## ICIAM 2011: Industrial Minisymposium: Contemporary Issues in Geophysical Inversion

**November 15, 2011**

**William W. Symes**

ICIAM 2011 featured several industry-focused minisymposia, two of which ("Finance and Risk Management" and "Imaging and Inversion") consisted of six sessions each extending over three days. This article describes the second of these, the official title of which was "Contemporary Issues in Geophysical Inversion." The organizers of this well-attended minisymposium were Felix Herrmann (University of British Columbia), Laurent Demanet (MIT), Timo Betcke (University of Reading), Maarten de Hoop (Purdue University), and the author. Inversion has become an extremely active topic of research among industrial and academic geoscientists, with many of the remaining core issues being mathematical or computational. This minisymposium provided an excellent introduction to some of the most exciting contemporary work on the subject.

In the geophysical context, "inversion" means recovery of a mechanical description (material density, elasticities, electromagnetic properties, . . . , as functions of position) of the earth's interior from measurements of auxiliary fields (seismic, electromagnetic, gravitational . . .) made near the surface. The highest spatial resolution is usually obtained from seismic measurements. Accordingly, seismology is the dominant exploration technology used to reveal the detailed structure of the earth at many scales: meters for civil and environmental engineering applications, kilometers for oil and gas prospecting, hundreds of kilometers for academic studies of the deep earth. Driven by the economic needs and resources of the oil industry, this technology has undergone dramatic changes in the last ten years: Images of the interior are now routinely produced in 3D, replacing the 2D slices that were the dominant end-result for the previous 30 years, and the volume of data has grown by orders of magnitude to support the enhanced physical fidelity of geophysical estimation.

In principle, seismic waves can be described to good approximation as solutions of the elastic wave equation or its anelastic extensions, in which the mechanical parameters mentioned above appear as coefficients. Thus, the seismic inversion task could be posed as a parameter identification problem for the wave equation. This point of view was first studied seriously in the 1980s; one very vocal proponent was Albert Tarantola (1949–2009), whose seminal book *Inverse Problem Theory *was recently republished by SIAM. The research of that era did not conclusively show an advantage of the "inverse" point of view over more conventional imaging techniques, which could be viewed as partial solutions based on drastic approximations to the physics of seismic wave propagation. In retrospect, the critical impediments seem clear (and were described by Tarantola and others): First, 3D computations were simply out of reach at that time, and viewing the earth as a 2D structure is an overwhelming error. Second, the relation between the elastic parameters, especially wave velocities, and the seismic wavefield is very nonlinear. Given the computational size of these problems, rapidly convergent Newton-like data-fitting algorithms have been the most favored approach, but these tend to fail (by becoming trapped at spurious stationary points) for extremely nonlinear objectives, such as those occurring in seismic inversion.

Over the last several years, computer hardware and software advances have begun to bring 3D inversion computations within reach. Several early field-scale evaluations have been spectacularly successful, appearing to justify the earlier optimism of Tarantola and others, and prompting a great deal of excitement in the industrial and academic geophysics communities. The speakers in this minisymposium provided a review of the state of the art and of prospective developments in this fast-moving field.

In hour-long talks, Wim Mulder of Shell Research and Omar Ghattas of the University of Texas–Austin kicked off the minisymposium with overviews of seismic inversion from industrial and academic perspectives. Mulder reviewed some of Shell's compelling 3D inversion exercises, and described a variety of attacks on the central issue of velocity-driven nonlinearity. The central observation in all of these ideas is that velocity information is inherent in cross-correlations between wavefields, an idea inherited from conventional seismic processing. Ghattas described the development of an extensive software infrastructure for inversion, encompassing innovations in simulation, optimization, and parallel computation. His group has recently produced some intriguing early results in inversion of whole-earth model data.

It is almost too early to discuss the practice of inversion, but not quite. The second session, titled "Inversion in Practice," included a talk by René-Edouard Plessix of Shell Global Solutions, who described in detail several of the publicly disclosed field inversions conducted by his organization. The improvements over older techniques were clear-cut and compelling. Several other speakers in this session discussed important aspects of applicable inversion techniques. Sam Gray of CGG-Veritas contrasted one-way wave propagation methods for downgoing waves, which dominated seismic processing for two decades, with outgoing one-way and two-way (full wave equation solution) methods, which have recently prevailed. Jean Virieux of Université Joseph Fourier reviewed methodology and examples for Fourier domain (Helmholtz)-based inversion methods, emphasizing the value of a hierarchical or continuation approach in time, space, and frequency. Oliver Dorn of the University of Manchester, in the only non-seismic talk of the minisymposium, described active-source electromagnetic inversion--a recently developed complement to exploration seismology.

The "classic" methods of exploration seismology are based on linearization and separation of scales, and these approximations still have much to teach us--indeed, the striking success of the seismic industry is direct evidence that they have great value. The third session centered on the use of these approximations. Paul Sava (Colorado School of Mines) and Tristan van Leeuven (UBC) discussed novel aspects of velocity analysis, that is, the estimation of the background velocity model, about which linearization takes place--the key issue, as it regulates the relation between time (data) and distance (earth model)! Mauricio Sacchi (University of Alberta) and Evren Yarman (WesternGeco–Schlumberger) described two rather different ways in which seismic images (approximate solutions of linearized inverse problems) can be compressed, or conditioned, to improve approximation and reduce the influence of noise.

The very nonlinear dependence of the seismic wavefield on wave velocity is the central difficulty of this subject: Once velocity is determined with accuracy, the remaining parameter identification problem is nearly linear (though generally ill-conditioned). The fourth session presented a number of approaches to ameliorating the nonlinearity. Paul Childs of Schlumberger Cambridge Research reviewed a variety of approaches and presented some evidence for their relative efficacy. Plessix, of Shell Global Solutions, described the use of spatial coordinate transformations to reduce the degree of nonlinearity. Dong Sun and the author, both of Rice University, explained related approaches motivated by conventional "velocity analysis" techniques, but extended to full wavefield simulation. All of these approaches show some promise.

Speakers in the last two sessions discussed numerical mathematics relevant to seismic inversion. Laurent Demanet and Christiaan Stolk (University of Amsterdam) applied numerical harmonic analysis to the solution of the linearized inverse problem, and to exploiting the sparseness of typical wavefields in these representations. These techniques, with their use of specially adapted localized and oscillatory frames to represent wavefields, are a natural and exciting development. Felix Herrmann used similar ideas, but in the context of the frequency domain (Helmholtz equation) formulation of the inverse problem. This formulation was also the subject of Lingyun Qiu's (Purdue) talk. Robert Ferguson (University of Calgary) took numerical harmonic analysis in a somewhat different direction, constructing efficient approximate propagators for downward wave propagation. Henri Calandra (Total) described the remarkable results achieved by the group of Changsoo Shin (Seoul National University), in which the Laplace transform of data overcomes, at least to some extent, the difficulties in data fitting arising from velocity nonlinearity. Johannes Huber (University of Basel) laid out a comprehensive approach to simulation and data fitting, using interior point optimization to incorporate regularizing constraints. Aleksandr Aravkin (UBC) compared various error measures related to assumed statistical distributions of data errors, and proposed Student's t-distribution as the source of an appropriate robust error measure.

These extremely brief descriptions cannot possibly do justice to the depth and scope of the material presented in these sessions, which gave the audience a comprehensive overview of a very lively and rapidly developing topic of great economic and scientific importance. Geophysical in-version will surely be ready for another extensive review at ICIAM 2015!

*William Symes is the Noah Harding Professor in the Department of Computational and Applied Mathematics and a professor of earth science at Rice University, where he is director of the Rice Inversion Project, a university–industry consortium for research on seismic inversion*.