the reflectivity is stored in the time domain and is strongly decoupled from the current value of the velocity field and cable depth. The increase in modeling complexity (an additional prestack migration is required for each function evaluation) is compensated by the enlargement of the attraction domain of the global minimum which allows the use of a local optimization technique. A numerical implementation with Born approximation and ray tracing is detailed, in which the derivatives of the travel time are computed via an adjoint state technique for more efficiency. Numerical results illustrate the behavior of the new objective function, and inversion of synthetic and real VHR data for kinematic parameters is performed.

Finite-difference Iterative Migration By Linearized Waveform Inversion In the Frequency Domain

We present an iterative migration technique for mapping seismic data to reflector amplitudes, obtained by formulating migration as an optimization problem. The method can be based on any kind of seismic modeling and provides true-amplitude images in a natural way. In this paper, we use a finite-difference solution of the linearized constant-density wave equation in the frequency domain. Because the constant-density acoustic equation cannot handle impedance contrasts, we use its linearized form, which is equivalent to the Born approximation. Examples are given for synthetic and real data and show that the iterative technique improves the amplitudes and resolution of reflectors.

VTI Full Waveform Inversion: a Parameterization Study With a Narrow Azimuth Streamer Data Example

Full Waveform Inversion (FWI) requires long offset data, notably diving/refracted waves, to update the velocity. Often, anisotropy is needed to model the kinematics of the short and long offsets. FWI may therefore require an anisotropic modeling to correctly represent the kinematics of all the waves. With a change of variables in the VTI (vertical transverse isotropic) wave equations, it can be shown that VTI FWI mainly depends on the NMO (normal moveout) velocity and the η parameter. There is a trade-off between depth and δ parameter. When using only the low frequencies of the seismic data and an acoustic modeling, FWI should primally focus on the kinematics of the waveform. A logical parameterization for VTI FWI is then the NMO velocity, the η parameter, and the δ parameter, since the kinematics is mainly parameterized by the NMO velocity and the η parameter in a smoothed medium. The η parameter may be replaced by the horizontal velocity. An elliptic FWI is therefore more or less equivalent to an isotropic FWI with a stretched depth. This is illustrated with several FWI on a narrow azimuth real data set recorded in the North Sea. We also show that taking into account anellipticity in the initial model improves the results of FWI, although there is a trade-off between velocity heterogeneities and anisotropy.