PARAMETRIC UNCERTAINTY ANALYSIS USING MARKOV CHAIN MONTE CARLO METHODS FOR URANIUM TRANSPORT SURFACE COMPLEXATION MODELS

Geoffery L. Miller December 16, 2010

This research is supported in part by theDOE grant DE-SC0002687 ORAU/ORNL High Performance Computing Program

Geoffery Miller1, Dan Lu1, Ming Ye1, Gary Curtis2, Bruno Mendes3, and David Draper3

1:FSU DSC 2:USGS 3:UCSC

The Column Experiment

Surface Complexation Modeling

Global Calibration

The Problem and Model2

Groundwater Contamination3

The Column Experiment4

An experimental model for uranium U(VI) reactive-transport through a column of homogeneous quartz

Kohler, et al. (1996) conducted 7 column experiments

Surface Complexation Modeling5

Kohler (1996) created 7 computational models

Model C4 was decided to be the best model with respect to model fit and complexity

Model C4 has 4 parameters Log(K1), Log(K2), Log(K3),

Log(Site Fraction)

Previous study of these models assume the parameters are distributed normally

Global Calibration6

Naïve Monte Carlo Analysis

Morris One-At-a-Time (MOAT) Method

Markov Chain Monte Carlo Analysis

Parametric Uncertainty7

Naïve Monte Carlo Results, N=1000

8

Prior distributions: 1: Uniform +/- 100% about calibration

2: Uniform +/- 10% about calibration

Morris One-At-a-Time Method9

Quantifies “Elementary Effects”of parameters on the model’s objective function

di(X) y(Xi ) y(X)

MOAT Results10

Optimal objective function is 91,000

All parameters seem to exhibit nonlinear or interacting effects, 106

None of the parameters can be discounted

The first functional group is the most sensitive

The site fraction is second

More accurate Prior distribution11

Still assume uniform and independent distributions

Determined from an independent isothermal tool

+/- 2 log units

Adaptive Markov Chain Monte Carlo

12

Adaptive Method13

Why use an adaptive method? High parameter sensitivity (seen from MOAT)

creates a small variance-covariance matrix around the calibration

Searching a small parametric region gives little information about the posterior parameter distribution function

Model runtime is 10-20 minutes

Adaptive Markov Chain Monte Carlo

Accept or Reject P2, "Flip a coin"

r = random(Uniform, 0, 1)1. if (a >= r) accept point P2, (P1 = P2),

Total_Accepted++

2. if (a < r) reject point P2, stay at point P1

14

Adaptive Markov Chain Monte Carlo

Performing Adaptations1. Acceptance_Ratio = Total_Accepted / Total_Proposed

2. scale(i+1) = Check ( scale(i) + factor*(Acceptance_Ratio – Target_Ratio) )

The Check function verifies the scaling parameter is not too large or too small

15

Parameter Posterior Probability Distributions

Likelihood Surfaces

Prediction Intervals

MCMC Results16

Posterior Parameter Distributions

17

Assessment of the Normal Assumption

18

Lilliefors Test

Null-hypothesis: the data comes from a normally distributed population

Rejected for all parameters

Likelihood Surfaces 1/219

LogK1

LogK1

Log f

Log f

Likelihood Surfaces 2/220

LogK2

LogK2

LogK3

LogK3

Prediction Intervals21

Prediction Intervals22

Parametric Uncertainty Analysis with MCMC

Other Kohler Models

Conclusions and Future Study

23

Parametric Uncertainty 1/224

Surface complexation model parameters do not follow normal distributions

Likelihood surfaces are not smooth

MCMC confirms MOAT results

MCMC confirms the calibration

Parametric Uncertainty 2/225

MCMC resulted in both a better predictive coverage and physically reasonable prediction intervals naïve Monte Carlo

MOAT results were confirmed by MCMC results

Prediction interval can give worst-case scenario for the long tail of the breakthrough of U(VI)

Other Kohler Models26

Models C5 and C6 have good or better calibrations than C4

C4 was only chosen for simplicity and also having a good fit

The parameters exhibit interacting effects, parameter correlations should be studied further

Thank you27

Extra Slides28

Testing Extreme Parameters29

Testing points of high likelihood far away from calibration for model fit

Testing Extreme Parameters30

Testing points of lower likelihood, but near the calibration, for predictive capability

MCMC Burn-In31

MCMC Convergence Mixing32

alpha0 chains 1:2

iteration

101 200 400 600

-1.5

-1.0

-0.5

0.0

0.5

MCMC Convergence Testing33

Gelman-Rubin Statistic

Calculates between and within chain variance

Determines a stable distribution when converging to a value less than 1.2

MCMC Convergence Results34

Adaptive MCMC Algorithm Notes 2/2

35

When Acceptance_Ratio is too high the scaling factor is increased Increasing the search area, as the chain is

considering too many points near a maximum

When Acceptance_Ratio is too low the scaling factor is decreased Decreasing the search area, so the chain is not

too divergent from maxima

Global Model Calibration36

Search a parameter space including all feasible parameter values in every dimension

Use an objective function, including measurement error, to minimize error between simulation and experiment

Avoid local minima with a multiple-start strategy using Latin Hypercube Sampling

Used BFGS nonlinear optimization from each start point

Adaptive MCMC Algorithm37

Prior Distribution from an isothermal tool

Proposal Distribution from an approximated variance-covariance matrix Scaled using an adaptive method

Starting point set to the global calibration point

Generates the posterior parameter distribution

MCMC Adaptive Scaling38

Scaling parameter achieved the target acceptance ratio 0.25

Multiple chains did not converge to one scaling parameter Longer MCMC run-time

Decrease scaling factor

Scaling ratio may not be appropriate for this problem

MCMC Convergence39

Convergence statistic used assumes that stationary distributions should be normal This is not the case for this study

MCMC convergence needs to be assessed for multi-modal, non-normal probability distribution functions

Gelman-Rubin is univariate, which may not be reasonable for parameters with high correlations

Model Runner Application40

Likelihood response surfaces41

UCODE Local Calibrations42

Regular Response Surface43

MC and MOAT Conclusions44

The parametric probability distributions must be understood to conclude the predictive capability of the model Investigate the posterior probability using Bayes’

Theorem

Introduce a likelihood function