A discrepancy-based penalty method for extended waveform inversion

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. R287-R298 ◽  
Author(s):  
Lei Fu ◽  
William W. Symes
Keyword(s):  
Born Approximation ◽  
Degrees Of Freedom ◽  
Linear Models ◽  
Waveform Inversion ◽  
Constant Density ◽  
Inversion Process ◽  
Waveform Data ◽  
Seismic Waveform ◽  
Optimization Formulation ◽  
Penalty Weight

Extended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model, thus permitting it to fit the data well throughout the inversion process. These extra degrees of freedom must be curtailed at the solution, for example, by penalizing them as part of an optimization formulation. For separable (partly linear) models, a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical (linear) degrees of freedom. The linear variables are eliminated in an inner optimization step, leaving a function of the outer (nonlinear) variables to be optimized. This variable projection method is convenient for computation, but it requires that the penalty weight be increased as the estimated model tends to the (physical) solution. We describe an algorithm based on discrepancy, that is, maintaining the data residual at the inner optimum within a prescribed range, to control the penalty weight during the outer optimization. We evaluate this algorithm in the context of constant density acoustic waveform inversion, by recovering background model and perturbation fitting bandlimited waveform data in the Born approximation.

Long-Wavelength Earth Model via Accelerated Full-Waveform Inversion

2021 ◽  
Author(s):  
Solvi Thrastarson ◽  
Dirk-Philip van Herwaarden ◽  
Lion Krischer ◽  
Martin van Driel ◽  
Christian Boehm ◽  
...  
Keyword(s):  
Waveform Inversion ◽  
Large Data ◽  
Earth Model ◽  
Full Waveform Inversion ◽  
Waveform Data ◽  
Long Wavelength ◽  
Computational Overhead ◽  
Full Dataset ◽  
Full Waveform ◽  
Seismic Waveform

<p>As the volume of available seismic waveform data increases, the responsibility to use the data in an effective way emerges. This requires computational efficiency as well as maximizing the exploitation of the information associated with the data.</p><p>In this contribution, we present a long-wavelength Earth model, created by using the data recorded from over a thousand earthquakes, starting from a simple one-dimensional background (PREM). The model is constructed with an accelerated full-waveform inversion (FWI) method which can seamlessly include large data volumes with a significantly reduced computational overhead. Although we present a long-wavelength model, the approach has the potential to go to much higher frequencies, while maintaining a reasonable cost.</p><p>Our approach combines two novel FWI variants. (1) The dynamic mini-batch approach which uses adaptively defined subsets of the full dataset in each iteration, detaching the direct scaling of inversion cost from the number of earthquakes included. (2) Wavefield-adapted meshes which utilize the azimuthal smoothness of the wavefield to design meshes optimized for each individual source location. Using wavefield adapted meshes can drastically reduce the cost of both forward and adjoint simulations as well as it makes the scaling of the computing cost to modelled frequencies more favourable.</p>

A theory-guided deep-learning formulation and optimization of seismic waveform inversion

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. R87-R99 ◽  
Author(s):  
Jian Sun ◽  
Zhan Niu ◽  
Kristopher A. Innanen ◽  
Junxiao Li ◽  
Daniel O. Trad
Keyword(s):  
Deep Learning ◽  
Seismic Data ◽  
Degrees Of Freedom ◽  
Data Science ◽  
Waveform Inversion ◽  
Optimization Method ◽  
Learning Rate ◽  
Physical Models ◽  
Seismic Waveform ◽  
Set Up

Deep-learning techniques appear to be poised to play very important roles in our processing flows for inversion and interpretation of seismic data. The most successful seismic applications of these complex pattern-identifying networks will, presumably, be those that also leverage the deterministic physical models on which we normally base our seismic interpretations. If this is true, algorithms belonging to theory-guided data science, whose aim is roughly this, will have particular applicability in our field. We have developed a theory-designed recurrent neural network (RNN) that allows single- and multidimensional scalar acoustic seismic forward-modeling problems to be set up in terms of its forward propagation. We find that training such a network and updating its weights using measured seismic data then amounts to a solution of the seismic inverse problem and is equivalent to gradient-based seismic full-waveform inversion (FWI). By refining these RNNs in terms of optimization method and learning rate, comparisons are made between standard deep-learning optimization and nonlinear conjugate gradient and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimized algorithms. Our numerical analysis indicates that adaptive moment (or Adam) optimization with a learning rate set to match the magnitudes of standard FWI updates appears to produce the most stable and well-behaved waveform inversion results, which is reconfirmed by a multidimensional 2D Marmousi experiment. Future waveform RNNs, with additional degrees of freedom, may allow optimal wave propagation rules to be solved for at the same time as medium properties, reducing modeling errors.

Ensemble-based seismic inversion for a stratified medium

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. R29-R39 ◽  
Author(s):  
Michael Gineste ◽  
Jo Eidsvik ◽  
York Zheng
Keyword(s):  
Waveform Inversion ◽  
Seismic Inversion ◽  
Stratified Medium ◽  
Elastic Parameters ◽  
Waveform Data ◽  
Optimization Task ◽  
Estimation Uncertainty ◽  
Seismic Waveform ◽  
Probabilistic Setting ◽  
Iterative Ensemble Smoother

Seismic waveform inversion is a nontrivial optimization task, which is often complicated by the nonlinear relationship between the elastic attributes of interest and the large amount of data obtained in seismic experiments. Quantifying the solution uncertainty can be even more challenging, and it requires considering the problem in a probabilistic setting. Consequently, the seismic inverse problem is placed in a Bayesian framework, using a sequential filtering approach to invert for the elastic parameters. The method uses an iterative ensemble smoother to estimate the subsurface parameters, and from the ensemble, a notion of estimation uncertainty is readily available. The ensemble implicitly linearizes the relation between the parameters and the observed waveform data; hence, it requires no tangent linear model. The approach is based on sequential conditioning on partitions of the whole data record (1) to regularize the inversion path and effectively drive the estimation process in a top-down manner and (2) to circumvent a consequence of the ensemble reduced rank approximation. The method is exemplified on a synthetic case, inverting for elastic parameters in a 1D medium using a seismic shot record. Our results indicate that the iterative ensemble method is applicable to seismic waveform inversion and that the ensemble representation indeed indicates estimation uncertainty.

scholarly journals The method of polarized traces for the 3D Helmholtz equation

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T313-T333 ◽  
Author(s):  
Leonardo Zepeda-Núñez ◽  
Adrien Scheuer ◽  
Russell J. Hewett ◽  
Laurent Demanet
Keyword(s):  
Helmholtz Equation ◽  
Message Passing ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Message Passing Interface ◽  
Domain Decomposition Method ◽  
Waveform Inversion ◽  
Representation Formula ◽  
Constant Density ◽  
Problem Frequency

We have developed a fast solver for the 3D Helmholtz equation, in heterogeneous, constant density, acoustic media, in the high-frequency regime. The solver is based on the method of polarized traces, a layered domain-decomposition method, where the subdomains are connected via transmission conditions prescribed by the discrete Green’s representation formula and artificial reflections are avoided by enforcing nonreflecting boundary conditions between layers. The method of polarized traces allows us to consider only unknowns at the layer interfaces, reducing the overall cost and memory footprint of the solver. We determine that polarizing the wavefields in this manner yields an efficient preconditioner for the reduced system, whose rate of convergence is independent of the problem frequency. The resulting preconditioned system is solved iteratively using generalized minimum residual, where we never assemble the reduced system or preconditioner; rather, we implement them via solving the Helmholtz equation locally within the subdomains. The method is parallelized using Message Passing Interface and coupled with a distributed linear algebra library and pipelining to obtain an empirical on-line runtime [Formula: see text], where [Formula: see text] is the total number of degrees of freedom, [Formula: see text] is the number of subdomains, and [Formula: see text] is the number of right-hand sides (RHS). This scaling is favorable for regimes in which the number of sources (distinct RHS) is large, for example, enabling large-scale implementations of frequency-domain full-waveform inversion.

scholarly journals Batch seismic inversion using the iterative ensemble Kalman smoother

2021 ◽  
Author(s):  
Michael Gineste ◽  
Jo Eidsvik
Keyword(s):  
Seismic Data ◽  
Time Windows ◽  
Seismic Inversion ◽  
Efficient Estimation ◽  
Solution Strategy ◽  
Adaptive Procedure ◽  
Waveform Data ◽  
Kalman Smoother ◽  
Seismic Waveform ◽  
Single Batch

AbstractAn ensemble-based method for seismic inversion to estimate elastic attributes is considered, namely the iterative ensemble Kalman smoother. The main focus of this work is the challenge associated with ensemble-based inversion of seismic waveform data. The amount of seismic data is large and, depending on ensemble size, it cannot be processed in a single batch. Instead a solution strategy of partitioning the data recordings in time windows and processing these sequentially is suggested. This work demonstrates how this partitioning can be done adaptively, with a focus on reliable and efficient estimation. The adaptivity relies on an analysis of the update direction used in the iterative procedure, and an interpretation of contributions from prior and likelihood to this update. The idea is that these must balance; if the prior dominates, the estimation process is inefficient while the estimation is likely to overfit and diverge if data dominates. Two approaches to meet this balance are formulated and evaluated. One is based on an interpretation of eigenvalue distributions and how this enters and affects weighting of prior and likelihood contributions. The other is based on balancing the norm magnitude of prior and likelihood vector components in the update. Only the latter is found to sufficiently regularize the data window. Although no guarantees for avoiding ensemble divergence are provided in the paper, the results of the adaptive procedure indicate that robust estimation performance can be achieved for ensemble-based inversion of seismic waveform data.

Using band-tridiagonal preconditioners for 1D seismic waveform inversion

2005 ◽  
Author(s):  
Milton J. Porsani ◽  
Saulo Pomponet de Oliveira
Keyword(s):  
Waveform Inversion ◽  
Seismic Waveform
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