Acceleration strategies for elastic full waveform inversion workflows in 2D and 3D

Full waveform inversion (FWI) is one of the most challenging procedures to obtain quantitative information of the subsurface. For elastic inversions, when both compressional and shear velocities have to be inverted, the algorithmic issue becomes also a computational challenge due to the high cost related to modelling elastic rather than acoustic waves. This shortcoming has been moderately mitigated by using high-performance computing to accelerate 3D elastic FWI kernels. Nevertheless, there is room in the FWI workflows for obtaining large speedups at the cost of proper grid pre-processing and data decimation techniques. In the present work, we show how by making full use of frequency-adapted grids, composite shot lists and a novel dynamic offset control strategy, we can reduce by several orders of magnitude the compute time while improving the convergence of the method in the studied cases, regardless of the forward and adjoint compute kernels used.The authors thank REPSOL for the permission to publish the present research and for funding through the AURORA project. J. Kormann also thankfully acknowledges the computer resources, technical expertise and assistance provided by the Barcelona Supercomputing Center - Centro Nacional de Supercomputacti ´on together with the Spanish Supercomputing Network (RES) through grant FI-2014-2-0009. This project has received funding from the European Union’s Horizon 2020, research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 644202. The research leading to these results has received funding from the European Union’s Horizon 2020 Programme (2014–2020) and from the Brazilian Ministry of Science, Technology and Innovation through Rede Nacional de Pesquisa (RNP) under the HPC4E Project (www.hpc4e.eu), grant agreement no. 689772.We further want to thank the Editor Clint N. Dawson for his help, and Andreas Fichtner and an anonymous reviewer for their comments and suggestions to improve the manuscript.Peer ReviewedPostprint (published version

Geometric and Level Set Tomography using Ensemble Kalman Inversion

Tomography is one of the cornerstones of geophysics, enabling detailed spatial descriptions of otherwise invisible processes. However, due to the fundamental ill-posedness of tomography problems, the choice of parametrizations and regularizations for inversion significantly affect the result. Parametrizations for geophysical tomography typically reflect the mathematical structure of the inverse problem. We propose, instead, to parametrize the tomographic inverse problem using a geologically motivated approach. We build a model from explicit geological units that reflect the a priori knowledge of the problem. To solve the resulting large-scale nonlinear inverse problem, we employ the efficient Ensemble Kalman Inversion scheme, a highly parallelizable, iteratively regularizing optimizer that uses the ensemble Kalman filter to perform a derivative-free approximation of the general iteratively regularized Levenberg–Marquardt method. The combination of a model specification framework that explicitly encodes geological structure and a robust, derivative-free optimizer enables the solution of complex inverse problems involving non-differentiable forward solvers and significant a priori knowledge. We illustrate the model specification framework using synthetic and real data examples of near-surface seismic tomography using the factored eikonal fast marching method as a forward solver for first arrival traveltimes. The geometrical and level set framework allows us to describe geophysical hypotheses in concrete terms, and then optimize and test these hypotheses, helping us to answer targeted geophysical questions

Lift and Relax for PDE-constrained inverse problems in seismic imaging

We present Lift and Relax for Waveform Inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the set of variables in the optimization problem to products of those variables, arranged as a moment matrix. This algebraic idea is a celebrated way to replace a hard polynomial optimization problem by a semidefinite programming approximation. Concretely, both the model and the wavefield are lifted from vectors to rank-2 matrices. The second technique (Relax) invites to consider the wave equation, not as a hard constraint, but as a soft constraint to be satisfied only approximately - a technique known as wavefield reconstruction inversion (WRI). WRI weakens wave-equation constraints by introducing wave-equation misfits as a weighted penalty term in the objective function. The relaxed penalty formulation enables balancing the data and wave-equation misfits by tuning a penalty parameter. Together, Lift and Relax help reformulate the inverse problem as a set of constraints on a rank-2 moment matrix in a higher dimensional space. Such a lifting strategy permits a good data and wave-equation fit throughout the inversion process, while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Numerical examples indicate that compared to FWI and WRI, LRWI can conduct successful inversions using an initial model that would be considered too poor, and data with a starting frequency that would be considered too high, for either method in isolation. Specifically, LRWI increases the acceptable starting frequency from 1.0 Hz and 0.5 Hz to 2.0 Hz and 2.5 for the Marmousi model and the Overthrust model, respectively, in the cases of a linear gradient starting model

Reverse-time migration-based reflection tomography using teleseismic free surface multiples

Converted and multiply reflected phases from teleseismic events are routinely used to create structural images of the crust–mantle boundary (Moho) and the elasticity contrasts within the crust and upper mantle. The accuracy of these images is to a large extent determined by the background velocity model used to propagate these phases to depth. In order to improve estimates of 3-D velocity variations and, hence, improve imaging, we develop a method of reverse-time migration-based reflection tomography for use with wavefields from teleseismic earthquakes recorded at broad-band seismograph arrays. Reflection tomography makes use of data redundancy—that is, the ability to generate numerous structural images of the subsurface with different parts of the wavefield. In exploration seismology (where it is known as migration velocity analysis) reflection tomography typically involves the generation of an extended image (e.g. offset- or angle-gathers), and the fitness of the background model is evaluated through the application of image-domain annihilators. In regional-scale passive source seismology, however, annihilation-based methods are inadequate because the sparse and irregular distribution of teleseismic sources is not likely to produce illumination over a sufficient range of angles. To overcome this problem we turn towards a source-indexed moveout scheme. Instead of extended image annihilation, we determine the success of the tomographic velocity model by cross correlating images produced with multiply scattered waves from different teleseismic sources. The optimal velocity model is the one that minimizes correlation power between windowed images away from zero depth shift. We base our inversion scheme on the seismic adjoint method and a conjugate gradient solver. For each image pair, the update direction is determined by correlations between downgoing wavefields with upgoing adjoint wavefields for both images. The sensitivity kernels used in this method is similar to those found in other forms of adjoint tomography, but their shapes are controlled by the spatial distribution of the error function. We present the method and a proof-of-concept with 2-D synthetic data

Nonlinear Waveform Inversion with Surface-Oriented Extended Modeling

This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/102263This thesis investigates surface-oriented model extension approach to nonlinear full waveform inversion (FWI). Conventional least-squares (LS) approach is capable of reconstructing highly detailed models of subsurface. Resolution requirements of the realistic problems dictate the use of local descent methods to solve the LS optimization problem. However, in the setting of any characteristic seismic problem, LS objective functional has numerous local extrema, rendering descent methods unsuitable when initial estimate is not kinematically accurate. The aim of my work is to improve convexity properties of the objective functional. I use the extended modeling approach, and construct an extended optimization functional incorporating differential semblance condition. An important advantage of surface-oriented extensions is that they do not increase the computational complexity of the forward modeling. This approach blends FWI technique with migration velocity analysis (MVA) capability to recover long scale velocity model, producing optimization problems that combine global convergence properties of the MVA with data fitting approach of FWI. In particular, it takes into account nonlinear physical effects, such as multiple reflections. I employ variable projection approach to solve the extended optimization problem. I validate the method on synthetic models for the constant density acoustics problem