Ludovic Métivier

Interlocked optimization and fast gradient algorithm for a seismic inverse problem

We give a nonlinear inverse method for seismic data recorded in a well from sources at several offsets from the borehole in a 2D acoustic framework. Given the velocity field, approximate values of the impedance are recovered. This is a 2D extension of the 1D inversion of vertical seismic profiles [18]. The inverse problem generates a large scale undetermined ill-conditioned problem. Appropriate regularization terms render the problem well-determined. An interlocked optimization algorithm yields an efficient preconditioning. A gradient algorithm based on the adjoint state method and domain decomposition gives a fast parallel numerical method. For a realistic test case, convergence is attained in an acceptable time with 128 processors.

A 2D nonlinear inversion of well-seismic data

Well-seismic data such as vertical seismic profiles are supposed to provide detailed information about the elastic properties of the subsurface at the vicinity of the well. Heterogeneity of sedimentary terrains can lead to far from negligible multiple scattering, one of the manifestations of the nonlinearity involved in the mapping between elastic parameters and seismic data. We present a 2D extension of an existing 1D nonlinear inversion technique in the context of acoustic wave propagation. In the case of a subsurface with gentle lateral variations, we propose a regularization technique which aims at ensuring the stability of the inversion in a context where the recorded seismic waves provide a very poor illumination of the subsurface. We deal with a huge size inverse problem. Special care has been taken for its numerical solution, regarding both the choice of the algorithms and the implementation on a cluster-based supercomputer. Our tests on synthetic data show the effectiveness of our regularization. They also show that our efforts in accounting for the nonlinearities are rewarded by an exceptional seismic resolution at distances of about 100 m from the well. They also show that the result is not very sensitive to errors in the estimation of the velocity distribution, as far as these errors remain realistic in the context of a medium with gentle lateral variations.

Optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling

We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic wave modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg-Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poissons ratios. Moreover, only 3.7 grid-points per minimum shear wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates wavefields having smaller error under the same discretization setups. Profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.

Fast Full Waveform Inversion with Source Encoding and Second Order Optimization Methods

In the context of full waveform inversion (FWI), second-order optimization methods, which take into account more precisely the effect of the Hessian such as the quasi-Newton l-BFGS method, have shown superior convergence properties than first-order methods. When using source encoding techniques, the regeneration of the random variables to assemble the sources at each FWI iteration plays a crucial role since it helps to reduce the so-called cross talk noise produced by the encodings. However, it is not clear how to combine the l-BFGS algorithm and encoding methods because, strictly speaking, l-BFGS needs previous iteration estimations, thus prohibiting the regeneration of the code at each iteration. We study how to combine second-order optimization methods with encoding techniques, considering two truncated matrix-free Newton algorithms (Gauss Newton and full Newton) and the l-BFGS algorithm with periodic restarts and we apply our method on the 2004 BP salt model.

Strategies for solving index one DAE with non-negative constraints

Liquid-liquid extraction modeling leads to solve an index one DAE system. For the sake of robustness, it is desirable to account for non-negative constraints. Based on the DASSL architecture (a classical index one DAE solver) we propose and compare three different strategies to implement these bound constraints. Each of these strategies corresponds to a different Newton modification: clipping, damping, or interior point method. The comparisons are made on two test cases: the Robertson ODE problem, and an example from liquid-liquid extraction modeling.

A Time-Domain Preconditioned Truncated Newton Approach to Visco-acoustic Multiparameter Full Waveform Inversion

A truncated Newton (TRN) method for time-domain full waveform inversion (FWI) in a visco-acoustic medium has been developed based on the 2nd-order adjoint state formulation. Time-domain gradient estimation and Hessian-vector product are managed by recomputing, without numerical instabilities, the incident wavefield at the same time as the adjoint wavefield for mitigating memory issues. Generic algorithm workflow has been proposed to switch between parameterizations, thanks to the chain rule. An efficient preconditioner adapted to the multiparameter configuration is developed to enhance the convergence rate of the inner conjugate gradient iterations. An additional user-defined scaling is introduced in the preconditioner to mitigate the weak sensitivity of specific parameters to waveform variations. The importance of the inverse Hessian for mitigating interparameter trade-off is validated on a toy example. Through a realistic 2D synthetic case based on a North Sea real data application, encouraging numerica...

A review of some methodological developments on full waveform inversion tackled in the SEISCOPE group

Full waveform inversion (FWI) is a data-fitting inverse problem aiming to delineate high-resolution quantitative images of the Earth.While its basic principle has been proposed in the eighties, the approach has been significantly developed and applied to 2D and 3D problems at various scales for the last fifteen years.Despite these successes, FWI is still facing some issues for applications in complex geological setups because of some lack of robustness and automatic workflow, while being computationally intensive.In this paper, after a short review of the basic FWI formulation and analysis of the FWI gradient, three recent methodological developments performed in the frame of the SEISCOPE project are presented.First, an algorithmic development is presented as a low-memory and computationally efficient approach for building the time-domain FWI gradient in 3D viscous media.Second, a reformulation of FWI is performed to handle reflections in their tomography regime while still using the diving waves, leading to the joint full waveform inversion (JFWI) approach.Finally, an optimal transport approach is proposed as an alternative to the classical difference-based misfit for mitigating the cycle-skipping issue.

Imaging challenging media by full waveform inversion of ultrasonic signals

Austenitic welds are important parts of the cooling system in nuclear power plants, which undergo extreme temperature and pressure variations that may cause defects to appear in the welded zone. If not attended, these may lead to a leakage of radioactive liquids. It is therefore crucial to assess the structural integrity of austenitic welds by detecting and imaging such defects. Ultrasonic methods are one of the reference methods in this matter. Current imaging techniques rely on phased array technology to focus the ultrasonic energy based on beamforming approaches such as the total focusing method. Because the elastic properties and exact geometry of austenitic welds are unknown, these methods fail to produce a reliable image of the inspected area. We introduce the full waveform inversion (FWI) of ultrasonic signals as a promising alternative imaging method. Adapted from Geophysics, the FWI uses a numerical model to simulate the experiment, and iterates by updating the model until the error between exper...

Frequency-domain modelling and inversion of electromagnetic data for 2D permittivity and conductivity imaging: An application to the Institut Fresnel experimental dataset

The need for quantitative imaging of the near subsurface leads to the development of inversion algorithms to infer ground properties from recorded data. The aim of this study is to validate an inversion method recently developed for the simultaneous imaging of dielectric permittivity and electrical conductivity from 2D ground-penetrating radar measurements. The validation is performed using electromagnetic data collected in a well-controlled laboratory environment. In this experiment, the knowledge of the targets enables a quality control of the inversion results. In addition, the free space environment and the measurement of the incident field simplify the choice of a starting model for the inversion, as well as the calibration of the data with respect to the source signature and to the geometrical spread. To perform accurate and efficient forward simulations, we use a frequency-domain finite-difference scheme whose stencil coefficients can be optimized for each simulated frequency. As the objects of interest are locally concentrated at the centre of the acquisition array, it is possible to restrict the computation domain to a small region enclosing the targets using an integral representation of the analytical incident field coming from the sources and of the scattered field that we analytically propagate towards the receivers. An analysis of the numerical errors done on synthetic data shows that this strategy provides an error level that is low enough not to perturb the inversion, while dramatically decreasing the computational cost compared to a full-domain simulation. The monoparameter reconstruction of a purely dielectric target recovers permittivity values in very good agreement with the expected ones, as well as a very satisfying data fit. We also validate our strategy for multiparameter inversion on targets involving both a purely dielectric cylinder and a purely metallic copper tube, although the optimization cannot recover the exact conductivity of copper.