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<strong class="journal-contentHeaderColor">Abstract.</strong> We present an efficient probabilistic workflow for the estimation of source parameters of induced seismic events in three-dimensional heterogeneous media. Our workflow exploits a linearized variant of the Hamiltonian Monte Carlo (HMC) algorithm. Compared to traditional Markov-Chain Monte Carlo (MCMC) algorithms, HMC is highly efficient in sampling high-dimensional model spaces. Through a linearization of the forward problem around the prior mean (i.e., the "best" initial model), this efficiency can be further improved. We show, however, that this linearization leads to a performance in which the output of an HMC chain strongly depends on the quality of the prior; in particular, because not all (induced) earthquake model parameters have a linear relationship with the recordings observed at the surface. To mitigate the importance of an accurate prior, we integrate the linearized HMC scheme into a workflow that (i) allows for a weak prior through linearization around various (initial) centroid locations, (ii) is able to converge to the mode containing the model with the (global) minimum misfit by means of an iterative HMC approach, and (iii) uses variance reduction as a criterion to include the output of individual Markov chains in the estimation of the posterior probability. Using a three-dimensional heterogeneous subsurface model of the Groningen gas field, we simulate an induced earthquake to test our workflow. We then demonstrate the virtue of our workflow by estimating the event's centroid (three parameters), moment tensor (six parameters), and the earthquake's origin time. We find that our workflow is able to recover the posterior probability of these source parameters rather well, even when the prior model information is inaccurate, imprecise, or both inaccurate and imprecise.

<strong class="journal-contentHeaderColor">Abstract.</strong> We present an efficient probabilistic workflow for the estimation of source parameters of induced seismic events in three-dimensional heterogeneous media. Our workflow exploits a linearized variant of the Hamiltonian Monte Carlo (HMC) algorithm. Compared to traditional Markov chain Monte Carlo (MCMC) algorithms, HMC is highly efficient in sampling high-dimensional model spaces. Through a linearization of the forward problem around the prior mean (i.e., the “best” initial model), this efficiency can be further improved. We show, however, that this linearization leads to a performance in which the output of an HMC chain strongly depends on the quality of the prior, in particular because not all (induced) earthquake model parameters have a linear relationship with the recordings observed at the surface. To mitigate the importance of an accurate prior, we integrate the linearized HMC scheme into a workflow that (i) allows for a weak prior through linearization around various (initial) centroid locations, (ii) is able to converge to the mode containing the model with the (global) minimum misfit by means of an iterative HMC approach, and (iii) uses variance reduction as a criterion to include the output of individual Markov chains in the estimation of the posterior probability. Using a three-dimensional heterogeneous subsurface model of the Groningen gas field, we simulate an induced earthquake to test our workflow. We then demonstrate the virtue of our workflow by estimating the event's centroid (three parameters), moment tensor (six parameters), and the earthquake's origin time. Using the synthetic case, we find that our proposed workflow is able to recover the posterior probability of these source parameters rather well, even when the prior model information is inaccurate, imprecise, or both inaccurate and imprecise.

We present an efficient probabilistic workflow for the estimation of source parameters of induced seismic events in three-dimensional heterogeneous media. Our workflow exploits a linearized variant of the Hamiltonian Monte Carlo (HMC) algorithm. Compared to traditional Markov-Chain Monte Carlo (MCMC) algorithms, HMC is highly efficient in sampling high-dimensional model spaces. Through a linearization of the forward problem around the prior mean (i.e., the "best" initial model), this efficiency can be further improved. We show, however, that this linearization leads to a performance in which the output of an HMC chain strongly depends on the quality of the prior; in particular, because not all (induced) earthquake model parameters have a linear relationship with the recordings observed at the surface. To mitigate the importance of an accurate prior, we integrate the linearized HMC scheme into a workflow that (i) allows for a weak prior through linearization around various (initial) centroid locations, (ii) is able to converge to the mode containing the model with the (global) minimum misfit by means of an iterative HMC approach, and (iii) uses variance reduction as a criterion to include the output of individual Markov chains in the estimation of the posterior probability. Using a three-dimensional heterogeneous subsurface model of the Groningen gas field, we simulate an induced earthquake to test our workflow. We then demonstrate the virtue of our workflow by estimating the event's centroid (three parameters), moment tensor (six parameters), and the earthquake's origin time. We find that our workflow is able to recover the posterior probability of these source parameters rather well, even when the prior model information is inaccurate, imprecise, or both inaccurate and imprecise.

Abstract. We present an efficient probabilistic workflow for the estimation of source parameters of induced seismic events in three-dimensional heterogeneous media. Our workflow exploits a linearized variant of the Hamiltonian Monte Carlo (HMC) algorithm. Compared to traditional Markov chain Monte Carlo (MCMC) algorithms, HMC is highly efficient in sampling high-dimensional model spaces. Through a linearization of the forward problem around the prior mean (i.e., the “best” initial model), this efficiency can be further improved. We show, however, that this linearization leads to a performance in which the output of an HMC chain strongly depends on the quality of the prior, in particular because not all (induced) earthquake model parameters have a linear relationship with the recordings observed at the surface. To mitigate the importance of an accurate prior, we integrate the linearized HMC scheme into a workflow that (i) allows for a weak prior through linearization around various (initial) centroid locations, (ii) is able to converge to the mode containing the model with the (global) minimum misfit by means of an iterative HMC approach, and (iii) uses variance reduction as a criterion to include the output of individual Markov chains in the estimation of the posterior probability. Using a three-dimensional heterogeneous subsurface model of the Groningen gas field, we simulate an induced earthquake to test our workflow. We then demonstrate the virtue of our workflow by estimating the event's centroid (three parameters), moment tensor (six parameters), and the earthquake's origin time. Using the synthetic case, we find that our proposed workflow is able to recover the posterior probability of these source parameters rather well, even when the prior model information is inaccurate, imprecise, or both inaccurate and imprecise.

The Hamiltonian Monte Carlo (HMC) algorithm is a Markov Chain Monte Carlo (MCMC) technique, which combines the advantages of Hamiltonian dynamics methods and Metropolis Monte Carlo approach, to sample from complex distributions. The HMC algorithm incorporates gradient information in the dynamic trajectories and thus suppresses the random walk nature in traditional Markov chain simulation methods. This ensures rapid mixing, faster convergence, and improved efficiency of the Markov chain. The leapfrog method is generally used in discrete simulation of the dynamic transitions. In this paper, we refer to this as the leapfrog–HMC. The primary goal of this paper is to present the HMC algorithm as a tool for rapid sampling of high dimensional and complex distributions, and demonstrate its advantages over the classical Metropolis Monte Carlo technique. We demonstrate that the use of an adaptive–step discretization scheme in simulating the dynamic transitions results in an algorithm which significantly outperforms the leapfrog–HMC algorithm. Relevance to reservoir parameter estimation and uncertainty quantification will be discussed.

Markov Chain Monte Carlo (MCMC) algorithms play an important role in statistical inference problems dealing with intractable probability distributions. Recently, many MCMC algorithms such as Hamiltonian Monte Carlo (HMC) and Riemannian Manifold HMC have been proposed to provide distant proposals with high acceptance rate. These algorithms, however, tend to be computationally intensive which could limit their usefulness, especially for big data problems due to repetitive evaluations of functions and statistical quantities that depend on the data. This issue occurs in many statistic computing problems. In this paper, we propose a novel strategy that exploits smoothness (regularity) of parameter space to improve computational efficiency of MCMC algorithms. When evaluation of functions or statistical quantities are needed at a point in parameter space, interpolation from precomputed values or previous computed values is used. More specifically, we focus on Hamiltonian Monte Carlo (HMC) algorithms that use geometric information for faster exploration of probability distributions. Our proposed method is based on precomputing the required geometric information on a set of grids before running sampling information at nearby grids at each iteration of HMC. Sparse grid interpolation method is used for high dimensional problems. Tests on computational examples are shown to illustrate the advantages of our method.

One way to quantify the uncertainty in Bayesian inverse problems arising in the engineering domain is to generate samples from the posterior distribution using Markov chain Monte Carlo (MCMC) algorithms. The basic MCMC methods tend to explore the parameter space slowly, which makes them inefficient for practical problems. On the other hand, enhanced MCMC approaches, like Hamiltonian Monte Carlo (HMC), require the gradients from the physical problem simulator, which are often not available. In this case, a feasible option is to use the gradient approximations provided by the surrogate (proxy) models built on the simulator output. In this paper, we consider proxy-aided HMC employing the Gaussian process (kriging) emulator. We overview in detail the different aspects of kriging proxies, the underlying principles of the HMC sampler and its interaction with the proxy model. The proxy-aided HMC algorithm is thoroughly tested in different settings, and applied to three case studies—one toy problem, and two synthetic reservoir simulation models. We address the question of how the sampler performance is affected by the increase of the problem dimension, the use of the gradients in proxy training, the use of proxy-for-the-data and the different approaches to the design points selection. It turns out that applying the proxy model with HMC sampler may be beneficial for relatively small physical models, with around 20 unknown parameters. Such a sampler is shown to outperform both the basic Random Walk Metropolis algorithm, and the HMC algorithm fed by the exact simulator gradients.

A rapid source fault estimation and quantitative assessment of the uncertainty of the estimated model can elucidate the occurrence mechanism of earthquakes and inform disaster damage mitigation. The Bayesian statistical method that addresses the posterior distribution of unknowns using the Markov chain Monte Carlo (MCMC) method is significant for uncertainty assessment. The Metropolis–Hastings method, especially the Random walk Metropolis–Hastings (RWMH), has many applications, including coseismic fault estimation. However, RWMH exhibits a trade-off between the transition distance and the acceptance ratio of parameter transition candidates and requires a long mixing time, particularly in solving high-dimensional problems. This necessitates a more efficient Bayesian method. In this study, we developed a fault estimation algorithm using the Hamiltonian Monte Carlo (HMC) method, which is considered more efficient than the other MCMC method, but its applicability has not been sufficiently validated to estimate the coseismic fault for the first time. HMC can conduct sampling more intelligently with the gradient information of the posterior distribution. We applied our algorithm to the 2016 Kumamoto earthquake (MJMA 7.3), and its sampling converged in 2 × 104 samples, including 1 × 103 burn-in samples. The estimated models satisfactorily accounted for the input data; the variance reduction was approximately 88%, and the estimated fault parameters and event magnitude were consistent with those reported in previous studies. HMC could acquire similar results using only 2% of the RWMH chains. Moreover, the power spectral density (PSD) of each model parameter's Markov chain showed this method exhibited a low correlation with the subsequent sample and a long transition distance between samples. These results indicate HMC has advantages in terms of chain length than RWMH, expecting a more efficient estimation for a high-dimensional problem that requires a long mixing time or a problem using nonlinear Green’s function, which has a large computational cost.Graphical

&amp;lt;p&amp;gt;In May 2019, an earthquake with a magnitude of 3.4 (local magnitude) hit the area of the Westerwijtwerd village in the province of Groningen, the Netherlands. The event is the result of the gas extraction in the Groningen gas field and is one of the largest events to date. To better understand the source characteristics of the event, we apply a probabilistic full-waveform inversion technique that we recently developed to the event's recordings. Specifically, we use a variant of the Hamiltonian Monte Carlo (HMC) algorithm. When sampling high-dimensional model spaces, HMC is proven to be more efficient than the generic Metropolis-Hasting algorithm. Compared to probabilistic inversions of tectonic events, two main challenges arise while applying the algorithm. First, the prior information of the event is usually incomplete and inaccurate. That is, the only available information is (an estimate of) the hypocenter and origin time. Second, the frequency content of the induced event's seismograms is higher than that of typical tectonic events. This implies a higher non-linearity, which in turn complicates the ability of a probabilistic inversion algorithm to sample the model spaces, particularly when considering the first challenge. Consequently, to address both challenges, first, we develop a procedure to estimate the necessary prior information and use it as input to the HMC variant. Second, we run our HMC algorithm iteratively to mitigate the non-linearity. Using the relatively detailed velocity model of the Groningen gas field, we eventually estimate ten posteriors of the source parameters. The latter being the hypocenter (three parameters), the moment tensor (six independent parameters), and the origin time. &amp;amp;#160;&amp;lt;/p&amp;gt;

Hamiltonian Monte Carlo methods for Subset Simulation in reliability analysis

In recent years, the Hamiltonian Monte Carlo (HMC) algorithm has been found to work more efficiently compared to other popular Markov chain Monte Carlo (MCMC) methods (such as random walk Metropolis–Hastings) in generating samples from a high-dimensional probability distribution. HMC has proven more efficient in terms of mixing rates and effective sample size than previous MCMC techniques, but still may not be sufficiently fast for particularly large problems. The use of GPUs promises to push HMC even further greatly increasing the utility of the algorithm. By expressing the computationally intensive portions of HMC (the evaluations of the probability kernel and its gradient) in terms of linear or element-wise operations, HMC can be made highly amenable to the use of graphics processing units (GPUs). A multinomial regression example demonstrates the promise of GPU-based HMC sampling. Using GPU-based memory objects to perform the entire HMC simulation, most of the latency penalties associated with transferring data from main to GPU memory can be avoided. Thus, the proposed computational framework may appear conceptually very simple, but has the potential to be applied to a wide class of hierarchical models relying on HMC sampling. Models whose posterior density and corresponding gradients can be reduced to linear or element-wise operations are amenable to significant speed ups through the use of GPUs. Analyses of datasets that were previously intractable for fully Bayesian approaches due to the prohibitively high computational cost are now feasible using the proposed framework.

Novel parameter update for a gradient based MCMC method for solid-void interface detection through elastodynamic inversion

SUMMARY The use of the probabilistic approach to solve inverse problems is becoming more popular in the geophysical community, thanks to its ability to address nonlinear forward problems and to provide uncertainty quantification. However, such strategy is often tailored to specific applications and therefore there is a need for common platforms to solve different geophysical inverse problems and showing potential and pitfalls of the methodology. In this work, we demonstrate a common framework within which it is possible to solve such inverse problems ranging from, for example, earthquake source location to potential field data inversion and seismic tomography. This allows us to fully address nonlinear problems and to derive useful information about the subsurface, including uncertainty estimation. This approach can, in fact, provide probabilities related to certain properties or structures of the subsurface, such as histograms of the value of some physical property, the expected volume of buried geological bodies or the probability of having boundaries defining different layers. Thanks to its ability to address high-dimensional problems, the Hamiltonian Monte Carlo (HMC) algorithm has emerged as the state-of-the-art tool for solving geophysical inverse problems within the probabilistic framework. HMC requires the computation of gradients, which can be obtained by adjoint methods. This unique combination of HMC and adjoint methods is what makes the solution of tomographic problems ultimately feasible. These results can be obtained with ‘HMCLab’, a numerical laboratory for solving a range of different geophysical inverse problems using sampling methods, focusing in particular on the HMC algorithm. HMCLab consists of a set of samplers (HMC and others) and a set of geophysical forward problems. For each problem its misfit function and gradient computation are provided and, in addition, a set of prior models can be combined to inject additional information into the inverse problem. This allows users to experiment with probabilistic inverse problems and also address real-world studies. We show how to solve a selected set of problems within this framework using variants of the HMC algorithm and analyse the results. HMCLab is provided as an open source package written both in Python and Julia, welcoming contributions from the community.

The Milne–Eddington (M-E) atmosphere model is commonly adopted in the inversion of the magnetic fields in the solar photosphere. By applying the Levenberg–Marquardt algorithm or training a neural network (NN) model, the magnetic field vector can be quickly inferred from the Stokes profile but lacks reliable and statistically well-defined confidence intervals for parameters. To address this, we present an efficient Bayesian inference method called NNHMC, combining the NN model with the Hamiltonian Monte Carlo (HMC) algorithm. The NN model is used to speedily synthesize batches of synthetic Stokes profiles, accelerating the inference process. The HMC algorithm significantly improves sampling efficiency in high-dimensional parameter spaces and can handle large-scale data sets in batches. The spectropolarimetric observation of an active region obtained by the Hinode/spectropolarimeter (SP) is used to demonstrate the capability of the NNHMC method. The strength, inclination, and azimuth of the magnetic field and the line-of-sight velocity inferred with the NNHMC method are very similar to those derived with the MERLIN code. Furthermore, this study provided posterior distributions and uncertainties for these parameters. A test on the same hardware and software platform shows a speed increase of up to 2.5 orders of magnitude with respect to the traditional Markov Chain Monte Carlo method (without the NN, using the M-E atmosphere model), establishing the NNHMC method as a highly effective tool for Stokes inversion based on Bayesian inference.

The classic 1953 paper of Metropolis et al. [2] introduced to us the world of Markov Chain Monte Carlo (MCMC). In their work, MCMC was used to simulate the distribution of states for a system of idealized molecules. Not long after this, in 1959, another approach to molecular simulation was introduced by Alder and Wainwright [14], in which they used a deterministic algorithm for the motion of the molecules. This algorithm followed Newton’s laws of motion, and it can be formalized in an elegant way using Hamiltonian dynamics. The two approaches, statistical (MCMC) and deterministic (molecular dynamics), coexisted peacefully for a long time. In 1987, an extraordinary paper by Duane et al. [15] combined the MCMC and molecular dynamics approaches. They called their method Hybrid Monte Carlo (HMC).