scholarly journals Multilevel Assimilation of Inverted Seismic Data With Correction for Multilevel Modeling Error

2021 ◽  
Vol 7 ◽  
Author(s):  
Mohammad Nezhadali ◽  
Tuhin Bhakta ◽  
Kristian Fossum ◽  
Trond Mannseth
Keyword(s):  
Monte Carlo ◽  
Multilevel Modeling ◽  
Seismic Data ◽  
Multilevel Model ◽  
Computational Cost ◽  
Parameter Estimates ◽  
Ensemble Size ◽  
Modeling Error ◽  
Parameter Uncertainties ◽  
Reservoir Models

With large amounts of simultaneous data, like inverted seismic data in reservoir modeling, negative effects of Monte Carlo errors in straightforward ensemble-based data assimilation (DA) are enhanced, typically resulting in underestimation of parameter uncertainties. Utilization of lower fidelity reservoir simulations reduces the computational cost per ensemble member, thereby rendering the possibility of increasing the ensemble size without increasing the total computational cost. Increasing the ensemble size will reduce Monte Carlo errors and therefore benefit DA results. The use of lower fidelity reservoir models will however introduce modeling errors in addition to those already present in conventional fidelity simulation results. Multilevel simulations utilize a selection of models for the same entity that constitute hierarchies both in fidelities and computational costs. In this work, we estimate and approximately account for the multilevel modeling error (MLME), that is, the part of the total modeling error that is caused by using a multilevel model hierarchy, instead of a single conventional model to calculate model forecasts. To this end, four computationally inexpensive approximate MLME correction schemes are considered, and their abilities to correct the multilevel model forecasts for reservoir models with different types of MLME are assessed. The numerical results show a consistent ranking of the MLME correction schemes. Additionally, we assess the performances of the different MLME-corrected model forecasts in assimilation of inverted seismic data. The posterior parameter estimates from multilevel DA with and without MLME correction are compared to results obtained from conventional single-level DA with localization. It is found that multilevel DA (MLDA) with and without MLME correction outperforms conventional DA with localization. The use of all four MLME correction schemes results in posterior parameter estimates with similar quality. Results obtained with MLDA without any MLME correction were also of similar quality, indicating some robustness of MLDA toward MLME.

scholarly journals Bayesian estimation of parameters in a regional hydrological model

2002 ◽  
Vol 6 (5) ◽  
pp. 883-898 ◽  
Author(s):  
K. Engeland ◽  
L. Gottschalk
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Hydrological Model ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Parameter Uncertainties ◽  
Daily Streamflow ◽  
Likelihood Model ◽  
Simulation Errors

Abstract. This study evaluates the applicability of the distributed, process-oriented Ecomag model for prediction of daily streamflow in ungauged basins. The Ecomag model is applied as a regional model to nine catchments in the NOPEX area, using Bayesian statistics to estimate the posterior distribution of the model parameters conditioned on the observed streamflow. The distribution is calculated by Markov Chain Monte Carlo (MCMC) analysis. The Bayesian method requires formulation of a likelihood function for the parameters and three alternative formulations are used. The first is a subjectively chosen objective function that describes the goodness of fit between the simulated and observed streamflow, as defined in the GLUE framework. The second and third formulations are more statistically correct likelihood models that describe the simulation errors. The full statistical likelihood model describes the simulation errors as an AR(1) process, whereas the simple model excludes the auto-regressive part. The statistical parameters depend on the catchments and the hydrological processes and the statistical and the hydrological parameters are estimated simultaneously. The results show that the simple likelihood model gives the most robust parameter estimates. The simulation error may be explained to a large extent by the catchment characteristics and climatic conditions, so it is possible to transfer knowledge about them to ungauged catchments. The statistical models for the simulation errors indicate that structural errors in the model are more important than parameter uncertainties. Keywords: regional hydrological model, model uncertainty, Bayesian analysis, Markov Chain Monte Carlo analysis

scholarly journals Multilevel and Quasi Monte Carlo methods for the calculation of the Expected Value of Partial Perfect Information

2021 ◽  
Author(s):  
Wei Fang ◽  
Zhenru Wang ◽  
Mike B Giles ◽  
Christopher H Jackson ◽  
Nicky J Welton ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Input Parameter ◽  
Computational Cost ◽  
Expected Value ◽  
Perfect Information ◽  
Parameter Estimates ◽  
Tree Model ◽  
Generalised Additive Model ◽  
Approximation Techniques ◽  
Quasi Monte Carlo

The expected value of partial perfect information (EVPPI) provides an upper bound on the value of collecting further evidence on a set of inputs to a cost-effectiveness decision model. Standard Monte Carlo (MC) estimation of EVPPI is computationally expensive as it requires nested simulation. Alternatives based on regression approximations to the model have been developed, but are not practicable when the number of uncertain parameters of interest is large and when parameter estimates are highly correlated. The error associated with the regression approximation is difficult to determine, while MC allows the bias and precision to be controlled. In this paper, we explore the potential of Quasi Monte-Carlo (QMC) and Multilevel Monte-Carlo (MLMC) estimation to reduce computational cost of estimating EVPPI by reducing the variance compared with MC, while preserving accuracy. In this paper, we develop methods to apply QMC and MLMC to EVPPI, addressing particular challenges that arise where Markov Chain Monte Carlo (MCMC) has been used to estimate input parameter distributions. We illustrate the methods using a two examples: a simplified decision tree model for treatments for depression, and a complex Markov model for treatments to prevent stroke in atrial fibrillation, both of which use MCMC inputs. We compare the performance of QMC and MLMC with MC and the approximation techniques of Generalised Additive Model regression (GAM), Gaussian process regression (GP), and Integrated Nested Laplace Approximations (INLA-GP). We found QMC and MLMC to offer substantial computational savings when parameter sets are large and correlated, and when the EVPPI is large. We also find GP and INLA-GP to be biased in those situations, while GAM cannot estimate EVPPI for large parameter sets.

scholarly journals Assessment of multilevel ensemble-based data assimilation for reservoir history matching

2019 ◽  
Vol 24 (1) ◽  
pp. 217-239
Author(s):  
Kristian Fossum ◽  
Trond Mannseth ◽  
Andreas S. Stordal
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Computational Cost ◽  
Ensemble Size ◽  
Matching Problems ◽  
Reservoir Models ◽  
Reservoir History Matching ◽  
Computational Resources

AbstractMultilevel ensemble-based data assimilation (DA) as an alternative to standard (single-level) ensemble-based DA for reservoir history matching problems is considered. Restricted computational resources currently limit the ensemble size to about 100 for field-scale cases, resulting in large sampling errors if no measures are taken to prevent it. With multilevel methods, the computational resources are spread over models with different accuracy and computational cost, enabling a substantially increased total ensemble size. Hence, reduced numerical accuracy is partially traded for increased statistical accuracy. A novel multilevel DA method, the multilevel hybrid ensemble Kalman filter (MLHEnKF) is proposed. Both the expected and the true efficiency of a previously published multilevel method, the multilevel ensemble Kalman filter (MLEnKF), and the MLHEnKF are assessed for a toy model and two reservoir models. A multilevel sequence of approximations is introduced for all models. This is achieved via spatial grid coarsening and simple upscaling for the reservoir models, and via a designed synthetic sequence for the toy model. For all models, the finest discretization level is assumed to correspond to the exact model. The results obtained show that, despite its good theoretical properties, MLEnKF does not perform well for the reservoir history matching problems considered. We also show that this is probably caused by the assumptions underlying its theoretical properties not being fulfilled for the multilevel reservoir models considered. The performance of MLHEnKF, which is designed to handle restricted computational resources well, is quite good. Furthermore, the toy model is utilized to set up a case where the assumptions underlying the theoretical properties of MLEnKF are fulfilled. On that case, MLEnKF performs very well and clearly better than MLHEnKF.

scholarly journals Multilevel and Quasi Monte Carlo Methods for the Calculation of the Expected Value of Partial Perfect Information

2021 ◽  
pp. 0272989X2110263
Author(s):  
Wei Fang ◽  
Zhenru Wang ◽  
Michael B. Giles ◽  
Chris H. Jackson ◽  
Nicky J. Welton ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Input Parameter ◽  
Computational Cost ◽  
Additive Model ◽  
Expected Value ◽  
Perfect Information ◽  
Parameter Estimates ◽  
Tree Model ◽  
Approximation Techniques ◽  
Quasi Monte Carlo

The expected value of partial perfect information (EVPPI) provides an upper bound on the value of collecting further evidence on a set of inputs to a cost-effectiveness decision model. Standard Monte Carlo estimation of EVPPI is computationally expensive as it requires nested simulation. Alternatives based on regression approximations to the model have been developed but are not practicable when the number of uncertain parameters of interest is large and when parameter estimates are highly correlated. The error associated with the regression approximation is difficult to determine, while MC allows the bias and precision to be controlled. In this article, we explore the potential of quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) estimation to reduce the computational cost of estimating EVPPI by reducing the variance compared with MC while preserving accuracy. We also develop methods to apply QMC and MLMC to EVPPI, addressing particular challenges that arise where Markov chain Monte Carlo (MCMC) has been used to estimate input parameter distributions. We illustrate the methods using 2 examples: a simplified decision tree model for treatments for depression and a complex Markov model for treatments to prevent stroke in atrial fibrillation, both of which use MCMC inputs. We compare the performance of QMC and MLMC with MC and the approximation techniques of generalized additive model (GAM) regression, Gaussian process (GP) regression, and integrated nested Laplace approximations (INLA-GP). We found QMC and MLMC to offer substantial computational savings when parameter sets are large and correlated and when the EVPPI is large. We also found that GP and INLA-GP were biased in those situations, whereas GAM cannot estimate EVPPI for large parameter sets.

Applying the Multilevel Monte Carlo Method for Heterogeneity-Induced Uncertainty Quantification of Surfactant/Polymer Flooding

SPE Journal ◽  
2016 ◽  
Vol 21 (04) ◽  
pp. 1192-1203 ◽  
Author(s):  
A.. Alkhatib ◽  
M.. Babaei
Keyword(s):  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Computational Cost ◽  
Spatial Uncertainty ◽  
Scale Model ◽  
Fine Scale ◽  
Multilevel Monte Carlo ◽  
Reservoir Model ◽  
Reservoir Models ◽  
Non Gaussian

Summary Reservoir heterogeneity can be detrimental to the success of surfactant/polymer enhanced-oil-recovery (EOR) processes. Therefore, it is important to evaluate the effect of uncertainty in reservoir heterogeneity on the performance of surfactant/polymer EOR. Usually, a Monte Carlo sampling approach is used, in which a number of stochastic reservoir-model realizations are generated and then numerical simulation is performed to obtain a certain objective function, such as the recovery factor. However, Monte Carlo simulation (MCS) has a slow convergence rate and requires a large number of samples to produce accurate results. This can be computationally expensive when using large complex reservoir models. This study applies a multiscale approach to improve the efficiency of uncertainty quantification. This method is known as the multilevel Monte Carlo (MLMC) method. This method comprises performing a small number of expensive simulations on the fine-scale model and a large number of less-expensive simulations on coarser upscaled models, and then combining the results to produce the quantities of interest. The purpose of this method is to reduce computational cost while maintaining the accuracy of the fine-scale model. The results of this approach are compared with a reference MCS, assuming a large number of simulations on the fine-scale model. Other advantages of the MLMC method are its nonintrusiveness and its scalability to incorporate an increasing number of uncertainties. This study uses the MLMC method to efficiently quantify the effect of uncertainty in heterogeneity on the recovery factor of a chemical EOR process, specifically surfactant/polymer flooding. The permeability field is assumed to be the random input. This method is first demonstrated by use of a Gaussian 3D reservoir model. Different coarsening algorithms are used and compared, such as the renormalization method and the pressure-solver method (PSM). The results are compared with running Monte Carlo for the fine-scale model while equating the computational cost for the MLMC method. Both of these results are then compared with the reference case, which uses a large number of runs of the fine-scale model. The method is then extended to a channelized non-Gaussian generated 3D reservoir model incorporating multiphase upscaling The results show that it is possible to robustly quantify spatial uncertainty for a surfactant/polymer EOR process while greatly reducing the computational requirement, up to two orders of magnitude compared with traditional Monte Carlo for both the Gaussian and non-Gaussian reservoir models. The method can be easily extended to other EOR processes to quantify spatial uncertainty, such as carbon dioxide (CO2) EOR. Other possible extensions of this method are also discussed.

A Bayesian Multilevel Modeling Approach to Time-Series Cross-Sectional Data

2007 ◽  
Vol 15 (2) ◽  
pp. 165-181 ◽  
Author(s):  
Boris Shor ◽  
Joseph Bafumi ◽  
Luke Keele ◽  
David Park
Keyword(s):  
Time Series ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Political Science ◽  
Multilevel Modeling ◽  
Multilevel Model ◽  
Multilevel Models ◽  
Cross Sectional ◽  
Better Than ◽  
Analysis Of Time Series

The analysis of time-series cross-sectional (TSCS) data has become increasingly popular in political science. Meanwhile, political scientists are also becoming more interested in the use of multilevel models (MLM). However, little work exists to understand the benefits of multilevel modeling when applied to TSCS data. We employ Monte Carlo simulations to benchmark the performance of a Bayesian multilevel model for TSCS data. We find that the MLM performs as well or better than other common estimators for such data. Most importantly, the MLM is more general and offers researchers additional advantages.

Predictors of grit: A multilevel model examination of demographics and school experiences

2018 ◽  
Author(s):  
Dana Wanzer
Keyword(s):  
Academic Success ◽  
School Culture ◽  
Multilevel Modeling ◽  
Predictive Validity ◽  
School Engagement ◽  
Multilevel Model ◽  
School Experiences ◽  
School Level ◽  
School Demographics ◽  
And Gender

Much of the research on grit has examined its predictive validity toward academic success; however, little research has treated grit as an outcome. This study uses multilevel modeling to examine how student-level demographics, school-level demographics, and students’ experiences in school predict grit. Results demonstrate that students’ experiences in school—including school engagement, relationships with adults and peers, and school culture—and self-reported GPA were most strongly related to grit, ethnicity was weakly related to grit, and gender and school demographics did not significantly relate to grit. Implications of this research on the potential malleability of grit are discussed.

Multilevel Monte Carlo by using the Halton sequence

2020 ◽  
Vol 26 (3) ◽  
pp. 193-203
Author(s):  
Shady Ahmed Nagy ◽  
Mohamed A. El-Beltagy ◽  
Mohamed Wafa
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Computational Cost ◽  
Random Numbers ◽  
Multilevel Monte Carlo ◽  
Halton Sequence ◽  
Random Generator ◽  
Mc Simulation ◽  
Multilevel Monte Carlo Method

AbstractMonte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

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, >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.

Restricted model domain time Radon transforms

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. A17-A21 ◽  
Author(s):  
Juan I. Sabbione ◽  
Mauricio D. Sacchi
Keyword(s):  
Inverse Problem ◽  
Seismic Data ◽  
Computational Cost ◽  
Synthetic Data ◽  
Model Space ◽  
Radon Transforms ◽  
Iterative Solver ◽  
Hard Thresholding ◽  
Restricted Model ◽  
Marine Data

The coefficients that synthesize seismic data via the hyperbolic Radon transform (HRT) are estimated by solving a linear-inverse problem. In the classical HRT, the computational cost of the inverse problem is proportional to the size of the data and the number of Radon coefficients. We have developed a strategy that significantly speeds up the implementation of time-domain HRTs. For this purpose, we have defined a restricted model space of coefficients applying hard thresholding to an initial low-resolution Radon gather. Then, an iterative solver that operated on the restricted model space was used to estimate the group of coefficients that synthesized the data. The method is illustrated with synthetic data and tested with a marine data example.

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