Hamiltonian Monte Carlo algorithms for target- and interval-oriented amplitude versus angle inversions

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R177-R194 ◽  
Author(s):  
Mattia Aleardi ◽  
Alessandro Salusti
Keyword(s):  
Monte Carlo ◽  
A Priori ◽  
Computational Cost ◽  
Model Space ◽  
Time Interval ◽  
Long Distance ◽  
Hamiltonian Monte Carlo ◽  
Zoeppritz Equations ◽  
Monte Carlo Algorithms ◽  
Amplitude Versus

A reliable assessment of the posterior uncertainties is a crucial aspect of any amplitude versus angle (AVA) inversion due to the severe ill-conditioning of this inverse problem. To accomplish this task, numerical Markov chain Monte Carlo algorithms are usually used when the forward operator is nonlinear. The downside of these algorithms is the considerable number of samples needed to attain stable posterior estimations especially in high-dimensional spaces. To overcome this issue, we assessed the suitability of Hamiltonian Monte Carlo (HMC) algorithm for nonlinear target- and interval-oriented AVA inversions for the estimation of elastic properties and associated uncertainties from prestack seismic data. The target-oriented approach inverts the AVA responses of the target reflection by adopting the nonlinear Zoeppritz equations, whereas the interval-oriented method inverts the seismic amplitudes along a time interval using a 1D convolutional forward model still based on the Zoeppritz equations. HMC uses an artificial Hamiltonian system in which a model is viewed as a particle moving along a trajectory in an extended space. In this context, the inclusion of the derivative information of the misfit function makes possible long-distance moves with a high probability of acceptance from the current position toward a new independent model. In our application, we adopt a simple Gaussian a priori distribution that allows for an analytical inclusion of geostatistical constraints into the inversion framework, and we also develop a strategy that replaces the numerical computation of the Jacobian with a matrix operator analytically derived from a linearization of the Zoeppritz equations. Synthetic and field data inversions demonstrate that the HMC is a very promising approach for Bayesian AVA inversion that guarantees an efficient sampling of the model space and retrieves reliable estimations and accurate uncertainty quantifications with an affordable computational cost.

scholarly journals Hamiltonian Monte Carlo inversion of seismic reflection data in the acoustic approximation

2021 ◽  
Author(s):  
Andrea Zunino ◽  
Klaus Mosegaard ◽  
Christian Boehm ◽  
Lars Gebraad ◽  
Andreas Fichtner
Keyword(s):  
Monte Carlo ◽  
Computational Cost ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Efficient Manner ◽  
Seismic Reflection Data ◽  
Hamiltonian Monte Carlo ◽  
Reflection Data ◽  
Synthetic Test ◽  
Acoustic Approximation

<p>The Hamiltonian Monte Carlo method (HMC) is gaining popularity in the geophysical community to fully address nonlinear inverse problems and related uncertainty quantification. We present here an application of HMC to invert seismic data in the acoustic approximation in the context of reflection seismology. We address a 2-D problem, in the form of a vertical cross section where both source and receivers are located near the surface of the model. To solve the forward problem we utilise the finite-difference method with PML absorbing boundary conditions. The observed data are represented by a set of shotgathers.</p><p>The crucial aspect for a successful application of the HMC lies in the capability of performing gradient computations in an efficient manner. To this end, we use the adjont state method to compute the gradient of the misfit functional, which has a computational cost of only about twice that of the forward computation, a very efficient strategy. From the collection of samples characterising the posterior distribution obtained with the HMC, we can derive quantities of interest using statistical analysis and assess uncertainties.</p><p>We illustrate an application of this methodology on a synthetic test mimicking the setup encountered in exploration problems.</p>

scholarly journals Basis-constrained Bayesian Markov-chain Monte Carlo difference inversion for geoelectrical monitoring of hydrogeologic processes

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. A37-A42 ◽  
Author(s):  
Erasmus Kofi Oware ◽  
James Irving ◽  
Thomas Hermans
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Computational Cost ◽  
Model Space ◽  
Time Lapse ◽  
Tracer Experiment ◽  
Hydrologic Process ◽  
Spatially Distributed ◽  
Geophysical Imaging

Bayesian Markov-chain Monte Carlo (McMC) techniques are increasingly being used in geophysical estimation of hydrogeologic processes due to their ability to produce multiple estimates that enable comprehensive assessment of uncertainty. Standard McMC sampling methods can, however, become computationally intractable for spatially distributed, high-dimensional problems. We have developed a novel basis-constrained Bayesian McMC difference inversion framework for time-lapse geophysical imaging. The strategy parameterizes the Bayesian inversion model space in terms of sparse, hydrologic-process-tuned bases, leading to dimensionality reduction while accounting for the physics of the target hydrologic process. We evaluate the algorithm on cross-borehole electrical resistivity tomography (ERT) field data acquired during a heat-tracer experiment. We validate the ERT-estimated temperatures with direct temperature measurements at two locations on the ERT plane. We also perform the inversions using the conventional smoothness-constrained inversion (SCI). Our approach estimates the heat plumes without excessive smoothing in contrast with the SCI thermograms. We capture most of the validation temperatures within the 90% confidence interval of the mean. Accounting for the physics of the target process allows the detection of small temperature changes that are undetectable by the SCI. Performing the inversion in the reduced-dimensional model space results in significant gains in computational cost.

Transdimensional seismic inversion using the reversible jump Hamiltonian Monte Carlo algorithm

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. R119-R134 ◽  
Author(s):  
Mrinal K. Sen ◽  
Reetam Biswas
Keyword(s):  
Monte Carlo ◽  
A Priori ◽  
Seismic Inversion ◽  
Monte Carlo Algorithm ◽  
Model Parameters ◽  
Reversible Jump ◽  
Step Method ◽  
Acceptance Probability ◽  
Hamiltonian Monte Carlo ◽  
Impedance Inversion

Prestack or angle stack gathers are inverted to estimate pseudologs at every surface location for building reservoir models. Recently, several methods have been proposed to increase the resolution of the inverted models. All of these methods, however, require that the total number of model parameters be fixed a priori. We have investigated an alternate approach in which we allow the data themselves to choose model parameterization. In other words, in addition to the layer properties, the number of layers is also treated as a variable in our formulation. Such transdimensional inverse problems are generally solved by using the reversible jump Markov chain Monte Carlo (RJMCMC) approach, which is a tool for model exploration and uncertainty quantification. This method, however, has very low acceptance. We have developed a two-step method by combining RJMCMC with a fixed-dimensional MCMC called Hamiltonian Monte Carlo, which makes use of gradient information to take large steps. Acceptance probability for such a transition is also derived. We call this new method “reversible jump Hamiltonian Monte Carlo (RJHMC).” We have applied this technique to poststack acoustic impedance inversion and to prestack (angle stack) AVA inversion for estimating acoustic and shear impedance profiles. We have determined that the marginal posteriors estimated by RJMCMC and RJHMC are in good agreement. Our results demonstrate that RJHMC converges faster than RJMCMC, and it therefore can be a practical tool for inverting seismic data when the gradient can be computed efficiently.

scholarly journals Emulation-accelerated Hamiltonian Monte Carlo algorithms for parameter estimation and uncertainty quantification in differential equation models

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
L. Mihaela Paun ◽  
Dirk Husmeier
Keyword(s):  
Monte Carlo ◽  
Sample Size ◽  
Posterior Distribution ◽  
Detailed Balance ◽  
Evaluation Study ◽  
Forward Model ◽  
Effective Sample Size ◽  
Hamiltonian Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Efficiency Measures

AbstractWe propose to accelerate Hamiltonian and Lagrangian Monte Carlo algorithms by coupling them with Gaussian processes for emulation of the log unnormalised posterior distribution. We provide proofs of detailed balance with respect to the exact posterior distribution for these algorithms, and validate the correctness of the samplers’ implementation by Geweke consistency tests. We implement these algorithms in a delayed acceptance (DA) framework, and investigate whether the DA scheme can offer computational gains over the standard algorithms. A comparative evaluation study is carried out to assess the performance of the methods on a series of models described by differential equations, including a real-world application of a 1D fluid-dynamics model of the pulmonary blood circulation. The aim is to identify the algorithm which gives the best trade-off between accuracy and computational efficiency, to be used in nonlinear DE models, which are computationally onerous due to repeated numerical integrations in a Bayesian analysis. Results showed no advantage of the DA scheme over the standard algorithms with respect to several efficiency measures based on the effective sample size for most methods and DE models considered. These gradient-driven algorithms register a high acceptance rate, thus the number of expensive forward model evaluations is not significantly reduced by the first emulator-based stage of DA. Additionally, the Lagrangian Dynamical Monte Carlo and Riemann Manifold Hamiltonian Monte Carlo tended to register the highest efficiency (in terms of effective sample size normalised by the number of forward model evaluations), followed by the Hamiltonian Monte Carlo, and the No U-turn sampler tended to be the least efficient.

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>

scholarly journals A Multilevel Simulation Method for Time-Variant Reliability Analysis

2021 ◽  
Vol 13 (7) ◽  
pp. 3646
Author(s):  
Jian Wang ◽  
Xiang Gao ◽  
Zhili Sun
Keyword(s):  
Monte Carlo ◽  
Reliability Analysis ◽  
High Reliability ◽  
Computational Cost ◽  
Simulation Method ◽  
Probability Of Failure ◽  
Cumulative Probability ◽  
Time Interval ◽  
Random Samples ◽  
Geometric Sequence

Crude Monte Carlo simulation (MCS) is the most robust and easily implemented method for performing time-variant reliability analysis (TRA). However, it is inefficient, especially for high reliability problems. This paper aims to present a random simulation method called the multilevel Monte Carlo (MLMC) method for TRA to enhance the computational efficiency of crude MCS while maintaining its accuracy and robustness. The proposed method first discretizes the time interval of interest using a geometric sequence of different timesteps. The cumulative probability of failure associated with the finest level can then be estimated by computing corrections using all levels. To assess the cumulative probability of failure in a way that minimizes the overall computational complexity, the number of random samples at each level is optimized. Moreover, the correction associated with each level is independently computed using crude MCS. Thereby, the proposed method can achieve the accuracy associated with the finest level at a much lower computational cost than that of crude MCS, and retains the robustness of crude MCS with respect to nonlinearity and dimensions. The effectiveness of the proposed method is validated by numerical examples.

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