Multimodal Markov chain Monte Carlo method for nonlinear petrophysical seismic inversion

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. M1-M13 ◽  
Author(s):  
Leandro Passos de Figueiredo ◽  
Dario Grana ◽  
Mauro Roisenberg ◽  
Bruno B. Rodrigues
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Seismic Inversion ◽  
Forward Model ◽  
Model Parameters ◽  
Petrophysical Properties ◽  
Mcmc Method ◽  
Data Set

One of the main objectives in the reservoir characterization is estimating the rock properties based on seismic measurements. We have developed a stochastic sampling method for the joint prediction of facies and petrophysical properties, assuming a nonparametric mixture prior distribution and a nonlinear forward model. The proposed methodology is based on a Markov chain Monte Carlo (MCMC) method specifically designed for multimodal distributions for nonlinear problems. The vector of model parameters includes the facies sequence along the seismic trace as well as the continuous petrophysical properties, such as porosity, mineral fractions, and fluid saturations. At each location, the distribution of petrophysical properties is assumed to be multimodal and nonparametric with as many modes as the number of facies; therefore, along the seismic trace, the distribution is multimodal with the number of modes being equal to the number of facies power the number of samples. Because of the nonlinear forward model, the large number of modes and as a consequence the large dimension of the model space, the analytical computation of the full posterior distribution is not feasible. We then numerically evaluate the posterior distribution by using an MCMC method in which we iteratively sample the facies, by moving from one mode to another, and the petrophysical properties, by sampling within the same mode. The method is extended to multiple seismic traces by applying a first-order Markov chain that accounts for the lateral continuity of the model properties. We first validate the method using a synthetic 2D reservoir model and then we apply the method to a real data set acquired in a carbonate field.

Efficient History Matching Using the Markov-Chain Monte Carlo Method by Means of the Transformed Adaptive Stochastic Collocation Method

SPE Journal ◽  
2019 ◽  
Vol 24 (04) ◽  
pp. 1468-1489 ◽  
Author(s):  
Qinzhuo Liao ◽  
Lingzao Zeng ◽  
Haibin Chang ◽  
Dongxiao Zhang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Collocation Method ◽  
Posterior Distribution ◽  
History Matching ◽  
Stochastic Collocation ◽  
Mcmc Method ◽  
Stochastic Collocation Method ◽  
Surrogate System

Summary Bayesian inference provides a convenient framework for history matching and prediction. In this framework, prior knowledge, system nonlinearity, and measurement errors can be directly incorporated into the posterior distribution of the parameters. The Markov-chain Monte Carlo (MCMC) method is a powerful tool to generate samples from the posterior distribution. However, the MCMC method usually requires a large number of forward simulations. Hence, it can be a computationally intensive task, particularly when dealing with large-scale flow and transport models. To address this issue, we construct a surrogate system for the model outputs in the form of polynomials using the stochastic collocation method (SCM). In addition, we use interpolation with the nested sparse grids and adaptively take into account the different importance of parameters for high-dimensional problems. Furthermore, we introduce an additional transform process to improve the accuracy of the surrogate model in case of strong nonlinearities, such as a discontinuous or unsmooth relation between the input parameters and the output responses. Once the surrogate system is built, we can evaluate the likelihood with little computational cost. Numerical results demonstrate that the proposed method can efficiently estimate the posterior statistics of input parameters and provide accurate results for history matching and prediction of the observed data with a moderate number of parameters.

scholarly journals SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

2016 ◽  
Vol III-7 ◽  
pp. 119-124 ◽  
Author(s):  
Q. H. Zhao ◽  
Y. Li ◽  
Y. Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Voronoi Tessellation ◽  
Random Variable ◽  
Model Parameters ◽  
Sar Image ◽  
Sar Images ◽  
Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

Transdimensional Markov Chain Monte Carlo joint inversion of direct current resistivity and transient electromagnetic data

2020 ◽  
Vol 224 (2) ◽  
pp. 1429-1442
Author(s):  
Ronghua Peng ◽  
Pritam Yogeshwar ◽  
Yajun Liu ◽  
Xiangyun Hu
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Parameter Uncertainty ◽  
Joint Inversion ◽  
Model Parameters ◽  
Mcmc Method ◽  
Complementary Information ◽  
Data Set ◽  
Electromagnetic Data ◽  
Transient Electromagnetic

SUMMARY Joint inversion of multiple geophysical data sets with complementary information content can significantly reduce the non-uniqueness inherent to each individual data set and, therefore, can improve subsurface characterization. Gradient-based joint inversion methods depend on the choice of model regularization and usually produce one single optimal model, and rely on linearization to estimate model parameter uncertainty. However, a quantitative evaluation of the parameter uncertainty of the derived model parameters is crucial for reliable data interpretation. In this study, we present a transdimensional Markov Chain Monte Carlo (MCMC) method for the joint inversion of direct current resistivity and transient electromagnetic data, which provides a rigorous assessment of the uncertainty associated with the derived model. The transdimensional property of the algorithm allows the number of unknown model parameters to be determined adaptively by the data. This usually favours models with fewer parameters through the parsimony criterion of the Bayesian method by choosing suitable prior distributions. In this paper, we demonstrate that the transdimensional MCMC method combines complementary information contained in each data set and reduces the overall uncertainty using synthetic examples. Furthermore, we successfully applied the new joint inversion scheme to field data from Azraq, Jordan. The transdimensional MCMC inversion results are in good agreement with the results obtained by deterministic inversion techniques. From the MCMC inversion results we identified the thickness of a basalt formation and a conductive zone, which were uncertain and not interpreted in prior studies, adding to the geological interpretation.

scholarly journals SAR IMAGE SEGMENTATION WITH UNKNOWN NUMBER OF CLASSES COMBINED VORONOI TESSELLATION AND RJMCMC ALGORITHM

2016 ◽  
Vol III-7 ◽  
pp. 119-124
Author(s):  
Q. H. Zhao ◽  
Y. Li ◽  
Y. Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Posterior Distribution ◽  
Voronoi Tessellation ◽  
Random Variable ◽  
Model Parameters ◽  
Sar Image ◽  
Sar Images ◽  
Number Of Classes

This paper presents a novel segmentation method for automatically determining the number of classes in Synthetic Aperture Radar (SAR) images by combining Voronoi tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy. Instead of giving the number of classes <i>a priori</i>, it is considered as a random variable and subject to a Poisson distribution. Based on Voronoi tessellation, the image is divided into homogeneous polygons. By Bayesian paradigm, a posterior distribution which characterizes the segmentation and model parameters conditional on a given SAR image can be obtained up to a normalizing constant; Then, a Revisable Jump Markov Chain Monte Carlo(RJMCMC) algorithm involving six move types is designed to simulate the posterior distribution, the move types including: splitting or merging real classes, updating parameter vector, updating label field, moving positions of generating points, birth or death of generating points and birth or death of an empty class. Experimental results with real and simulated SAR images demonstrate that the proposed method can determine the number of classes automatically and segment homogeneous regions well.

Gaussian mixture Markov chain Monte Carlo method for linear seismic inversion

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R463-R476 ◽  
Author(s):  
Leandro Passos de Figueiredo ◽  
Dario Grana ◽  
Mauro Roisenberg ◽  
Bruno B. Rodrigues
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Elastic Properties ◽  
Posterior Distribution ◽  
Seismic Data ◽  
Gaussian Mixture ◽  
Gaussian Component ◽  
Well Test ◽  
Mcmc Method

We have developed a Markov chain Monte Carlo (MCMC) method for joint inversion of seismic data for the prediction of facies and elastic properties. The solution of the inverse problem is defined by the Bayesian posterior distribution of the properties of interest. The prior distribution is a Gaussian mixture model, and each component is associated to a potential configuration of the facies sequence along the seismic trace. The low frequency is incorporated by using facies-dependent depositional trend models for the prior means of the elastic properties in each facies. The posterior distribution is also a Gaussian mixture, for which the Gaussian component can be analytically computed. However, due to the high number of components of the mixture, i.e., the large number of facies configurations, the computation of the full posterior distribution is impractical. Our Gaussian mixture MCMC method allows for the calculation of the full posterior distribution by sampling the facies configurations according to the acceptance/rejection probability. The novelty of the method is the use of an MCMC framework with multimodal distributions for the description of the model properties and the facies along the entire seismic trace. Our method is tested on synthetic seismic data, applied to real seismic data, and validated using a well test.

scholarly journals A Bayesian Estimation and Predictionof Gompertz Extension Distribution Using the MCMC Method

2020 ◽  
Vol 19 (1) ◽  
pp. 142-160
Author(s):  
Arun Kumar Chaudhary ◽  
Vijay Kumar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Real Data ◽  
Simulation Method ◽  
Mcmc Methods ◽  
Mcmc Method ◽  
Data Set ◽  
Bayes Estimates ◽  
Check Method

 In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of the Gompertz extension distribution based on a complete sample. We have developed a procedure to obtain Bayes estimates of the parameters of the Gompertz extension distribution using Markov Chain Monte Carlo (MCMC) simulation method in OpenBUGS, established software for Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods. We have obtained the Bayes estimates of the parameters, hazard and reliability functions, and their probability intervals are also presented. We have applied the predictive check method to discuss the issue of model compatibility. A real data set is considered for illustration under uniform and gamma priors.  

Seismic inversion and uncertainty quantification using transdimensional Markov chain Monte Carlo method

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. R321-R334 ◽  
Author(s):  
Dehan Zhu ◽  
Richard Gibson
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Uncertainty Quantification ◽  
Seismic Inversion ◽  
Inversion Algorithm ◽  
Seismic Attribute ◽  
Data Set ◽  
Location Uncertainty ◽  
Parameter Values

We applied a transdimensional stochastic inversion algorithm, reversible jump Markov chain Monte Carlo (rjMCMC), to angle-stack seismic inversion for characterization of reservoir acoustic and shear impedance with uncertainty quantification. The rjMCMC is able to infer the number of parameters for the model as well as the parameter values. In our case, the number of parameters depends on the number of model layers for a given data set. We also use this method in uncertainty quantification because a transdimensional sampling helps prevent underparameterization or strong overparameterization. An ensemble of models with proper parameterization can improve parameter estimation and uncertainty quantification. Our new results in uncertainty analysis indicate that (1) the uncertainty in seismic inversion, including uncertainty in earth properties and their locations, is related to the discontinuity of property across an interface, and (2) there is a trade-off between property uncertainty and location uncertainty. A stronger discontinuity will induce more property uncertainty but less location uncertainty at the discontinuity interface. Therefore, we further use the inversion uncertainty as a novel seismic attribute to assist in delineation of subsurface discontinuity interfaces and quantify the magnitude of the discontinuities, which further facilitates quantitative interpretation and stratigraphic interpretation.

scholarly journals Efficient Markov Chain Monte Carlo Methods for Decoding Neural Spike Trains

2011 ◽  
Vol 23 (1) ◽  
pp. 46-96 ◽  
Author(s):  
Yashar Ahmadian ◽  
Jonathan W. Pillow ◽  
Liam Paninski
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Mutual Information ◽  
Posterior Distribution ◽  
Spike Trains ◽  
Mcmc Methods ◽  
Average Error ◽  
Model Parameters ◽  
Wide Range

Stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity. We discuss Bayesian decoding methods based on an encoding generalized linear model (GLM) that accurately describes how stimuli are transformed into the spike trains of a group of neurons. The form of the GLM likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is log concave so long as the prior is. This allows the maximum a posteriori (MAP) stimulus estimate to be obtained using efficient optimization algorithms. Unfortunately, the MAP estimate can have a relatively large average error when the posterior is highly nongaussian. Here we compare several Markov chain Monte Carlo (MCMC) algorithms that allow for the calculation of general Bayesian estimators involving posterior expectations (conditional on model parameters). An efficient version of the hybrid Monte Carlo (HMC) algorithm was significantly superior to other MCMC methods for gaussian priors. When the prior distribution has sharp edges and corners, on the other hand, the “hit-and-run” algorithm performed better than other MCMC methods. Using these algorithms, we show that for this latter class of priors, the posterior mean estimate can have a considerably lower average error than MAP, whereas for gaussian priors, the two estimators have roughly equal efficiency. We also address the application of MCMC methods for extracting nonmarginal properties of the posterior distribution. For example, by using MCMC to calculate the mutual information between the stimulus and response, we verify the validity of a computationally efficient Laplace approximation to this quantity for gaussian priors in a wide range of model parameters; this makes direct model-based computation of the mutual information tractable even in the case of large observed neural populations, where methods based on binning the spike train fail. Finally, we consider the effect of uncertainty in the GLM parameters on the posterior estimators.

scholarly journals Markov Chain Monte Carlo Parameter Estimation of the ODE Compartmental Cell Growth Model

2018 ◽  
Vol 13 (2) ◽  
pp. 376-391 ◽  
Author(s):  
T. Luzyanina ◽  
G. Bocharov
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Profile Likelihood ◽  
Live Cells ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Data Set ◽  
T Lymphocyte Proliferation

We use a Markov chain Monte Carlo (MCMC) method to quantify uncertainty in parameters of the heterogeneous linear compartmental model of cell population growth, described by a system of ordinary differential equations. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set specifies the following characteristics of the kinetics of human T lymphocyte proliferation assay in vitro: the total number of live cells and dead but not disintegrated cells and the number of cells divided j times. Our goal is to compare results of the MCMC analysis of the uncertainty in the best-fit parameter estimates with the ones obtained earlier, using the variance-covariance approach, the profile-likelihood based approach and the bootstrap technique. We show that the computed posterior probability density functions are Gaussian for most of the model parameters and they are close to Gaussian ones for other parameters except one. We present posterior uncertainty limits for the model solution and new observations.

scholarly journals Parameter stability and consistency in an alongshore-current model determined with Markov chain Monte Carlo

2008 ◽  
Vol 10 (2) ◽  
pp. 153-162 ◽  
Author(s):  
B. G. Ruessink
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Parameter Stability ◽  
Small Uncertainty ◽  
Best Fit ◽  
Stability And Consistency

When a numerical model is to be used as a practical tool, its parameters should preferably be stable and consistent, that is, possess a small uncertainty and be time-invariant. Using data and predictions of alongshore mean currents flowing on a beach as a case study, this paper illustrates how parameter stability and consistency can be assessed using Markov chain Monte Carlo. Within a single calibration run, Markov chain Monte Carlo estimates the parameter posterior probability density function, its mode being the best-fit parameter set. Parameter stability is investigated by stepwise adding new data to a calibration run, while consistency is examined by calibrating the model on different datasets of equal length. The results for the present case study indicate that various tidal cycles with strong (say, &gt;0.5 m/s) currents are required to obtain stable parameter estimates, and that the best-fit model parameters and the underlying posterior distribution are strongly time-varying. This inconsistent parameter behavior may reflect unresolved variability of the processes represented by the parameters, or may represent compensational behavior for temporal violations in specific model assumptions.

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