Machine Learning at Scale for Full Waveform Inversion at PGS

Seismic exploration and imaging in the ultra-deep Gulf of Mexico is incredibly challenging: The hydrocarbon-bearing sedimentary rocks are deep and buried under evaporite salt domes and geological layers that come in complex shapes and feature strong acoustic impedance and velocity contrasts. Proper imaging requires application of advanced velocity model-building techniques to address not only velocity variations, but also seismic velocity anisotropy. Then wave propagation must be performed with numerical schemes that accurately model reflections, refractions and even converted waves. While such features are known (and advanced) textbook examples of wave propagation in complex media, computing them requires massive supercomputer resources and clever algorithms.

It is for that reason that PGS — a Norwegian seismic exploration company with a major presence in Houston — acquired Abel a few years ago. Abel is a Cray® XC40™ system, at one point the largest commercial supercomputer on the Top500 list and still #16 today. PGS can hold one of the largest offshore seismic surveys ever in Abel’s memory — the Triton survey in the ultra-deep Gulf of Mexico, close to a petabyte in processed size. By leveraging Abel’s fast interconnect, PGS is quickly bringing into production a new class of algorithms. First among these is an enhanced and improved reverse time migration (RTM) imaging algorithm that uses the full wave field in imaging, not just a synthesized source (snapshots). PGS is also using Abel to process very long, continuous records with energy from multiple sources, bucking long-standing industry practice. All this was amply demonstrated by PGS at the latest Society of Exploration Geophysicists’ (SEG) 86th annual meeting this October in Dallas.

“But,” says Dr. Sverre Brandsberg-Dahl, global chief geophysicist for imaging and engineering at PGS, “imaging is only part of the problem; using these images to calculate the subsurface’s rocks’ acoustic and elastic properties — a process known in the industry as Full Waveform Inversion — is equally challenging; if not more so, because it’s a multi-dimensional, ill-posed optimization problem that is far from automated and requires lots of skilled resources’ intervention — sometimes more art than science in many cases. So, we decided to have Abel take a crack at it by using analytical machine learning techniques.” (By the way, of Niels Henrik Abel, the Norwegian mathematician after whom the supercomputer is named, it was said at a young age that he would “discover Magellanian thoroughfares to large portions of a vast analytical ocean” — so perhaps this development was pre-ordained). With the help of research from Prof. M. de Hoop at Rice University, PGS applied machine learning optimization techniques such as regularization and steering to determine the velocity model of a well established, challenging benchmark synthetic data set — the BP velocity estimation benchmark. It is the essence of machine learning that such model building is as automated as possible and that a computer, in this case Abel, can find hidden and unexpected insights in complex data with as little explicit  (programming) help as possible.

The results were impressive and much better than expected: Starting from a very simple initial model, Abel learned how to best steer refracted and diving waves for deep model updates and reproduce the sharp salt boundaries typical in the Gulf of Mexico. This is illustrated by two figures and a short movie, provided to us courtesy of PGS. In the top half of Figure 2 is shown the exact velocity model, as a function of depth, in a 2-D slice of the subsurface. Colors are representative of sound velocities; dark red is indicative of very large velocities as would be found in salt bodies and diapirs. On average, velocities increase as a function of depth, but there are very noticeable inversions of that profile and sharp and irregular lateral variations. At the bottom of that figure is the starting initial velocity model, a very smooth velocity profile.

Figure 1 True and starting velocity model for the BP velocity estimation benchmark

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The hope is that the FWI algorithm will take this velocity model and quickly converge to the true model by some sort of a least squares fit. In “conventional” FWI, this does indeed occur with some level of success — see the next figure — but many artifacts can be observed.

Figure 2 True velocity model versus the one obtained in current FWI calculations

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In particular, the areas underneath the salt are poorly modeled, there is ringing, and the velocity inversion profile is overdone.

However, the movie below (double click on it) shows how the PGS machine learning approach dramatically improves the quality of the resulting model. It shows how the Cray supercomputer “learns” the velocities for a substantially clearer final image. Most, if not all, of the artifacts of the conventional FWI results have been removed.

 

For aficionados of machine learning, we’ve thrown in the mathematical expression of the misfit function in terms of its model parameters and weights.

Figure 3 PGS’ machine learning approach to FWI. The model m is to be obtained via constrained minimization of a regularized and steered misfit function given data d and a reference model mr.

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Where does this leave us? There now clearly seems to be a promising path towards applying machine learning technologies to some of the hardest problems in seismic exploration. In fact, further thought could well lead to the conviction that machine learning is the only path forward to “solving” FWI. Indeed, the data acquired in seismic exploration is quite complex, much more so than the usual pixel data. The model is multidimensional; the subsurface is divided into billions of cells and the rock properties at each of those cells are given in terms of 4th order tensors with 21 independent entries. Poroelasticity and non-linear wave propagation effects will further complicate any direct inversion effort. It is all the more gratifying to see how supercomputers, typically used in challenging “direct physics” modeling and simulation, are equally well positioned to tackle very large machine learning problems.

 

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