Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

We have developed and implemented a robust and practical scheme for anisotropic 3D acoustic full-waveform inversion (FWI). We demonstrate this scheme on a field data set, applying it to a 4C ocean-bottom survey over the Tommeliten Alpha field in the North Sea. This shallow-water data set provides good azimuthal coverage to offsets of 7 km, with reduced coverage to a maximum offset of about 11 km. The reservoir lies at the crest of a high-velocity antiformal chalk section, overlain by about 3000 m of clastics within which a low-velocity gas cloud produces a seismic obscured area. We inverted only the hydrophone data, and we retained free-surface multiples and ghosts within the field data. We invert in six narrow frequency bands, in the range 3 to 6.5 Hz. At each iteration, we selected only a subset of sources, using a different subset at each iteration; this strategy is more efficient than inverting all the data every iteration. Our starting velocity model was obtained using standard PSDM model bui...

https://library.seg.org/doi/pdf/10.1190/geo2012-0338.1

Anisotropic 3D full-waveform inversion

Building high-resolution models of several physical properties of the subsurface by multiparameter full waveform inversion (FWI) of multicomponent data will be a challenge for seismic imaging for the next decade. The physical properties, which govern propagation of seismic waves in visco-elastic media, are the velocities of the P- and S-waves, density, attenuation, and anisotropic parameters. Updating each property is challenging because several parameters of a different nature can have a coupled effect on the seismic response for a particular propagation regime (from transmission to reflection). This is generally referred to as trade-off or crosstalk between parameters. Moreover, different parameter classes can have different orders of magnitude or physical units and footprints of different strength in the wavefield, which can make the inversion poorly conditioned if it is not properly scaled.

A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice

Seismic velocity is one of the most important parameters used in seismic exploration. Accurate velocity models are the key prerequisites for reverse time migration and other high-resolution...

Deep-learning inversion: A next-generation seismic velocity model building method

The GPR (Ground Penetrating Radar) conference in Hong Kong year 2016 marked the 30th anniversary of the initial meeting in Tifton, Georgia, USA on 1986. The conference has been being a bi-annual event and has been hosted by sixteen cities from four continents. Throughout these 30 years, researchers and practitioners witnessed the analog paper printout to digital era that enables very efficient collection, processing and 3D imaging of large amount of data required in GPR imaging in infrastructure. GPR has systematically progressed forward from “Locating and Testing” to “Imaging and Diagnosis” with the Holy Grail of ’Seeing the unseen’ becoming a reality. This paper reviews the latest development of the GPR’s primary infrastructure applications, namely buildings, pavements, bridges, tunnel liners, geotechnical and buried utilities. We review both the ability to assess structure as built character and the ability to indicate the state of deterioration. Finally, we outline the path to a more rigorous development in terms of standardization, accreditation, and procurement policy.

A review of Ground Penetrating Radar application in civil engineering: A 30-year journey from Locating and Testing to Imaging and Diagnosis

Full waveform inversion using the conventional L2 distance to measure the misfit between seismograms is known to suffer from cycle skipping. An alternative strategy is proposed in this study, based on a measure of the misfit computed with an optimal transport distance. This measure allows to account for the lateral coherency of events within the seismograms, instead of considering each seismic trace independently, as is done generally in full waveform inversion. The computation of this optimal transport distance relies on a particular mathematical formulation allowing for the non-conservation of the total energy between seismograms. The numerical solution of the optimal transport problem is performed using proximal splitting techniques. Three synthetic case studies are investigated using this strategy: the Marmousi 2 model, the BP 2004 salt model, and the Chevron 2014 benchmark data. The results emphasize interesting properties of the optimal transport distance. The associated misfit function is less prone to cycle skipping. A workflow is designed to reconstruct accurately the salt structures in the BP 2004 model, starting from an initial model containing no information about these structures. A high-resolution P-wave velocity estimation is built from the Chevron 2014 benchmark data, following a frequency continuation strategy. This estimation explains accurately the data. Using the same workflow, full waveform inversion based on the L2 distance converges towards a local minimum. These results yield encouraging perspectives regarding the use of the optimal transport distance for full waveform inversion: the sensitivity to the accuracy of the initial model is reduced, the reconstruction of complex salt structure is made possible, the method is robust to noise, and the interpretation of seismic data dominated by reflections is enhanced.

/pdf/measuring-the-misfit-between-seismograms-using-an-optimal-2z5yytg0qq.pdf

Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion

Full waveform inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield This method consists in minimizing the distance between

/pdf/full-waveform-inversion-and-the-truncated-newton-method-3ipjdovvf6.pdf

Full Waveform Inversion and the Truncated Newton Method

The use of optimal transport distance has recently yielded significant progress in image processing for pattern recognition, shape identification, and histograms matching. In this study, the use of this distance is investigated for a seismic tomography problem exploiting the complete waveform; the full waveform inversion. In its conventional formulation, this high resolution seismic imaging method is based on the minimization of the L2 distance between predicted and observed data. Application of this method is generally hampered by the local minima of the associated L2 misfit function, which correspond to velocity models matching the data up to one or several phase shifts. Conversely, the optimal transport distance appears as a more suitable tool to compare the misfit between oscillatory signals, for its ability to detect shifted patterns. However, its application to the full waveform inversion is not straightforward, as the mass conservation between the compared data cannot be guaranteed, a crucial assumption for optimal transport. In this study, the use of a distance based on the Kantorovich–Rubinstein norm is introduced to overcome this difficulty. Its mathematical link with the optimal transport distance is made clear. An efficient numerical strategy for its computation, based on a proximal splitting technique, is introduced. We demonstrate that each iteration of the corresponding algorithm requires solving the Poisson equation, for which fast solvers can be used, relying either on the fast Fourier transform or on multigrid techniques. The development of this numerical method make possible applications to industrial scale data, involving tenths of millions of discrete unknowns. The results we obtain on such large scale synthetic data illustrate the potentialities of the optimal transport for seismic imaging. Starting from crude initial velocity models, optimal transport based inversion yields significantly better velocity reconstructions than those based on the L2 distance, in 2D and 3D contexts.

/pdf/an-optimal-transport-approach-for-seismic-tomography-2w9uymx03x.pdf

An optimal transport approach for seismic tomography: application to 3D full waveform inversion

Full waveform inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing the distance between predicted and recorded data. The predicted data are computed as the solution of a wave-propagation problem. Conventional numerical methods for the solution of FWI problems are gradient-based methods, such as the preconditioned steepest descent, the nonlinear conjugate gradient, or more recently the $l$-BFGS quasi-Newton algorithm. In this study, we investigate the desirability of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves. For multiparameter reconstruction, the inverse Hessian operator also offers the possibility of better removing trade-offs due to coupling effects between parameter classes. The truncated Newto...

/pdf/full-waveform-inversion-and-the-truncated-newton-method-2eabz33atr.pdf

Ludovic Métivier

Papers

A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice

Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion

Full Waveform Inversion and the Truncated Newton Method

An optimal transport approach for seismic tomography: application to 3D full waveform inversion

Full Waveform Inversion and the Truncated Newton Method