Multiparameter viscoelastic full waveform inversion of shallow seismic surface waves with a preconditioned truncated-Newton method

2D full waveform inversion (FWI) of shallow seismic Rayleigh waves has become a powerful method for reconstructing viscoelastic multiparameter models of shallow subsurface with high resolution. The multiparameter reconstruction in FWI is challenging due to the potential presence of crosstalk between different parameters and the unbalanced sensitivity of Rayleigh-wave data with respect to different parameter classes. Accounting for the inverse Hessian using truncated Newton methods based on second-order adjoint methods provides as an effective tool to mitigate crosstalk caused by the coupling between different parameters. In this study, we apply a preconditioned truncated Newton method (PTN) to shallow-seismic FWI to simultaneously invert for multiparameters near-surface models (P- and S-wave velocities, attenuation of P and S waves, and density). We firstly investigate scattered wavefields caused by these parameters to evaluate the coupling between them. Then we investigate the performance of PTN on shallow-seismic FWI of Rayleigh wave for reconstructing all five parameters simultaneously. The application to spatially correlated and uncorrelated models demonstrate that PTN helps to mitigate the crosstalk and improves the resolution of the multiparameter reconstructions, especially for the weak parameters with small sensitivity such as attenuation and density parameters. The comparison with the classical preconditioned conjugate gradient method highlights the improved performance of PTN and thus the benefit of accounting for the information included in the Hessian

Visco-acoustic full waveform seismic inversion: from a DG forward solver to a Newton-CG inverse solver

In this paper we present a holistic framework for full waveform inversion (FWI) in the visco-acoustic regime. FWI entails the reconstruction of material parameters (such as density and sound speed) from measurements of reflected wave fields (seismograms). We derive a discontinuous Galerkin (DG) solver for the visco-acoustic wave equation and incorporate it into an inverse solver. For the DG discretization we provide a block diagonal preconditioner for the efficient computation of the time steps by GMRES which yields a convergence estimate in space and time. Numerical tests illustrate these results. Furthermore, we set up an inverse solver of well established Newton-CG type, and we express the required Fréchet derivative and its adjoint in the DG setting. Reconstructions from simulated cross-well seismograms highlight the challenges of FWI and demonstrate the performance of the scheme. Some of the inversion experiments use seismograms generated by an independent FDTD forward solver to avoid an inverse crime

3D elastic time-frequency full-waveform inversion

In this work I present the theory and implementation of a 3D elastic full waveform inversion. The implementation is based on the adjoint-state method using a time-frequency approach. Synthetic applications in transmission acquisition geometry show the ability of the method to reconstruct elastic parameters of different 3D subsurface structures. Optimisations with the diagonal Hessian matrix for preconditioning and the L-BFGS scheme enable the inversion of more complex surface geometry data

Inner product preconditioned trust-region methods for frequency-domain full waveform inversion

Full waveform inversion is a seismic imaging method which requires to solve a large-scale minimization problem, typically through local optimization techniques. Most local optimization methods can basically be built up from two choices: the update direction and the strategy to control its length. In the context of full waveform inversion, this strategy is very often a line search. We here propose to use instead a trust-region method, in combination with non-standard inner products which act as preconditioners. More specifically, a line search and several trust-region variants of the steepest descent, the limited memory BFGS algorithm and the inexact Newton method are presented and compared. A strong emphasis is given to the inner product choice. For example, its link with preconditioning the update direction and its implication in the trust-region constraint are highlighted. A first numerical test is performed on a 2D synthetic model then a second configuration, containing two close reflectors, is studied. The latter configuration is known to be challenging because of multiple reflections. Based on these two case studies, the importance of an appropriate inner product choice is highlighted and the best trust-region method is selected and compared to the line search method. In particular we were able to demonstrate that using an appropriate inner product greatly improves the convergence of all the presented methods and that inexact Newton methods should be combined with trust-region methods to increase their convergence speed

Strategies for visco-acoustic waveform inversion in the Laplace-Fourier domain, with application to the Nankai subduction zone

Waveform inversion is a non-linear and ill-posed inverse problem, with the objective of utilizing the full information content of recorded seismic waveforms. A Laplace-Fourier domain implementation allows a natural `multiscale\u27 approach that mitigates the non-linearity and ill-posedness by inverting low-frequency, early arrival data in the initial stages of inversion. High-frequency components, and late arrivals are incorporated at a later stage. This allows the development of robust inversion strategies capable of handling large wide-angle crustal surveys, leading to reliable, high-resolution velocity and attenuation models of crustal structures. I apply waveform inversion to extract a P-wave velocity model of the active megasplay fault system in the seismogenic Nankai subduction zone offshore Japan, using controlled-source Ocean Bottom Seismograph data. The resulting velocity model includes detailed thrust structures, and low velocity zones not previously identified. The connection of large low-velocity zones in the inner and outer wedge suggests a significant distribution of overpressured regions in the vicinity of the megasplay fault, with the potential to strongly influence coseismic rupture propagation. I identify six-fold key strategies for successful waveform inversion; i) the availability of low-frequency and long offset data, ii) a highly accurate starting model, iii) a hierarchical approach in which phase spectra are inverted first, and amplitude information is only incorporated in the final stages, iv) a Laplace-Fourier approach, v) careful preconditioning of the gradient, vi) strategies for source estimation. Chequerboard tests and point-scatter tests demonstrate the resolution and the limitations of the acoustic implementation. I also compare four misfit functionals for optimization, and demonstrate that velocity information may be reliably extracted from phase alone, and that amplitude information is secondary in updating the velocity model. Finally I develop inversion strategies for retrieving both velocity and attenuation models. Cross-talk between these two classes of parameter estimates arises from the lack of parameter scaling in the gradient of the objective function, and primarily affects the attenuation model. I show the cross-talk can be suppressed by the combination of an appropriate attenuation damping parameter, and by the use of smoothing constraints. Initial velocity-only inversions also help in reducing the effects of cross-talk in subsequent velocity-attenuation inversion

2D multi-parameter viscoelastic shallow-seismic full waveform inversion: reconstruction tests and first field-data application

2D full-waveform inversion (FWI) of shallow-seismic wavefields has recently become a novel way to reconstruct S-wave velocity models of the shallow subsurface with high vertical and lateral resolution. In most applications, seismic wave attenuation is ignored or considered as a passive modelling parameter only. In this study, we explore the feasibility and performance of multi-parameter viscoelastic 2D FWI in which seismic velocities and attenuation of P- and S-waves, respectively, and mass density are inverted simultaneously. Synthetic reconstruction experiments reveal that multiple crosstalks between all viscoelastic material parameters may occur. The reconstruction of S-wave velocity is always robust and of high quality. The parameters P-wave velocity and density exhibit weaker sensitivity and can be reconstructed more reliably by multi-parameter viscoelastic FWI. Anomalies in S-wave attenuation can be recovered but with limited resolution. In a field data application, a small-scale refilled trench is nicely delineated as a low P- and S-wave velocity anomaly. The reconstruction of P-wave velocity is improved by the simultaneous inversion of attenuation. The reconstructed S-wave attenuation reveals higher attenuation in the shallow weathering zone and weaker attenuation below. The variations in the reconstructed P- and S-wave velocity models are consistent with the reflectivity observed in a GPR profile

Enhancement of In-Plane Seismic Full Waveform Inversion with CPU and GPU Parallelization

Full waveform inversion is a widely used technique to estimate the subsurface parameters with the help of seismic measurements on the surface. Due to the amount of data, model size and non-linear iterative procedures, the numerical computation of Full Waveform Inversion are computationally intensive and time-consuming. This paper addresses the parallel computation of seismic full waveform inversion with Graphical Processing Units. Seismic full-waveform inversion of in-plane wave propagation in the finite difference method is presented here. The stress velocity formulation of the wave equation in the time domain is used. A four nodded staggered grid finite-difference method is applied to solve the equation, and the perfectly matched layers are considered to satisfy Sommerfeld’s radiation condition at infinity. The gradient descent method with conjugate gradient method is used for adjoined modelling in full-waveform inversion. The host code is written in C++, and parallel computation codes are written in CUDA C. The computational time and performance gained from CUDA C and OpenMP parallel computation in different hardware are compared to the serial code. The performance improvement is enhanced with increased model dimensions and remains almost constant after a certain threshold. A GPU performance gain of up to 90 times is obtained compared to the serial code

Probabilistic waveform inversion: Quest for the law

Full-waveform inversion (FWI) is an algorithm (and a part of the measuring procedure in a wide sense) with the aim to find the governing law of a physical system using the partially measured physical fields with limited computational resources. A law is a forward theory equipped with the model parameters and the data parameters. The main characteristic of the law is the realizability assumption: the law explains all subsets of the measured data parameters and predicts all subsets of the unmeasured (in the given experiment) data parameters. To find the law, we have to guess a law (a forward theory and parametrization), measure some data parameters and check the realizability assumption. To put it more precisely, I formulate a new probabilistic setting for inverse problems and full-waveform inversion. Instead of using the Bayes\u27 theorem, the Tarantola-Valette conjunction or the principle of maximum entropy based on the prior information for the averaged quantities, I propose a principle of minimum relative information using the prior information for the non-averaged quantities. The Tarantola-Valette formula is obtained as a special case under the assumption that the theoretical and prior measures exist. Using the realizability assumption as a prior information, the principle of minimum relative information provides the parametric probabilistic solution with the arbitrary misfit functions. Maximization of the parametric probabilistic solution leads to a multiobjective minimization problem. All global Pareto optima are the sample points of the probabilistic solution with the highest values of the volumetric measure. Unfortunately, even a local multiobjective minimization problem is computationally intractable for FWI with many millions of model parameters. To make it computationally attractive for large-scale FWI and to find at least a few local solutions of the multiobjective minimization problem, I implement the bilevel multiobjective waveform inversion (BMWI) using a single randomly chosen shot gather at each iteration. BMWI is a stochastic, nested algorithm with an adaptive parabolic line search and multiscale strategy. The computational cost per iteration is five forward modellings only. BMWI can worsen some of the single-shot misfit functions and the different random runs of BMWI converge to different points in the model manifold. I interpret these inverted models as the sample points of the probabilistic solution. I estimate the solution, uncertainty and sensitivity using the sample estimates of the mean, standard deviation and initial deviation of the sample points, respectively. Using the numerical examples with the Marmousi-2 model, I illustrate the potential of BMWI for automatic uncertainty and sensitivity analysis with just two-three sample points. To test the idea with real-world data, I apply stochastic single-shot BMWI in a 2D acoustic finite-difference approximation to a 2D line of pressure data acquired in a shallow-water river delta with ocean bottom cables. I use minimal data preprocessing (only a new 3D-to-2D transform which is strictly valid in a linear-gradient medium), the linear gradient starting models and the diagonal preconditioners with a negligible regularization. I estimate the theoretical uncertainties due to the neglected 3D effects using the 3D-to-2D transforms. The uncertainties estimated by the random sequences of BMWI are higher than the uncertainties related to the 3D-to-2D transforms. I provide the estimates of the solution, uncertainty and sensitivity using up to fourteen sample points inverted with the different linear-gradient starting models, the differently 3D-to-2D-transformed real data sets and the different random sequences of descent directions. The uncertainty of sound velocities is the lowest in the central semicircle with the radius 3 km equal to half the length of the ocean bottom cable. The uncertainty of mass densities is the highest in the central semicircle. The sensitivity of the measuring procedure with respect to sound velocity and mass density is the highest in the central semicircle representing a footprint of the acquisition geometry. Outside the central semicircle the parameters are not falsifiable in the specified setting. Full-waveform inversion is the quest for the unique governing law of the physical system under study. If the governing law is deterministic and the sample mean, standard deviation and initial deviation of the sample points represent the insufficient description of the solution, uncertainty and sensitivity, then the measuring procedure in a wide sense has to be improved