A Gibbs sampler for inequality‐constrained geostatistical interpolation and inverse modeling

Michalak, A. M.

doi:10.1029/2007wr006645

Cited by 38 publications

(31 citation statements)

References 56 publications

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“…Thus, the mean of the conditional realizations might not be equal to the best estimate from the BLUE method. Michalak and Kitanidis (2003) and Michalak (2008) propose a Gibbs sampler and Markov chain Monte Carlo technique to enforce non‐negativity. These authors point out that the best estimate would not differ significantly, but may have an impact on uncertainty.…”

Section: Methodsmentioning

confidence: 99%

Increasing Confidence in Mass Discharge Estimates Using Geostatistical Methods

Cai¹,

Wilson

Cardiff

et al. 2011

Ground Water

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Mass discharge is one metric rapidly gaining acceptance for assessing the performance of in situ groundwater remediation systems. Multilevel sampling transects provide the data necessary to make such estimates, often using the Thiessen Polygon method. This method, however, does not provide a direct estimate of uncertainty. We introduce a geostatistical mass discharge estimation approach that involves a rigorous analysis of data spatial variability and selection of an appropriate variogram model. High-resolution interpolation was applied to create a map of measurements across a transect, and the magnitude and uncertainty of mass discharge were quantified by conditional simulation. An important benefit of the approach is quantified uncertainty of the mass discharge estimate. We tested the approach on data from two sites monitored using multilevel transects. We also used the approach to explore the effect of lower spatial monitoring resolution on the accuracy and uncertainty of mass discharge estimates. This process revealed two important findings: (1) appropriate monitoring resolution is that which yielded an estimate comparable with the full dataset value, and (2) high-resolution sampling yields a more representative spatial data structure descriptor, which can then be used via conditional simulation to make subsequent mass discharge estimates from lower resolution sampling of the same transect. The implication of the latter is that a high-resolution multilevel transect needs to be sampled only once to obtain the necessary spatial data descriptor for a contaminant plume exhibiting minor temporal variability, and thereafter less spatially intensely to reduce costs.

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Section: Methodsmentioning

confidence: 99%

Increasing Confidence in Mass Discharge Estimates Using Geostatistical Methods

Cai¹,

Wilson

Cardiff

et al. 2011

Ground Water

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show abstract

“…The MCMC method was also used to generate conditional realizations in parameter/function estimation (Michalak and Kitanidis 2003). Oliver et al (1997), Michalak (2008) and Fu and Gomez-Hernandez (2009) applied MCMC methods in various groundwater applications. Vrugt et al (2008) designed the differential evolution adaptive Metropolis (DREAM) algorithm especially for efficient sampling of the posterior distribution of hydrological models, and Vrugt et al (2009) later compared this algorithm with the generalized likelihood uncertainty estimation (GLUE) method.…”

Section: Introductionmentioning

confidence: 99%

Parameter estimation in nonlinear environmental problems

Liu

Cardiff

Kitanidis

2010

Stoch Environ Res Risk Assess

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Popular parameter estimation methods, including least squares, maximum likelihood, and maximum a posteriori (MAP), solve an optimization problem to obtain a central value (or best estimate) followed by an approximate evaluation of the spread (or covariance matrix). A different approach is the Monte Carlo (MC) method, and particularly Markov chain Monte Carlo (MCMC) methods, which allow sampling from the posterior distribution of the parameters. Though available for years, MC methods have only recently drawn wide attention as practical ways for solving challenging highdimensional parameter estimation problems. They have a broader scope of applications than conventional methods and can be used to derive the full posterior pdf but can be computationally very intensive. This paper compares a number of different methods and presents improvements using as case study a nonlinear DNAPL source dissolution and solute transport model. This depth-integrated semianalytical model approximates dissolution from the DNAPL source zone using nonlinear empirical equations with partially known parameters. It then calculates the DNAPL plume concentration in the aquifer by solving the advection-dispersion equation with a flux boundary. The comparison is among the classical MAP and some versions of computer-intensive Monte Carlo methods, including the Metropolis-Hastings (MH) method and the adaptive direction sampling (ADS) method.

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“…Unfortunately, the simultaneous fulfillment of three or all four conditions requires specific effort. Additionally, different specific properties of the distribution, such as nonnegativity of the values, trigger further problems (Michalak 2008).…”

Section: To Have the Temporally Dependent Head Field H W (X T) Corrementioning

confidence: 99%

Random Mixing: An Approach to Inverse Modeling for Groundwater Flow and Transport Problems

Bàrdossy

Hörning

2015

Transp Porous Med

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This paper presents a novel methodology for inverse modeling of groundwater flow and transport problems in a Monte Carlo framework, i.e., multiple solutions to the inverse problem are generated. The methodology is based on the concept of random mixing of spatial random fields. The conditional target hydraulic transmissivity field is obtained as a linear combination of unconditional spatial random fields. The corresponding weights of the linear combination are selected such that the spatial variability of the hydraulic transmissivities as well as the actual observed transmissivity values are reproduced. The constraints related to the hydraulic head and contaminant concentration observations are nonlinear. In order to fulfill these constraints, a specific property of the presented approach is used. A connected domain of fields fulfilling all linear constraints is identified. This domain includes an infinite number of realizations, and in this domain, the head and concentration deviations are minimized using standard continuous optimization techniques. The methodology uses spatial copulas to describe the spatial dependence structure. A combination with multiple point statistics allows inversion under specific structural constraints.

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A Gibbs sampler for inequality‐constrained geostatistical interpolation and inverse modeling

Cited by 38 publications

References 56 publications

Increasing Confidence in Mass Discharge Estimates Using Geostatistical Methods

Increasing Confidence in Mass Discharge Estimates Using Geostatistical Methods

Parameter estimation in nonlinear environmental problems

Random Mixing: An Approach to Inverse Modeling for Groundwater Flow and Transport Problems

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