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.

Identification of Contact Failures in Multilayered Composites With the Markov Chain Monte Carlo Method

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
L. A. Abreu ◽  
H. R. B. Orlande ◽  
J. Kaipio ◽  
V. Kolehmainen ◽  
R. M. Cotta ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Heat Conduction Problem ◽  
Thermal Contact ◽  
Inverse Heat Conduction Problem ◽  
Thermal Contact Conductance ◽  
Contact Conductance ◽  
Spatially Distributed

This paper deals with the solution of an inverse heat conduction problem, aiming at the identification of the interface thermal contact conductance, which can be directly associated to the quality of the adhesion between layers of multilayered composite materials. The inverse problem is solved within the Bayesian framework, with a Markov chain Monte Carlo method. A total variation prior is used for the spatially distributed contact conductance. The feasibility of the approach is evaluated with simulated temperature measurements for cases with contact failures of different sizes.

scholarly journals Another look at the treatment of data uncertainty in Markov chain Monte Carlo inversion and other probabilistic methods

2020 ◽  
Vol 222 (1) ◽  
pp. 388-405
Author(s):  
F J Tilmann ◽  
H Sadeghisorkhani ◽  
A Mauerberger
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Measurement Error ◽  
Probability Distribution ◽  
Unknown Parameter ◽  
Model Space ◽  
Data Uncertainty ◽  
Uncertainty Distribution

SUMMARY In probabilistic Bayesian inversions, data uncertainty is a crucial parameter for quantifying the uncertainties and correlations of the resulting model parameters or, in transdimensional approaches, even the complexity of the model. However, in many geophysical inference problems it is poorly known. Therefore, it is common practice to allow the data uncertainty itself to be a parameter to be determined. Although in principle any arbitrary uncertainty distribution can be assumed, Gaussian distributions whose standard deviation is then the unknown parameter to be estimated are the usual choice. In this special case, the paper demonstrates that a simple analytical integration is sufficient to marginalise out this uncertainty parameter, reducing the complexity of the model space without compromising the accuracy of the posterior model probability distribution. However, it is well known that the distribution of geophysical measurement errors, although superficially similar to a Gaussian distribution, typically contains more frequent samples along the tail of the distribution, so-called outliers. In linearized inversions these are often removed in subsequent iterations based on some threshold criterion, but in Markov chain Monte Carlo (McMC) inversions this approach is not possible as they rely on the likelihood ratios, which cannot be formed if the number of data points varies between the steps of the Markov chain. The flexibility to define the data error probability distribution in McMC can be exploited in order to account for this pattern of uncertainties in a natural way, without having to make arbitrary choices regarding residual thresholds. In particular, we can regard the data uncertainty distribution as a mixture between a Gaussian distribution, which represent valid measurements with some measurement error, and a uniform distribution, which represents invalid measurements. The relative balance between them is an unknown parameter to be estimated alongside the standard deviation of the Gauss distribution. For each data point, the algorithm can then assign a probability to be an outlier, and the influence of each data point will be effectively downgraded according to its probability to be an outlier. Furthermore, this assignment can change as the McMC search is exploring different parts of the model space. The approach is demonstrated with both synthetic and real tomography examples. In a synthetic test, the proposed mixed measurement error distribution allows recovery of the underlying model even in the presence of 6 per cent outliers, which completely destroy the ability of a regular McMC or linear search to provide a meaningful image. Applied to an actual ambient noise tomography study based on automatically picked dispersion curves, the resulting model is shown to be much more consistent for different data sets, which differ in the applied quality criteria, while retaining the ability to recover strong anomalies in selected parts of the model.

scholarly journals On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

2014 ◽  
Vol 16 (2) ◽  
pp. 467-490 ◽  
Author(s):  
Jun Li ◽  
Philippe Vignal ◽  
Shuyu Sun ◽  
Victor M. Calo
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Sample Size ◽  
Computational Cost ◽  
Sampling Interval ◽  
Ensemble Average ◽  
Cycle Number ◽  
Stochastic Error ◽  
General Rules

AbstractIn Markov Chain Monte Carlo (MCMC) simulations, thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them.

scholarly journals Bayesian Markov-Chain-Monte-Carlo Inversion of Time-Lapse Crosshole GPR Data to Characterize the Vadose Zone at the Arrenaes Site, Denmark

2012 ◽  
Vol 11 (4) ◽  
pp. vzj2011.0153 ◽  
Author(s):  
Marie Scholer ◽  
James Irving ◽  
Majken C. Looms ◽  
Lars Nielsen ◽  
Klaus Holliger
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Vadose Zone ◽  
Time Lapse

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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