Seismic Tomography by Monte Carlo Sampling

Dębski, Wojciech

doi:10.1007/s00024-009-0006-3

Cited by 17 publications

(7 citation statements)

References 37 publications

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“…Letβ ridge (λ) := (X ′ X + λI d ) −1 X ′ y be the corresponding ordinary ridge regression (ORR) estimate with shrinkage parameter λ [Hoerl and Kennard (1970), Swindel (1976), Dȩbski (2010)]. For given hyperparameters η, φ and ψ, the full conditional of β is β | η, φ, ψ ∼ N d (Ω −1 β ξ β , Ω −1 β ) with Ω β := ηQ(ψ) + φX ′ X and ξ β := ηQ(ψ)β 0 + φX ′ y.…”

Section: 2mentioning

confidence: 99%

“…While probabilistic seismic tomography using Markov chain Monte Carlo (MCMC) methods has been given considerable attention by the geophysical (seismological) community, these applications have been restricted to linear or nonlinear problems of much lower dimensionality assuming Gaussian errors Tarantola (1995, 2002), Sambridge and Mosegaard (2002)]. For example, Dȩbski (2010) compares the damped least-squares method (LSQR), a genetic algorithm and the Metropolis-Hastings (MH) algorithm in a low-dimensional linear tomography problem involving copper mining data. He finds that the MCMC sampling technique provides more robust estimates of velocity parameters compared to the other approaches.…”

mentioning

confidence: 99%

“…Our approach is also applicable to other kinds of travel time tomography, such as cross-borehole tomography or mining-induced seismic tomography [Dȩbski (2010)]. Other types of tomography, such as X-ray tomography in medical imaging, can also be recast as a linear matrix problem of large size with a very sparse forward matrix.…”

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confidence: 99%

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A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

Zhang¹,

Czado²,

Sigloch³

2013

Ann. Appl. Stat.

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We apply a linear Bayesian model to seismic tomography, a highdimensional inverse problem in geophysics. The objective is to estimate the three-dimensional structure of the earth's interior from data measured at its surface. Since this typically involves estimating thousands of unknowns or more, it has always been treated as a linear(ized) optimization problem. Here we present a Bayesian hierarchical model to estimate the joint distribution of earth structural and earthquake source parameters. An ellipsoidal spatial prior allows to accommodate the layered nature of the earth's mantle. With our efficient algorithm we can sample the posterior distributions for large-scale linear inverse problems and provide precise uncertainty quantification in terms of parameter distributions and credible intervals given the data. We apply the method to a full-fledged tomography problem, an inversion for upper-mantle structure under western North America that involves more than 11,000 parameters. In studies on simulated and real data, we show that our approach retrieves the major structures of the earth's interior as well as classical least-squares minimization, while additionally providing uncertainty assessments.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

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A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

Zhang¹,

Czado²,

Sigloch³

2013

Ann. Appl. Stat.

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“…Examples of such methods include the adaptive Metropolis (AM), Metropolis-adjusted Langevin (MALA) and hybrid (or Hamiltonian) Monte Carlo (HMC) methods, amongst many others. Random-walk Metropolis-Hastings MCMC methods on the full state space are typically very slow to converge, not uncommonly requiring 10 4 to 10 5 iterations for a single independent sample [8,9], with each iteration requiring simulation of high-dimensional data over a high-dimensional image space. We do not implement such a calculation here as the marginal then conditional algorithm we present next is several orders of magnitude cheaper.…”

Section: Posterior Inferencementioning

confidence: 99%

Fast Sampling in a Linear-Gaussian Inverse Problem

Fox¹,

Norton²

2016

SIAM/ASA J. Uncertainty Quantification

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We solve the inverse problem of deblurring a pixelized image of Jupiter using regularized deconvolution and by sample-based Bayesian inference. By efficiently sampling the marginal posterior distribution for hyperparameters, then the full conditional for the deblurred image, we find that we can evaluate the posterior mean faster than regularized inversion, when selection of the regularizing parameter is considered. To our knowledge, this is the first demonstration of sampling and inference that takes less compute time than regularized inversion in an inverse problems. Comparison to random-walk Metropolis-Hastings and block Gibbs MCMC shows that marginal then conditional sampling also outperforms these more common sampling algorithms, having better scaling with problem size. When problem-specific computations are feasible the asymptotic cost of an independent sample is one linear solve, implying that sample-based Bayesian inference may be performed directly over function spaces, when that limit exists.1 A classification of Bayesian image representations and prior models as low-level, intermediate-level, and high-level is given in [19].

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“…The main advantages of MC methods are that they provide a suite of well-fitting models as well as estimates of the uncertainties of the obtained models. However, while a variety of MC algorithms (in particular genetic algorithms and simulated annealing) have been successfully applied to different geophysical problems [for example to waveform fitting; see , , and references therein], only a few attempts have been made to apply them to tomographic problemsparticularly to surface-based refraction geometries (Pullammanappallil & Louie 1994;Weber 2000;Debski 2010Debski , 2013Bottero et al 2016). This seems to be mainly due to the typically large number of model parameters, the large number of models necessary to be tested, and the usually 'expensive' traveltime calculation.…”

Section: Introductionmentioning

confidence: 99%

Bayesian inversion of refraction seismic traveltime data

Ryberg

Haberland

2017

Geophysical Journal International

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S U M M A R YWe apply a Bayesian Markov chain Monte Carlo (McMC) formalism to the inversion of refraction seismic, traveltime data sets to derive 2-D velocity models below linear arrays (i.e. profiles) of sources and seismic receivers. Typical refraction data sets, especially when using the far-offset observations, are known as having experimental geometries which are very poor, highly ill-posed and far from being ideal. As a consequence, the structural resolution quickly degrades with depth. Conventional inversion techniques, based on regularization, potentially suffer from the choice of appropriate inversion parameters (i.e. number and distribution of cells, starting velocity models, damping and smoothing constraints, data noise level, etc.) and only local model space exploration. McMC techniques are used for exhaustive sampling of the model space without the need of prior knowledge (or assumptions) of inversion parameters, resulting in a large number of models fitting the observations. Statistical analysis of these models allows to derive an average (reference) solution and its standard deviation, thus providing uncertainty estimates of the inversion result. The highly non-linear character of the inversion problem, mainly caused by the experiment geometry, does not allow to derive a reference solution and error map by a simply averaging procedure. We present a modified averaging technique, which excludes parts of the prior distribution in the posterior values due to poor ray coverage, thus providing reliable estimates of inversion model properties even in those parts of the models. The model is discretized by a set of Voronoi polygons (with constant slowness cells) or a triangulated mesh (with interpolation within the triangles). Forward traveltime calculations are performed by a fast, finite-difference-based eikonal solver. The method is applied to a data set from a refraction seismic survey from Northern Namibia and compared to conventional tomography. An inversion test for a synthetic data set from a known model is also presented.

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Seismic Tomography by Monte Carlo Sampling

Cited by 17 publications

References 37 publications

A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

A Bayesian linear model for the high-dimensional inverse problem of seismic tomography

Fast Sampling in a Linear-Gaussian Inverse Problem

Bayesian inversion of refraction seismic traveltime data

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