Extended Model Space Full Waveform Inversion via Minimization of Virtual Scattering Source

2019 ◽  
Author(s):  
D. Lee ◽  
S. Pyun
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Extended Model ◽  
Full Waveform Inversion ◽  
Full Waveform

Image-guided sparse-model full waveform inversion

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. R189-R198 ◽  
Author(s):  
Yong Ma ◽  
Dave Hale ◽  
Bin Gong ◽  
Zhaobo (Joe) Meng
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Widespread Application ◽  
Low Frequencies ◽  
Sparse Model ◽  
Image Guided ◽  
Full Waveform ◽  
Recorded Data

Multiple problems, including high computational cost, spurious local minima, and solutions with no geologic sense, have prevented widespread application of full waveform inversion (FWI), especially FWI of seismic reflections. These problems are fundamentally related to a large number of model parameters and to the absence of low frequencies in recorded seismograms. Instead of inverting for all the parameters in a dense model, image-guided full waveform inversion inverts for a sparse model space that contains far fewer parameters. We represent a model with a sparse set of values, and from these values, we use image-guided interpolation (IGI) and its adjoint operator to compute finely and uniformly sampled models that can fit recorded data in FWI. Because of this sparse representation, image-guided FWI updates more blocky models, and this blockiness in the model space mitigates the absence of low frequencies in recorded data. Moreover, IGI honors imaged structures, so image-guided FWI built in this way yields models that are geologically sensible.

scholarly journals Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 45
Author(s):  
Marwan Charara ◽  
Christophe Barnes
Keyword(s):  
Inverse Problem ◽  
Spatial Correlation ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Model Space ◽  
Elastic Model ◽  
Full Waveform Inversion ◽  
Spatial Correlations ◽  
Full Waveform

Full-waveform inversion for borehole seismic data is an ill-posed problem and constraining the problem is crucial. Constraints can be imposed on the data and model space through covariance matrices. Usually, they are set to a diagonal matrix. For the data space, signal polarization information can be used to evaluate the data uncertainties. The inversion forces the synthetic data to fit the polarization of observed data. A synthetic inversion for a 2D-2C data estimating a 1D elastic model shows a clear improvement, especially at the level of the receivers. For the model space, horizontal and vertical spatial correlations using a Laplace distribution can be used to fill the model space covariance matrix. This approach reduces the degree of freedom of the inverse problem, which can be quantitatively evaluated. Strong horizontal spatial correlation distances favor a tabular geological model whenever it does not contradict the data. The relaxation of the spatial correlation distances from large to small during the iterative inversion process allows the recovery of geological objects of the same size, which regularizes the inverse problem. Synthetic constrained and unconstrained inversions for 2D-2C crosswell data show the clear improvement of the inversion results when constraints are used.

Seismic full-waveform inversion using minimization of virtual scattering sources

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R299-R311
Author(s):  
Donguk Lee ◽  
Sukjoon Pyun
Keyword(s):  
Newton Method ◽  
Scattering Problem ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Simulated Data ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Velocity Inversion ◽  
Full Waveform

Full-waveform inversion (FWI) is a powerful tool for imaging underground structures with high resolution; however, this approach commonly suffers from the cycle-skipping issue. Recently, various FWI methods have been suggested to address this problem. Such methods are mainly classified into either data-space manipulation or model-space extension. We developed an alternative FWI method that belongs to the latter class. First, we define the virtual scattering source based on perturbation theory. The virtual scattering source is estimated by minimizing the differences between observed and simulated data with a regularization term penalizing the weighted virtual scattering source. The inverse problem for obtaining the virtual scattering source can be solved by the linear conjugate gradient method. The inverted virtual scattering source is used to update the wavefields; thus, it helps FWI to better approximate the nonlinearity of the inverse scattering problem. As the second step, the virtual scattering source is minimized to invert the velocity model. By assuming that the variation of the reconstructed wavefield is negligible, we can apply an approximated full Newton method to the velocity inversion with reasonable cost comparable to the Gauss-Newton method. From the numerical examples using synthetic data, we confirm that the proposed method performs better and more robust than the simple gradient-based FWI method. In addition, we show that our objective function has fewer local minima, which helps to mitigate the cycle-skipping problem.

Constrained full-waveform inversion by model reparameterization

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. R117-R127 ◽  
Author(s):  
Antoine Guitton ◽  
Gboyega Ayeni ◽  
Esteban Díaz
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Real Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Coherent Noise ◽  
Low Frequencies ◽  
Full Waveform ◽  
Geological Information ◽  
Ill Posed

The waveform inversion problem is inherently ill-posed. Traditionally, regularization schemes are used to address this issue. For waveform inversion, where the model is expected to have many details reflecting the physical properties of the Earth, regularization and data fitting can work in opposite directions: the former smoothing and the latter adding details to the model. We propose constraining estimated velocity fields by reparameterizing the model. This technique, also called model-space preconditioning, is based on directional Laplacian filters: It preserves most of the details of the velocity model while smoothing the solution along known geological dips. Preconditioning also yields faster convergence at early iterations. The Laplacian filters have the property to smooth or kill local planar events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models. We illustrate with 2D synthetic and field data examples how preconditioning with nonstationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted and when sharp velocity contrasts are present. Adding geological information with preconditioning could benefit full-waveform inversion of real data whenever irregular geometry, coherent noise and lack of low frequencies are present.

scholarly journals Elastic Full-Waveform Inversion Using Migration‑Based Depth Reflector Representation in the Data Domain

2020 ◽  
Author(s):  
Vladimir Cheverda
Keyword(s):  
Seismic Data ◽  
Optimization Problem ◽  
Waveform Inversion ◽  
Nonlinear Least Squares ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Brute Force ◽  
Data Inversion ◽  
Full Waveform ◽  
Least Squares Optimization

Full-waveform seismic data inversion has given rise to hope for the simultaneous and automated execution of tomography and imaging by solving a nonlinear least-squares optimization problem. As previously recognized, brute force minimization by classical methods is hopeless if the data lacks low temporal frequencies. The article developed a reliable numerical method for recovering smooth velocity using model space decomposition. We present realistic synthetic examples to test the presented algorithm.

Auto-tuning Hamiltonian Monte Carlo

2020 ◽  
Author(s):  
Andreas Fichtner ◽  
Lars Gebraad ◽  
Christian Boehm ◽  
Andrea Zunino
Keyword(s):  
Monte Carlo ◽  
Inverse Problems ◽  
Mass Matrix ◽  
Waveform Inversion ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Hamiltonian Monte Carlo ◽  
Traveltime Tomography ◽  
Full Waveform ◽  
Auto Tuning

<p>Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that exploits derivative information in order to enable long-distance moves to independent models, even when the model space dimension is high (Duane et al., 1987). This feature motivates recent research aiming to adapt HMC for the solution of geophysical inverse problems (e.g. Sen & Biswas, 2017; Fichtner et al., 2018; Gebraad et al., 2020).</p><p>Here we present applications of HMC to inverse problems at variable levels of complexity. At the lowest level, we study linear inverse problems, including, for instance, linear traveltime tomography. Though this is not the class of problems for which Monte Carlo methods have been developed, it allows us to understand the important role of HMC tuning parameters. We then demonstrate that HMC can be used to obtain probabilistic solutions for two important classes of inverse problems: 2D nonlinear traveltime tomography and 2D elastic full-waveform inversion. In both scenarios, no super-computing resources are needed for model space dimensions from several thousand to ten thousand.</p><p>By far the most important, but also most complex, tuning parameter in HMC is the mass matrix, the choice of which critically controls convergence. Since manual tuning of the mass matrix is impossible for high-dimensional problems, we develop a new HMC flavour that tunes itself during sampling. This rests on the combination of HMC with a variant of the L-BFGS method, well-known from nonlinear optimisation. L-BFGS employs a few Monte Carlo samples to compute a matrix factorisation <strong>LL</strong><sup>T</sup>which dynamically approximates the local Hessian <strong>H</strong>, while the sampler traverses model space in a quasi-random fashion. The local curvature approximation is then used as mass matrix. Following an outline of the method, we present examples where the auto-tuning HMC produces almost perfectly uncorrelated samples for model space dimensions exceeding 10<sup>5</sup>.</p><p> </p><p><strong>References</strong></p><p>[1] Duane et al., 1987. "Hybrid Monte Carlo", Phys. Lett. B., 195, 216-222.</p><p>[2] Sen & Biswas, 2017. "Transdimensional seismic inversion using the reversible-jump Hamiltonian Monte Carlo algorithm", Geophysics, 82, R119-R134.</p><p>[3] Fichtner et al., 2018. "Hamiltonian Monte Carlo solution of tomographic inverse problems", Geophys. J. Int., 216, 1344-1363.</p><p>[4] Gebraad et al., 2020. "Bayesian elastic full-waveform inversion using Hamiltonian Monte Carlo", J. Geophys. Res., under review.</p>

A gradient-based Markov chain Monte Carlo method for full-waveform inversion and uncertainty analysis

Geophysics ◽  
2020 ◽  
Vol 86 (1) ◽  
pp. R15-R30
Author(s):  
Zeyu Zhao ◽  
Mrinal K. Sen
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Inference ◽  
Posterior Distribution ◽  
Waveform Inversion ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Gradient Based

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the diagonal approximate Hessian of the local log posterior. Our algorithm is called the gradient-based Markov chain Monte Carlo (GMCMC) method. The GMCMC FWI method can quantify inversion uncertainties with estimated posterior distribution given sufficiently long Markov chains. By directly sampling the posterior distribution, we obtain a global view of the model space. Theoretically speaking, statistical assessments do not depend on starting models. Our method is applied to the 2D Marmousi model with the frequency-domain FWI setting. Numerical results suggest that our method can be readily applied to 2D cases with affordable computational efforts.

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