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WIND project

Seismic Imaging by Waveform Inversion of Node Data


Who are we?

This is the site of the WIND project devoted to seismic imaging from ultra-long offsets data by slope tomography and Full Waveform Inversion (FWI) . We are interested in sparse stationary-recording acquisitions, in particular sea bed acquisitions carried out with multi-component Ocean Bottom Seismometers or Nodes (OBS/OBN) but also land acquisitions . This project started in 2020 and is supported by a consortium of oil companies. The WIND project is hosted by the Geosciences Geoazur Institute located on the French riviera under the supervision of University of Côte d'Azur (UCA), CNRS, IRD and Observatory of Côte d'Azur (OCA). Stéphane Operto coordinates the scientific activities of a multidisciplinary team gathering applied mathematicians, HPC engineers and computational/applied geophysicists. You will find on this site a basic FWI tutorial as a prelude of the presentation of the WIND researches, the recent publications of our team related to the project, the composition of our scientific team, the conditions to join the WIND project, and the open positions. The intranet page is a restricted access area reserved for our sponsors.

You can download the WIND proposal here.

For more information, contact us.

What is FWI?

FWI is a high-resolution seismic imaging method which has experienced an explosive rice of popularity since a decade thanks to the development of high-performance computing and the design of new acquisition technologies (broadband and wide azimuth). Basically, it is a constrained optimization problem which seeks to fit the oscillatory waveforms of seismic signals (the observations) to reconstruct the physical properties of the subsurface (the unknown parameters) using as constraint a mathematical model of wave propagation (namely, the wave equation, a linear partial differential equation (PDE) describing wave propagation in the earth). Its area of application, initially confined to exploration geophysics, has extended to earthquake seismology, from regional to global scales Tromp (2020), civil engineering from seismic and ground penetrating radar (GPR) data, medical imaging and helioseismology. For more details, see the papers & books of A. Tarantola for the seminal contributions on this topic from the Bayesian perspective, Pratt et al. (1998) and other publications of G. Pratt for the frequency-domain formulation of FWI, and Virieux&Operto (2009) , Operto et al (2013) and Virieux et al. (2017) for recent reviews in exploration geophysics (a synthesis is presented in the FWItuto page of this site).

Application context and objectives

Our overall objective is to develop a suitable FWI technology for sparse ultra-long offset stationary-recording seismic acquisitions such as multi-component ocean bottom node ( OBN ) surveys in marine environments or land acquisitions. We are interested in such long-offset acquisition geometries because it is well acknowledged that the resolution power of FWI and its ability to reconstruct all the physical properties governing elastic wave propagation is fully exploited when the acquisition design allows for a broad angular illumination of the subsurface manifested by the recording of a wide variety of wave types, from transmitted to reflected waves. Simply put, the offsets should be long enough such that the transmitted diving waves undershoot the deepest targeted structures (as a sketch illustration of this specification, the WIND logo shows the recording of compressional body waves which have continuously propagated in a two-layer model from the transmission to the reflection regimes; this is typically the piece of information with which FWI should be feed to build broadband subsurface model). To reach the necessary versatility to record waves at long offsets, the receiver device should be independent to the source device and designed with autonomous instruments. This requirement has prompted the oil industry to recently investigate further sea-bottom node acquisition for deep offshore exploration (for example, see the presentations of the ACQ5 session at 2019 SEG annual meeting) at the expense of towed-streamer acquisition. Beyond industrial applications, our motivation is also related to the seismic exploration of the deep crust for more academic-oriented applications. In this context, towed-streamer acquisitions will not have the necessary sensitivity to build a reliable macro velocity model by reflection-based velocity analysis techniques due to insufficient hyperbolic normal move out at crustal depths. Moreover, the sources may be not powerful enough to record deep reflections (at depths exceeding the streamer length) at pre-critical angles with a sufficient signal-to-noise ratio. In this context, sparse long-offset node acquisition is the only alternative to perform reliable deep crustal imaging from post-critical reflections, diving waves and head waves (without considering expensive two-vessels experiments).


Long-offset acquisition geometries raise at least three methodological issues.

The first is related to the sparsity of the areal acquisition . Recording waves at long offsets requires to cover wide areas (a few thousands of square kilometers) with a finite number of instruments. The best trade off needs to be found between the sampling of the receiver device, the spread of the receiver device, and the multi-fold coverage for signal-to-noise ratio issue. Sub-sampling of wide areas by parsimonious pool of instruments will translate into sub-sampling of the spectral components of the earth, leading to spatial wraparound artifacts in the seismic images. Conversely, a deficit of long offset coverage for the benefit of dense acquisition sampling will narrow the spectral bandwidth of the reconstructed subsurface medium, preventing the reconstruction in depth of the long to intermediate wavelengths and the resolution of steep structural dips in complex geological environments. It will also degrade the redundancy of the wavenumber sampling provided by different pairs of frequency and scattering angles and hence increase the ill-posedness of multi-parameter reconstruction. In this context, seismic acquisition design should be cast as an optimization problem and compressive sensing strategies should be used to mitigate the curse of dimensionality in seismic acquisition by sampling it below the Nyquist rate. These are key issues that we want to address during WIND .

The second is a high-performance computing issue related to the forward problem . Areal geometries require to repeatedly simulate wave propagation in huge numerical meshes containing up to billions of unknowns with hundreds to thousands of excitation terms (seismic sources). Therefore, choosing the optimal forward engine providing the best compromise between computational efficiency, accuracy of the wavefield solution, versatility of the discretization and physical representation of the subsurface medium is a key issue as well as the design of strategies allowing one to mitigate the computational burden of multi-source simulation (blending and encoding, subsampling, ...). Seismic wave simulation can be performed either in the time-space or frequency-space domains. The two formulations are rather orthogonal in terms of numerical methods, the first one being a time evolution problem generally solved with explicit (matrix free) time stepping, the second being a boundary value problem requiring the solution of large and sparse linear systems with multiple sparse right-hand sides. Therefore, both have their own pros and cons depending the specifications of the acquisition geometriy, the size of the computational mesh, the number of seismic sources, and the physics that needs to be considered. Our goal is to implement seismic wave simulation and FWI in both domains to be able to use the most suitable formulation for a given case study.

The third challenge is related to the nonlinearity and ill-posedness of FWI. FWI is nonlinear PDE-constrained optimization problem because the measurements are non linearily related to the unknown parameters. Due to the dimension of the data and model space, this nonlinear optimization problem is generally solved with local optimization techniques based upon gradient methods, in particular in 3D where global optimization methods remain out of reach. The FWI objective function exhibits many local minima when (1) the distance between the measurements and the numerical prediction is the least-squares norm of their differences and (2) the multi-variate optimization problem (the variables are the wavefields and the parameters) is reduced to a monovariate parameter-estimation problem after projection (or elimination) of the wavefields. This results because the domain of validity of the linearization of the reduced-space inversion (based upon the singled-scattering Born approximation) requires the simulated data to match the recorded counterparts with a kinematic error which does not exceed half a period. This half period is directly related to the width of the Fresnel zones in the spatial domain on which data residuals are back-projected during FWI. Also, a time error of half a period corresponds to a phase shift of π beyond which the phase is wrapped leading to a cycle ambiguity. For this reason, fitting the wrong cycle of monochromatic data during FWI is referred to as cycle skipping . Due to the oscillatory nature of seismic waves, the condition that needs to be satisfied to prevent cycle skipping becomes increasingly challenging to fulfill as the number of propagated wavelengths increases: This is typically the case when data lacks low frequencies (below 1Hz) or the maximum offset increases (the price to be paid to reach a sufficient aperture illumination from surface acquisition). To tackle this non linearity issue, we develop in parallel two approaches: First, we develop high-resolution slope tomography based on eikonal solver and the adjoint-state method to build kinematically-accurate initial velocity model for FWI such that the occurrence of cycle skipping is delayed as much as possible. The strength of slope tomography is to rely on locally-coherent events, allowing for dense picking. In this sense, slope tomography shares with FWI the idea to fully exploit the information content of seismic data. Second, we develop a variant of FWI, originally called Wavefield Reconstruction Inversion (WRI) by T. van Leeuwen and F. Herrmann, which aims to extend its search space through a relaxation of the wave equation during the early stages of the inversion such that the observations are matched with inaccurate subsurface models. Then, the subsurface parameters are updated by minimizing the wave-equation violations. This is an orthogonal strategy compared to classical FWI, where the wavefields are estimated by solving the wave equation exactly, before updating the parameters by minimizing the data residuals. We have shown how the WRI technology can be implemented efficiently with the alternating-direction method of multiplier (ADMM) . FWI is also ill-posed (non uniqueness of the solution) because surface acquisitions provide an incomplete angular illumination of the subsurface in depth, in particular in large-contrast media where salt body, basaltic sills or carbonate layers tend to channel waves at shallow depths. A second source of ill-posedness is related to the joint reconstruction of parameters of different nature, which can have a coupled signature in the recorded data. In this context, FWI needs to be stabilized by injecting some priors through regularization . These priors can be provided by some approximate knowledge of the statistical properties of the earth, some empirical relationships tying together parameter classes of different nature, or local measurements provided by well logs. During WIND, we propose some methodologies to interface in a versatile way these priors with the optimization algorithm.

Why should you support WIND?

The ambition of the WIND project is to cover as much as possible all the aspects of the FWI problematic , from theoretical issues related to numerical solving of PDE, nonlinear optimization, regularization to 3D large-scale application on real data through algorithmic issues (high-performance computing in parallel environments). We launch the WIND project because our team performed several achievements on all these aspects these last years and we manage to create a network of multi-disciplinarity collaborations gathering expertises in the fields of applied mathematics, computer science and applied geophysics.

You will find a review and some illustrations of our research activity here .

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